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The Repayment of Local 
 and Other Loans 
 
 Sinking Funds 
 
The Repayment of Local 
 and Other Loans 
 
 Sinking Pounds 
 
 BY 
 
 EDWARD HARTLEY BURNER, A.C.A. 
 
 Lecturer on Municipal Accounts at the Manchester Municipal 
 School of Commerce 
 
 THE RONALD PRESS COMPANY 
 
 198 Broadway, New York 
 1913 
 

 <h 
 
 "^^^t 
 
PREFACE. 
 
 In writing this book my primary object has been to render it 
 practically useful to those engaged in connection with the loan 
 debt of public authorities and privately owned undertakings. 
 It is the result of actual experience extending over many years, 
 but the problems which have arisen have been isolated examples 
 and have not occurred in any regular sequence, but often at 
 long intervals. For this reason each problem has been treated 
 independently in order that any particular adjustment may be 
 made by reference to a similar example fully worked out and 
 described. This entails a certain amount of repetition in order 
 that each chapter may contain a brief summary, and a reference 
 to the results, of previous and subsequent investigations. In 
 order to carry this out in its entirety, each adjustment has been 
 reduced to a series of stages briefly stated, and is accompanied 
 by detailed statements of the method adopted, and actual 
 calculations upon standard forms which I have specially 
 prepared. But I have gone further than this, and have grouped 
 the problems under the heads of the various factors governing 
 a sinking fund, namely, the amount in the fund, the period of 
 repayment, the rate of accumulation and the rate of income to 
 be received upon the present investments. This orderly 
 arrangement of isolated practical examples has compelled a 
 theoretical method of treatment which has been adopted 
 throughout. In my own earlier experience I very soon realised 
 that even after making the fullest use of the published tables 
 of compound interest, the ordinary methods of arithmetic were 
 utterly inadequate and that all calculations must be made by 
 logarithms; consequently a full knowledge of the use of a log. 
 table has been assumed. The next practical difficulty which 
 arose was that the ordinary published tables of compound 
 interest very often did not contain the required rate per cent., 
 and it therefore became necessary to make the calculation by 
 other means. For this reason I have included in the earlier 
 chapters a brief summary of the mathematical principles, 
 showing the derivation of the formulae upon which all the tables 
 are constructed. In order, however, to render this method by 
 formula generally available I have reduced all such calcula- 
 tions to simple rules, and in Chapter X, dealing with the 
 
 290048 
 
vi PREFACE 
 
 various staudard calculation forms, minute instructions are 
 given for finding tlie actual values of the whole of the factors 
 relating to any rate per cent. I have, in fact, endeavoured to 
 state the methods in such a manner that a knowledge of the 
 meaning of the formula is not required. 
 
 Throughout the book the final results are expressed in 
 decimal form. This course has been adopted partly in order 
 to save time, but primarily to make the book applicable to any 
 currency of a decimal nature; and it is recommended that all 
 pro forma accounts should be prepared in this form. 
 
 The methods of adjustment described as the deductive 
 method, the annual increment (ratio) method, and the annual 
 increment (balance of loan) metliod, are not new. The terms 
 have merely been used for convenience of reference. 
 
 The actual compilation of the book has occupied considerable 
 time, and it has not been written in the consecutive order in 
 which it is now presented to the reader. It has involved the 
 preparation of many standard forms and statements, and very 
 many calculations Avhich are not given in the text. Every effort 
 has been made to ensure absolute accuracy of detail, but in a 
 few cases the final decimal figure in the result obtained may 
 not agree with the result found by another method, and the 
 same applies to the final figure of some of the logs. These 
 small differences are not, however, of any practical importance. 
 The cross-references to other results have been carefully 
 compared, but they are very numerous and a few errors may 
 be found. 
 
 The methods adopted, and the results obtained, have been 
 in all cases verified by an alternative method of proof, even 
 where it is not shown in detail in the text. The summaries of 
 methods, and the rules and formulse given in italics at the head 
 of the various chapters have been in all cases caref Tilly compared 
 with the individual statements and with each other, and a 
 uniform wording has, as far as possible, been adopted 
 throughout. 
 
 As stated in the Introduction, no attempt has been made to 
 include anything in the nature of a full statement of the 
 statutory obligations as to tlie repayment of the loan debt of 
 local authorities or the policy of Parliament with relation 
 thereto, except so far as they affect the actual method of 
 repayment. 
 
 In the final chapters dealing with the life of tlie asset, the 
 equation of the period of repayment and the incidence of 
 taxation a serious attempt has been made to elucidate this 
 
PREFACE vii 
 
 difficult subject. It has been treated niaiuly from the mathe- 
 matical standpoint, both as regards the annual instalment and 
 interest upon the loan. It is a matter about which there is 
 naturally some divergence of opinion as to its practical 
 application, but I hope that the considerations here set forth 
 will be of assistance in arriving at a definite solution. 
 
 The object of the book is entirely practical. I hope it will 
 materially assist those who are approaching the consideration of 
 such problems for the first time, and that it will also be useful 
 to those who are already fully acquainted with the subject, if 
 only because it attempts to deal with it in a consecutive and 
 orderly manner. 
 
 In conclusion, I wish to tender my thanks to my friend and 
 colleague, Mr. Arthur Holme, A.C.A., who has assisted me in 
 the revision of the manuscript and the correction of the proofs, 
 and also to my son, Edward Gordon Turner, who has rendered 
 me very valuable aid in the preparation of the final manuscript 
 and the verification of the various calculations in the text. 
 
 E.H. T. 
 
 Manchester, 
 
 November, 191'J. 
 
TO AMEEICAN READERS. 
 
 In the early part of the year 1906, I was appointed the 
 British Accounting Expert to the National Civic Federation of 
 New York in the Enquiry into the Municipal and Private 
 Ownership of Public Utilities (Municipal Trading) in Great 
 Britain. 
 
 The enquiry was made by a Commission of Twenty-one, 
 under the Chairmanship of Melville E. Ingalls, Chairman of 
 the Board of Directors, Big Four Railroad, Cincinnati. The 
 details of the enquiry were under the supervision of Dr. Milo 
 R. Maltbie, now a Member of the Public Service Commission of 
 the State of New York. The Report of the Commission is 
 contained in three volumes, issued in New York in 1907. 
 
 The enquiry in Great Britain was confined to the results of 
 municipal and private ownership and operation of gas works, 
 street railways or tramways, and electric lighting and power 
 undertakings. In the United States the enquiry was extended 
 to water works. 
 
 The Commission examined 24 public and private under- 
 takings in London, Birmingham, Dublin, Glasgow, Leicester, 
 Liverpool, Manchester, Newcastle-on-Tyne, Norwich and 
 Sheffield. 
 
 The following is a full list of the Committee on Investiga- 
 tion : — 
 
 *Melville E. Ingalls, Chairman (Chairman Board of Directors, 
 Big Four Railroad), Cincinnati. 
 Albert Shaw, V ice-Chairman (Editor " RevicAv of Reviews "), 
 
 New York City. 
 Talcott William (Editorial Writer, the Press), Philadelphia. 
 W. D. Mahon (President Association Street Railway Em- 
 ployes), Detroit. 
 ^Professor Frank J. Goodnow (Columbia University), New York 
 
 City. 
 *Walton Clark (Third Vice-President The United Gas Improve- 
 ment Company), Philadelphia. 
 ^Professor Edward W. Bemis (Superintendent Water Works), 
 
 Cleveland. 
 *Professor John H. Gray (University of Minnesota), Minneapolis. 
 Walter L. Fisher (Special Traction Counsel for City of Chicago 
 and ex-President Municipal Voters' League), Chicago. 
 
X PREFACE 
 
 *Timotliy Healey (President International Brotberliood 
 
 Stationary Firemen), JN^ew York City. 
 *William J. Clark (General Manager Foreign Department, 
 
 General Electric Company), New York City. 
 H. B. F. Macfarland (President Board of Commissioners, 
 
 District of Columbia), Washington. 
 Daniel J. Keefe (President International Longshoremen's 
 
 Association), Detroit. 
 *Professor Frank Parsons (President National Public Ownership 
 
 League), Boston. 
 *Professor John R. Commons (Wisconsin University), Madison, 
 
 Wisconsin. 
 *J. W. Sullivan (Editor " Clothing Trades' Bulletin "), New 
 
 York City. 
 *F. J. McNulty (President International Brotherhood of 
 
 Electrical Workers), Washington. 
 *Albert E. Winchester (General Superintendent City of South 
 
 Norwalk Electric Works), South Norwalk, Conn. 
 *Charles L. Edgar (President The Edison Electric and 
 
 Illuminating Company), Boston. 
 *Dr. Milo R. Maltbie (member of the Public Service Commis- 
 sion), New York City; now a member of the Public 
 
 Service Commission of the State of New York. 
 Leo S. Rowe (University of Pennsylvania), Philadelphia. 
 *Edward A. Moffet, Secretary (Editor "Bricklayer and Mason"), 
 
 Indianapolis, Ind. 
 
 The investigation in Great Britain was made by the members 
 of the Committee indicated by an asterisk. 
 
 The following summary of the financial details of the 
 undertakings examined will give some idea of the magnitude of 
 the enquiry. 
 Loan Capital : — 
 Outstanding : 
 
 Municipal £18,8GG,648 
 
 Private 6,144,945 
 
 £25,011,593 
 
 Repaid by means of Sinking Fund : 
 
 Municipal 3,868,301 
 
 £28,879,894 
 Share Capital or Stocly, Private 12,146,504 
 
 Total Capital raised £41,026,398 
 
PREFACE xi 
 
 Capital expended on WorJcs : — 
 
 Municipal £24,517,370 
 
 Private 18,266,368 
 
 Gross Revenue for the year under Examination:- 
 
 Municipal £6,091,821 
 
 Private 3,956,936 
 
 £42,783,738 
 
 £10,048,757 
 
 The examination by the experts was commenced early in 
 March, 1906, and continued without interruption until the end 
 of July, 1906. The members of the Commission were engaged 
 in Great Britain from 30th May to the end of -July, 1906. 
 
 The expert assistance was divided between the Technical 
 and Accounting sides. In each class of undertaking an 
 American and a British technical expert made a joint investiga- 
 tion, valuation, and report. 
 
 The British technical experts were : — 
 Gas Works, Mr. William Newbigging, Consulting Engineer, of 
 
 Manchester. 
 Tramways (Street Railroads), Mr. J. H. Woodward, Consulting 
 
 Engineer, London. 
 
 The American technical experts were : — 
 Electricity Supply, Mr. Albert E. Winchester, General Supt. 
 
 City of South Norwalk Electric Works, South Norwalk, 
 
 Conn. 
 Gas Works, Mr. J. B. Klumpp, Inspecting Engineer of the 
 
 United Gas Improvement Company, Philadelphia. 
 Tramways (Street Railroads), Mr. Norman McD. Crawford, 
 
 Consulting Engineer, Hartford, Conn. 
 
 The accounting side of the whole of the undertakings 
 examined was entrusted to myself and Mr. R. C. James, the 
 Chief Accountant to the United Gas Improvement Company, of 
 Philadelphia. During the investigation all the experts were in 
 constant communication and worked together, and each under- 
 taking was, as far as possible, examined by them at the same 
 time. This resulted in a very minute comparison of American 
 and British technical and accounting practice. During the 
 months we were associated many friendships were formed, but 
 
xii PREFACE 
 
 the greater value of the enquiry to all the experts engaged is 
 undoubtedly the wider appreciation of American and British 
 practice. On the technical and practical side, many hitherto 
 diverging views were reconciled or reduced to an agreed mean, 
 but as regards the accounting and financial side no such 
 difference was found. In America, as in Great Britain, all 
 problems relating to large financial operations follow the same 
 economic laws, and the only variations on each side of the 
 Atlantic are due to the actual conditions now existing. On 
 this side the growth of large municipal undertakings, concerned 
 with the provision of public utilities, has been a gradual process 
 extending over many years requiring the constant supervision 
 of Parliament. On the other side the municipal operation of 
 public utilities is still in its infancy, and is therefore an urgent 
 practical problem, as evidenced by the enquiry, made at very 
 considerable expense, by the National Civic Federation of New 
 York. 
 
 In connection with pure methods of accounting, it must be 
 acknowledged tliat as regards uniformity the British system is 
 not so far advanced as the American. Standard methods have 
 there loujx been adopted for the accounts of gas works, electric 
 lighting undertakings, and street railways, and include not 
 only the form of the final Revenue Account and Balance Sheet, 
 but also methods of analysis and book-keeping in minute detail. 
 It is true that in Great Britain standard forms of accounts for 
 gas works are contained in the Gas Works Clauses Act. The 
 Board of Trade have, under their power in the Electric Lighting 
 Acts, prescribed standard forms of account for electric lighting 
 undertakings. A standard form of account for tramways has 
 been drawn up by the Tramway Institute and the Institute of 
 Municipal Treasurers and Accountants, and as the result of an 
 enquiry by a Government Departmental Committee (report 
 dated 1907) further amended standard forms have been 
 suggested for all tlie above-mentioned utilities. The British 
 standard forms, however, are deficient to the extent that they 
 lack that minute attention to detail which is the characteristic 
 of the American systems. A standard form of revenue, or 
 profit and loss, account may be very useful for comparing the 
 final results of the year's operations, but unless there is absolute 
 uniformity of detail in the items charged to each head of 
 operating expense it is impossible to make any reliable 
 comparison between the operating costs of two or more under- 
 takings. This fact was very clearly shown in the enquiry by 
 the Commission in Great Britain. 
 
PREFACE xiii 
 
 Turning now to the immediate subject of this book, namely, 
 the repayment of the loan debt of local authorities and privately 
 owned undertakings, the result of the enquiry proved that as 
 regards the actual methods of repayment the practice is the 
 same in both countries. Another outstanding feature of the 
 enquiry was the minute attention paid by the legal, labour and 
 economic experts upon the Commission to the statutory and 
 other obligations imposed by Parliament upon local authorities 
 in Great Britain. This was due to the fact that the municipal 
 operation of public utilities in this country is a much older 
 institution than in the United States. The report of the 
 Commission contains much valuable information in considerable 
 detail as to these statutory obligations and is a useful resume 
 of a very complicated question, well worth perusal by municipal 
 experts in this country. This book does not in any way attempt 
 to deal with these matters in an exhaustive manner, and they 
 are only mentioned in so far as they relate to the actual methods 
 of repayment of loan debt. In writing the book I have borne in 
 mind the results of the enquiry and have so arranged it that it 
 will apply to all problems in whatever currency. The formula 
 relating to a geometrical progression does not recognise any 
 geographical limits, consequently the methods based thereon 
 may be applied equally in the United States, as in Great 
 Britain. Owing to the interest in municipal ownership 
 and operation in the United States, the practical application of 
 the statutory obligations as regards the repayment of loan debt 
 by British municipalities, even as briefly stated, will, it is 
 hoped, be useful. 
 
 As far as possible the terms used have been chosen in order 
 to avoid any misunderstanding on either side. The word 
 " Corporation " in Great Britain denotes both a Local Authority 
 and a privately owned undertaking. The term " Local 
 Authority " has therefore been used throughout, although it 
 may not in a few cases be strictly correct. It includes all 
 public authorities having control of public moneys for the 
 public good and empowered to raise such moneys by way of an 
 annual rate based upon an assessment of the annual value of the 
 property. It is not the practice in Great Britain to levy a local 
 rate or assessment based upon the capital value. All such rates 
 are levied at so many pence in the pound {£) sterling of annual 
 value. 
 
 In Great Britain there are several kinds of privately owned 
 companies or corporations, which may be divided into two 
 groups. First, those in which the liability of the members is 
 
xiv PREFACE 
 
 unlimited, wliicli, however, are not numerous. Secondly, those 
 companies in whicli the liability of the members is limited to 
 the actual amount for the time being unpaid upon the shares 
 held by them. Such companies may be divided into two classes, 
 namely, those which derive their powers of operation from, and 
 are incorporated by, special Act of Parliament; and, secondly, 
 those incorporated under the Limited Liability Acts (The 
 Companies Acts, 1862 — 13)08). In this book all such companies 
 liave been included under the generic term " Private Under- 
 takings " as distinguished from " Local Authorities." 
 
 The capital of all privately owned companies or undertakings 
 in Great Britain is provided by the members and is raised in 
 various ways. In the case of companies incorporated under the 
 Limited Liability Acts the capital is invariably raised by the 
 issue of a definite number of shares of a uniform nominal value, 
 now as a general rule of £1 each. In the case of companies 
 incorporated by special Act of Parliament the capital is some- 
 times raised by the issue of shares, similar to companies incor- 
 porated under the Limited Liability Acts, but often by the 
 issue of stock. All capital raised by all privately owned 
 undertakings, whether by shares or stock, may be issued with 
 certain defined priorities or preferences both as to dividend and 
 also as to repayment upon the final winding up of the company. 
 In all cases considered, the capital provided by the members 
 of a privately owned undertaking has been included under the 
 generic term of " Share Capital or Stock." 
 
 The term " Loan Debt " has been used to denote all moneys 
 borrowed by a public authority or private undertaking and 
 secured by way of mortgage upon the assets, including in the 
 term assets the power of a local authority to levy a rate or 
 assessment. In the case of a private undertaking such loan 
 debt is always repayable, on a winding up, in priority to the 
 share capital or stock, and may or may not be repayable, during 
 the life of the undertaking, by means of a sinking fund to be 
 built up out of the annual profits. In the case of all public 
 authorities in Great Britain, Parliament now invariably imposes 
 the obligation to repay such loan debt by means of a sinking 
 fund, or other alternative method, Avithin a period having a 
 more or less definite relation to the life of the asset created out 
 of tlie loan, and such annual provision for repayment almost 
 witliout (xccption operates immediately upon the borrowing of 
 the money, and is charged against the annual rate or assessment 
 levied by the h)cal authority. In the case of a loan raised to 
 ptovido works of a revenue-earning character, such as gas works, 
 
PREFACE 
 
 XV 
 
 etc., the annual redemption cliarges, as well as tlie interest upon 
 tlie loan, are charged against the profits of the undertaking, 
 and any deficiency is made good out of the annual rate or 
 assessment levied upon the whole of the community. 
 
 The foregoing remarks will sufficiently explain the terms 
 used in the book, but if fuller details are required the reader is 
 referred to the Financial Appendix to the Report of the Com- 
 mission (Vol. II, Part II, p. 628), prepared by Mr. James and 
 myself. 
 
 E.H.T. 
 Manchester (Enr/.), 
 
 November, 1912. 
 
SUMMARY OF SECTIONS. 
 
 PAGE 
 
 I. Mathematical Principles. 
 
 Chapters II to X 19 
 
 II. The Methods of Repayment of Loan Debt. 
 
 Chapters XI to XIII 107 
 
 III. The Annual Instalment. 
 
 Chapters XIV to XXI 143 
 
 IV. The Annual Increment, 
 
 Chapters XXII to XXVII 257 
 
 V. The Date of Borrowing. 
 
 Chapters XXVIII to XXX 343 
 
 VI. The Life of the Asset and its Relation to 
 
 the Redemption Period. 
 
 Chapter XXXI 375 
 
 VII. The Equation of the Period of Repayment. 
 
 Chapter XXXII 387 
 
 VIII. The Equation of the Incidence of Taxation. 
 
 Chapters XXXIII to XXXV 407 
 
CONTENTS. 
 
 CHAP. P'^*^'^ 
 
 I. Introduction " i 
 
 SECTION I. 
 Mathematical Principles. 
 
 CHAP. 
 
 II. Logarithms 
 
 III. Simple and Compound Interest 28 
 
 Compound interest a geometrical progression. 
 Derivation of the formula, 
 
 A = PR^ 
 
 relating to compound interest, from the algebraical formula, 
 
 I — ar""^ 
 relating to a geometrical progression. Explanation of 
 terms. Difference between the amount of /,i, and of £1 per 
 annum, at the end of one year. " Present Value " com- 
 pared with " Practical Discount." 
 
 IV. Compound Interest as Applied to a Sum of Money - - 36 
 
 Table I. The amount of £1 in any number of years. 
 The formula, A = P RN and rules deduced therefrom. 
 Calculation by the arithmetical method. Compilation of 
 tables. Thoman's method and formula. Author's Standard 
 Calculation form. No. i. 
 
 V. Compound Interest as Applied to a Sum of Money {contd.) - 42 
 
 Table II. The present value of £1 due at the end of any 
 number of years. 
 
 ^ ~ R^ 
 
 Derivation of the formula and rules deduced therefrom. 
 Tables of ratios and logs, of R, and r; calculations for 
 periods other than years. Thoman's method and fornnila. 
 Author's Standard Calculation form, No. 2. 
 
XX CONTENTS 
 
 chap, page 
 
 \1. Compound Interest as Applied to an Annual or other 
 
 Periodic Payment '5° 
 
 Table III. The amount of £i per annum in any number of 
 years. 
 
 'C^) 
 
 M = A2/( 
 
 Derivation of the formula and rules deduced therefrom. 
 Thoman's method and formula. Author's vStandard Calcu- 
 lation form, No. 3. 
 
 VII. Compound Interest as Applied to an Annual or other 
 
 Periodic Payment {coutd.) 62 
 
 Table IV. The present value of £1 per annum for any 
 number of years. 
 
 P = Ay 
 
 
 Derivation of the formula and rules deduced therefrom. 
 Thoman's method and formula. Author's Standard Calcu- 
 lation form, No. 4. 
 
 VIII. Compound Interest as Applied to an Annual or other 
 
 Periodic Payment (contd.) - - - - - ■ 67 
 
 Table V. The anuuit}- which £1 will purchase for an}- 
 number of 3'ears, or of which £1 is the present value. 
 
 A'/-(|^) 
 
 Derivation of the formula and rules deduced therefrom. 
 Thoman's method and formula. Author's Standard Calcu- 
 lation form. No. 5. 
 
 IX. Tii()^l\n's LoGARniiMic Tables of Compound Interest and 
 
 Annuities - - - - - - - - - -73 
 
 Explanation of the symbols used in- Thoman, and their 
 relation to the abo\e tallies and formula;. 
 
 X. Standard Calculation Forms, prepared by the Author, 
 
 relating To TIIK ABOVE TABLES, WITH E-XPLANATIONS AND 
 INSTRUCTIONS AS TO MAKING THE CALCULATIONS - - 8o 
 
CONTENTS 
 
 SECTION II. 
 The Methods of Repayment of Loan Debt. 
 
 CHAP. PAGE 
 
 XI. The Repayment of the Loan Debt of Local Authorities 
 
 AND commercial AND FINANCIAL UNDERTAKINGS - - 109 
 
 Alternative methods allowed by the Public Health Act, 1S75, 
 and other Acts. Comparison of methods as regards the 
 actual repayment to the lender and the annual charge 
 against revenue or rate. 
 
 The Instalment Method, by equal annual instalments of 
 principal repaid to the lender. vStatement showing the final 
 repayment of the loan. 
 
 XII. The Repayment of the Loan Debt of Local Authorities, 
 
 ETC. (contd.) - - -114 
 
 The Annuity Method, by an equal annual instalment of 
 principal and interest combined, repayable to the lender. 
 Methods of calculation. FormuhTC and rules deduced there- 
 from. vStatement showing the final repayment of the loan. 
 Comparison with the sinking fund iiistal;nent, and the 
 equal annual instalment of principal only. 
 
 XIII. The Repayment of the Loan Debt of Local Authorities, 
 
 ETC. (contd.) - - - 126 
 
 The Sinking Fund Method, by setting aside and accumu- 
 lating an equal annual instalment to provide the principal 
 only at the end of the redemption period. 
 
 1. The Accumulating Sinking Fund. Methods of calcula- 
 tion. Formula and rules deduced therefrom. Statement 
 showing the final repayment of the loan. 
 
 Author's Standard Calculation form. No. 3X. 
 
 2. The Non-accumulating vSinking Fund, compared with 
 the instalment and accumulating sinking fund methods. 
 Statement showing the final repayment of the loan. State- 
 ment showing the comparison between the three methods 
 already described. 
 
CONTENTS 
 
 SECTION III. 
 The Annual Instalment- 
 
 CHAP. PAGE 
 
 XI\'. Sinking Fund Problems in general, relating to : - ' 145 
 
 1. The amount in tlie fund. 
 
 2. The rate per cent, 
 
 (a) of income to be received upon the present investments 
 
 representing the fund. 
 
 (b) the future rate of accumulation. 
 
 3. The redemption period. 
 
 4. The rate per cent, and the redemption period in 
 
 combination. 
 Definition and explanation of the terms " present invest- 
 ments " and " annual increment." 
 
 XV. Sinking Fund Problems relating to the amount in the 
 
 Fund. A deficiency in the fund - - - ■ - 154 
 Calculation of a t3^pical fund to be used as an example. 
 How a deficiency ma}' arise, and how it may be ascertained. 
 Summary of the methods of making the adjustment in the 
 annual instalment. 
 
 XVI. Sinking Fund Problems relating to the amount in the 
 
 Fund. A deficiency in the fund {contd.) - - - 171 
 
 Variation i. To be corrected b}- an additional annual 
 instalment to be set aside during the whole of the unexpired 
 portion of the repayment period. 
 
 Variation 2. To be corrected by an additional annual 
 instalment to be set aside during the earlier part only of 
 the unexpired portion of the repayment period. 
 
 XVII. vSiNKiNG Fund Problems relating to the amount in the 
 
 Fund (contd.). A surplus in the fund - - - - 186 
 
 Variation i. Arising in consequence of an excessive past 
 accumulation of tlie fund. 
 
 \'ariation 2. Arising in consequence of the payment into 
 the fund of the proceeds of sale of part of the assets 
 representing the securit}^ for the loan, or a realised jn'ofit 
 upon the sale of an investment representing the fund. 
 
CONTENTS 3^^"^ 
 
 PAGE 
 ^YVIIl. Sinking Fund Problems relating to the amount in the 
 
 Fund (contd.) --''''' ^99 
 
 A surplus in the fund of a coniniercial or financial under- 
 taking arising on the withdrawal of part of the loan from 
 the operation of the fund, owing to the conversion of such 
 part of the loan into ordinary share capital or stock of the 
 undertaking. 
 
 Variation 3. In which the original annual instalment was 
 found by calculation based upon a specified period of 
 repayment and rate of accumulation. 
 
 Variation 4. In which the original annual instalment is 
 a stated sum, and is not based, except in a general way, 
 upon any period of repayment or rate of accumulation. 
 
 XIX Sinking Fund Problems relating to the rate per cent. 
 OF income upon the present investments representing 
 
 THE amount in THE FUND ; AND .^LSO THE FUTURE RATE OF 
 ACCUMULATION OF THE FUND. - " " " " '^^S 
 
 Variation A. In ^vhich there is a variation in the rate of 
 accumulation without any variation in the rate of income 
 upon the present investments, or in the period of repayment. 
 
 XX. Sinking Fund Problems relating to the rates per cent. 
 
 OF INCOME AND ACCUMULATION (COUtd.) " ' " "236 
 
 Variation B. In which there is a variation in the rate of 
 income upon the present investments without any variation 
 iu the rate of accumulation, or in the period of repayment. 
 The subject is further considered in Chapter XXVII. 
 
 XXI. Sinking Fund Problems relating to the rates per cent. 
 
 OF INCOME AND ACCUMULATION [COntd.) - " ' "247 
 
 Variation C. In which there is a variation in the rate of 
 accumulation, and also in the rate of income upon the 
 present investments, but without any variation m the 
 period of repayment. 
 
xxiv CONTENTS 
 
 Section IV. 
 The Annual Increment. 
 
 CHAP. PAGE 
 
 XXII. Sinking Fund Problems relating to the rate per cent. 
 
 of ACCUMULATION 259 
 
 The above variations further considered, namel}-, a variation 
 in the rate of accumulation with or without a variation in 
 the rate of income upon the present investments, but 
 without any variation in the period of repayment. 
 Comparison of the results already obtained in terms of the 
 annual instalment, with those obtained by means of the 
 annual increment, and the varying rates of accumulation. 
 
 XXIII. Sinking Fund Problems relating to the rate per cent. 
 
 OF accumulation (contd.) - - 277 
 
 Derivation of a rule and formula relating to a variation in 
 the rate per cent, of accumulation based upon the foregoing 
 results, by the annual increment (ratio) method. 
 
 XXIV. Sinking Fund Problems relating to the redemption 
 
 PERIOD ----------- 283 
 
 The deductive method of finding the amended annual 
 instalment due to a variation in the period onlj-. The 
 annual increment (ratio) method. 
 
 XXV. Sinking Fund Problems rel.\ting to the redemption 
 
 PERIOD (contd.) - - - 295 
 
 Derivation of a rule and formula relating to a variation in 
 the period of repayment based upon the foregoing results, 
 by the annual increment (ratio) method. 
 
 XXVI. Sinking Fund Problems rel.ating to the rate per cent. 
 
 OK accu.mulation and the redemption period in 
 combination -.- 200 
 
 The deductive method of ascertaining the amended annual 
 instalment due to a variation in both the above factors in 
 combination. 
 
 Derivation of a rule and formula relating to a dual variation 
 of this nature based upon the foregoing results, bj^ the 
 annual increment (ratio) method. 
 
CONTENTS XXV 
 
 CHAP. PAGE 
 
 XXVII. Sinking Fund Problems relating to the rate per 
 
 CENT. OF income UPON THE PRESENT INVESTMENTS REPRE- 
 SENTING THE FUND ------- -322 
 
 (In continuation of Chapter XX) but in which the rate of 
 income yielded by such investments is not uniform during 
 the whole of the unexpired portion of the repayment period. 
 
 1. In which the future variation in the rate of income is 
 known, and is definite both as to time and amount. 
 
 2. In which the future variation in the rate of income is 
 anticipated, but is uncertain both as to time and amount. 
 
 SECTION V. 
 
 The date of Borrowing, and its relation to the Redemption 
 Period. 
 
 WITHOUT ANY COMPLICATION AS REGARDS THE LIFE OR 
 
 DURATION OF CONTINUING UTILITY OF THE ASSET CREATED OUT 
 
 OF THE LOAN. 
 
 XXVIII. Loan borrowed over several years in one sum in 
 
 EACH year, each YEAR'S BORROWINGS BEING REPAYABLE 
 
 IN A PRESCRIBED PERIOD FROM THE D.\TES OF BORROWING - 345 
 
 1. By means of one sinking fund only. 
 
 2. By separate sinking funds for each year's borrowings. 
 
 XXIX. Loan borrowed over several years in one sum in each 
 
 YEAR, REPAYABLE IN ONE SUM ON A CERTAIN SPECIFIED DATE 354 
 
 1. Where the date of repayment is known at the time the 
 money is borrowed. 
 
 2. Where the date of repayment is fixed after the sinking 
 fund has been in operation for a number of years, and an 
 adjustment of the fund is required. 
 
 XXX. Loan borrowed in one or more years, in varying 
 
 AMOUNTS and AT VARIOUS DATES IN EACH YEAR, AND IT IS 
 REQUIRED THAT THE REVENUE OR RATE ACCOUNT OF EACH 
 YEAR SHALL BE CHARGED WITH A PROPORTIONATE PART OF 
 THE ANNUAL SINKING FUND INSTALMENT - - " - 365 
 
CONTENTS 
 
 SECTION VI. 
 
 CHAP. PAGE 
 
 XXXI. The Life of the Asset, and its relation to the 
 
 Redemption Period - - - - - - 377 
 
 SECTION VII. 
 XXXII. The Equation of the Period of Repayment 
 
 OF LOANS REPAYABLE AT VARIOUS DATES, WHICH ARE 
 REQUIRED TO BE REDEEMED ON ONE UNIFORM DATE - 389 
 
 1. Where the loans are authorised in respect of outlays of 
 varying character, each having a different life or period of 
 continuing utility and consequent repa3nnent. 
 
 2. Where the necessity to find the equated period of repay- 
 ment arises on the consolidation of existing loans. 
 
 The arithmetical method of finding the equated period 
 known as the " equation of payments." The true or 
 mathematical method. The error in the generally adopted 
 arithmetical method. 
 
CONTENTS 
 
 SECTION VIII. 
 The Equation of the Incidence of Taxation. 
 
 CHAP. PAGE 
 
 XXXIII. Comparison of the total annual loan charges to 
 
 REVENUE OR RATE BEFORE, AND AFTER, THE EQUATION OF 
 THE PERIOD OF REPAYMENT, SHOWING THE UNEQUAL 
 INCIDENCE OF TAXATION, IF THE ANNUAL INSTALMENT, AND 
 INTEREST UPON THE TOTAL LOAN, BE SPREAD EQUALLY OVER 
 THE EQUATED PERIOD - - - ^09 
 
 XXXIV. The Annual Instalment 418 
 
 The various methods of adjusting the annual charges to 
 revenue or rate during the equated period in proportion 
 
 to the life or duration of continuing utility of the asset 
 created out of the loan, viz. : — 
 
 By charging the revenue or rate account of each year of the 
 equated period with the annual instalment chargeable 
 against each year before equation, and in addition thereto 
 a supplementary annual instalment : — 
 
 (a) to be spread equally over the equated period ; or 
 
 (b) to be proportionate year by year to the annual instal- 
 ments before equation. 
 
 XXXV. Interest upon the Loan ------ -436 
 
 The method of adjusting the annual interest charges to 
 revenue or rate during the equated period in proportion to 
 the life or duration of continuing utility of the asset created 
 out of the loan : — 
 
 By charging the revenue or rate account of each year of the 
 equated period with the annual amount of interest payable 
 before equation, and in addition thereto a supplementary 
 annual amount proportionate year by year to the annual 
 interest charges before equation. A general Summary of 
 the results obtained in Chapters XXXIII, XXXIV, and 
 XXXV, illustrated by diagrams. 
 
Introduction 
 
CHAPTER I. 
 
 INTRODUCTION. 
 
 This book is the outcome of many years' professional work 
 ill connection with the accounts of municipal corporations, other 
 local authorities, and privately owned commercial and financial 
 undertakings. It deals only with the loan debt of such public 
 authorities and private undertakings, and includes, in addition 
 to the actual borrowing and repayment of the loan, the 
 numerous problems which arise in connection with the Sinking 
 Fund, relating to the amount in the fund, the rate of accumula- 
 tion, and the period of repayment. The concluding sections 
 contain chapters upon, (1) the relation between the life of the 
 asset and the period of repayment; (2) the methods of finding 
 the equated period of repayment ; and (3) the equation of the 
 incidence of taxation after the equation of the period, both as 
 regards the annual instalment and interest upon the loan. 
 In the last three chapters this difficult subject is treated in an 
 exhaustive manner. 
 
 Subject Matter. The book does not pretend, in any way, 
 to be a treatise upon the law relating to the loan debt of local 
 authorities, or to give full particulars of the various statutory 
 obligations, as regards repayment, imposed by Parliament ; nor 
 does it include a full statement of the general practice of 
 Parliament and the government departments having control of 
 such matters. All such statutory obligations are of a very 
 variable nature and are contained in many general and special 
 Acts of Parliament and provisional orders of the Local Govern- 
 ment Board. The general practice cannot be said to be based 
 upon any well-defined principles, but it should be stated that 
 the Local Government Board endeavours, as far as it is 
 empowered, to impose a uniform system, especially as to the 
 periods to be allowed for the repayment of loans raised for 
 public works having longer or shorter periods of duration or 
 continuing utility. 
 
 Although the actual practice varies considerably in detail, 
 the methods to be adopted in the solution of the various 
 problems all follow certain well-defined mathematical rules; 
 consequently the primary object is to demonstrate briefly the 
 mathematical principles involved and afterwards to apply such 
 c 
 
2 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 principles to a number of typical examples. As between the 
 mathematical and practical sides of the subject, preference has 
 been, and must necessarily be, given to the mathematical 
 portion, because this is definite, and may be exactly stated. 
 The practical variations from the ideal mathematical conditions 
 are so numerous that only typical examples have been con- 
 sidered, although every effort has been made to include all the 
 principal problems which are likely to arise. 
 
 Mathematical Principles. Since all problems, relating to 
 the future redemption or repayment of a present loan, to be 
 spread over a period of years, involve questions of compound 
 interest, it is first necessary to investigate the mathematical 
 principles governing the annual or other periodic accumiilation 
 of a present sum of money, and also of a sum of money payable 
 or receivable at the end of each of a number of equal and 
 definitely recurring periods of time. All such problems follow- 
 the algebraical rule relating to a geometrical progression, as 
 distinguished from the rule relating to an arithmetical progres- 
 sion which concerns only simple interest. The primary object 
 therefore is to convert the simple algebraical formula relating 
 to a geometrical progression into a formula which may be 
 applied to the subject matter of this book, namely, compound 
 interest. Simple interest is purely a matter of arithmetical 
 calculation and does not arise in any way in the problems to 
 be discussed. On the other hand, compound interest involves 
 a mathematical method of calculation and affects all problems 
 which will be hereafter considered. 
 
 The algebraical formula relating to any geometrical pro- 
 gression, as regards the last and first terms in any series of 
 numbers is : — 
 
 and this formula may be converted into a standard formula 
 relating to the accumulation of a sum of money now in hand, 
 namely : — 
 
 A=PEN 
 
 as described in Chapter III. The derivation of the formula 
 relating to the accumulation of an annuity or other periodic 
 payment, namely : — 
 
 M = A,/(?f^) 
 
 is described in Chapter YI. 
 
INTRODUCTION 3 
 
 FoRMUL.E AND SYMBOLS. The svmbols adopted by the author 
 differ somewhat from those given in the books on algebra, and 
 in other mathematical works. They have, however, been 
 chosen after much consideration in order to afford some 
 indication of the factors they represent. The very full treat- 
 ment which is given to the various formulae is due to the fact 
 that they are indispensable if a calculation has to be made at 
 any rate per cent, not included in the published tables of 
 compound interest. A detailed explanation of the symbols and 
 formulae is contained in Chapter X, dealing with the standard 
 calculation forms prepared by the author. 
 
 Logarithms. Throughout the book, the method of calcula- 
 tion is entirely by logarithms, since any attempt to arrive at 
 the results by arithmetical methods would involve a serious 
 waste of time, and a greater liability to errors in computation. 
 The use of logarithms is fully explained in the usual 
 arithmetical works, and also in the introduction to most of the 
 published tables of logarithms, but a short chapter (No. II) has 
 been included in order to make the book self-contained. There 
 is not anything at all difficult in the use of a table of logs. : 
 which is merely a very much neglected " ready-reckoner." 
 There are several good tables giving seven-figure logarithms 
 of the numbers from 1 to 108000. 
 
 Mathematical Tables. There are many published tables 
 of compound interest, which may be used to facilitate the 
 various calculations, and which may be divided into three 
 groups, namely : — 
 
 (1) Tables giving the actual values of £1, and of £1 per 
 annum, for various periods at stated rates per cent, per annum. 
 These tables are valuable in proportion to the number of rates 
 per cent, for which the actual values are given. In using all 
 such tables, a table of logs, is also required. In England, the 
 one most generally used is known as Inwood's Tables (21st 
 
 ' edition, 1880), and the new edition by Schooling (1899).* 
 
 (2) Tables in tvhich the actual values are not given, but 
 ivhich contain their logarithndc equivalents. The tables of 
 M. Fedor Thoman are of this type, and they are especially 
 valuable, because they are worked out for many intermediate 
 rates per cent, not given in Inwood's and other similar tables 
 of compound interest, and also because they enable one to 
 
 * In America, Tables of Compound Interest, Discount, Sinking Funds, 
 Annuities, etc., by Charles E. Sprague. 
 
4 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 dispense with a table of logs., except as regards the actual sums 
 of money involved in the calculation. They are particularly 
 useful because all values are reduced to two factors only, 
 namely, E^, and a", by various combinations of which all the 
 calculations may be made. The derivation and use of these 
 tables are fully explained in Chapter IX. 
 
 (3) Tables icorked out on the "ready-reckoner" principle, 
 giving, for example, the sinking fund instalments, or the equal 
 annual instalments of principal and interest combined, for £1, 
 and multiples of £1, for various periods of years, at various 
 rates per cent. Such tables may be very useful to some, but 
 they have not any educational value whatever, and it is 
 doubtf vil if they effect any actual saving of time when compared 
 with the other types of tables, especially Thoman's, which are, 
 when possible, always used by the author. The practical value 
 of tables of this kind is limited by the number of rates per cent, 
 actually worked out in detail, and the same applies to Inwood's 
 and Thoman's tables. 
 
 If a problem be required to be worked out at any rate per 
 cent, not given in any published table, such problem is 
 impossible of solution by anyone not acquainted with the 
 mathematical principles wpon which all such tables are based. 
 The object throughout has been to reduce these mathematical 
 principles to the very simplest form, and to give such minute 
 instructions, and to provide such standard forms of calculation, 
 that anyone acquainted with the ordinary rules of arithmetic, 
 and the use of a table of logs., may obtain the result required. 
 
 Standard Calculation Foems. A special feature of the 
 book is the series of standard forms, which have been specially 
 prepared by the author, and by means of which all the calcula- 
 tions in the book have been made. They are fully described in 
 Chapter X. The advantage of using these forms is that one 
 or all of the three methods given on each form may be 
 adopted ; and it is generally advisable to make the calculation 
 in two ways in order to prove the accuracy of the result and 
 also to avoid any possible error due to a misprint in the mathe- 
 matical table used. The three methods shown in each form 
 are : — 
 
 A. by the mathematical formula. 
 
 B. by tlie published tables of com])Ound interest. 
 
 C. l)y Thoman's Logarithmic Tables. 
 
 and ill ;il1 cases tlio calculations are made by logarithms. The 
 arithmetical method, based iipon the published tables, is subject 
 
INTRODUCTION 5 
 
 to error, and is therefore unreliable. A supply of these forms 
 is invaluable to anyone requiring to make many calculations 
 of this nature, owing to their uniformity and also because they 
 avoid any reference as to the particular method to be adopted. 
 The formula, after a time, suggest the method. As a general 
 rule the author uses, in the first instance, method C, by 
 Thoman's Tables, as being the shorter, and also because these 
 tables include a greater number of fractional rates per cent., 
 than the ordinary published tables of compound interest. The 
 factors being expressed in their log. values, a reference to the 
 log. table is saved. The result is generally proved by logs, by 
 method B, using the ordinary published tables of compound 
 interest. In very few cases is it necessary to use method A, by 
 formula, when the rate per cent, is worked out in Thoman's or 
 other tables, but where the calculation is required at a rate per 
 cent, not given in either table, method A, by formula, is the 
 only one available. It is therefore necessary to become fully 
 acquainted with the method by formula, and to use it to prove 
 the result obtained by Thoman's method C, where it cannot be 
 proved by method B, owing to the fact that the particular rate 
 per cent, is not included in the table available. When it is 
 required to ascertain the number of years with accuracy, the 
 use of the formula is imperative, and the same applies to 
 problems in which the rate per cent, is required. Yery 
 minute instructions as to the use of the forms are given in 
 Chapter X, which contains also ten standard forms by which to 
 ascertain the rate per cent, or the number of years. 
 
 Pro forma Accounts of Sinking Funds. Throughout the 
 book the author has repeatedly laid great stress upon the 
 supreme importance of following up the original calculation of 
 the annual instalment by at once preparing a pro forma account 
 showing year by year, how the fund should accumulate until 
 maturity. To make these accounts fully answer their object 
 they should be copied into a book kept solely for the purpose 
 of preserving a permanent record of all such accounts, and not 
 in the current ledger. This course will save endless trouble in 
 future years. If any adjustment be made in the fund at any 
 future time an amended pro forma account should be prepared 
 and a reference made to the original account. If a copy of each 
 calculation be forwarded to the Local Government Board it 
 will materially assist the officials and simplify, if it does not 
 entirely avoid, much subsequent correspondence between the 
 Board and the local authority. Several pro forma accounts 
 
6 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 have been prepared relating to examples given in the book, not 
 only with regard to a normal sinking fund, but also as to an 
 adjustment of the fund due to a variation in the period of 
 repayment, the rate of accumulation and the income from 
 investments. It is in such cases of adjustment that the provision 
 of an account of this nature, showing the effect of the various 
 changes until maturity, becomes particularly valuable. 
 
 The Repaymext of Loan Debt. Having in the earlier 
 chapters described the methods of ascertaining the working 
 formulse and rules relating to the various classes of calculations, 
 actual problems are next considered, beginning with the re- 
 payment of the loan debt of local authorities, taking as a basis 
 the three alternative methods laid down in Sec. 234 of the 
 Public Health Act, 1875. This section is a very concise 
 statement of such methods of repayment, especially when 
 supplemented by the non-accumulating sinking fund first 
 mentioned in the model clauses inserted by the Local Govern- 
 ment Board in provisional orders about the year 1893 and 
 which have since been applied to many special Acts. The 
 effects of the methods above mentioned are then fully discussed 
 both as regards the lender and the rate or revenue account of 
 the undertaking, illustrated by examples worked out in detail. 
 
 These three alternative methods apply equally to the re- 
 payment of the debt of privately owned commercial and 
 financial undertakings, although the conditions in such cases 
 are much more elastic and variable than is the case with local 
 authorities. This is fully discussed in Chapter XIII. 
 
 Problems eelatixg to Sixkixg FrxDS. The remainder of 
 the book is occupied by the discussion of actual problems relating 
 to sinking funds proper, since the instalment and annuity 
 methods do not involve the accumulation of any such fund but 
 provide for the actual periodical repayment to the lender. Very 
 few complications are likely to arise in the case of the instalment 
 method, and any variations in the annuity method will follow 
 the general rules as to a simple annuity. Such problems 
 concern the amount in the fund at any time, the rate of 
 accumulation of the fund, the rate of income to be received 
 upon the present investments representing the fund, the period 
 of repayment, or a combination of any or all of these factors. 
 
 The amount in the fund at any time may be the correct 
 calculated amount which should stand to the credit of the fund, 
 or may vary therefrom, resulting in a deficiency or a surplus. 
 
INTRODUCTION^ 7 
 
 A deficiency in the fund may be due to a fall in value, or a loss 
 upon the realisation, of an investment representing the fund, 
 but may also be caused by the accumulation of many minor past 
 deficiencies in the annual income received from the investments ; 
 and, although it does not now often occur, may be due to a 
 deficiency in the annual instalments set aside in past years. 
 Cases have occurred, within the knowledge of the author, where 
 the provision of a sinking fund in relation to an old loan has 
 been entirely overlooked. 
 
 A surplus in the fund may arise iu several ways ; either by 
 an increased rate of. accumulation or by the payment into the 
 fund of the proceeds of sale of part of the assets representing 
 the security for the loan, or a realised profit upon the sale of 
 an investment. In the case of commercial and financial under- 
 takings, a surplus may arise upon the withdrawal of part of the 
 loan from the operation of the fund. Two typical examjjles of 
 this nature are very fully discussed in Chapter XVIII. 
 
 In all such problems it is first necessary to ascertain the 
 actual position of the fund at the time the adjustment is required 
 to be made, and this may be expressed in terms of the present 
 investments and the future annual increment to accrue to the 
 fund. The problem may be simplified by treating, as one 
 factor, the " Annual Increment '' of the fund which consists of 
 the annual instalment and the income to be received from the 
 present investments, whether the rate per cent, of such income 
 is the same as the rate of accumulation or is different. Any 
 variation in such rates may continue during the whole of the 
 unexpired portion of the repayment period or for a portion of 
 the period only. The term " Annual Increment " is fully 
 discussed in Chapters XIY. and XXII. 
 
 The principal causes giving rise to a necessity to make an 
 adjustment of the fund are variations in the rates jDer cent, of 
 accumulation or of income upon the present investments and 
 variations in the period of repayment, or a combination of both 
 rate per cent, and period. Of the two causes a variation in 
 either of the rates per cent, is the most probable for many 
 obvious reasons, and it is very important that this should be 
 carefully observed and immediately corrected, in order to avoid 
 the necessity at some future time of having to make a substantial 
 adjustment due to the accumulation of small errors. The 
 longer a deficiency is allowed to accumulate the greater becomes 
 the resulting burden imposed upon the correspondingly reduced 
 number of the final years of the redemption period. 
 
8 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation of a Typical Sinking Fund. Iu order to 
 provide an example wliicli may be used to illustrate the wliole 
 of the above problems, Chapter XY, Calculation (XY) 1, shows 
 the method of ascertaining- the annual instalment relating to a 
 loan of £26,495 repayable in 25 years with an assumed rate of 
 accumulation of 3| per cent, per annum. (xA^uthor's standard 
 calculation form, No. 3 x.) 
 
 Methods of Adjustment. Throughout the book the fact 
 that the particular method adopted is not the most direct one 
 has been left entirely out of consideration, provided it has an 
 educational value. In all cases, however, the shorter and more 
 direct method has been shown and the results by the two 
 methods compared. In the case of the adjustment due to a 
 deficiency in the fund, in Chapter XY., four methods are given, 
 which are summarised at the head of that chapter. The 
 adjustment of a deficiency has been treated in this exhaustive 
 manner, far beyond its relative importance, in order to present 
 to the student a practical example illustrating the interdepend- 
 ence of the present value and future amount of £1 and of 
 ■£1 per annum. In the above example, as well as in later ones, a 
 statement has been prepared showing the various stages by 
 which the amended annual instalment is ascertained, and this is 
 followed in all cases by a further statement showing the final 
 repayment of the loan by the operation of the sinking fund and 
 the amended annual instalment rendered necessary by the 
 variation in the original conditions. 
 
 Wherever required, the method has been reduced to a series 
 of stages briefly stated giving a reference to the individual 
 calculations. In the earlier parts of the book the actual details 
 of the calculations are given in full or in the Appendix, but in 
 later chapters only the final results are given owing to 
 consideration of space and also because similar examples have 
 previously been worked out. 
 
 The Antjtual Increment Methods. The adjustments next 
 considered are those due to a variation in the rate per cent, 
 either of accumulation or income from investments, a variation 
 in the period of repayment, or a combination of the two factors 
 of rate per cent, and period. As in the case of a deficiency or a 
 surplus in the fund, the amended annual instalment is first 
 ascertained hj the deductive method fully described in 
 Chapter XIX, which is based upon the consideration of the 
 whole of the factors governing the fund. The same result is 
 also shown by the annual increment (balance of loan) method 
 
INTRODUCTION 9 
 
 fully described in Chapter XXII. In the case, however, of a 
 variation in the rate of accumulation accompanied by a 
 variation in the rate of income from investments the latter 
 factor is eliminated by merging it in the annual increment and 
 dealing only with that annual sum. The varying rate of 
 accumulation then becomes the only outstanding factor, and it 
 is therefore possible to deduce a method which has been called 
 "the annual increment (ratio) method," depending upon the 
 ratio existing between the original and amended rates of 
 accumulation. The whole of the calculations by the annual 
 increment (ratio) method relating to a variation in the rate per 
 cent, only, and also to a variation of the period of repayment 
 only, bear a strong family likeness and are capable of being 
 reduced to simple rules and formulse, and this has been shown 
 in detail. Having in Chapter XXVI discussed a combined 
 variation in both factors of rate per cent, and period and having 
 again deduced a formula therefrom the whole of the formulae 
 so obtained have been reduced to simple rules. 
 
 The Dates of Boreowing and Hepayment. Up to this 
 point all possible causes of the adjustment of a sinking fund 
 have been exhausted, but the subject has been treated from the 
 purely mathematical or actuarial standpoint, namely, that all 
 loans are borrowed in one sum at the beginning of the financial 
 year and that the annual instalments are set aside at the end of 
 such year. The actual practical conditions are next dealt with, 
 namely, that the loan is, as a rule, borrowed over a period of 
 years, in various amounts and at various dates in any year, and 
 is repayable sometimes over a period of years, but often on a 
 given date. The subject is further complicated by the fact that 
 varying periods are allowed for the repayment of loans 
 sanctioned for different classes of outlay depending upon the 
 life, or duration of continuing utility, of the individual works. 
 And this varying period of repayment may be, and often is, 
 complicated by practical variations in the dates of borrowing. 
 This part of the subject has therefore been divided, by dealing 
 first with loans authorised for outlay of one character only 
 where the problem is not complicated by different periods of 
 repayment due to the life of the asset. The problems relating 
 to the dates of borrowing are sub-divided as follows : — 
 
 (a) Where the loan is borrowed over several years, in one 
 sum in each year, and is repayable over a term of years 
 in a prescribed period from the several dates of borrowing. 
 
 Chapter XXVIII. 
 
lo REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 (6) Where the loan is borrowed over several years, in one 
 sum in each year, and is repayable in one sum on a 
 certain specified date. Chapter XXIX. 
 
 (c) Where the loan is borrowed in one or more years in 
 varying amounts and at varying dates in each year and 
 is repayable in one sum on a certain specified date, and 
 it is further required that the revenue or rate account 
 of each year of borrowing shall be charged with a 
 proportionate part of the annual sinking fund instalment. 
 
 Chapter XXX. 
 
 In the case of loans borrowed over a series of years, where 
 the repayment is spread over a period equal to the extended 
 years of borrowing, the amounts borrowed in each year may be 
 treated as individual loans, and the only points to be considered 
 are administrative, and relate to the number of sinking funds, 
 namely, whether it is preferable to keep a separate sinking fund 
 for each year's borrowings or to keep only one fund for the 
 total loan. This is fully discussed in Chapter XXYIII, and, 
 as there stated, cannot be applied to the redemption of stock. 
 
 In the case of loans borrowed over a period of years, raised 
 by the issue of stock redeemable on a fixed date the several 
 sinking fund instalments, although commencing at various 
 dates, yet mature on the same date. The enquiry is still 
 confined to loans in respect of outlay of one nature and having 
 a uniform period of repayment. This class has been sub- 
 divided into two groups, and is fully discussed in Chapter 
 XXIX. 
 
 1. Where the date of repayment is known at the time the 
 
 money is borrowed. 
 
 2. Where the date of repayment is fixed after the sinking 
 
 fund has been in operation for a number of years, and 
 
 an adjustment becomes necessarj'. 
 The apportionment of a part of a full year's instalment to 
 be charged against the revenue or rate account of the year in 
 which the money is borrowed is treated fully in Chapter XXX. 
 As a rule, this may be, and generally is, ignored; but the 
 particular circumstances in connection with a large loan may 
 render it advisable to make a charge of this nature. There are 
 several interesting features in the method which is illustrated by 
 the example in Chapter XXX, and which may be compared 
 with the instalments to be set aside when the year of borrowing 
 is not charged with any such proportional annual instalment, 
 as in Chapter XXIX. Stated briefly, the effect is to ante-date 
 
INTRODUCTION ii 
 
 the charge to revenue or rate and to impose an increased 
 burden upon the years of borrowing. The third year of the 
 sinking fund period is charged with the same amount under 
 each method because the repayment period is assumed to 
 commence at the conclusion of the first year of borrowing. 
 Such increased annual burden during the earlier years operates 
 by way of relief to the remainder of the repayment period, 
 but only to a slight extent. This problem furnishes another 
 example of an adjustment being required in consequence of 
 irregular contributions to the fund during the earlier years. 
 
 The Life of the Asset, and the Equation of the Period 
 OF Repayment. Having dealt with problems relating solely to 
 the adjustment of the sinking fund, owing to causes of a purely 
 actuarial or mathematical nature, there is still to be considered 
 the more difficult subject of the variation in the periods allowed 
 for the redemption of loans for large public works, where each 
 component part of the outlay has a different life or duration of 
 continuing utility. This variation in the redemption period 
 has not any disturbing effect when the loan is authorised for 
 one class of outlay only, or is in respect of several classes of 
 outlay, each having the same period of repayment. But loans 
 are now often authorised for large public works which include 
 various classes of outlay, each class having its oAvn redemption 
 period, based upon its duration of continuing utility, and also 
 forming a variable proportion of the total cost, and it is 
 required that the total loan shall be repaid on the same date. 
 In such cases it becomes necessary to ascertain the equated 
 period of repayment. The same necessity arises on the consoli- 
 dation of existing loans repayable at various future dates, but 
 in such cases the problem is further complicated by the amounts 
 then standing to the credit of the individual sinking funds, 
 the value of the investments representing each fund, the rate of 
 income arising therefrom, and also the incidence of the present 
 redemption charges upon different departments of the local 
 authority. It may be stated generally that the problem of the 
 equation of the period of repayment applies equally to all such 
 cases and that in fixing the equated date of repayment there are 
 two interests to be considered, namelj^, the loanholder, who looks 
 only for the due payment of his principal, and the annual 
 interest thereon, and the individual ratepayer who is required 
 to provide his proper portion of the annual amount which 
 Parliament, or the government department concerned, has laid 
 down in principle as the annual wastage of the assets created 
 
12 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 out of the loan. This annual wastage of the asset is imported 
 into the problem owing to the fact that the period allowed for 
 the repayment of the loan is based upon the life, or duration of 
 continuing utility, of the asset. As regards the loauholder, the 
 problem is a simple one. In all cases he will be repaid, at some 
 future date, the amount which he originally advanced to the 
 local authority, and, in addition, he will receive until the 
 repayment of the loan, interest at the rate per cent, originally 
 fixed. The only question remaining therefore, so far as he is 
 concerned, is the relation between the rate of interest agreed to 
 be paid by the local authority and the rate per cent, obtainable 
 upon the open market when the loan is proposed to be repaid. 
 This arises only upon the consolidation of existing loans repay- 
 able at fixed future dates where the effect of consolidation is to 
 vary, and generally to anticipate, the date at which the loan 
 was originally repayable. As regards the loanholder, therefore, 
 the most important factor is the period during which he will 
 continue to receive interest upon the loan at the present rate 
 payable by the local authority. If the present rate so payable 
 be a high one and the current rate, to be obtained upon the open 
 market, at the time when the local authority propose to repay 
 the loan, be expected to be lower, the loanholder will naturally 
 object to any variation of the- original conditions, and, per 
 contra, he will gladly accept an earlier repayment of the loan 
 if thereby he may expect to obtain a higher rate of interest 
 upon his investment. Consequently the loanholder must be 
 consulted, and his consent obtained, before any change be made 
 in the original conditions upon the consolidation of loans. This 
 uncertainty as to the future rate of interest is one of the reasons 
 why Parliament has, for some years past, refused to sanction 
 the issue of an irredeemable stock in consequence of the difficulty 
 in applying the amount in the sinking fund to its proper 
 purpose. The stock can thus only be redeemed by purchase 
 upon the open market, and the premium paid upon such 
 occasions cannot be taken out of the sinking fund, but must 
 be charged against the revenue or rate account of the current 
 year. 
 
 The ratepayer, on the contrary, is in a very different 
 position, in that the money paid to the loanholder by way of 
 interest \\\wu tlie loan, and the annual sums set aside out of 
 revenue or rate to redeem the debt are paid by him. But the 
 ratepayer comes and goes, whilst the loanholder goes on for 
 ever, or at least until his loan is repaid. The loanholder 
 naturallv cinisidcrs the value of his investment and tlie interest 
 
INTRODUCTION 13 
 
 to be derived therefrom, and the state of the money market both 
 at the present time and in the future are to him very important 
 factors. The ratepayer, on the contrary, considers only the 
 annual amount paid by him by way of rate, and compound 
 interest is to him a negligible, if not an unknown term. In 
 addition, he is never consulted individually as to the annual 
 amount of rate which he may be called upon to pay. He may 
 be invited to attend a meeting called to approve or disapprove 
 of a Bill to be laid before Parliament to authorise the spending 
 of money on capital account, but he is generally ignorant of 
 the matter, and is too busy trying to earn the amount he has 
 to pay by way of rate, to attend any such meetings. The result 
 is that the final adjustment is left entirely to the officials of the 
 local authority subject only to the control of Parliament or the 
 Local Government Board, and the next .step therefore is to 
 investigate the methods generally adopted in order to arrive at 
 the equated period of repayment and the consequent amended 
 annual sinking fund instalment to be charged to revenue or 
 rate. 
 
 Before doing so, however, it should be pointed out that 
 any necessity to fix the equated period did not arise to any 
 great extent until it became the common practice of local 
 authorities to issue stock or to consolidate existing loans re- 
 payable at various dates. Prior to that time, any variation 
 in the periods of repayment alloAved for different classes of 
 outlay was met by keeping separate funds for each amount of 
 loan having the same repayment period and allowing each fund 
 to' mature at the due date. The relation between the life of 
 the asset and the consequent annual loan charge upon the 
 revenue or rate accounts of successive years is fnlly discussed in 
 Chapter XXXII, where it is found that the variation in the 
 periods of repayment allowed is not of itself a cause of an 
 equation being required, which depends upon a combination of 
 two factors, namely, the variable period of repayment and the 
 obligation to repay A^arious loans on one instead of on different 
 dates. The problem arising on the consolidation of stocks or 
 loans repayable at various dates is exactly similar in principle 
 althouo-h arising in a somewhat different manner, but is further 
 complicated by the amount in the fund at the time of making 
 the adjustment. 
 
 The Equation of the Peeiod of Repayment. The 
 equation of the period of repayment has been considered from 
 two points of view, namely, one relating to the method of 
 
14 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 ascertaining the equated date and tlie other to the incidence of 
 the annual burden upon the revenue or rate account. These 
 two points are fully treated in Chapters XXXII, XXXIII and 
 XXXIV. The method generally adopted to find the equated 
 period is the arithmetical one known as the " equation of pay- 
 ments, ■" which is fully described in Chapter XXXII. It is 
 there proved, by two examples worked out in detail, that the 
 equated period as generally adopted is not the true equated 
 period and that the effect of adopting it is to extend the period 
 of repayment beyond the true or mathematically equated period. 
 This may not be important in many cases, but may be extremely 
 so in the case of very large loans ; and, if it be necessary to make 
 such an adjustment at all, it is surely imperative that the 
 principle upon which it is made is scientifically accurate. 
 
 Having described the proper method of finding the true 
 equated period, Chapter XXXIII is occupied with an 
 examination, illustrated by the actual example used in Chapter 
 XXXII, of the effect of the generally adopted practice of fixing 
 the amended annual sinking fund instalment by spreading the 
 burden equally over the whole of the equated period as if it 
 related to an original loan, repayable in such period, the whole 
 of the loan representing outlay of one character only, having a 
 life or period of utility of that length. The method is a simple 
 one, but is wrong in principle although it has received the 
 approval of many years' adoption. 
 
 If the preliminary stages in the sanction of a loan be care- 
 fully reviewed it will be recognised that much thought and care 
 are expended in determining the proper periods to be allowed 
 for the repayment of loans authorised for different classes of 
 outlay. The whole question is still in a transition state, and 
 great divergence exists in the conditions now in force ; the only 
 recognised factors being that in future the annual charges for 
 redemption of the debt shall bear a definite relation to the life 
 of the asset, with a further extension of the principle, that even 
 in the case of works of almost permanent utility the repayment 
 shall not extend beyond a certain number of years. This latter 
 requirement is to protect the interests of future generations of 
 ratepayers. The relation between the period of repaynu'ut and 
 the life or duration of continuing utility of the asset is imposed 
 in order to ensure that the present generation shall contribute, 
 year by year, the proper portion of the wastage of the asset. In 
 the case of a loan raised for works comprising outlays of varying 
 nature with varying periods of repayment and where separate 
 sinking funds are kept in respect of each class of outlay the 
 
INTRODUCTION 15 
 
 principle is carried out exactly because the earlier years bear 
 the heaviest burden, as they should do, owing to the fact that 
 classes of outlay having a short life Avill be worn out and 
 require replacing at the end of the period. Under these condi- 
 tions the loan is entirely repaid by the time the works cease to 
 be of utility or are worn out. 
 
 The Equation of the Peeiod of Repayment, and the 
 Incidence of Taxation. The Annual Instalment. Under 
 the present practice on equation the above principle is departed 
 from, and the burden is spread equally over the equated period 
 with a total disregard to any period of utility, and as demon- 
 strated in Chapter XXXIII, there is actually considerable relief 
 to the early years of the equated period as well as an entire 
 removal of any burden during the years of the original period 
 beyond the equated period. As a consequence, the whole of 
 this relief is imposed as an additional burden of considerable 
 magnitude upon the final years of the equated period. There 
 is here a total reversal of the generally accepted principle of 
 spreading the repayment of the loan over the period represented 
 by the life of the asset, accompanied by an absolute injustice to 
 a section of the ratepayers. By adopting the equated method 
 in general use, as applied to works consisting of various classes 
 of outlay, and also on the consolidation of loans, the present 
 generation relieve themselves of a liability to contribute their 
 fair share of the burden which has been fixed after careful 
 enquiry by Parliament; and thereby impose an extra burden 
 upon future years. In addition they also postpone the repay- 
 ment of the loans with shorter periods which would have been 
 repaid during the earlier years, and the result is that money 
 cannot properly be reborrowed to replace assets with a shorter 
 life than the equated period because, when they are worn out, 
 the original loan has not been repaid by means of the sinking 
 fund. A remedy for this state of affairs, so far as the annual 
 instalment only is concerned, is pointed out in Chapter XXXIV, 
 namely, by spreading the burden over the equated period, not 
 by an equal annual instalment, as is the present practice, but 
 by instalments of varying amounts approximating to those 
 originally imposed which were based upon the life of the asset. 
 The principle of this method is to ascertain, first, the amount 
 of loan which will be provided at the end of the equated period, 
 by the accumulation of the annual instalments as originally 
 fixed. The amount of such instalments at the end of the 
 number of years for which they would have been set aside under 
 
i6 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 the original conditions slionld be ascertained, and if any of 
 these periods are shorter than the equated period, the amount 
 at the end of such periods should be further accumulated until 
 the end of the equated period. The difference between the 
 amount of loan so ascertained and the total amount of the loan 
 ultimately repayable will represent the amount to be provided 
 by supplementary annual instalments to be spread over the 
 equated period, due to the fact that the equated period is 
 shorter than the original periods allowed for the repayment of 
 the parts of the loan with longer periods, and that the relief 
 afforded by the equation to these later years should be borne 
 equitably by each year of the equated period. Strictly 
 speaking, such supplementary annual instalments should be 
 graded in some manner proportionate to the original annual 
 instalments which were based upon the life of the asset, and 
 although the calculation is fullv described it is somewhat 
 intricate, and the justice of the case will generally be met by 
 spreading this supplementary annual instalment equally over 
 the equated period. 
 
 The Equation of the Peeiod of Repayment, and the 
 Incidence of Taxation. Interest upon the Loan. The result 
 of spreading the annual instalment over the equated period in 
 proportion to the instalments before equation is shown in Table 
 XXXIA', J, where the original annual instalments are corrected 
 in this manner. On referring to Table XXXIII, B, it Avill be 
 seen that the effect of equating the period is to throw a heavy 
 additional burden upon the final year's of the equated period 
 in respect of interest upon the loan. In Chapter XXXIY, 
 a method is described of distributing the redemption charge 
 (the annual instalment) equitably over the equated period, and 
 in Chapter XXXV a similar course is adopted with regard to 
 the interest upon the loan, with the result shown in Table 
 XXXY, C. By combining the correctly equated annual instal- 
 ments shown in Table XXXIV, J, with the correctly equated 
 annual interest charges shown in Table XXXV, C, the total 
 annual loan charges during the equated period may be ascer- 
 tained as shown in Table XXXV, F. The subject is so 
 important that the result has been shown in graphic form, 
 which is fully described and explained in Chapter XXXV. 
 In order to express in actual values the effect of the above 
 adjustment both as to the instalment and interest upon the 
 loan it may be stated in terms of annual rate. It has been 
 "•iven in evidence before a Parliamentarv Committee that in one 
 
INTRODUCTION 17 
 
 particular case of consolidation of loans the immediate effect of 
 an equation of the period of repayment was a saving of three- 
 pence in the £ in the annual rate. In this connection it should 
 be remembered that as shown in Table XXXIII, C, there is 
 not any difference in the annual charges for interest upon the 
 loan during the early years of the equated period, before and 
 after equation, but that the decrease is entirely in the annual 
 instalment. Adopting the figure of threepence in the pound as 
 a standard, the result in the present case would be as follows, 
 including the interest upon the loan as well as the annual 
 instalment. The following figures may be converted into 
 American currency by adopting the equivalent of 2^ cents to 
 the dollar instead of 6d. in the pound : — 
 
 Decreased rate Increased rate 
 
 Original redemption period. per £. of per £ of 
 
 Annual Value. Annual Value. 
 
 ( 5 years ... 652 pence 
 Equated period ...-10 years ... 3-20 pence 
 
 ( 8 years ... ... 11 "85 pence. 
 
 Post-equated period ... 6 years ... 9'76 pence 
 
 16 years ... 2 79 pence 
 
 Actual Calculations. The method adopted throughout 
 the book is to insist upon a very careful scrutiny of the present 
 and future conditions and also of the actuarial and mathe- 
 matical principles involved. It is very important to prove all 
 ascertained results, both as to method and accuracy of computa- 
 tion, seeing that the actual working out of the fund will occupy 
 many years and the effect of any present error will be serious 
 if it be allowed to accumulate for any length of time. Mere 
 repetition of the actual calculation is not sufficient. A far 
 preferable method is to work out the operation of the fund 
 year by year by the arithmetical method as shown in the pro 
 forma accounts already referred to. This should be done in all 
 cases without exception before the problem is finally disposed 
 of, but the method is laborious and much time is wasted if the 
 original calculation be wrong. The best way is to prove the 
 result by mathematical means which are much shorter, either 
 by adopting an alternative method, of which many instances 
 are given in the various chapters, or by comparing the amended 
 with the original annual instalment and accounting for the 
 difference. The actual arithmetical calculation of the pro 
 forma sinking- fund accounts mav be left to a iunior official, 
 but it will save him considerable time if the senior first ascer- 
 
i8 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 tains the amount in the fund at the end of every five or ten 
 years by means of the tables or otherwise as described in the 
 various chapters. The whole of the calculations in the book 
 have been verified in this manner and in many cases the method 
 of proof is shown in detail. In cases, however, where it is not 
 shown the verification has been made and is only omitted for 
 want of space. 
 
Section I. 
 Mathematical Principles. 
 
CHAPTER II. 
 LOGARITHMS. 
 
 Advantage of use of logs. History. Connection between 
 
 logs. and arithmetical and geometrical progressions. 
 
 Definition. Various arithmetical calculations by logs. 
 
 Logs, of numbers between even multiples of 10. 
 
 Characteristic. Mantissa. Method of dividing a log. 
 
 with a negative characteristic. Method of dividing one 
 
 log. by another. 
 In a work of this nature, dealing with calculations which 
 are based upon the higher branches of mathematics, it is 
 obvious that the ordinary methods of arithmetic are inadequate, 
 and that the aid of logarithms must be invoked even if the 
 fullest use be made of the various published tables of compound 
 interest. There is a limit to the number of rates per cent. 
 which may be included in any table, and it is often required to 
 make a calculation at a rate per cent, not worked out. In such 
 cases it is necessary to revert to the original formulae, all of 
 which involve raising numbers, containing as many as five or 
 six figures, to the power of the number of years, and the method 
 of continued multiplication becomes too laborious and uncertain. 
 Even when using the tables it is always necessary to multiply or 
 divide by the numbers (containing five or six figures) given in 
 the tables, and a great saving of time and labour is effected by 
 doing this by the aid of logarithms. But beyond the little time 
 expended in becoming familiar with the method of using such a 
 table, there is not any greater difficulty than in using any 
 ordinary commercial ready reckoner. 
 
 This chapter deals only generally with the subject of 
 logarithms, and as the use of this book cannot be complete 
 without a copy of Inwood's or other similar tables, and a 
 reliable table of logs., for a fuller acquaintance, reference must 
 be made to the introductory chapter which will be found in most 
 log. tables, or else to some good advanced arithmetic. 
 
 Logarithms were invented by John Napier, Baron of 
 MerchivSton, in Scotland, who published his first work in 
 Edinburgh in 1614. This work contained only the logarithms 
 of natural sines, and are not what are now known as Naperian 
 or hyperbolic logarithms, which are used in mathematical 
 investigations only and are not the logarithms in common use 
 
22 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 to-day. jS^apier died in IGIT, and a further work by bim, 
 edited by bis son, was published in 1619, 
 
 The first published table of decimal or common logarithms 
 was published by Henry Briggs, Professor of Geometry at 
 Gresham College, London, and afterwards Savilian Professor 
 of Geometry at Oxford, Avho visited jN"apier in 1615. Briggs 
 published his table in 1617 (after the death of Napier), and 
 these logarithms which are in common use to-day are calculated 
 to a base of 10. Briggs' first tables contained only the 
 logarithms of numbers from unity to 1,000 to 14 places of 
 decimals. The arithmetical calculation of logarithms as used 
 by Briggs, is a very laborious process, and it was not until 
 1628 that the table was extended for numbers from unity to 
 101,000 by Briggs and Adrian Vlacq, of Gouda, in Holland. 
 But the arithmetical method of Briggs Avas afterwards super- 
 seded by shorter methods depending upon more advanced 
 mathematical rules, and there are now, as the result of all this 
 labour, very accurate tables which are in universal use, and by 
 means of which very intricate calculations may be made by 
 very simple methods. 
 
 The principle upon Avhicli a logarithm is based is exceedingly 
 simple, and is founded upon the relation existing between an 
 arithmetical and a geometrical progression. An arithmetical 
 progression is a series of numbers each of which is found by 
 adding a constant number to the previous term in the series, the 
 constant number so added being called the ratio. A geometrical 
 progression is a series of numbers each of which is found by 
 multiplying the previous term in the series by a constant 
 number, such constant multiplier being called the ratio. 
 Taking a series of numbers in geometrical progression, with a 
 ratio of 10, which is the one adopted in the Briggean or decimal 
 or common logarithms, and commencing the series with unity, 
 the following series is obtained : — 
 
 Geometrical 
 Progression, 1. 10. 100. 1,000. 10,000. 100,000. 1,000,000. 
 
 Taking another series of numbers in arithmetical progression, 
 witli a ratio of 1, and commencing the series with 0, the 
 f ollow^ing series is obtained : — 
 
 Arithmefical 
 Pro;jr('s.siui,, 0. 1. 2. 3. 4. 5. 6. 
 
 The above geometrical progression will now be re-written 
 expressing eacli term by the power of 10 wliich it represents, 
 and under it the above arithmetical progression, as follows : — 
 
LOGARITHMS 23 
 
 Geoinctricnl 
 Progression, 10° 10^ 10= 10^ 10* 10^ 10^ 
 
 Arithmetical 
 Progression, 12 3 4 5 6 
 
 It will be at once noticed that the index of each term in the 
 geometrical progression is the same as the corresponding term 
 in the arithmetical progression. 
 
 If it be assumed that the terms in the arithmetical series 
 are the logarithms of the corresponding terms in the first 
 geometrical series, this is exactly what Briggs did when he 
 adopted 10 as the basis of his system, as follows : — 
 
 The log. of 1. 
 
 = 
 
 10° 
 
 = 
 
 
 
 
 10. 
 
 = 
 
 101 
 
 = 
 
 1 
 
 
 100. 
 
 = 
 
 102 
 
 = 
 
 
 
 'V 
 
 
 1,000. 
 
 = 
 
 103 
 
 = 
 
 a 
 
 
 10,000. 
 
 = 
 
 104 
 
 = 
 
 4 
 
 
 100,000. 
 
 = 
 
 105 
 
 = 
 
 5 
 
 
 1,000,000. 
 
 = 
 
 106 
 
 = 
 
 6, 
 
 and so on, 
 
 and therefrom the definition of a common logarithm may be 
 expressed, viz., the logarithm of a number [to tJic base 10) is 
 the poiver or index to ivhich 10 has to be raised to yroduce that 
 number, for example : the logarithm of 10,000, to the base 10, 
 is the power or index 4, to which 10 has to be raised to produce 
 10,000, but as 10 is the common base to which all numbers 
 are reduced, the indices, or logarithms, only are required, and 
 the decimal part of this index is all that is given in the log. 
 tables. 
 
 The logarithms of numbers which are even powers of 10 may 
 be ascertained in the above simple manner, and attention has 
 been drawn to the enormous labour involved in calculating the 
 logarithms of the intermediate numbers. It is not necessary to 
 enquire deeper into the methods, but only to accept the tables 
 which are the product of that labour. The logs, which have 
 been already found may be used to illustrate the various 
 advantages of their use, taking familiar arithmetical calcula- 
 tions, using the actual numbers in the ordinary way, and then 
 repeating the calculations, using the logs, of the numbers 
 instead of the actual numbers, as follows : — 
 
 Miiltiplieatiori : 
 
 10x1,000 = 10,000. by logs., 1 + 3 = 4, or log. 10,000 
 Division : 
 
 100,000-100= 1,000. by logs., 5-2 = 3, or log. 1,000 
 
24 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Involution : 
 
 1003 =1,000,000. by logs., 2x3 = 6, or log. 1,000,000 
 
 Evolution : 
 
 V 1,000,000 = 100. by logs., 6-3 = 2, or log. 100 
 
 It will be seen therefore that by the use of logs. — 
 
 Multiplication'! / addition \ 
 
 Division | of ordinary \ subtraction I 
 
 T 1 ,• / 1 " -becomes-, -,.. -,. ,. (respective 
 
 Involution numbers j muitipiication ^ 
 
 Evolution / I division ' ^ 
 
 It is a common practice when multiplying together two 
 powers of 10, such as (10 x 1,000) to write down 1, and add 
 4 cyphers, thus, 10,000, being the sum of the cyphers in the 
 two numbers. This is actually a logarithmic method of calcula- 
 tion often used by people who do not know anything about 
 logarithms. This is the principle of the algebraical theory of 
 indices, of which the following is an illustration : — 
 
 Multiplication : x x x^ = x'^i+S) = .x* 
 
 Division : x^-^x" = .x*^" -* = x^ 
 
 Involution : {x'^ Y = x^^^^^ = x^ 
 
 Evolution: jx^ =x^''^^^ = x" 
 
 The above operations correspond to the previous illustrations 
 in which the actual powers of 10 were used. In the algebraical 
 form above, 10 has been replaced by a:, with the result that 
 similar logs, are obtained in each set of examples. 
 
 Up to this point only whole numbers have been considered 
 which are even multiples of 10, and of which the logs, are 
 whole numbers above unity, and it has been ascertained that the 
 logarithm of 10 is 1, that of 100 is 2, and so on for any even 
 power of 10. It is therefore obvious that the logs, of numbers 
 less than 10 must be fractions. This also applies to numbers 
 between 10 and 100, the logs, of which must be between 1 and 2, 
 and equally to numbers between any two consecutive powers of 
 10, which logs, consist of a whole number and a fractional part, 
 the whole number being the log. of the next loAver power of 10. 
 
 A logarithm then consists of two parts — the integral part, 
 which is called the Characteristic, and the fractional or decimal 
 part, which is called the Mantissa, and all logs, are expressed 
 in decimals, usually to 7 places. 
 
 The Mantissa (or fractional part) is always positive, and is 
 
LOGARITHMS 25 
 
 always tlie same for any one combination of figures, irrespective 
 of the place of the decimal point. 
 
 The Characteristic represents merely the position of the 
 decimal point in the number which the log. represents, and 
 changes only after passing each power of 10. The characteristic 
 is in all cases the power to which 10 has to be raised to produce 
 the next lower power of 10. 
 
 The logarithm tables contain only the mantissa part of the 
 logarithm corresponding to the particular combination of figures 
 forming the number of which the log. is required, and since 
 these figures may, according to the position of the decimal 
 point, represent either whole numbers or fractions, the 
 characteristic is positive in the case of a whole number, and 
 negative in the case of a fractional number. 
 
 It will be noticed, on referring to the logs, of the powers 
 of 10, referred to above, that the log. of 1,000,000 is 6, or one 
 less than the number of integral figures (seven) in the number, 
 and similarly with the other powers of 10, and the rule applies 
 generally, as will be seen by taking the logarithm of the 
 number 26495. 
 
 In the table of logs, opposite 26495 are the figures 423,1639 
 which is the mantissa of the log. of any number containing 
 the figures 26495 in this order, whether preceded or followed 
 by any number of cyphers. The actual position of the decimal 
 point determines the characteristic or integral part of the log. 
 as follows : — 
 
 Log. 26495 =4-423 1639 
 
 2649-5 =3-423 1639 
 
 264-95 =2-423 1639 
 
 26-495 =1-423 1639 
 
 2-6495 =0-423 1639 
 
 •26495 =1-423 1639 
 •026495 =2-423 1639 
 •0026495 = 3-423 1639, and so on. 
 
 On comparing the above logs, with the logs, of the powers 
 of 10 previously given, it will be noticed, for instance, that 
 264.95 (being above 100 and below 1,000) has the characteristic 
 
26 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 2 as previously explained, and it will be further noticed that 
 as the decimal point in the original number is moved place by 
 place to the left (equivalent to dividing the previous number 
 by 10) the characteristic of the logarithm is reduced by 1. 
 But in the case of 2'6-l:95 the characteristic becomes 0, and as 
 the number is further divided by 10 and the decimal point 
 moved still further to the left, it becomes -1, -2, —3, and so 
 on. The characteristic being the only negative part of the log., 
 the minus sign is placed over it instead of to the left. 
 
 A fflance at the above lof>-s. will show that the characteristic 
 follows two rules, viz. : — 
 
 (1) In the case of nuiiihers yreater than unity, the charac- 
 teristic is one les.s titan the 7iiimher of integral figures 
 in the number, and is always positive; and 
 
 (2j In the case of numbers less titan unity, tJte characteristic 
 is one more than the nutnber of cyphers after the decimal 
 point in the number, or is the same number as the place 
 from the decimal point which the first significant figure 
 occupies; and is alivays negative. 
 
 The usual published tables of common logarithms give the 
 mantissa for each number from unity to 108,000, and the logs, 
 of all numbers containing 5 figures may be found at one 
 reference. If the number of which the log. is required contains 
 more than 5 figures, the corrected log. is found by reference 
 to one of the tables of proportional parts given in the margin 
 of the tables, but all the published tables describe so fully how 
 this is done that it is not necessary to repeat it here. 
 
 There are also several other practical operations required 
 which are fully explained in the tables, amongst others, (1) 
 finding the antilog. or the number corresponding to any 
 logarithm, and (2) the method of dealing with logs, having 
 negative characteristics, either by addition or subtraction, 
 which follows the ordinary rules of algebra. 
 
 Special attention should, however, be given to the rules as 
 to multiplying or dividing a log. with a negative characteristic. 
 The following method of dividing such a log. is used by the 
 author in order to find the value of the factor R, and differs 
 from the method given in the tables, but is simpler. It is as 
 follows : — - 
 
LOGARITHMS 
 
 27 
 
 Having obtained the log. of EN (N = 20 years), viz., 2-987 8003 
 it is required to divide the log. by 20, in order 
 to obtain log R-, 
 
 proceed by adding 20, 20" 
 
 = 18-987 8003 
 
 Divide this log. by 20= 0949 3900 
 and deduct 1, to correct the addition of 20, 
 
 divided by 20, = l' 
 
 Leaving tlie required log. 1'949 3900 
 
 It is sometimes required to divide one log. by another, as in 
 Calculation XXXII, E., in order to find the number of years, N, 
 in an equated period at a given rate per cent., knowing tbe 
 value of the factors E^ ^nd E. If both the logs, are positive or 
 negative, they may be treated as ordinary numbers and tlie 
 corresponding logs, found in the usual way, but if their charac- 
 teristics are, one plus and the other minus in sign, they must be 
 reduced to the same sign. 
 
28 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER III. 
 
 SIMPLE AND COMPOUND INTEREST. 
 
 Simple Interest. An arithmetical progression. Formula. 
 Tables. Incidental use of the Tables. 
 
 Compound Interest. A geometrical progression. Derivation 
 OF THE Formula, A = P R^, relating to Compound 
 Interest, from the algebraical formula, l = ar ^"~'^\ 
 relating to a geometrical progression. Explanation 
 OF terms. Difference between the amounts of £1 and 
 
 OF £1 per annum at the end of 1 YEAR. " PrESENT 
 
 Value " compared with " Practical Discount." 
 
 Simple Interest, an Arithmetical Progression. Simple 
 interest is an arithmetical progression, and the amount of any 
 sum of money, at the end of any given term, may be ascertained 
 by continued addition of the interest upon the sum for one year, 
 or other period, at the stated rate per cent. It is the method 
 in general use in all commercial and financial transactions, 
 although in cases where balances in an account current are 
 struck at stated j^eriods, it may partake of the nature of 
 compound interest. The main feature of this method is that 
 the calculations may relate to varying sums, varying times, and 
 varying rates per cent., and are expressed by the formula : — - 
 
 Principal X rate per cent, per annum x years 
 
 Interest, 
 
 100 
 
 and the ascertained amount of interest is stated in the same 
 terms as the principal, whether pounds sterling, shillings 
 sterling, dollars or other currency. All such calculations are 
 extremely simple, and many tables are published giving the 
 amounts of interest on varying amounts of principal for varying 
 periods, whether days or years. The above formula is the one 
 used to calculate the amount of interest for one or more years. 
 If it be required to calculate the amount of interest for any 
 number of days at a given rate per cent, per annum, the 
 formula becomes : — 
 
 _ , , Piiiicipwl X rate i)er colli, per aiinuiu X iiunibfM' of diivs 
 
 Interest, = 1 _ i_ - 
 
 100 X 865 
 
SIMPLE AND COMPOUND INTEREST 29 
 
 The utility of any table of simple interest is limited only 
 by its size, and it is very easy by means of tbe above formula 
 to ascertain any required sum not given in the table, Tliere are 
 several modifications of tbis method to suit individual or special 
 requirements which do not, however, require special mention. 
 
 Whilst on the question of simple interest, there is an 
 interesting method of using such tables which may not be 
 generally known. It is often required to ascertain the amount 
 of rent, or other annual sum for a given number of days. If, 
 for instance, it is required to ascertain the amount of 97 days' 
 rent at £865 per annum, proceed as follows : — Multiply the 
 annual rent £865, by 20 = £17,300; refer to the tables and 
 ascertain 97 days' interest upon £17,300 at 5 per cent. This 
 will be the amount of 97 days' rent. 
 
 Similarly, the annual rent may be multiplied by 25 and 
 interest upon the product ascertained at 4 per cent., but the 
 above method is the simplest as it involves multiplying by 2 
 only. As a matter of fact any other equivalent multiplier and 
 rate per cent., having 100 for their product, may be used. 
 This method may be applied to ascertain the proportion of the 
 annual sinking fund instalment to be set aside in respect of a 
 loan borrowed at various dates in one year as afterwards pointed 
 out in Chapter XXX. 
 
 Compound Interest, a Geometrical Progression. Com- 
 pound interest differs from simple interest in that it is a 
 geometrical progression in which the rate per cent, is always 
 uniform during the whole period, and the periods are all equal, 
 whether years, half years, months, or otherwise. There are 
 several published tables of compound interest, and many tables 
 have been calculated for special purposes. The one most 
 generally used in England is by William Inwood (18th Edition 
 published 1880), commonly referred to as " Inwood's Tables." 
 A new and much improved edition was issued in 1899, revised 
 and extended by Mr. William Schooling. 
 
 Tables of this character are extremely useful, and provide 
 for the majority of calculations required to be made by Local 
 Government and municipal authorities, actuaries, accountants, 
 bankers and valuers, and the officials of commercial and financial 
 undertakings. 
 
 Derivation of the Formitlj.. It is a very interesting 
 study to analyse the tables mathematically and to derive each 
 
30 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 tahle from the simple algebraical formula used to find the 
 last of a series of numbers in a geometrical progression, viz. : — 
 
 I = ar'^- ^ 
 where a = the first term, 
 / = the last term, 
 r = the constant factor or ratio, 
 w = the number of terms in the progression. 
 
 A geometrical progression consists of a series of numbers 
 which increase or decrease by a constant factor or common 
 ratio, and many problems may be solved by means of the 
 algebraical formulae relating to such a progression, namely, the 
 sum of a series, either finite or to infinity, the insertion of a 
 number of geometric means between two numbers, and finding 
 the last term of a series. Problems involving compound 
 interest, however, include only the first term, the ratio, and the 
 last term, all of which may be determined by means of the 
 algebraical formula with only slight modification. The factors 
 (which remain unchanged except as regards the actual vSymbol) 
 are as follows : — • 
 
 n = fhc frst tenii of the i^rofjrc^f^inn, which corresponds to the 
 principal sum (P) at the beginning of the number of 
 years. 
 
 l=t]ie last terin of the progression, which corresponds to the 
 amount (A) of the principal sum (P) at the end of the 
 number of years. 
 
 r = the common ratio, or the number by which each term in the 
 progression is multiplied in order to find the succeeding 
 term. In the formulse relating to compound interest 
 this is expressed by the symbol (R) because when dealing 
 with annuities, a symbol is required to represent a new 
 factor (R-1) which is denoted by (r), and which will be 
 explained later. 
 11 = file nuinher of terms in the progression, and is the only 
 factor in the algebraical formula requiring any alterati<m 
 in the sense in M-hich it is used. In both formuht the 
 ratio acts in exactly the same manner, or once during 
 each interval in the progression, and it acts upon each 
 term except the last. In any progression, the number 
 of intervals between the terms is one less than the number 
 of terms, or, as it may be expressed : 
 
 (??) intcrvals=(» -1) terms. 
 
SIMPLE AND COMPOUND INTEREST 31 
 
 In the case of compound interest, tlie intervals are years, 
 or other equal periods of time, consequently the 
 algebraical formula is altered by substituting (N) years 
 for ('i — 1) terms, using the capital (N) to denote the 
 number of years in order to distinguish it from the small 
 (7?) which denotes the number of terms in the algebraical 
 formula . 
 
 Substituting the amended symbols as above, 
 
 I = ar'"' ' ^ becomes, A = P R^, 
 
 and the above symbols have the following meaning throughout 
 the book : — 
 
 A = the auwtint, or the ultimate sum to which the present sum 
 (P) will accumulate in (N) years at the ratio or constant 
 factor (R). This symbol will be used to denote this 
 factor whether it represents the ultimate sum required to 
 be found at the end of the period ; or the given sum due 
 at the end of the period, of which it is required to find 
 the present value (P). 
 
 The use of the word " amount " is different from the usual 
 meaning attached to it in ordinary language, and it is very 
 necessary to distinguish it from a sum of money. A very much 
 better word would be " accumulate." 
 
 'P = thc ptincipal or j^resent value, and denotes a sum of 
 money in hand, or due, now. It also denotes : 
 (1) the present value of a definite sum of money (A) due at 
 
 the end of a stated period of years, and 
 (2) the present value of an annuity or other periodic sum 
 (A^) payable or receivable at the end of each of a stated 
 number of years or periods. 
 
 These two factors (A) and (P) are intimately related. (P) is 
 the first term, and (A) the last term, of a geometrical progres- 
 sion. (P) is the present value of (A) due at the end of a stated 
 term, and (A) is the amount to which (P) will accumulate 
 dviring that period. 
 
 'R = thc ratio or common factor, and denotes the rate of increase 
 (expressed in terms of unity) in each term of the pro- 
 gression. It does not denote the rate per cent, per 
 annum, although it is derived directly from the rate per 
 cent. It is, in all cases, £1 increased by interest upon 
 
33 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 £1 for one year at the rate per cent, in question; in the 
 case of 5 per cent, it is 105, as will be clearly shown in 
 Calculation (IV) 1, and so on for every other rate 
 per cent. The ratios corresponding to each rate per cent, 
 from I to 7 per cent, are given later in Table No. Y. A, 
 together with the corresponding logarithms. In calcula- 
 tions involving compound interest the actual rate per 
 cent, is never used, but only in its relation to £1 by way 
 either of a ratio (R) or of the interest upon £1 for one 
 year (r). 
 
 'N = the number of years, or other equal periods, and, as already 
 explained, must not be confounded with [n) in the 
 algebraical formula for a geometrical progression which 
 denotes the number of terms in the progression. This 
 number of terms includes the first term, but in the case 
 of compound interest the number of years is one less 
 than the number of terms in the progression; therefore 
 (N) years =(n-l) terms. In the case of annuities, a 
 modification of the above formula will be required, the 
 derivation of which from the formula (A = P R^) ^{i\ ^g 
 fully explained. This modified formula will contain 
 additional symbols, namely — 
 
 Ay ^ the annuity or other periodic sum, to be set aside, paid or 
 received at the end of each year or period. 
 
 ^l = the amount of the annuity or other periodic sum (A//) 
 accumulated for a given number of years or periods (N) 
 at a given rate per cent. This symbol bears the same 
 relation to a periodic sum (A?/) as (A) bears to a present 
 sum (P). 
 
 r = the interest upon £1 for one year or period at the stated rate 
 per cent. It is found from the above ratio or common 
 factor by deducting unity therefrom. The values of this 
 factor for the various rates per cent., and the corre- 
 sponding logs., are shown in Table No. Y. A, which will 
 be given later. 
 
 The above formula, A = P R^, with its various modifications, 
 may be used to find factors which are sufficient to solve all 
 questions of compound interest in relation to sinking funds and 
 annuities. The actual values of each factor are capable of 
 l>eing tabulated for varying terms at varying rates per cent. ; 
 and to make them generally useful tlu> results are stated in the 
 
SIMPLE AND COMPOUND INTEREvST 33 
 
 published tables in terms of £1 so that any problem as to other 
 amounts may be solved by multiplying or dividing the actual 
 figure in the problem by the amounts given in the published 
 tables. In the old edition of Inwood these tables, I. to V., 
 are given separately; but in the new edition, Tables I. to lY. are 
 shown in four separate columns in one table. Throughout the 
 book they will be referred to as Tables I. to Y., and anyone 
 using the new edition will refer to the corresponding column in 
 the table on pages 50 to 85. 
 
 The Difference between the Amounts of £1 and of £1 
 PER Annum at the End of One Year. It is important to 
 remember that in all calculations involving (P) the sum of 
 money which it represents is due or in hand at the beginning of 
 the first year of the period. In the case of annuities, the annual 
 sum is assumed to be set aside, paid, or received at the end of 
 the first and every subsequent year of the period. This is very 
 important, sufficiently so to justify the following extracts from 
 the tables : — 
 
 Table 1. The amount (A) of (P) £1 at the end 
 
 of one year at 5 per cent, is £1'05 
 
 Table 11. The present value (P) of (A) £1 due 
 
 at the end of one year at 5 per cent, is £0'9524 
 
 Table 111. The amount (M) of [Ky) £1 per annum 
 
 at the end of one year at 5 per cent, is £100 
 
 Table IV . The present value (P) of (Ai/) £1 per 
 
 annum for one year at 5 per cent, is £09524 
 
 Prom the above it will be seen that the amount of £1 at the 
 end of one year (£105) is greater than the amount of £1 per 
 annum at the end of one year (£1) because the £1 is in hand 
 and bears interest during the first year, whereas the annuity 
 of £1 per annum is not due until the end of the year. But on 
 comparing the present value of £1 due at the end of one year, 
 and the present value of £1 per annum due at the end of one 
 year, they are the same (viz., £0-9524) because they are both 
 due at the same time. 
 
 Problems may arise involving a variation from this principle 
 when dealing with purchases on the deferred payment system. 
 In such cases, the annual instalment of principal and interest 
 combined is generally payable at the end of the first and 
 
34 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 subsequent years, in the above manner, but it sometimes 
 liappens that the agreement provides that the first payment shall 
 be made at the beginning of the first year which makes an 
 important alteration in the method. Such problems, however, 
 rarely arise in connection with the sinking funds of local 
 authorities or of commercial or financial undertakings, and will 
 not be further considered. 
 
 Practical Discount as Compaeed with Present Value, 
 Discount of Bills, &c. The above extracts show that £100 at 
 5 per cent, at the end of one year will amount to £105, and that 
 £105 due at the end of one year at 5 per cent, is worth noAv 
 £100. The difference between the two amounts viz., £5, is the 
 mathematical or true discount, and is based upon the present 
 value. In practical finance the method adopted in discounting 
 bills is to deduct interest at the rate per cent, from the amount 
 of the bill payable at the end of the period, and as this amount 
 is always greater than the present value, practical discount, 
 as it is called, is always greater than the mathematical or true 
 discount. For instance, a bill for £105 due at the end of one 
 year, and discounted by the bank at 5 per cent., is worth now 
 £99"75, ascertained as follows: — 
 
 Amount of the bill £10500 
 
 Less the practical discount at 5 per cent, for one year £5^25 
 
 or a net value of £99'75 
 
 If the customer leaves this sum on deposit with the 
 bank, at 5 per cent, he will at the end of the 
 year be credited with 5 per cent, upon £99"T5 or £4-9875 
 
 and will then receive ... £104*7375 
 as compared with the amount of the bill £105" 
 
 a difference of £0-2625 
 
 In other words, he would lose and the bank woubl gain £02625 
 although the bank liav<> liad the use of th(> money for the whole 
 of the year. 
 
SIMPLE AND COMPOUND INTEREST 35 
 
 The bank would gain the difference between the 
 
 practical discount of £5"^5 
 
 and the true or mathematical discount of £500 
 
 £0-25 
 
 And in addition, interest upon this amount for one 
 
 year at 5 per cent., or ••■ £001-^5 
 
 £0-2625 
 
 This proves that the present values as given in the tables of 
 compound interest are not available for discounts which are 
 merely arithmetical calculations, and for which special tables 
 are constructed. 
 
36 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER ly. 
 
 COMPOUND INTEREST AS APPLIED TO A SUM OF 
 
 MONEY. 
 
 TABLE I. The amount of £1 in any number of years. 
 
 The formula, A = P Ji^, and rules deduced therefrom. 
 Calculation by the arithmetical method. Compilation 
 OF Tables. Thoman's method and formula. 
 
 Author's .Standard Calculation Form, No. 1. 
 
 Formulae. 
 
 A. 'To find the Amount of £1 in any number of years, as 
 
 given in the published tables : — 
 Formula, A = RN 
 
 by logs. : Log. (Amount of £l) = Log. E^ 
 
 B. To find the Amount of any sum of money in any number 
 
 of years : — 
 
 Formula, A = P RN 
 
 by logs: Log. [Amount of principal sum) = Log. 
 [principal sum) + Log. R^ 
 
 The above formuhe, and methods by logs, apply equally to 
 Thoman's Formulce and Tables^ which are fully described in 
 Chapter IX. 
 
 General Rules deduced from the above formulae. 
 
 To find the amount of any sum of money in any number of 
 years. Author's Standard Calrulaf ion Form, No. 1. 
 
 Rule 1. If the rate per cent, be not given in Table /, or in 
 Thoman's Tables : — 
 
 Proceed by the formula relating to Table I . 
 
 Calculation {IV) 3 A. 
 
 Rule 2. If the rate per ceiit. be given in Table I : — 
 
 Multiply the amount given in the table, by the 
 given sum. The product is the amount required. 
 
 Calculation {/]')3 B. 
 
THE AMOUNT OF ONE POUND 37 
 
 Rule 3. If the rate per cent, be given in Thoman's Tables : — 
 
 To the log. of the given sum, add the log. of R^ as 
 
 given by Tlioman. The sum of the logs, is the log. 
 
 of the amount required. Calculation (IV) 3 C. 
 
 To find the rate per cent., or number of years, proceed as 
 
 shown in the standard forui for the purpose, given in 
 
 Chapter X. 
 
 The formula, A = P li^, will now be applied to the solution 
 of problems involving compound interest in relation to a sum 
 of money, whether now in hand or payable or receivable at 
 any future date. The published tables are as follows : — 
 
 Table I. The amount of £1 in any number of years. 
 
 Table II. The present value of £1 due at the end of any 
 number of years. 
 Each table will be considered in detail to show the method 
 of compilation by means of the above formula, but in the 
 present case the arithmetical method of calculation will first 
 be given in full, in order to point out the relation between the 
 two methods. 
 
 The Arithmetical Method. In the following calculation, 
 IV (1), at the end of the first year, interest at 5 per cent, is 
 added to the principal sum in hand at the beginning of the 
 year. At the end of the following, and each subsequent year, 
 interest is added to the amount of principal and interest 
 combined, at the beginning of the year. The amount of added 
 interest increases each year, but if each item of interest be 
 compared with the sum upon which it is based, it will be seen 
 that in all cases they bear the same ratio, namely, 005 to 1. 
 On comparing the amount of principal and interest at the end 
 of any year, with the similar amount at the end of the succeed- 
 ing year, it will be observed that they are always in the ratio 
 of 1 to 105. 
 
 In other words, although an amount of interest has been 
 added each year, the amount of principal and interest at the 
 end of each year might have been obtained by multiplying 
 the amount at the end of the previous year by 1-05, or the 
 ratio R. This is therefore a geometrical progression increasing 
 at a ratio of 1-05. This calculation will be referred to again in 
 Chapter YI, when considering the derivation of the formula 
 relating to an annual or other periodic payment, and the 
 discussion of the matter in that chapter may be referred to at 
 this stage with advantage. 
 
38 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (IV) 1. 
 
 To find the Amouut of a given Sum at the end of a given term. 
 
 Table I. 
 
 Required the Amount of £1 at the end of 5 years at 5 per cent., 
 compound interest. 
 
 By Arithmetical Calculation. 
 
 0000 
 (r) -0500 
 
 Principal Sum at the beginning of the first year. . . 
 1. First year's Interest thereon = 
 
 (1x1-05) 
 
 2. Second year's Interest thereon 
 
 (1-05 X 105) 
 
 3. Third year's Interest thereon 
 
 (1-1025 X 1-05) 
 
 4. Fourth year's Interest thereon 
 
 (1-1576x1-05) 
 
 5. Fifth year's Interest thereon 
 
 (1-2155x1-05) 
 
 = (R) 1 
 
 0500 
 0525 
 
 1025 
 0551 
 
 1576 
 0579 
 
 2155 
 0608 
 
 1-2763 
 
 which is the required amount at the end of the 5th year; and 
 agrees with the amount given in Table I. A further amplifica- 
 tion of this calculation will be made in Chapter YI. 
 
 The Mathematical Method. It very rarely happens that 
 calculations of compound interest are required for so short a 
 period as 5 years ; generally they are for very much longer periods. 
 Consequently the arithmetical method as shown in the above 
 Calculation (lY) 1, becomes cumbrous and liable to error, 
 and it is imperative to adopt a shorter method, namely, the 
 algebraical or mathematical one, based upon the formula, 
 A = P RN. Here it is required to find the amount A, knowing 
 that : — 
 
 P = l, R = 105 and N = 5. 
 
 The equation therefore becomes : 
 
 A = RN or A= (105)5. 
 
 but to raise R, or 1"05 to ihv 5th power or perliaps to the 20th, 
 30th, or 60th power is a much longer task than to make the 
 
THE AMOUNT OF ONE POUND 39 
 
 original calculation by the arithmetical method, as in the 
 previous example, and recourse is had to logarithms, which 
 have been fully described in Chapter II. The calculation will 
 be made upon standard calculation form No. 1 by method (A) 
 therein contained, and it will be found that the resulting 
 amount agrees with the value given in Table I in the published 
 tables. It will also be seen that the resulting log. of the 
 required amount agrees with the log. of H^ in Thoman's tables. 
 
 The above methods will now be applied to the following 
 example in order to demonstrate that the calculation by means 
 of logarithms and the above formula is quite as simple, not 
 only for any longer period, but at any rate per cent., whereas 
 the calculation by the arithmetical method will be longer in 
 proportion to the number of years, and will consequently involve 
 a greater possibility of error in the arithmetical computation. 
 
 " Required the amount of £500 at the end of 20 years at 
 5 per cent, per annum compound interest." Calculation (IV) 3. 
 
 As in the previous calculation, relating to £1 only, the 
 result will be ascertained by the same methods, viz: — 
 
 A. by the formula, A=P EN Eule 1 
 
 B. by the published table No. I, giving the amount 
 
 of £1 at the end of any number of years Rule 2 
 
 C. by Thoman's tables Rule 3 
 
 in each case adopting the logarithmic method of calculation. 
 The above rules and formulse are fully set out in the heading to 
 this chapter. 
 
 Thoman's Method and Formula. Altliough Thoman's 
 method applies more particularly to calculations involving 
 annuities or other periodic payments, these tables may with 
 advantage be utilised to solve problems relating to the amount 
 and present value of £1, owing to the fact that the actual logs, 
 of RN are there given, instead of having to be taken from the 
 loo;, tables. The full consideration of Thoman's tables is 
 contained in Chapter IX. 
 
40 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (IV) 2. 
 
 Standard Calculation Form, No. 1. 
 
 To find the future amount of a present sum, and thereby prove 
 the accuracy of the published table. Table I. 
 
 Required the amount of £1 at the end of 5 years at 5 per cent., 
 per annum, compound interest. 
 
 (A) Bj 
 
 ^ Fornmla. A = P R^ 
 
 Rulel, 
 
 Chapter IV. 
 
 Log.. 
 
 ' Log. Ratio 
 
 multiply Log. R by 
 
 R 
 
 105 
 5 
 
 00211893 
 5 
 
 R^ 
 
 RN 
 
 (1-05)5 
 
 0-1059465 
 
 Log. Present Sum 
 add Log. RN above 
 
 P 
 
 RN 
 
 r 
 
 0- 
 
 0-1059465 
 
 
 A 
 
 
 0-1059465 
 
 Required future amount, £1-27628, which agrees with the result 
 obtained by the arithmetical method, Calculation (IV) 1, 
 and also with the amount given in Table I. 
 
 (B) By Table I. A = 
 
 P RN 
 
 Rule 2, 
 
 ChapterlV. 
 
 Table I. 5 years, 5 per cent. 
 Amount of £1 
 
 add Log. Present Sum 
 
 RN 
 
 p 
 
 1-27628 
 
 
 
 A 
 
 1-27628 
 
 
 Required future amount, £1 
 
 •27628, 
 
 as given in 
 
 Table I. 
 
 (C) By Thoman's Table. A = 
 5 per cent. 5 years. 
 
 P RN 
 
 Rule 3, 
 
 Chapter IV. 
 
 Log. Present Sum 
 add Log. RN 
 
 P 
 
 RN 
 
 1- 
 
 0- 
 
 0-1059465 
 
 
 A 
 
 
 01059465 
 
 Required future amount, £1-27628. This log. is given in 
 Thoman's Table. 
 
THE AMOUNT OF ONE POUND 
 
 41 
 
 Calculation (IV) 3. 
 
 Standard Calculation Form, No. I. 
 
 To find the future amount of a present sum. 
 
 Table I. 
 
 Required tlie amount of £500 at the end of 20 years at 5 per 
 cent, per annum, compound interest. 
 
 (A) By Fornuda. A = : 
 
 [> l.N 
 
 Rulel, 
 
 Chapter lY. 
 
 Log-. 
 
 rLo^. Ratio 
 
 multiply Log. R by 
 
 Log. Present Sum 
 add Log. RN above 
 
 R 
 
 N 
 
 1-05 
 20 
 
 00211893 
 20 
 
 R'^ 
 
 RN 
 
 (1-05)20 
 
 0-4237860 
 
 
 P 
 
 RN 
 
 500 
 
 2-6989700 
 0-4237860 
 
 
 A 
 
 
 3-1227560 
 
 Required future amount, £1326-65. 
 
 (B) By Table I. A = 
 
 P RN 
 
 Rule 2, 
 
 Chapter lY. 
 
 Table I. 20 years, 5 per cent. 
 Amount of £1 
 
 add Log. Present Sum 
 
 RN 
 
 p 
 
 2-6533 
 500 
 
 0-4237860 
 2-6989700 
 
 
 A 
 
 
 3-1227560 
 
 Required future amount, £1326-65. 
 
 (C) 
 
 By Thoman's Table . A = P RN 
 5 per cent, 20 years. 
 
 Rule 3, 
 
 Chapter lY. 
 
 
 Log. Present Sum 
 add Log. RN 
 
 P 
 
 RN 
 
 500 
 
 2-6989700 
 0-4237860 
 
 
 A 
 
 
 31227560 
 
 Required future amount, £1326' 65. 
 
42 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTEE V. 
 
 COMPOUND INTEREST AS APPLIED TO A SUM OF 
 MONEY (Continued). 
 
 TABLE II. The present value of i^i, due at the end of any 
 number of years. 
 
 Derivation of the formula, P= ^^ and rules deduced 
 
 THEREFROM. COMPILATION OF T AISLES. TaBLE OF EaTIOS. 
 AND LOGS.. OF E, AND r. CALCULATIONS FOR PERIODS OTHER 
 THAN YEARS. ThOMAN's METHOD AND FORMULA. 
 
 Author's Standard Calculation Form, No. 2. 
 
 Formulae. 
 
 A. To find the present value of £1, due at the end of any 
 
 numher of years, as given in the puhlished tables: — 
 
 Formula, P=^p^- 
 
 hy logs.: Log. [present value of £T) — 
 Log. 1{ = 0)- Log. R^ 
 
 B. To find the present value of any s^im of money, due at 
 
 the end of any numher of years : — 
 
 Form ula, P = pj^ 
 
 by logs.: Log. (present value) = Log. [amount d'ue 
 at end of period) -Log. E^ 
 
 The above formulce, and methods by logs., apply equally to 
 Thoman's formula' and tables, which are fully described in 
 Chapter IX. 
 
 General Rules deduced from the above formulae. 
 
 To find the present value of any sum of money, due at the 
 end of any number of years. 
 
 Author's Standard Calculation Form, No. 2. 
 
THE PRESENT VALUE OF ONE POUND 43 
 
 Rule 1. If the rate per cent, be not given in Table II, or in 
 Thoman's Tables: — 
 
 Proceed by the formula relating to Table II. 
 
 Calculation (F) 2 A. 
 
 Rule 2. If the rate per cent, be given in Table II: — 
 
 Multiply the amount given in the table, by the 
 given sum. The product is the present value 
 required. Calculation (Y) 2 B. 
 
 Rule 3. If the rate per cent, be given in Thoman's Tables : — 
 
 From the log. of the given sum, deduct the log. of 
 
 RN as given by Thoman. The remainder is the log. 
 
 of the present value reqtiired. Calculation (\') 2C. 
 
 To find the rate per cent, or number of years, proceed as 
 
 shown in the standard form for the purpose, given in Chapter X. 
 
 Derivation of the Formula. Having ascertained the 
 methods of finding the accumulated amount of any sum of 
 money at the end of any number of years at any rate per cent., 
 the converse will now be considered, namely, the present value 
 of any sum due at the end of a given number of years. These 
 two factors are intimately related. In Calculation (IV) 3 it 
 was found that a present sum of £500 at 5 per cent, compound 
 interest will in 20 years amount to £1326-65, but this denotes 
 also that £500 at 5 per cent, is the present value of £1326-65 
 payable at the end of 20 years, consequently the formula 
 A = P E^ will give two results, or reciprocals, namely, 
 Table I. The amount of a given sum, P, in any 
 
 number of years, N, =A. 
 
 Table II. The present value of a given sum, A, due 
 
 at the end of any number of years, N, =P. 
 
 The formula for finding the present value of a given sum, 
 instead of being, 
 
 A = P RN, becomes P= ^^^ 
 
 or, in other words, the present value of a sum due at a future 
 date may be ascertained by dividing the amount due at the 
 end of the number of years by the ratio, R, raised to the power 
 equal to the number of years, N. In the case of £1 as in 
 
 Table II, the formula becomes P= .^ since A, the future 
 amount, is £1. 
 
44 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 This formula, P= ^pj^ may now be used to ascertain the 
 
 amounts given in Table II, and, as in the previous example, 
 the calculation will be made by three different methods, namely, 
 
 A, by formula Rule 1. 
 
 B, by the published Table II, giving the 
 
 present value of £1 due at the end of any 
 
 number of years Rule 2 . 
 
 C, by Thoman's Tables Eule 3. 
 
 in each case adopting the logarithmic method of calculation. 
 The above rules and formulae are fully set out in the heading 
 of this chapter. 
 
 Thoman's Method and Formula. In considering the 
 methods of finding the amounts of £1 in any number of years 
 as given in Table I, attention was drawn to the advantage of 
 using Thoman's tables. It was found that the calculation 
 by this method is similar to the calculation by Table I, but in 
 the case of Table II, relating to the present value of a future 
 
 sum, it is necessary to make use of the reciprocal of R^, or p^. 
 
 The only difference between the two tables is that in the case of 
 Table I the log. of R^ is added to the log. of the present sum, 
 whereas in the case of Table II the same log. is deducted from 
 the log. of the future given sum of which it is required to hnd 
 the present value. Calculation (V) 1. 
 
 The same formula will next be used in order to ascertain 
 the present value of £1326" 65 due at the end of 20 years at 
 5 per cent, compound interest, and thereby prove the converse 
 of Calculation (lY) 3, adopting the same methods, namely, 
 by formula : by the published Table II : and by Thoman's 
 method. Calculation (Y) 2. 
 
 Calculations for Periods other than Years. In cases 
 where it is required to calculate compound interest for periods 
 oth(>r than years, and the rate per cent, is expressed as per 
 annum, it is necessary to take a rate per cent, proportionate to 
 the period of a year. For instance, if it be required to calculate 
 the amount of a sum of money rolling \i\) half yearly, douhle 
 the nu lit her of years anil take one-half the rate per cent, per 
 annum, as follows: — 
 
THE PREvSENT VALUE OF ONE POUND 45 
 
 £1 at the end of 10 years at 10 per cent, per annum 
 
 will amount to (yearly breaks) £2'5937 
 
 £1 at the end of 10 years at 10 per cent, per annum 
 
 will amount to (half-yearly breaks) 
 
 = 20 years at 5 per cent £2"6533 
 
 £1 at the end of 10 years at 10 per cent, per annum 
 
 will amount to (quarterly breaks) 
 
 = 40 years at 2i per cent £2"6851 
 
 This is a very useful method to adopt when it is required 
 to ascertain the effect of compounding the interest at various 
 periods, and the rule applies equally to calculations involving 
 annuities. x\ll that is necessary is to deal with the number of 
 periods at the corresponding rate per cent, per period, based 
 upon the rate per cent, per annum. 
 
 The Factors E (Eatio) and r (The Interest of £1 for One Year) 
 AND THE Corresponding Logarithms. 
 In order to simplify the method by formulae and logs, the 
 following Table, Xo. Y. A., has been prepared. It gives the 
 ratio (E) and the corresponding logs, for 49 rates from i per 
 cent, to T per cent. It also contains the A-alues and correspond- 
 ing logs, of the factor {r) which is the interest upon £1 for one 
 year. There is not anything difficult in the compilation of the 
 table, which is here given only for convenience of reference. 
 The logarithms corresponding to any rate per cent, may be 
 ascertained from the log. tables at the time of making the 
 calculation, but since many of the ratios contain six figures, it 
 involves the use of the proportional parts of the logarithms, and 
 a reference to this table will save time. When dealing with 
 annuities, the logs, of (E^) and {r) are required in each 
 calculation, and as they have always to be looked for in 
 different parts of the log. tables, it is a convenience to have 
 them in one place. If it is necessary to make a calculation at 
 any intermediate rate per cent, not included in this table all 
 that is required is to find the ratio, which is one pound, 
 increased hij interest vpon one jjound, for one year, at the given 
 rate per cent., and then the corresponding log. The factor (/•), 
 as will be seen from the table, is ascertained by deducting 1 
 from the ratio so found. The logs, of both are found from the 
 tables of logs, in the usual way, paying due attention to the 
 sign of the " characteristic" of the log. of (r). The logs, of 
 (EN) are given in Thoman's tables for many rates per cent, for 
 a large number of years. 
 
46 
 
 REPxWMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (V) 1. 
 
 Standard Calculation Forni^ ,\o. 2. 
 
 To find the present value of a sum due at the end of any number 
 of years, and thereby prove the accuracy of the published 
 table. Table II. 
 
 Required the present value of £1, due at the end of 20 years, 
 at 5 per cent, per annum compound interest. 
 
 (A) By Formula. "^"IP 
 
 Rule 1, 
 
 Chapter V. 
 
 Log. Ratio 
 Log-, viultiply Log. R by 
 R^ 
 
 i; 
 
 RN 
 
 105 
 20 
 
 (1-05)20 
 
 00211893 
 20 
 
 0-4237860 
 
 Log. Future Sum 
 deduct Log. R^ above 
 
 A 
 
 RN 
 
 1- 
 
 0- 
 
 0-4237860 
 
 
 P 
 
 
 1-5762140 
 
 Required present value, £037689, Avhich agrees with the 
 amount given in Table II. 
 
 (B) By Table II. 
 
 R^ 
 
 Rule 2, Chapter Y. 
 
 Table II. 20 years, 5 per cent. 
 Present value of £1 
 
 add Log. Future Sum 
 
 1 
 
 RN 
 A 
 
 0-37689 
 
 0-37689 
 
 Required present value, £0-37689, as given in Table II. 
 
 (C) By Thoman's Table. 
 
 W 
 
 Rule 3, Chapter V 
 
 5 per cent. 20 years. 
 
 
 
 
 Log. Future Sum 
 deduct Log. 
 
 A 
 
 RN 
 
 1- 
 
 0- 
 
 0-4237860 
 
 
 P 
 
 
 1-5762140 
 
 Required present value, £0-37689. This log. is given in 
 Thoman's Table. 
 
THE PRESENT VALUE OF ONE POUND 
 
 47 
 
 Calculation (V) 2. 
 
 Standard Calculation Form, No. 2. 
 
 To find the present value of a sum due at the end of any number 
 of years. Table II. 
 
 Required the present value of £132665, due at the end of 
 20 years, at 5 per cent, per annum compound interest. 
 
 (A) By Formula. ^^W 
 
 Rule 1, 
 
 Chapter Y . 
 
 Log. Ratio 
 Log. multiply Log. R by 
 
 V 
 
 RN 
 
 105 
 20 
 
 (1-05)20 
 
 0211893 
 20 
 
 0-4237860 
 
 Log. Future Sum 
 deduct Log. R^ above 
 
 A 
 
 RN 
 
 1326-65 
 
 3-1227560 
 0-4237860 
 
 Required preseni 
 
 P 
 
 value 
 
 , £50000. 
 
 2-6989700 
 
 (B) By Table II. ^=W 
 
 Rule 2, 
 
 Chapter Y. 
 
 Table II. 20 years, 5 per cent. 
 Present value of £1 
 
 add Log. Future Sum 
 
 1 
 
 R^ 
 A 
 
 0-37689 
 1326-65 
 
 1-5762140 
 3-1227560 
 
 
 P 
 
 
 2-6989700 
 
 Required presen 
 
 ; value 
 
 , £50000. 
 
 
 (C) By Thoman's Table. P= -^ 
 5 per cent. 20 years. 
 
 Rule 3, 
 
 Chapter Y . 
 
 Log. Future Sum 
 deduct Log. 
 
 A 
 
 RN 
 
 1326-65 
 
 3-1227560 
 0-4237860 
 
 
 P 
 
 
 2-6989700 
 
 Required present value, £50000. 
 
48 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE V, A. 
 Giving the values of (R) and (r) for the following rates per 
 cent, (from 5^ to 7 per cent.) and the corresponding log 
 of each value. 
 (R) = the amount of £1 plus one year's interest at any rate 
 per cent. 
 
 = {l + r) 
 
 (r) = the interest upon £1 for one year at any rate per cent. 
 
 = (Il-l). 
 
 The Logarithms of (R^) are given in Thoman's Tables under 
 each rate per cent. 
 
 Rate 
 
 % 
 
 R 
 
 10025 
 
 1005 
 
 10075 
 
 Ratio = R 
 
 Loo-. R 
 
 000108438 
 
 000216606 
 000324505 
 
 101 000432137 
 
 1-015 000646604 
 
 1-01625 000700056 
 
 1-0175 000753442 
 
 1-01875 000806762 
 
 1 
 
 8 
 
 
 1 
 4 
 
 
 3 
 
 8 
 
 
 1 
 2 
 
 
 5 
 
 8 
 
 
 # 
 
 
 -02 
 
 •02125 
 
 •0225 
 
 •02375 
 
 -025 
 
 •02625 
 
 •0275 
 
 •02875 
 
 •03 
 
 ■03125 
 •0325 
 •0:{375 
 
 ■o;{5 
 •03625 
 
 000860017 
 000913207 
 000966332 
 001019391 
 001072387 
 001125317 
 0-01178183 
 001230985 
 
 ■01283722 
 01:536396 
 ■Ol;589006 
 ■01441552 
 •014940:55 
 ■01546454 
 
 Rate Interest on £1 for 1 year = j' 
 
 % r Log. r 
 
 i 00025 33979400 
 
 i 0005 :!• 6989700 
 
 I 00075 3-8750613 
 
 •01 
 •015 
 ■01625 
 •0175 
 
 •01875 
 
 002 
 
 002125 
 
 00225 
 
 002375 
 
 0025 
 
 002625 
 
 00275 
 
 0^02875 
 
 003 
 
 003125 
 
 00:325 
 
 0^ 0:3375 
 
 00:55 
 003625 
 
 2^0000000 
 2^1760913 
 2^21085:54 
 22430380 
 22730013 
 
 23010;500 
 2^3273589 
 23521825 
 2^3756636 
 23979400 
 24191293 
 2-4393327 
 2^4586378 
 
 2-4771213 
 
 2^4948500 
 2^51 188:34 
 2^5282738 
 2^5440680 
 2^5593080 
 
THE PRESENT VALUE OF ONE POUND 
 
 49 
 
 Rate 
 
 Ratio = R Rate Interest on £1 for 1 year = r 
 
 R Log. R % r Log. r 
 
 10375 
 1-0.3875 
 
 104 
 
 104125 
 
 10425 
 
 1-04375 
 
 1-045 
 
 104625 
 
 1-0475 
 
 1-04875 
 
 1-05 
 
 1-05125 
 
 1-0525 
 
 1-05375 
 
 1055 
 
 105625 
 
 1-0575 
 
 1-05875 
 
 106 
 
 1-06125 
 
 1-0625 
 
 1-06375 
 
 1-065 
 
 1-06625 
 
 1-06750 
 
 1-06875 
 
 1-07 
 
 001598811 
 001651104 
 
 0-01703334 
 0-01755501 
 0-01807606 
 0-01859649 
 001911629 
 0-01963547 
 002015403 
 002067197 
 
 0-02118930 
 002170601 
 002222210 
 002273759 
 0-02325246 
 0-02376672 
 0-02428038 
 002479342 
 
 0-02530587 
 0-02581770 
 0-02632894 
 002683957 
 0-02734961 
 0-02785904 
 002836788 
 0-02887613 
 0-02938378 
 
 3 
 
 4 
 
 0-0375 
 
 2-5740313 
 
 7 
 
 8 
 
 0-03875 
 
 2-5882717 
 
 4 
 
 0-04 
 
 2-6020600 
 
 1 
 
 8 
 
 004125 
 
 2-6154240 
 
 1 
 4 
 
 0-0425 
 
 2-6283889 
 
 3 
 
 S 
 
 0-04375 
 
 2-6409781 
 
 1 
 2 
 
 0-045 
 
 2-6532125 
 
 5 
 
 8 
 
 0-04625 
 
 2-6651117 
 
 3 
 
 4 
 
 0-0475 
 
 2-6766936 
 
 7 
 8 
 
 0-04875 
 
 2-6879746 
 
 5 
 
 0-05 
 
 2-6989700 
 
 1 
 
 8 
 
 0-05125 
 
 2-7096939 
 
 1 
 
 4 
 
 0-0525 
 
 2-7201593 
 
 3 
 
 8 
 
 0-05375 
 
 2-7303785 
 
 1 
 2 
 
 0-055 
 
 2-7403627 
 
 5 
 
 8 
 
 0-05625 
 
 2-7501225 
 
 3 
 
 4 
 
 00575 
 
 2-7596678 
 
 7 
 8 
 
 0-05875 
 
 2-7690079 
 
 6 
 
 0-06 
 
 2-7781513 
 
 1 
 
 8 
 
 0-06125 
 
 2-7871061 
 
 1 
 
 4 
 
 00625 
 
 2-7958800 
 
 3 
 
 S 
 
 006375 
 
 2-8044802 
 
 1 
 2 
 
 0065 
 
 2-8129134 
 
 5 
 
 8 
 
 006625 
 
 2-8211859 
 
 3 
 
 4 
 
 0-06750 
 
 2-8293038 
 
 7 
 8 
 
 0-06875 
 
 2-8372727 
 
 7 
 
 007 
 
 2-8450980 
 
50 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER VI. 
 
 CUMPOUXD INTEREST AS APPLIED TO AN ANNUAL 
 OR OTHER PERIODIC PAYMENT. 
 
 TABLE in. The amount of ;^i per annum in any number 
 of years. 
 
 Geneeal remarks as to annuities. The relation between 
 
 THE AMOUNTS OF £1, AND OF £1 PER ANNUM. ThE 
 
 arithmetical METHOD FURTHER CONSIDERED. DERIVATION 
 
 (J^N 2\ 
 1 AND RULES DEDUCED 
 
 THEREFROM. COMPILATION OF TaBLES. ThOMAn's METHOD 
 
 and formula. 
 
 Author's Standard Calculation Form, No. 3. 
 
 Formulae. 
 
 A. To find the amount, of £1 ycr oniinm in any nnmher of 
 
 years, as given in the inihlished tables : — 
 
 /RN ix 
 
 (7) Formula, M= (J^^) 
 
 by logs. : Log. {amount of £1 'per aniium) = 
 Log. (RN-l)-Lo^.r 
 (2) By Thoma7i\s method: — - 
 
 RN 
 
 Formula, M= — - 
 
 a" 
 
 by logs. : Log. (amount of £1 per amium) = 
 Log. RN+lO-Lo^r. a« 
 
 B. To find the amount of any annuity in any number of 
 
 years : — 
 
 (7) Formula, M = 
 
 ^H"^) 
 
 by logs.: Log. (amount of annuity) = Log. annuity + 
 Log. {l{^-\)-Log. T 
 {2) By Tlionian's method: — 
 
 RN 
 
 Form ula, M = A-?/ 
 
 by logs. : Log. (amount of annuity) = Log. annuity + 
 Log. W^+10-Log. a« 
 
THE AMOUNT OF ONE POUND PER ANNUM 51 
 
 The present chapter deals only uiitli the formula 
 M = Ay ( — ^ — )' Th Oman's method and formulec, are fully 
 described in Chapter IX. 
 
 General Rules deduced from the above formulae. 
 
 To find the ainount of any annuity in any number of years. 
 Author's Standard Calculation Form, No. 3. 
 
 Rule 1. If the rate per cent, be not given in Table III, or in 
 
 TJioman's Tables: — 
 
 Proceed by the formula relating to Table III. 
 
 Calculation (VI) 2 A. 
 Rule 2. If the rate per cent, he given in Table III : — ■ 
 
 Multiply the amount given in the table, by the 
 given annuity. The product is the amount required. 
 
 Calculation{VI)2B. 
 
 Rule 3. If the rate per cent, be given in Thoman's Tables : — 
 To the log. of the given annuity, add the log. of R^ 
 as given by ThoTnan. Add 10 to the sum of the two 
 logs., and deduct therefrom the log. of a^ as given 
 by Thoman. The remainder is the log. of the 
 required amount. Calculation {^I) 2C. 
 
 To find the rate per cent., or mirnber of years ^ proceed as 
 shown in the standard form for the purpose^ given in Chapter X. 
 
 Annuities or other Periodic Payments. All problems 
 relating to annual sums involve calculations of a more complex 
 character than the steady accumulation of a given sum of 
 money. Matters are complicated by the intrusion of a factor 
 representing an equal annual or other periodic sum, to be set 
 aside, received or paid, at the end of each year, and 
 accumulated at a given rate per cent., for a given number of 
 periods. Such an equal annual or other periodic sum is called 
 an annuity, but in this connection it should be borne in mind 
 that actuarially the term annuity includes any definite sum of 
 money to be paid or received at the end of any given number 
 of regular intervals. There is room for a better word, but it 
 does not matter so long as it is known what the term includes. 
 In the following pages the word annuity will be used to denote 
 any equal sum payable at the end of regular periods, except 
 that in the case of sinking funds, the word " instalment " or 
 " annual increment " will be substituted. 
 
52 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 As in the case of a principal sum, an annuity or otlier 
 periodic payment may be expressed in terms of its " amount " 
 or " present value " whicli are given in Tables III and IV 
 respectively. 
 
 The Factors E (Eatio) aub r (the Interest of £1 for One Year). 
 In all calculations involving a geometrical progression tbe 
 predominant factor is the ratio which, in the algebraical 
 formula, is expressed by the symbol, r. A pure geometrical 
 progression relates only to a series of numbers, increasing in a 
 definite ratio, similar to the annual accumulation, by way of 
 compound interest, of a given sum of money as described in 
 Chapter IV, dealing with Table I. The algebraical formula 
 for a pure geometrical progression does not provide for any 
 further addition to each term of the progression. In the case 
 of compound interest, however, the problem may be complicated 
 by the annual or other periodic addition of a definite sum, 
 namely, the annuity, and it is necessary therefore to amend the 
 formula, A=P B,^ , by dividing the factor, E, or ratio, into two 
 parts, namely, the actual algebraical ratio and the equal annual 
 addition to each term of the progression, representing the 
 constant sum or annuity to be added to each term. In the 
 algebraical formula the ratio is expressed by the symbol, r. 
 In the formula relating to compound interest two symbols are 
 used, namely : — 
 
 E = the common ratio existing between the successive terms 
 of the progression irrespective of any periodic equal 
 additions to the progression. This factor, E, in the 
 formula? relating to compound interest is the equivalent 
 of the algebraical factor (r). 
 
 7' = the annual or other periodic sum added to each term of 
 the progression, and which, as regards the formula 
 relating to unity, represents the annual interest of £1 
 for one year. 
 
 In this manner the accumulation of an annual sum by way of 
 compound interest, cannot properly be considered a pure 
 geometrical progression. It is rather the sum of several 
 arithmetical progressions in echelon, which accounts for the 
 difficulty in determining the rate per cent, by means of the 
 formula, as will be seen on reference to the standard form for 
 the purpose given in Chapter X. 
 
THE AMOUNT OF ONE POUND PER ANNUM 53 
 
 The Eelation between the Amount of £1 and of £1 feu 
 Annum. It is necessary to derive a formula relating to the 
 amounts and the present values of £1 per annum as given in 
 the published tables, which formula, although based thereon, 
 is of a somewhat more complicated character than the simple 
 formula relating to Tables I and II. The additional symbols 
 which will be required have already, in anticipation, been 
 explained in Chapter III. 
 
 Before proceeding to find such a formula the subject will be 
 considered from the point of view of the accumulation of a 
 single sum now in hand, as illustrated by Calculation (IV) 1 
 in Chapter lY. It is possible to ascertain the sum to which 
 an annuity will amount at the end of a stated period, by 
 treating each of the annual payments separately, and finding 
 the sums to which they will respectively amount at the end 
 of the period, by the method already considered in relation to 
 Table I. The total of these separate results will represent the 
 sum to which the whole annuity will amount at the end of the 
 period (see columns 1 to 4 in the following table). 
 
 The method is a cumbrous one, and therefore not practical, 
 but the working of such a calculation is given in order to 
 demonstrate the relation between Table I, giving the amounts 
 of £1, and Table III, giving the amounts of £1 per annum. 
 It will also emjDhasise Avliat has been already pointed out in 
 Chapter III, namely, the difference betAveen the amounts of £1 
 and of £1 per annum at the end of any equal number of years, 
 as also shown in columns 5 and 6 in the following- table. 
 This difference is due to the fact that in all calculations of this 
 nature the sum of money of which it is required to find the 
 amount at the end of a term of years, as in Table I, is assumed 
 to be in hand and to commence to accumulate at once, whereas, 
 in the case of an annuity, the annual or other periodic payments 
 are assumed to be made at the end of each year or period, at 
 which date they begin to accumulate. An annuity of £1 for a 
 given number of years may, therefore, be considered as a series 
 of sums of money, each of which is deferred, both as to the 
 date of payment and of accumulation, for 1, 2, 3, 4, etc., years. 
 Taking as an example an annuity of £1 for 10 years to accumu- 
 late at 5 per cent, per annum, the sum to which each separate 
 payment will amount at the end of the 10 years will be ascer- 
 tained by the method adopted in Calculation (IV) 1, and after 
 deriving the formula relating to the amounts of an annuity, as 
 given in Table III, the same example will be worked out, by 
 means of the formula, in Calculation (VI) 1. 
 
54 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE VI, A. 
 
 Showing the method of fiiiding the amount of an annuity from 
 the figures given in' Table I, relating to a principal sum 
 and illustrating the relation between the amounts of £1 
 and of £1 per annum at the end of one year, 
 
 Rate of accumulation, 5 per cent. 
 
 
 
 TABLE I. 
 
 
 TABLE IIL 
 
 
 TABLE 
 
 I. 
 
 Amount 
 
 set Will 
 aside accu- 
 
 at Ululate 
 end of for 
 year, years. 
 
 Amount 
 of each 
 annual 
 sum at 
 end of 
 10th year. 
 
 Total at 
 
 end of 
 
 each year 
 
 of the 
 amounts 
 in Col. 3. 
 
 Amount of 
 
 £1 per annum 
 
 at end of 
 
 each year 
 
 from 
 
 1 to 10 years. 
 
 Amount of £1 
 
 at end of 
 
 each year 
 
 from 
 
 1 to 10 years. 
 
 Total at 
 end of 
 
 each year 
 of the 
 
 amounts 
 
 in Col, 6. 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 
 (5) 
 
 
 (6) 
 
 (7) 
 
 10 
 
 
 
 1-0000 
 
 1-0000 
 
 1 
 
 1-0000 
 
 
 — 
 
 — • 
 
 9 
 
 1 
 
 10500 
 
 2-0500 
 
 9 
 
 2-0500 
 
 1 
 
 1-0500 
 
 10500 
 
 8 
 
 2 
 
 11025 
 
 3-1525 
 
 3 
 
 3-1525 
 
 2 
 
 1-1025 
 
 21525 
 
 7 
 
 3 
 
 11576 
 
 4-3101 
 
 4 
 
 4-3101 
 
 3 
 
 1-1576 
 
 3-3101 
 
 6 
 
 4 
 
 1-2155 
 
 5-5256 
 
 5 
 
 5-5256 
 
 4 
 
 1-2155 
 
 4-5256 
 
 5 
 
 5 
 
 1-2763 
 
 6-8019 
 
 6 
 
 6-8019 
 
 5 
 
 1-2763 
 
 5-8019 
 
 4 
 
 6 
 
 1-3401 
 
 81420 
 
 i 
 
 8-1420 
 
 6 
 
 1-3401 
 
 7-1420 
 
 3 
 
 T 
 
 1-4071 
 
 9-5491 
 
 8 
 
 9-5491 
 
 7 
 
 1-4071 
 
 8-5491 
 
 2 
 
 8 
 
 1-4775 
 
 11-0266 
 
 9 
 
 110266 
 
 8 
 
 1-4775 
 
 100266 
 
 1 
 
 9 
 
 1-5513 
 
 12-5779 
 
 10 
 
 12-5779 
 
 9 
 
 1-5513 
 
 11-5779 
 
 
 12-5779 
 
 11-5779 
 
 
 
 
 
 
 In the above table : — 
 
 Column 1, contains the year at the end of which each annual 
 sum is set aside, and Column 2 the number of years 
 for which it afterwards accumulates. 
 
 Column 3, is taken item by item from Table I (with the 
 exception of the first item of £1) and shows the 
 amount of each separate annual sum (beginning 
 with the 10th) at the end of the 10th year as if it 
 were accumulated separately. The total of Column 3 
 is the accumulated amount of the whole of the 
 annual sums obtained in this manner, and agrees 
 with the amount given in Table III and found by 
 Calculation (YI) 1. 
 
 Column 4, gives, at, the end of each successive year, the total of 
 the previous items in Column 3, which is the amount 
 of all the annual sums set aside up to the end of that 
 
THE AMOUNT OF ONE POUND PER ANNUM 55 
 
 year. The items in this coliiinn correspond, year by 
 year, Avith the amounts of an annuity of £1 given in 
 Column 5, which is copied item for item, from 
 Table III, which gives the amounts of an annuity 
 of £1 for any number of years. 
 
 Column 6, contains the amounts of £1 at the end of each year 
 from 1 to 10 years, copied, item by item, from 
 Table I. These figures correspond with eacli item, 
 except the first, in Column 3. 
 Column 7, contains the total at the end of each successive year 
 of the previous items in Column 6, and might have 
 been obtained by adding together the figures given 
 in Table I. On comparing the totals of Columns 3 
 and 6, it will be seen that the total of Column G, 
 at the end of the 10th year, is less by £1 than the 
 total of Column 3. Similarly, if at the end of any 
 year the totals in Column 5 are compared with the 
 totals in Column 7, the same difference will be 
 found. 
 Consequently, if it be required to ascertain the accumulated 
 amount of an annuity of £1 for 10, or any other number of 
 years, at 5 per cent, per annum from Table I, which gives the 
 amounts of £1, it may be found by adding together the 
 successive amounts given in Table I for 9, or one less than the 
 specified number of years, and increasing the sum so obtained 
 by £1. The sum of the 9 amounts is the amount of nine years' 
 accumulation of £1 per annum, and the £1 so added is the 
 last annual sum, which does not accumulate at all owing to its 
 being set aside on the last day of the last year of the term. 
 
 Derivation of the Formula. The above-described arith- 
 metical method of finding the amount of an annuity for any 
 number of years depends upon treating each annual sum as a 
 separate entity, but does not treat the annuity qua annuity, 
 and, further, it does not give any clue to a rule or formula 
 by which the result may be obtained by direct mathematical 
 calculation. Many problems contain factors involving the 
 accumulation of £1, and also of £1 per annum, and it is 
 advisable therefore that all formulae should be expressed in the 
 same or similar terms. The formula relating to the aiuount 
 of £1 in any number of years, namely, A = P E^, has already 
 been ascertained, and it will now be used in order to deduce 
 therefrom a formula relating to the accumulation of periodic 
 sums. The practical application of that formula will be first 
 
=,6 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 considered, and will be based upon the arithmetical Calculation 
 {L\) 1, as afterwards proved by means of the formula, in 
 Calculation (lY) 2. As explained in Chapter lY, at the end 
 of the first year, interest at 5 per cent, per annum was added 
 to the original principal sum of £1, and at the end of each 
 subsequent year interest at 5 per cent, per annum was added to 
 the amount of principal and interest at the beginning of such 
 year. In Calculation (lY) 1, the interest added each year was 
 treated as one sum, and was not divided in order to differentiate 
 between the interest added yearly in respect of the original 
 principal sum as distinguished from the interest added yearly 
 upon the interest added in previous years. In the following 
 table (No. YI, B) such a distinction has been made, and the 
 results obtained in Calculation (lY) 1 are repeated in Column 2. 
 The interest added each year has been divided as between the 
 principal and the interest previously added, and Column 3 
 contains the constant annual amount of interest upon the 
 original principal of £1, which is (/■) in the list of symbols 
 given in Chapter III. Columns 4, 5, 6, and 7 contain each 
 year's accumulated interest upon each annual amount of 
 interest (r) upon the original principal of £1. The table is as 
 
 follows : — 
 
 TABLE YI, B. 
 
 Showing the amount of £1, for 5 years at 5 per cent, per annum. 
 Calculation (lY) 1. Showing also the annuity of (r) = 005, 
 and its accumulations. 
 
 1 
 
 At 
 
 end 
 
 of 
 
 year. 
 
 2 
 
 Amount 
 
 of £L 
 
 5 years 
 
 5%. 
 
 3 
 
 Annual 
 
 Interest 
 
 on £1 
 
 Principal. 
 
 4 5 6 
 
 Accumulation of (r) Annu 
 Interest on £1 at end of 
 
 7 
 al 
 
 8 
 Total 
 Accumu- 
 lation 
 
 2nd 
 
 3rd 
 
 4tli 
 
 5th 
 
 of (r). 
 
 
 10000 
 
 
 
 
 
 
 
 1 
 
 ■0500 
 
 •0500 
 
 •0025 
 
 •0026 
 
 •0028 
 
 •0029 
 
 •0108 
 
 
 10500 
 
 
 2 
 
 0525 
 
 •0500 
 
 
 •0025 
 
 •0026 
 
 •0028 
 
 •0079 
 
 
 11025 
 
 
 3 
 
 •0551 
 11576 
 
 •0500 
 
 — 
 
 " 
 
 •0025 
 
 •0026 
 
 •0051 
 
 4 
 
 •0579 
 
 •0500 
 
 " 
 
 
 
 •0025 
 
 •0025 
 
 
 12155 
 
 
 5 
 
 •0608 
 
 •0500 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 1-2763 
 
 •2500 
 
 •0025 
 
 •0051 
 
 •0079 
 
 •0108 
 
 •0263 
 
 1^0000 Original Sum 
 •2500 Annual Interest 
 •0263 Accumulations of Annual Interest 
 
 ~m63 
 
THE AMOUNT OF ONE POUND PER ANNUM 57 
 
 The original principal sum of £1 may now be left out of 
 the calculation, and be considered only as the origin of an 
 annual sum or annuity, of £005 to be accumulated for 5 years 
 at 5 per cent, per annum compound interest. It is in fact, at 
 5 per cent, the present value of a perpetual annuity of £0'05. 
 The results obtained in the above table will now be translated 
 into terms of the formula A = P 11^, writing against each factor 
 in the arithmetical result the corresponding symbol in the 
 formula, but as the formula is being considered in its relation to 
 £1 only, there will be substituted for P its equivalent 1, with 
 the following result, viz., A = R^. 
 
 The above Table YI, B, expressing the results of Calcula- 
 tion IV (1) may be analysed as follows: — 
 
 Actual 
 
 results. Formula. 
 
 Amount of £1 in 5 years at 5 per cent. ... 1'2763 R^ 
 Deduct, the principal sum of* which 
 this is the amount at the end of 
 5 years (Table I) l" 1 
 
 leaving ... 02763 EN-1 
 
 which is the accumulated 
 amount of the annvial interest 
 upon £1 at 5 per cent., or 
 105-1 = 005 005 E-1 
 
 which is the annuity which will in 5 years at 5 per cent., 
 amount to £0"2763, as shown in the above table, No. YI, B. 
 
 The formula relating to the accumulation of an annual sum 
 is derived from the foregoing results as follows : — 
 
 It has been ascertained by means of the formula 
 A = P 11^ relating to the accumulation of £1 
 as given in Table I, that the amount (A), 
 of £1 (P) at the end of any number of years is 11^ 
 
 and by deducting therefrom the original sum, P, 
 
 or its equivalent, which in this case is 1 
 
 a constant is obtained which will apply to any 
 
 rate per cent., namely 11^ — 1 
 
58 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 This constant represents the accumulated amount 
 of the annual interest upon £1, resulting 
 from the accumulation of the original 
 principal sum P, at the ratio R, for N years. 
 
 In the above example, the ratio, which is P, plus 
 
 one year's interest, is R or 1"05 
 
 and by deducting therefrom the original principal 
 
 sum P, or 1. the remainder is Il-lor0"05 
 
 which represents the interest upon £1 for one year 
 and is constant for any rate per cent. 
 
 Expressing the above in terms of the calculation in Table VI, B, 
 it is found that : — 
 
 (1{N_1) or 0'2T6y is the accumulated amount of an annual 
 sum of (Ii-1) or 0-05 for (N) or 5 years, at a ratio (R) 1-05, 
 which is the equivalent of 5 per cent, per annum. 
 
 Stated in the form of a proportion the problem becomes : — • 
 If 0-05 per annum or (R-1) amounts to 0-2763 or (R^-l), 
 what sum will £1 per annum amount to under the same 
 conditions, as follows : — 
 
 0-2763 RN-1 
 
 = = 5-526 
 
 0-05 R-1 
 
 which agrees with the amount given in Table III, and provides 
 a formula which may be used to calculate the amount of £1 per 
 annum for any number of years at any rate per cent. To find 
 the amount of any other annual sum all that is required is to 
 multiply the result obtained in the above manner by the annual 
 sum in question. 
 
 It is not possible to simplify the above factor (RN- 1) because 
 RN varies with each number of years, but (R-1) may be 
 expressed by a simple symbol because it is always constant 
 for eacli rate per cent. It may be found by deducting unity 
 from R or by dividing the rate per cent, by 100. Tlie factor 
 (R-1) is denoted by the symbol (/■) to show at once its relation 
 to, and variation from, the factor (R) from whicli it is derived. 
 The amount of £1 per annum is denoted by the symbol (M) 
 to distinguish it from (A) the amount of £1. 
 
THE AMOUNT OF ONE POUND PER ANNUM 59 
 
 The formula therefore becomes : — 
 
 (1) as to £1 per annum : 
 
 ^r /RN-1 
 
 M: 
 
 and 
 
 (2) as to any annual sum (Ay) : 
 
 --^^C~^) 
 
 and the symbols have the meanings described in Chapter III. 
 
 The annuity or other periodic sum in all cases, as already 
 pointed out, is presumed to be paid or received, set aside or 
 invested, at the end of the first and each succeeding year, which 
 is the usual method in all annuity calculations. If it be set 
 aside at the beginning of the year the calculation is somewhat 
 different. 
 
 Calculations. Having found the above formula relating 
 to the amount of £1 per annum in any number of years, two 
 calculations will now be made by its aid, upon the author's 
 standard form Xo. o. Both will include the three methods of 
 which the general rules are stated at the head of this chapter, 
 namely, by formula, by the published tables, and by Thoman's 
 method and tables. The first calculation will deal only with 
 an annuity of £1, and will show the method of computing the 
 amounts given in Table III, Calciilation (VI) 1. The second 
 calculation will deal with an annuity of stated amount, and 
 will illustrate the method to be adopted in actual practice. 
 Calculation (VI) 2. 
 
6o 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (VI) 1. 
 
 Standard Calculation Form, No. 3. 
 
 To find the amount of an annuity in any number of years, and 
 
 thereby prove the accuracy of the published table. 
 
 Table III. 
 Required the amount of £1 per annum for 10 years at 5 per cent. 
 
 per annum compound interest. 
 
 (A) By Formula. ^i = Ky (^^^^^ ^^^1© 1, Chapter YI. 
 
 Log. 
 R^— 1^ 
 
 Log. Ratio I R 
 
 multiply Log. R by N 
 
 Convert Log. to 
 
 ordinary number 
 deduct unity 
 
 Log. of this is 
 
 Log. Annuity 
 add. Log . RN - 1 above 
 
 deduct Log. r 
 
 105 
 10 
 
 00211893 
 10 
 
 n^ 
 
 (l-05jio 
 
 0-2118930 
 
 RN 
 
 -1 
 
 1-6289 
 1- 
 
 
 RN-l 
 
 0-6289 
 
 1-7985779 
 
 A:y 
 
 RN-l 
 
 1- 
 
 00000000 
 1-7985779 
 
 A2/(RN- 
 r 
 
 -1) 
 •05 
 
 1-7985779 
 2-6989700 
 
 M 
 
 
 1-0996079 
 
 Required future amount, £12-5779. 
 
 (B) By Table III. M = A.y (^ —^\ Rule 2, Chapter YI 
 
 Table III. 10 years, 5 per cent. 
 Amount of £1 per annum 
 Add Log. Annuity 
 
 RN-l 
 
 Ky 
 
 12-5779 
 
 M 
 
 Required future amount, £12-5779. This amount is given in 
 Table III. 
 
 (C) By Thoman's Table. M = Ky (^ \ Rule 3, Chapter YI. 
 
 5 per cent. 10 years. 
 
 Log Annuity 
 Add Log. RN in 
 
 Table +10 
 
 deduct Log. a" 
 
 Required future amount, £12- 57 79. 
 
 A,y 1- 
 
 RN 
 
 0-0000000 
 10-2118930 
 
 A2/RN 
 
 10-2118930 
 9-1122851 
 
 M 
 
 1-0996079 
 
THE AMOUNT OF ONE POUND PER ANNUM 
 
 Calculation (VI) 2. 
 
 Standard Calculation Form, No. 3. 
 
 To find the amount of an annuity in any number of years. 
 
 Table III. 
 Required the amount of £500 per annum for 10 years at 5 per 
 cent, per annum compound interest. 
 
 (A) By Formula . M = Ai/ (" ~-^ ) I^^il© 1 , Chapter VI . 
 
 ' Log. Ratio 
 
 viultiply Log. R by 
 
 Log. I 
 RN -I"* Convert Log. to 
 
 ordinary number 
 deduct unity 
 
 I Log. of this is 
 
 Log Annuity 
 add Log . RN - 1 above 
 
 deduct Loff. r 
 
 1-05 
 10 
 
 00211893 
 10 
 
 RN 
 
 (ro5)i« 
 
 0-2118930 
 
 RN 
 
 -1 
 
 1-6289 
 1- 
 
 
 RN-1 
 
 0-6289 
 
 T- 7985779 
 
 Ay 
 
 RN-1 
 
 500 
 
 2-6989700 
 1-7985779 
 
 Ai/(RN- 
 /■ 
 
 -1) 
 
 2-4975479 
 2-6989700 
 
 M 
 
 
 3-7985779 
 
 Required future amount, £6288-94 
 
 (B) By Table III. U = Ay (^^^ — ^ Rule 2, Chapter YI 
 
 Table III. 10 years, 5 per cent. 
 Amount of £1 per annum 
 add Log. Annuity 
 
 RN-1 
 
 12-5779 
 500 
 
 10996079 
 
 A.y 
 
 2-6989700 
 
 M 
 
 
 3-7985779 
 
 Required future amount, £6288-94 
 
 (C) By Thoman's Table. M^At/ ("^^ Rule 3, 
 5 per cent. 10 years. 
 
 Chapter VI. 
 
 Log Annuity 
 Add Log. RN in 
 
 Table +10 
 
 Ay 
 
 RN 
 
 500 
 
 2-6989700 
 10-2118930 
 
 deduct Log. a" 
 
 Ai/RN 
 
 a^ 
 
 
 12-9108630 
 91122851 
 
 
 M 
 
 
 3-7985779 
 
 Required future amount, £6288-94 
 
62 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTEH VII. 
 
 COMPOUND INTEREST AS APPLIED TO AN ANNUAL 
 OR OTHER PERIODIC PAYMENT {Cojitinued). 
 
 TABLE IV. The present value of £i per annum for any 
 number of years. 
 
 -%(^) 
 
 FORMUL.^ USED IN CALCULATIONS AND RULES DEDUCED 
 THEREFROM. DERIVATION OF FORMULA AND APPLICATION TO 
 COMPILATION OF TABLES AND TO CALCULATIONS. CALCULATIONS 
 TO DEMONSTRATE THE THEORETICAL CONCLUSIONS BOTH AS 
 REGARDS THE PUBLISHED TABLES AND PRACTICAL EXAMPLES. 
 ThOMAN's method and FORMULA. 
 
 Author's Standard Calculation Form, No. 4. 
 
 Formulae. 
 
 A. To find file present vahie of £1 per annum for any 
 ninnher of years, as given in the puhlished tables : — 
 
 (7) Formula, P= (^-jpr^^ 
 
 by logs.: Log. [Freserit value of £1 per annum) = 
 Log. {W^- 1) -Log. RN-Log. r. 
 
 {2) By Th Oman's 'method: — 
 
 Formula, P= -— 
 
 a" 
 
 by logs. : Log. {Present value of £1 per annum) — 
 
 10 -Log. rt'i 
 
 B. To fnd the present value of any annuity for any number 
 
 of years : — 
 
 /RN_1\ 
 
 (7) Formula, V = Ky\^^^^^ ^ 
 
 by logs. : Log. [Present value of annuity) = Log. 
 Annuity + Log. {ll^-l)-Lug. W^-Log. r 
 
 [2] By Tho)na7i's method: — 
 
 Formula, P= —;— 
 
 a'' 
 
 by logs.: Log, [Present value of annuity) = Log . 
 annuity + 10 — Log. a"- 
 
THE PRESENT VALUE OF ONE POUND PER ANNUM 63 
 The 'present chapter deals only with the formula 
 
 (T> N 1 X 
 — y Th Oman's method and formula' are fidly 
 
 described in Chapter IX. 
 
 General Rules deduced from the above formulae. 
 
 To find the present value of any annuity for any number of 
 years. Author's Standard, Calc\iJation Form, No. 4. 
 
 Rule 1. If the rate per cent, be not given in Table IV or in 
 Thoman's Tables: — 
 
 Proceed by the formula relating to Table IV. 
 
 Calculation {VII) 2 A. 
 
 Rule 2. If the rate per cent, be given in Table IV : — ■ 
 
 Multiply the amount given in the table, by the 
 given annuity. The product is the present value 
 required. Calculation {VII) 2 B. 
 
 Rule 3. If the rate per cent, be given in Thomans Table : — 
 To the log. of the given annuity add 10, and deduct 
 therefrom the log. of a"- as given by Thoman. The 
 remainder is the log. of the present value required. 
 
 Calculation {VII) 2C. 
 
 To fnd the rate per cent., or number of years, proceed as 
 shown in the standard form for the purpose, given in 
 Chapter X. 
 
 Derivation of the Formula. In order to find tlie formula 
 relating to the present value of an annuity due at the end of 
 each year of a given term, the most direct method is to consider 
 an annuity of £1 in order to demonstrate the principle involved 
 and to arrive at the necessary modification in the previous 
 formula relating to the amount of an annuity. It will be 
 readily seen that the present value of an annuity for any 
 number of years is the same as the present value of the sum to 
 which that annviity will amount in the same period at the same 
 rate per cent. It has been shown in Chapter \1 that the 
 amount of an annuity of £1 may be found by the formula : — 
 
 r 
 and that the present value of £1 due at the end of any number 
 
64 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 of years is found by tlie formula, relating to Table II and 
 described in Chapter Y, namely : — 
 
 p=A 
 
 but A — 1 , therefore P = ^j^^ . 
 
 Consequently by multiplying these two formulae together 
 the required formula for the present purpose is obtained as 
 follows : — - 
 
 R^ - 1 1 R^ — 1 
 
 P = (present value of £1 per annum) = ^ "dn = ""^dn — 
 
 and the formula to find the present value of any annuity, Ky, 
 for any number of years becomes: — 
 
 ' . . . '-M^) 
 
 There is a similarity between the formuli3e relating to 
 Tables III and IV, namely, that they are both based upon the 
 
 RN - 1 
 
 factor ; — . In both cases, as will be seen by an inspection 
 
 of the standard calculation forms. Rule 1, the method consists 
 in adding to the log. of the annuity the log. of R^-l^ and 
 deducting from the sum of the logs, the log. of r. This gives 
 the desired result in the case of Table III relating to the 
 amount of an annuity, but in Table IV relating to the present 
 value of an annuity, the log. of R^ is previously deducted from 
 the log. of the annuity, which is ecjuivalent to saying that 
 the present values in Table IV may be found by dividing the 
 
 amounts in Table III by R^ ; but the values of -jj^ are given in 
 
 Table II, therefore the amounts in Table IV are equal to the 
 amounts in Table III, multiplied by the amounts in Table II, 
 or divided by the amounts in Table I. 
 
 Calculations. Having found the above formula relating 
 to the present value of an annuity of £1 for any number of 
 years, two calculations will now be made by its aid upon the 
 author's standard calculation form, No. 4, Both cases will 
 include the tliree methods of which the general rules are stated 
 at the head of this chapter, namely, by formula, by the 
 published tables, and by Thoman's method and tables. The 
 first calculation will deal only with an annuity of £1, and will 
 show the method of computing tlie amounts given in Table IV. 
 Calculation (VII) 1. 
 
 The second calculation will deal with an annuity of stated 
 amount, and will illustrate the method to be adopted in actual 
 practice. Calculation (VII) 2. 
 
THE PRESENT VALUE OF ONE POUND PER ANNUM 65 
 
 Calculation (VII) 1. 
 
 Standard Calculation Form, No. 4. 
 
 To find the present value of an annuity for any number of years, 
 
 and thereby prove the accuracy of the published table. 
 
 Table ly. 
 Required the present value of £1 per annum for 10 years at 
 5 per cent, per annum, compound interest. 
 
 (A) By Formula. P = A2/('^j^) Rule 1, Chapter VII. 
 
 
 ' Log. Ratio 
 
 multiply Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 ^ Log. of this is 
 Log. Annuity 
 add Log. (RN - 1) above 
 
 deduct Log. R^ above 
 deduct Log. r 
 
 R 
 
 N 
 
 RN 
 
 1-05 
 10 
 
 (1-05)10 
 
 0-0211893 
 0-2118930 
 
 Log . 
 W -1 
 
 RN 
 
 -1 
 
 1-6289 
 1- 
 
 
 
 RN-1 
 
 0-6289 
 
 1-7985779 
 
 
 Ay 
 
 RN-1 
 
 1- 
 
 0-0000000 
 1-7985779 
 
 
 RN 
 
 
 1-7985779 
 0-2118930 
 
 
 r 
 
 
 1-5866849 
 2-6989700 
 
 
 P 
 
 
 0-8877149 
 
 Required present value, £7-72174. 
 
 (B) By Table lY. P = A.y (^^£r^) ^^^1© 2, Chapter VII. 
 
 Table IV. 10 years, 5 per cent. 
 Present Value £1 per annum 
 add Log. Annuity 
 
 RN-1 
 
 RNr 
 A.y 
 
 7-72174 
 
 Required present value, £7-72174. This amount is given in 
 
 Table IV. 
 
 (C) By Thoman's Table 
 
 Ky 
 
 Rule 3, Chapter VII. 
 
 5 per cent. 10 years. 
 
 
 
 
 Log. Annuity 
 
 A.y 
 
 1- 
 
 0- 
 
 a^^lO 
 deduct Log. 
 
 a« 
 
 
 100000000 
 9-1122851 
 
 
 P 
 
 
 0-8877149 
 
 Required present value, £7-72174. 
 
66 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (VII) 2. 
 
 Standard Calculation Form., No. 4. 
 
 To find tlie present value of an annuity for any number of years. 
 
 Table IV. 
 Required the present value of £500 per annum for 10 years 
 at 5 per cent, per annum, compound interest. 
 
 (A) By Formula. P = Ay (^^n^) R"le 1, Chapter YII. 
 
 Log 
 RN_i 
 
 ^ Log. Ratio R 105 
 
 nniltiply Log. R by N 10 
 
 -1 Convert Log, 
 
 to ordinary number 
 deduct unity 
 
 Log-, of this is 
 
 Log. Annuity 
 
 add Log. (RN - 1) above RN _ l 
 
 deduct Log. R^ above RN 
 
 deduct Log. r 
 
 Required present value, £8860-86" 
 
 0021189:3 
 10 
 
 RN 
 
 (1-05)10 
 
 0-2118930 
 
 RN 
 
 -1 
 
 1-6289 
 1- 
 
 
 RN_1 
 
 0-0289 
 
 r7985779 
 
 Ay 
 RN-l 
 
 500 
 
 2-6989700 
 1-7985779 
 
 RN 
 
 
 2-4975479 
 0-2118930 
 
 r 
 
 
 2-2856549 
 2-6989700 
 
 P 
 
 
 3-5866849 
 
 (B ) By Table lY. P = Ay /"^^^^^ Rule 2, Chapter YII 
 
 Ti 
 
 iblelY. 10 years, 
 
 Present Yalue £1 
 
 Add Log 
 
 5 per cent, 
 per annum 
 Annuity 
 
 RN- 
 
 -1 
 
 7-72r 
 
 500 
 
 '4 
 
 0-8877149 
 2-6989700 
 
 3-5866849 
 
 
 RN; 
 
 Ay 
 
 P 
 
 Required present value, £;)860-867 
 
 (C) By Thoman's Table. P: 
 
 5 per cent. 10 years. 
 
 Ay 
 a" 
 
 Rule 3, Chapter YII. 
 
 Log. Annuity 
 
 Ay 
 
 500 
 
 2-6989700 
 
 add 10 
 deduct Log. 
 
 a"" 
 
 
 12-6989700 
 9-1122851 
 
 
 P 
 
 
 3-5866849 
 
 Required present value, £3860867. 
 
THE ANNUITY ONE POUND WILL PURCHAvSE 67 
 
 CHAPTER Till. 
 
 COMPOUN'D INTEREST AS APPLIED TO AN ANNUAL 
 OE OTHER PERIODIC PAYMENT {Contimied). 
 
 TABLE V. The annuity which £1 will purchase for any 
 number of years, or of which £1 is the present value. 
 
 --^ (^d 
 
 FOEMUL.E AND RULES DEDUCED THEEEFROM. GENERAL 
 
 REMARKS AS TO TaBLE V AND ITS RELATION TO AN EQUAL 
 
 annual instalment, of i'rincip.al and interest combined. 
 This table gives the actual values of Thoman's log. 
 FACTOR, a". Derivation of formula and application to 
 compilation of tables and to calculations. Calculations 
 to demonstrate the theoretical conclusions both as 
 regards the published tables and practical examples. 
 Thoman's method and formula. 
 
 Author's Standard Calculation Form, No. 5. 
 
 Formulae. 
 
 A. To find the annuity ivhich £1 will imrchase for any 
 number of years, or of wliich £1 is the ijresent value, as 
 given in the inihltshed tables : — 
 
 (1) Formula, Ai/= ( ]p"^ J 
 
 by logs.: Log. (Annuity £1 will i)urchase) = Log . R^ 
 + Log.T-Log.(W^-V) 
 
 [2) By Thoman's method: — 
 
 Formula, Ay = a^ 
 
 by logs.: Log. (Annuity £1 icill pxir chase) = Log. 
 a«-10 
 
68 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 B. To fnd the annuity ivhich may be ijurcliased with any 
 given sum for any number of years : — 
 
 (1) Formula, Ay = V (jpr^l) 
 
 by logs. : Log. [required annuity) = Log. {jjrincifal 
 sum) + Log. W^ + Log. r- Log. (RN-l) 
 
 (2) By Tho man's method : — 
 
 Formula, Ay = V a" 
 
 by logs. : Log. (required annuity) = Log. [yrinci'pal 
 sum) + Log. «"• — 10 
 
 The present chapter deals only ivith the formula 
 
 Thoman's method and formulce are fully described in Chapter 
 IX. 
 
 General Rules deduced from the above formulae. 
 
 To find the annuity which may be purchased with any given 
 sum for any number of years. 
 
 Author's Standard Calculation Form, No. 5. 
 
 Rule 1. If the rate per cent, be not given in Table Y or in 
 Thoman's Tables:- — ■ 
 
 Proceed by the formula relating to Table V . 
 
 Calculation {VIII) 2 A. 
 
 Rule 2. If the rate per cent, be given in Table V : — 
 
 Multiply the annuity given in the table, by the given 
 sum. The product is the reqtiired annuity which 
 may be purchased. Calculation [VIII) 2 B. 
 
 Rule 3. If tJie rate per cent, be given in Thoman's Tables : — 
 To the log. of the given sum, add the log. of a" as 
 given by Thoman. Deduct 10 from the sum of the 
 two logs. The remainder is the log. of the required 
 annuity which may be purchased. 
 
 Calculation (VIII) 2 C. 
 
 To find the rate per cent., or number of years, proceed as 
 shown in the standard form for the purpose^ given in 
 Chapter X. 
 
 Tables III and IV, containing the amounts and present 
 values of £1 per annum, correspond to Tables I and II relating 
 to tbe amounts and present values of £1. Tliere is a further 
 
THE ANNUITY ONE POUND WILL PURCHASE 69 
 
 Table, No. Y, given in Inwood and other published tables, 
 which is useful in order to ascertain the annuity which may be 
 purchased with a given sum of money, because anyone 
 contemplating the purchase of an annuity generally has a 
 definite sum to invest in this manner. Consequently it is 
 required to know the annuity which £1 will purchase for any 
 number of years, and from this can be ascertained by simple 
 multiplication the annuity which any given sum will purchase. 
 But the principal value of this table lies in the fact that 
 the amounts there given represent the respective annuities of 
 which £1 is the present value. The importance of this will be 
 recognised when it is remembered that this is the principle 
 underlying the repayment of debt by an equal annual instal- 
 ment of principal and interest combined, as laid down in 
 Section 234(4) of the Public Health Act, 1875. Table Y 
 represents Thoman's factor {a"'), and is a connecting link 
 between £1 and £1 per annum considered both in regard to 
 future amount and present value. By its aid the cumbersome 
 factor E^-l, previously referred to, may be avoided in cases 
 where the rate per cent, is included in Thoman's tables. 
 
 Deeivatiox of the Foemfla. The formula relating to this 
 table may be found by simple proportion without resorting to 
 any algebraical calculation. It has been ascertained by 
 Calculation (YII) 1, that £7-7217 is the present value of an 
 annuity of £1, and such values are given in Table lY. It is 
 required to find the annuity of which £1 is the present value. 
 
 It is obvious that it will be ^ „,.-,^ of £1. 
 
 < w21< 
 
 Consequently, by dividing unity by the present value of an 
 
 annuity of £1, as given in Table lY, the result is the annuity 
 
 which may be purchased by £1, and may be expressed by the 
 
 following rule : — 
 
 To find the annuity which £1 will purchase for any number 
 of years, first ascertain by Table IV the present value 
 of an annuity of £1 for the same period at the same 
 rate per cent.; and divide 1 by the present value so 
 found. The quotient tvill be the annuity ivhich may be 
 purchased by £1. 
 
 This rule simply means that to ascertain the annuity which 
 £1 will purchase, unity is divided by the values given in 
 
70 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Table 1\, but if it be reduced to terms of tlie annuity formula 
 
 it becomes : — 
 
 ^ 1 R^ r 
 
 The annuity of which £1 is tbe present [ ^^TTTi *^^' j^n _ x 
 
 value, as given in Table Y I p jj 
 
 Table Y. Calculations. Having found the above formula 
 relating to the annuity which £1 will purchase for any number 
 of years or the annuity of which £1 is the present value, two 
 calculations will now be made by its aid upon the author's 
 standard form, No. 5. Both cases will include the three 
 methods of which the general rules are stated at the head of this 
 chapter, namely, by formula, by the published tables, and by 
 Thoman's method and tables. The first calculation will deal 
 only with the annuity which £1 will purchase, or of which £1 
 is the present value, and will show the method of computing the 
 amounts given in Table Y. Calculation (YIII) 1. 
 
 The second calculation will deal with a stated amount to be 
 invested in an annuity, or of Avhieh it is required to ascertain 
 the future equivalent expressed in an annual payment, and 
 will illustrate the method to be adopted in actual practice. 
 Calculation (YIII) 2. 
 
THE ANNUITY ONE POUND WILL PURCHAvSE 
 
 71 
 
 Calculation (VIII) 1. 
 
 Statulard Calculation Form, No. 5. 
 
 To find the annuity which a prevsent sum will purchase for any 
 number of years, and also the equal annual instalment of 
 principal and interest combined, and thereby to prove the 
 accuracy of the published table. Table Y. 
 
 Eequired the annuity Avliich £1 will purchase for 10 years at 
 5 per cent, per annum, compound interest. 
 
 (A) By Formula. A// = P ( j^^-^j) I^^^lt' 1, Chapter Till. 
 
 Log'. Ratio 
 
 )nu.ltiply Log. R by 
 
 Jog . Convert IjO^. 
 
 W -1 
 
 to ordinary number 
 deduct unity 
 
 Log. Present Sum 
 add Log. RN above 
 Loff. r 
 
 deduct Log. (R^ - 1) above 
 
 R 
 
 105 
 10 
 
 0-0211893 
 10 
 
 RN 
 
 (1-05)10 
 
 0-2118930 
 
 RN 
 -1 
 
 1-6289 
 1- 
 
 
 RN-1 
 
 0-6289 
 
 i- 7985779 
 
 P 
 RN 
 
 /' 
 
 1- 
 
 1-6289 
 0-05 
 
 0- 
 
 0-2118930 
 2-6989700 
 
 RN-1 
 
 
 2-9108630 
 1-7985779 
 
 Ay 
 
 1-1122851 
 
 Required annuity, £0-129546. 
 
 (B) By Table Y. Ay = V (j^^^) Rule 2, Chapter YIII 
 
 Table Y. 10 years, 5 per cent. 
 Annuity £1 will purchase 
 add Loo'. Present Sum 
 
 RNr 
 
 RN 
 P 
 
 0-1295 
 
 A) 
 
 Required annuity, £0-1295. Tliis amount is given in Table Y. 
 
 (C) By Thoman's Table. A.y = P r/" Rule 3, Chapter YIII. 
 5 per cent. 10 years. 
 
 Log. Present Sum 
 add Los". «"■ 
 
 deduct 10 
 
 IP 
 
 I an 
 
 1- 
 
 0- 
 
 91122851 
 
 9-1122851 
 
 A;?/ 
 
 
 11122851 
 
 Required annuity, £0-129546. 
 
72 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (VIII) 2. 
 Standard Calculation Form, No. 5. 
 To find tlie annuity which a present sum Avill purchase for any 
 
 number of years. Tabled. 
 
 To find the annuity which may be purchased with £6288-94 for 
 
 10 years at 5 cent, per annum, compound interest. 
 
 (A) By Formula A2/ = P (f^-3^1) I^^^le 1, Chapter YIII. 
 
 
 Log. Ratio 
 
 multiijly Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 R 
 
 N 
 
 105 
 10 
 
 0-0211893 
 10 
 
 
 RN 
 
 (1-05)10 
 
 0-2118930 
 
 R^^ -1 
 
 RN 1-6289 
 -1 1- 
 
 
 RN- 
 
 -1 0-6289 
 
 1-7985779 
 
 
 Log. Present Sum 
 add Log. RN above 
 Log. r 
 
 P 
 
 RN 
 
 r 
 
 6288-94 
 1-6289 
 0-05 
 
 3-7985779 
 0-2118930 
 2-6989700 
 
 deduct Log. (R^ - 1) above 
 
 RN- 
 
 -1 0-6289 
 
 2-7094409 
 1-7985779 
 
 
 Ay 
 
 
 2-9108630 
 
 Required annuity, £814-447 
 
 (B) By Table Y. Ay^V (j^^;,) Rule 2, Chapter YIII. 
 
 Table Y. 10 years, 5 per cent. 
 Annuity £1 will purchase 
 add Log. Present Sum 
 
 RNr 
 
 RN-1 
 
 01295 
 
 6288-94 
 
 1-1122851 
 3-7985779 
 
 Ay 
 
 2-9108630 
 
 Required annuity, £814-447 
 
 (C) By Thoman's Table. A.y = P a« Rule 3, Chapter YIII. 
 5 per cent. 10 years. 
 
 Log. Present Sum 
 add Log. a" 
 
 P 
 
 a"- 
 
 6288-94 
 
 3-7985779 
 9-1122851 
 
 
 12-9108630 
 
 deduct 10 
 
 Ay 2-9108630 
 
 Required annuity, £814-447. 
 
THOMAN'S TABLES 73 
 
 CHAPTER IX. 
 
 THOMAN'S LOGARITHMIC TABLES OF COMPOUND 
 INTEREST AND ANNUITIES. 
 
 Explanation of Thoman's symbols, R^, and, a", and their 
 
 RELATION, SEPARATELY OR IN COMBINATION, TO THE FORMULA 
 ALREADY ASCERTAINED. ThOMAN's METHOD OF STATING LOG. 
 
 of a"- by adding 10 to the log. 
 
 Author's Standard Calculation Forms, 1 to 5. 
 
 Symbols used by Thoman : — 
 
 'R^=the amount of £1 in any number of years. 
 a"' = the annuity lohicli £1 ivill 'purchase for any number 
 of years. 
 
 an = 
 
 Rn _ 1 [Table V. 
 
 Comparison with previous formulae : - 
 
 Thoman's 
 Table General Logarithmic 
 
 Chapter. No. Giving Values of Formulas. Formulse. 
 
 IV. 1. Amount of £1 R^ R"" 
 
 F. II. Present value of £1 ^on ""dn 
 
 VI. III. Amount of £1 per annum 
 
 RN -1 RN 
 
 r a"- 
 
 RN - 1 1 
 
 VII. IV. Present value of £1 per annum ... -^^ — -^ 
 
 RNr 
 
 VIII. V . Annuity which £1 will purchase ^ .^ - t'-" 
 
 The logarithmic equivalents of the above formulae and the 
 rules based thereon are given at the head of the chapter dealing 
 with each table. 
 
74 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Inchided iu Inwood, and iu other published tables of 
 compound interest, are a series of valuable tables by M, Fedor 
 Thoman, of the Soc. Credit Mobilier, of Paris, the author of 
 " Logarithmic Tables of Interest, etc." These tables are of 
 great assistance in the solution of many problems; and the 
 whole of the formulae already obtained by derivation from the 
 algebraical formula for a geometrical progression will now be 
 compared with the simplified formulse given by Thoman. A 
 glance at the above comparative table will show that the whole 
 of the formula already obtained by derivation from the formula 
 A = P E^, may be expressed by some modification of the factors 
 E^', and /■, which have already been fully explained iu previous 
 chapters. The table at the end of Chapter Y contains the 
 values of E, and /•, for many rates per cent, likely to be 
 required in practice, and also gives the corresponding logs, of 
 these values ; and it has been explained how to find by means 
 of the formulse the values and logs, of E, and /•, for any 
 intermediate rates per cent, not inchided in the above table. 
 The values of E^' may be obtained by multiplying the log. of E 
 by the number of years, as shown in Calculation (IT) 2 and 
 others. 
 
 Problems involving an annual or other periodic payment, as 
 in Tables III, lY, and Y, cause the introduction of a variation 
 of E^', namely (E^^-1), which imports a new calculation which, 
 although not of itself difficult, is inconvenient because it is 
 necessary to convert the log. of E^\ to an ordinary number 
 before deducting unity, and afterwards to find the log. of the 
 remainder. This might be avoided by preparing tables of 
 (E^-l), and the corresponding logs., for each rate per cent, for 
 any number of years. 
 
 In the above formulse relating to an annual sum in Tables 
 • III, lY, and Y, the factor (EN-1) is always associated with i\ 
 or (E^^ ;•), and E^ is the factor relating to the amount and 
 present value of £1 as shown by Tables I and II. By 
 combining the factors (E^-l), and E^, a connecting link is 
 obtained between Tables I and II relating to £1, and Tables 
 III, lY, and Y, relating to the amount and present value of 
 £1 per annum ; and this is the principle underlying Thoman's 
 method . 
 
 They are merely tables, and do not enunciate any new 
 principle, but by giving under each rate per cent, for various 
 numbers of years the logs, of two factors only, they enable any 
 calculation to be made, at the rates included in the tables, 
 without any further reference to the ordinary published tables 
 
THOMAN'S TABLES 75 
 
 I to Y. Tliomau's tables have two advantages over the ordinary 
 tables of compound interest in that (1) they are worked out for 
 fractional eighths per cent, up to 6 per cent., and (2) they give 
 the logs, direct and thereby avoid any reference to the log. 
 tables ; but since they are worked out for a limited number only 
 of rates per cent, they do not dispense entirely with the method 
 of calculation by means of the formulae previously stated, and 
 these methods will therefore be included in subsequent chapters 
 as well as Thoman's method. 
 
 The factors included in Thoman's tables are R^ and a". 
 EN is the factor governing Tables I and II, Avithout any 
 alteration, and Thoman's tables may be referred to instead of 
 finding the logs, of these values by the methods shown in the 
 various calculations, using the ordinary tables and logs. 
 
 a« is used by Thoman to denote the annuity which £1 will 
 purchase or of which £1 is the present value. 
 
 The logs, of a^ in Thoman's tables are, purely for convenience 
 of calculation and perhaps for facilitating printing, given in a 
 different form to the logs, of E^ in the same tables. In Table I (E^) 
 the values are all greater than unity, hence the characteristics 
 of the logs, of these values are always positive. In Table V 
 the values are all decimals of unity, and the logs, of these 
 values have negative characteristics. Thoman adds 10 to the 
 characteristics of the logs, of the values of a'' in Table Y, and 
 thus, bearing in mind that any calculation can be made by 
 means of E^ and a'\ it is possible to eliminate the troublesome 
 negative characteristic altogether. All that is required is to 
 correct the characteristic of the final log. by adding 10 in the 
 case of Tables III and lY, and deducting 10 in the case of 
 Table Y, to or from the resulting log. before ascertaining the 
 antilog., or numerical equivalent. Thoman's logs, of a« may be 
 treated in the ordinary way, by using the mantissa given in the 
 table and converting the characteristic there giA-en to the proper 
 minus quantity, i.e., 10 minus the given characteristic. For 
 the sake of clearness the method of deducting or adding 10 
 from or to the log. has been adopted in the whole of the 
 standard calculation forms prepared by the author. 
 
 There are two methods of connecting the ordinary tables and 
 formula- relating to £1, and £1 per annum, namely, either by 
 means of Table I and III, dealing with the respective amounts 
 (as adopted by Thoman to derive the factor «« as shown by 
 Table Y) or by means of Tables II and III, leading to Table lY, 
 Avhich is the reciprocal of Table Y, as shoAvn when dealing with 
 the latter table in Chapter YIII. 
 
76 REPAYIvrENT OF LOCAL AND OTHER LOANS 
 
 Thoman's method Avill now be applied in order to derive 
 Table Y, or tbe formula relating thereto, from: — 
 
 Table I, . the amount of £1 for any number of years, 
 
 and, Table III, the amount of £1 per annum in any number of 
 
 years, 
 taking, in each case, a period of 10 years, and a rate of 
 accumulation of 5 per cent, per annum, as follows: — 
 
 Table I, amount of £1 1-6289 or RN 
 
 Table 777, amount of £1 per annum ... 12"5779 or 
 
 The annuity which £1 will purchase is obtained by dividing 
 16289 by 12'5TT9, and the formula corresponding thereto may 
 also be obtained by dividing the corresponding formulae as 
 follows : — 
 
 Table I 1-6289 _ R^ 
 lable V -.rj^,.^blein 12-5779 W -I ' 
 
 r 
 Stated in actual values : — 
 
 1-6289 
 Table V = ,^":^^ = 0-1295 
 125/ 79 
 
 as may be found by actual calculation, or obtained by direct 
 reference to Table Y. 
 
 Stated in terms of the above formulae : — 
 
 Table V = rra ^ or 
 
 RN -1 R^ -1 
 I' 
 
 which is the formula relating to Table Y, as shown in 
 Chapter YIII, giving the annuity which £1 will purchase 
 for any number of years. But Thoman's symbol a», although 
 expressed in logarithmic form, represents the same factor; 
 therefore the formula relating to Table Y, as derived in Chapter 
 YIII, from the simple formula A = P RN, may be replaced by 
 Thoman's symbol «", with the result that : — 
 
 Table V = ^^ _x =^^" «^" Thoman. 
 
 The above formula, relating to Table Y, contains the three 
 factors, RN, r, and (RN-1)^ already referred to and fully 
 explained in previous chapters, in order to express the relation 
 between the respective amounts and present values of £1, and 
 
THOMAN'S TABLES 77 
 
 of £1 per annum. Thoman, by adopting tlie symbol a", 
 eliminates the factor (R^ — 1) altogether. 
 
 The factor (R^ — 1) may, however, be ascertained from 
 Thoman's tables if required as follows : — - 
 
 or by logarithms : 
 
 Log. (EN -1) = Log. RN + Log. r-Log.a". 
 
 and Thoman's a^ may be found from the above factors, as 
 follows : — 
 
 Log. o« = Log. RN + Log. 7' -Log. (R^-l). 
 
 The whole of the formulae previously ascertained by 
 derivation from the simple formula A = P R^, may now be 
 expressed in terms of Thoman's factors of R^, and a", as 
 follows : — 
 
 Table I. RN The amount of £1 : — 
 
 will be expressed by Thoman's factor R^ 
 
 Table 11. zg-^ The present value of £1: — 
 
 will be expressed by Thoman's factor ^^ 
 
 Table 111. ( ) The amount of £1 per annum: — 
 
 It has been proved that : — 
 Table Y = _™^ 
 And transposed, it will be seen that : — 
 
 Ta^l^ III = TaHeT 
 but Table I = RN 
 and Table Y = a^ 
 Therefore Table III will be expressed 
 
 by Thoman's symbols -— — 
 

 78 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Table IV. ( —ry^ — ) The ijiesent value of £1 per annum : - 
 
 It has been proved in Chapter Till, 
 that Table lY, is the reciprocal 
 of Table Y; and it has been 
 shown above, that Table Y, is 
 expressed by Thomau's symbol, 
 
 Therefore Table lY will be expressed 
 by Thonian's factor 
 
 Table V. ( -ns — ^ ) ^^'^ annuitij irhich £1 icill purchase : — 
 
 This, as shown above, is the equiva- 
 lent of Thoman's factor a"^ 
 
 The above f ormiilse by Thomau may be stated logarithmically 
 as follows, using' the logs, of a"- increased by 10, as given in 
 Thoman's tables: — 
 
 Table T = R^ =Loo- R^' 
 
 Table // = J^ = Log 1 = 0- Log R^' 
 
 Table 111= ^= Log R^ _ Log a" (add 10) 
 
 Table /F = — = Log 1 = 0- Log a" (add 10) 
 a^^ 
 
 Table V = a'' = Log a'' (deduct 10) 
 
 and any problem may be solved by adding to, or deducting 
 from, the log. of the given sum or annuity the logs, of E-^ and 
 a"' as given by Thoman, as shown in the various standard 
 calculation forms prepared by the author given in Chapter X. 
 
 In using the above log. formulae it is important to bear in 
 mind the previous remarks as to the addition of 10 to the 
 log. of a"' in the tables by Thoman, and the following examples 
 will make the matter clear. It affects only Tables III, lY 
 and Y. 
 
 Taking the figures relating to a period of 10 years and a rate 
 •of accumulation of 5 per cent, in each case, the above log. 
 formulae will now be applied to find the amounts given in 
 those tables, which, of course, relate to £1 only: — 
 
THOMAN'vS TABLES 79 
 
 Table HI. Required the amount of £1 per annum for 10 rears 
 at 5 per cent. 
 
 Calculation (VI) 1. 
 
 Log. EN+10 ... -10-211 S930 
 deduct Los:, a'^ = 9112 2851 
 
 1-099 6079 
 
 which is the Loff. of £12-5779 
 
 Table IV. Required the present value of £1 per annum for 
 10 years at 5 per cent. 
 
 Calculation (YII) 1. 
 
 Log. 1 = 0+10... =10000 0000 
 dedtictLos.a^ = 9112 2851 
 
 0-887 7149 
 
 ■which is the Log. of £7-7217 
 
 I 
 
 The above examples show, that in using the logs, of a"' as 
 given in Thoman's tables in conjunction with the log. formulae 
 relating to Tables III and IT, the log. of fl" (which is increased 
 by 10 in the tables), is deducted, and consequently 10 must be 
 added to the resulting log., but it is immaterial where 10 is 
 added. In the above example 10 has been added to the log. 
 of H^ as given in Thoman's tables. 
 
 In the case of Table Y, the factor a^ represents the values 
 given in the table, and as Thoman's logs, are increased by 10, 
 it only remains to deduct 10 therefrom in order to find the true 
 log. of the annuity required. 
 
 In previous chapters dealing with Tables I to Y and in the 
 following chapters dealing with other calculations, the formulae 
 and rules relating to the method by Thoman will be found at 
 the head of each chapter, without any further explanation. 
 
8o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER X. 
 
 STANDARD CALCULATION FORMS, PREPARED BY 
 THE AUTHOR. 
 
 The five published tables of compound interest. The 
 three methods of calculation in each form and the 
 corresponding rules relating thereto. meaning of 
 all symbols and factors, and the various methods of 
 finding the actual log . values. six standard forms, with 
 references to the above and list of problems to which 
 
 EACH MAY BE APPLIED. AlSO STANDARD CALCULATION FORMS 
 TO FIND THE EXACT OR APPROXIMATE RATE PER CENT. OR 
 NUMBER OF YEARS, IN CONNECTION WITH EACH OF THE FIVE 
 TABLES OF COMPOUND INTEREST AND THE SINKING FUND 
 INSTALMENT. 
 
 The FIVE STANDARD TABLES OF COMPOUND INTEREST RELATING 
 TO THE AMOUNT AND PRESENT VALUE OF £1, AND OF £1 
 PER ANNUM. 
 
 Table I. The amount of £1 in any number of years. 
 
 Chapter 7T', Form 7. 
 
 Table II. The present value of £1 due at the end of any 
 number of years. Chapter F, Form II. 
 
 Table III. The amount of £1 per annum in any number of 
 years. 
 
 Chapters VI and XIII, Forms III and Ilia;. 
 
 Table IV. The present value of £1 per annum for any number 
 of years. Chapter VII, Form IV. 
 
 Table V. The anmiity which £1 ivill purchase for any 
 number of years. Chapter VIII, Form V. 
 
 In these tables the actual values only are generally given, 
 and not their logarithmic equivalents which are found in the 
 tables of M. Fedor Thoman. 
 
STANDARD CALCULATION FORMS 8i 
 
 In actual practice, extending over many years, the autlior 
 nas repeatedly felt the want of a uniform system of making the 
 various calculations, and at the same time a means of avoiding 
 frequent references in order to ascertain the particular method 
 to be adopted. 
 
 The absence of such a standard was not felt so much when 
 the problems occurred only occasionally, but during the com- 
 pilation of this book such a large number of calculations had 
 to be made that some such standard became imperative if only 
 as a means of saving the labour involved in re-writing the mere 
 framework of each. The result is the sis standard forms used 
 throughout the work which are useful not only as blanks to be 
 filled up, but also because they contain the formula relating 
 to each of the above five tables of compound interest. Each 
 standard form includes three methods of calculation — namely 
 (A) by formula, (B) by the published mathematical tables, and 
 (C) by Thoman's tables, and these methods are based upon the 
 three rules stated both by formula and in words at the head of 
 the chapter dealing with each of the five tables of compound 
 interest. In order to bring together the whole of the formulae, 
 symbols and rules, as well as the methods adopted, a series of 
 notes have been made showing, firstly, the meaning of all the 
 symbols and factors, and, secondly, the method of finding the 
 various factors used in the calculations. The object of doing 
 this is to render this chapter a complete guide to anyone 
 requiring to make similar calculations, who is not sufficiently 
 interested in the subject to become more fully acquainted with 
 the derivation of the formulae. Finally, in order to indicate 
 the particular form required a fairly comprehensive list has 
 been prepared of the problems which have in the book been 
 solved by the use of each standard form. The forms are 
 numbered to correspond with the five tables of compound interest 
 given in the published tables. Each form contains a short 
 heading of the problem as stated in the tables. It is here 
 advisable to mention that form 3x for finding the annual 
 sinking fund instalment is based upon Table III; and form 5 
 may be used to find the equal annual instalment, of principal 
 and interest combined, to repay a given loan in a stated number 
 of years. 
 
 If any of the calculations involve recurring periods less than 
 one year, the necessary alteration may be made by taking the 
 number of periods and a correspondingly reduced rate per cent., 
 in other words, the rate per cent, per annum in the published 
 tables becomes the rate per cent, per period. This is where the 
 
82 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 metliod by foriuula Avill be found invaluable, inasmuch as in 
 many cases the rate per cent, per period will probably not be 
 found worked out in any of the published tables. 
 
 The Three Methods of Calculation ix Each Form. 
 
 A. By formula — Rule 1. The method of making the calcula- 
 
 tions in this manner is fully explained in the following 
 notes and in the standard forms. 
 
 B. By the puhlished tables of compound Interest — Rule 2. The 
 
 first step is to ascertain from the tables the actual values 
 relating to £1 under similar conditions as to period and 
 rate per cent. This amount, multiplied by or divided 
 into the sum in respect of which the calculation is to be 
 made, gives the result required. 
 
 C. By Thoman's Tables — Rule 3. This method is fully 
 
 described in the standard forms, and consists merely of 
 various combinations of the logs, of R^ and <7" and of 
 the amounts in the problem. Thonian's tables are fully 
 described in Chapter IX. If the results be recjuired to 
 be correct to the utmost decimal point the method by 
 Thoman should be adopted because these tables give the 
 actual log. values. 
 
 A full explanation of the rules in each form is given at the 
 head of the chapter dealing with the subject matter of each 
 form. 
 
 The Rate Per Cent, and the Number of Years. In 
 addition to the six standard forms, the author has prepared ten 
 forms showing the methods of determining the rate per cent, 
 and the number of years in connection with each of the five 
 tables of compound interest and the sinking fund instalment. 
 These latter forms each contain particulars of an example, 
 worked out in full in the book and to which a reference is 
 made. 
 
 It will be noticed that the results obtained are in several 
 ■cases approximate only, especially as regards the rate per cent. 
 as expressed by the factors R and r, which, however, can be 
 determined to any required degree of accuracy only by methods 
 which are far too technical to be included in a work of this 
 nature. For all practical purposes an approximation of the 
 rate per cent, is sufficient, and this may generally be obtained 
 
STANDARD CALCULATION FORMS 83 
 
 from the published tables of conipoiuid interest giving either 
 the actual values or their logarithmic equivalents. This 
 difficulty in finding the exact rate per cent, arises only in the 
 case of an annuity or other periodic payment. In the case of 
 the amount and present value of a sum of money [Tables I 
 and II] the calculation is a simple one depending upon the value 
 of 11^, If the value of this be known as well as one of the 
 factors, the other may be found as shown in the following 
 standard forms for finding the rate per cent, and number of 
 
 /RN — 1\ 
 
 years. The factor relating to all annuities is ( ; — j and 
 
 represents the amount of £1 per annum at the end of N years. 
 The rate per cent, cannot be determined exactly from this 
 factor, and the method by approximation is too long. The 
 practical method of finding the approximate rate per cent, of 
 accumulation of an annuity is to reduce the actual example to 
 terms of an annuity of £1 as given in the various published 
 tables of compound interest. Having done so, a reference is 
 made to the tables and the nearest figure ascertained. This 
 figure is adopted in the calculation, and if necessary the result- 
 ing error is calculated and corrected. An example of this may 
 be found on referring to Chapter XXXII, where the effect of 
 taking the equated period at 23, instead of 23' 136 years, is 
 fully explained and accounted for. 
 
 Other Methods of Calculatiox. In addition to the 
 problems which may be solved by means of the standard forms 
 included in this chapter there are several others which occur in 
 the course of the book, and which are fully explained as they 
 arise. In all cases the calculation has been made in such a 
 form that it may be adapted to any similar problem, and it is 
 not necessary to repeat the forms here. The folloAving list 
 includes the whole of these special calculations : — 
 
 (1) A stated sum is required to be set aside and accumulated 
 
 at compound interest for a stated number of years. At 
 the end of that time the annual sum ceases, but the 
 amount then in the fund continues to accumulate for a 
 further stated term. The rate of accumulation may be 
 varied or not at the end of the first period. It is required 
 to ascertain the amount in the fund at the end of the 
 second period. Statement XVI. D. 1. 
 
 (2) A stated amount is required to be provided at the end of a 
 
 prescribed period by the accumulation of an annual sum 
 
B4 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 to be set aside during- the early years of such period only. 
 The amount then in the fund will continue to accumulate 
 until the end of the prescribed period. It is required to 
 ascertain the annual amount to be set aside during the 
 first part of the prescribed period. 
 
 Calculation (XXXIY) G. 
 (3) A stated annual sum to be set aside for a prescribed 
 period and accumulated at a definite rate per cent., is 
 sufficient to provide a stated amount at the end of that 
 period. Before the expiration of the period a change is 
 made in the conditions affecting either the period or the 
 rate per cent., or both, but not the amount to be provided. 
 It is required to ascertain by one calculation the equiva- 
 lent future annual sum under the amended conditions. 
 
 Statement XXVI. D. 
 
 Symbols. The following is a complete list of the symbols 
 used in the various formulae, and in the standard calculation 
 forms, with the meaning of each symbol, all of which have been 
 fully described in Chapter III : — 
 
 A denotes the amount or ultimate sum to which the present 
 sum P will accumulate in X years at the ratio or common 
 factor R. It represents both the ultimate sum, to which 
 a stated present sum will amount at the end of the period, 
 as well as the stated sum, due at the end of the period, 
 of which the present value P is required. 
 
 P denotes the principal sum in hand ; and represents also 
 the present value of a definite sum of money A due at the 
 end of a stated period of years ; it denotes also the present 
 value of an annuity or other periodic sum, Ay, payable 
 at the end of each of a stated number of years or periods 
 X. 
 
 R denotes the ratio or common factor existing between each 
 term of the progression, or the amounts of £1 at the end 
 of each succeeding year. It is in all cases £1 increased 
 by interest upon £1 for one year at the rate per cent, in 
 question. It corresponds with the algebraical factor r. 
 r denotes the interest upon £1 for one year or period at the 
 stated rate per cent. It is always less than the above 
 factor E, bv unity. 
 
 R-l = r. 
 
 The actual rate per cent, is never used in calculations 
 involving compound interest, but is always expressed in 
 
STANDARD CALCULATION FORMS 85 
 
 its relation to £1 ouly, as II, and /■, above. This term r 
 is not tlie equivalent of the algebraical factor r, in a 
 geometrical progression. 
 
 N denotes the number of years, or other equal periods, and 
 must not be confounded with the factor n in the 
 algebraical formula for a geometrical progression which 
 represents the number of terms in the progression. For 
 this reason it is expressed by a capital letter. This term 
 is the equivalent of the algebraical term n — 1. 
 
 Ay denotes the annuity or other periodic sum to be paid, set 
 aside or received at the end of each year or period, N. 
 
 M denotes the sum to which the annuity or other periodic 
 sum Ay will amount, if accumulated for a stated number 
 of years or periods, N, at a stated rate per cent. 
 
 Formulae, The above symbols are combined in various 
 ways in the formulse given in the book resulting in various 
 factors, and the following list has been prepared in order to 
 show the meaning of such factors, and also the manner in 
 which they may be found, not only in actual values, but also in 
 their logarithmic equivalents. 
 
 The numbers in brackets in the author's standard calculation 
 forms in this chapter refer to the following notes : — 
 
 Note Symbol or factor Remarks 
 
 (1) E, = ratio. This may be found by adding to £1, 
 
 interest upon £1 for one year; and 
 the log. of E- may be found from the 
 log. tables or from the special table 
 of those logs, given in Chapter V. 
 
 (2) 7-= interest upon This may be found by deducting unity 
 
 £1 for one year. from the value of R above, and the 
 log. of r may be found, as in note (1). 
 
 (3) RN. This symbol corresponds with the 
 
 symbol r« of Thoman. The log. of R^ 
 is found by multiplying the log. of R 
 by the number of years or periods. 
 The logs, of RN are given in 
 Thoman's tables. The actual values 
 of R^ are given in Table I. 
 
86 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Note Symbol or factor Kemaiks 
 
 1 This factor Avill very rarely, if ever, be 
 
 (3a) EN. required to be found by calculation. 
 
 The actual values are g-iveu in Table 
 II. The log. of this factor may be 
 found by deducting the log. of H^ 
 from 0. 
 
 (4) IlN_i^ The actual values of this factor are not 
 
 given in Inwood or other published 
 tables, although they may be found 
 by deducting unity from the values 
 given in Table I. The log. of this 
 factor is not given in Thoman's tables, 
 but may be found by converting the 
 log. of EN there stated into its 
 equivalent ordinary number or anti- 
 log., deducting unity therefrom, and 
 finding the log. of the remainder. 
 The log. of E^-l so found may be 
 proved by Inwood, by deducting 
 unity from the amount given in 
 Table I, and finding the log. of the 
 remainder. The log. of E^-l may 
 be found by Thoman's tables as 
 follows : — 
 
 Log. EN + log. r+lO-log. a", 
 log. 7' being found, as explained in 
 note (2) above. 
 
 (5) RN_i, tpjj^g actual values of this factor are 
 
 r given in Table III in Inwood. The 
 
 logs, of this factor may be found by 
 
 Thoman's tables as follows: — 
 
 Log. EN + 10 -log. a'K 
 
 (6) EN-1. The actual values of this factor are 
 
 EN r. given in Table IV in Inwood or other 
 
 similar tables. The logs, of this factor 
 may be found from Thoman's tables 
 by deducting the log. of iV\ there 
 given, from 10. 
 
STANDARD CALCULATION FORMS 87 
 
 Note Symbol or factor Remarks 
 
 R^ ^" The actual values of this factor are 
 
 (7) li^ — 1. given in Table Y in Inwood or other 
 
 similar tables. The logs, of this 
 factor may be found from Thoman's 
 tables by deducting 10 from the log. 
 of a" there given. 
 
 (8) <2". This is a term employed by Thoman 
 
 to denote the annuity, £a per annum, 
 which £1 will purchase for n years; 
 and the actual values of Avhich are 
 given in Table Y. The logs, given 
 in Thoman's table are as explained in 
 Chapter IX, the true logs, of a" 
 increased by 10. The relations between 
 a" and the above symbols are ex- 
 plained briefly in the foregoing notes 
 and fully in Chapter IX. This factor 
 is extremely useful for finding the 
 equal annual instalment of prin- 
 cipal and interest combined (the 
 annuity method, Chapter XII). 
 
 Standard Calculation Form, No, 1. 
 
 Table I. To find the future amount of a present sum. 
 
 Chapter lY. 
 This form has been used in the solution of problems of the 
 following nature : — 
 
 Calculation. 
 
 To find the amount of loan which will be provided 
 by the future accumulation of the present 
 investments representing a sinking fund ... (^'^) 4. 
 
 To find the amount of loan which will be un- 
 provided for if an ascertained deficiency in a 
 sinking fund remains uncorrected (XY) 6. 
 
 To find the amount of loan which will be provided 
 by the future accumulation of the proceeds 
 of sale of assets paid into the fund (XYII) 3. 
 
REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Calculation Form, No. 1, 
 
 Table I. To find the future amount of a present sum. 
 The following rules are explained at the head of Chapter lY. 
 Here state the general nature of the problem. Calculation 
 
 No. 
 
 Here state full details of the actual problem. 
 
 (A) By Formuh 
 
 A = P RN 
 
 Eulel. 
 
 Log. ^ 
 
 Log. Ratio (1) R 
 
 MulHijly Log. R by N 
 
 (3) . RN 
 
 Log. Present Sum P 
 
 fl^^Log.RNaboTe(3) RN 
 
 Y allies . 
 
 Required future amount, £ 
 
 Loqs. 
 
 (B) By Table I. 
 
 A= PRN 
 
 Rule 2. 
 
 Table I. years per cent. 
 
 Amount of £1 (3) 
 
 add Loff. Present Sum 
 
 RN 
 
 P 
 
 A 
 
 Required future amount, £ 
 
 (C) By Thoman's Table. A - P RN 
 
 per cent. years 
 
 Rule 3. 
 
 Log. Present Sum 
 add Log. RN (3) 
 
 P 
 
 RN 
 
 Required future amount, £ 
 
STANDARD CALCULATION FORMS 
 
 89 
 
 The Amount and Present Value of One Pound. 
 
 Tables I and II. 
 To find the number of years 
 
 Standard Forms, 1 and 2. 
 
 based on Calculation (XVI) 5. 
 
 Given factors : 
 
 Present sum 
 
 Amount thereof .. 
 Rate per cent. 
 
 Ratio 
 
 Interest of £1 
 
 Details of Method : 
 
 find 
 
 find, and deduct .. 
 
 P 
 A 
 
 R 
 
 r 
 
 . Log. A 
 . Log. P 
 
 . Log. RN 
 
 . Log. R. 
 
 9463-00 
 1123907 
 31 
 1035 
 0-035 
 
 11239-07 
 9463-00 
 
 1-035 
 
 4-0507305 
 3-9760288 
 
 difference . . 
 
 00747017 
 
 find 
 
 0-0149403 
 
 To find the number of years, divide the above log. of R^ by 
 the above log. of R, as described in Chapter XXXII, 
 and the quotient is the number of years required, in 
 this case, 5 years. 
 
 The Amount and Present Value of One Pound. 
 Tables I and II. Standard Forms, 1 and 2. 
 
 To find the rate per cent : 
 
 based on Calculation (XV) 4. 
 
 Given factors : 
 
 Present sum P 
 
 Amount thereof ... A 
 
 Number of years ... N 
 
 Details of method : 
 
 find Log. A 
 
 find, and deduct ... Log. P 
 
 difference ... Log. R^ 
 
 divide this log. by 
 
 the number of years N 
 
 which is the log. of ... R 
 
 946300 
 
 14799-71 
 
 13 
 
 14799-71 
 9463-00 
 
 13 
 1-035 
 
 4-1702533 
 3-9760288 
 
 0-1942245 
 0-0149403 
 
 To find the rate per cent., deduct unity from the above ratio 
 and multiply the remainder by 100, or ^ per cent. 
 
90 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Calculation Form, No. 2, 
 
 Table II. To find the present yalne of a sum due at a future 
 date. Chapter Y. 
 
 This form has been used in the solution of problems of the 
 following nature: — 
 
 Calculation. 
 
 To find the sum now payable ^Yhich is the equiva- 
 lent of a given loan payable at the end of a 
 prescribed number of years. 
 
 vSee Chapter XXXII. 
 
 Givea a stated sum, to find the accumulated amount 
 of an annual instalment, to be set aside for a 
 limited period only; the amount so found to 
 accumulate for a further stated period, and 
 then amount to the stated sum. " The method 
 by step" (XVI) 3. 
 
 The annual instalment is then found by 
 means of standard form, No. 3x (XA I) 4. 
 
 The methods of finding the rate per cent, and the number 
 of years are similar to those given under Table I. 
 
STANDARD CALCULATION FORMS 
 
 91 
 
 Standard Calculation Form, No. 2, 
 
 Table II. To find the present value of a sum due at a future 
 date. 
 
 The following rules are explained at the head of Chapter Y. 
 
 Here state the general nature of the problem. Calculation 
 
 No. 
 Here state full details of the actual problem. 
 
 (A) By Formula. P = 
 
 A 
 EN 
 
 Eulel. 
 
 /-Log. Eatio (1) 
 Log MultiplijLog.'Rhj 
 
 I (3) 
 
 E 
 
 Values. 
 
 Logs. 
 
 EN 
 
 Log. Future vSum 
 deduct Log. E^ above 
 
 (3) 
 
 A 
 EN 
 
 
 P 
 
 Eequired present v. 
 
 ilue, 
 
 £ 
 
 • 
 
 (B) By Table II. P = 
 
 A 
 
 EN 
 
 Eule 2. 
 
 Table II. years per cent. 
 
 Present value of £1 (3a) 
 
 add Log. Future Sum 
 
 1 
 
 EN 
 
 A 
 
 
 P 
 
 Eequired present v 
 
 alue, 
 
 £ 
 
 
 (C) By Thoman's Table. P = 
 percent. years 
 
 A 
 EN 
 
 Eule 3. 
 
 Log. Future Sum 
 deduct Log. (3) 
 
 A 
 EN 
 
 
 P 
 
 Eequired present value, £ 
 
92 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Calculation Form, No. 3. 
 
 Table III. To find tlie amount of an annuity. Chapter VI. 
 
 This form has been used in the solution of problems of the 
 following nature : — ■ 
 
 Calculation. 
 
 To find the amount which should stand to the 
 credit of a sinking fund at any time during 
 the repayment period (■^^^) 2. 
 
 To find the amount of loan which will be provided 
 by the future accumulation of : — 
 
 (1) the original annual sinking fund 
 
 instalment (XY) 5. 
 
 (2) the additional, augmented, or reduced 
 
 annual sinking fund instalment (^^^I) 2. 
 
 (3) the income from the present invest- 
 
 ments representing the fund (XIX) 1. 
 
 (4) the annual increment of the fund ... (XIX) 4. 
 
STANDARD CALCULATION FORMvS 
 
 9S 
 
 Standard Calculation Form, No, 3. 
 Table III. To find the amount of an annuity. 
 The following rules are explained at the head of Chapter YI. 
 Here state the general nature of the problem. Calculation 
 
 Here state full details of the actual problem. 
 
 Required future amount, £ 
 (B) By Table III. M = A^y (^^~^^ 
 
 Table III. years percent. W^-l 
 
 Amount of £1 per annum (5) 
 add Log. Annuity 
 
 ^y 
 
 M 
 
 Required future amount, £ 
 
 No. 
 
 (A) By 
 
 Fornnila. 
 
 M = A, (1^^- 
 
 ^) 
 
 Rulel. 
 
 
 Log. Ratio 
 Multiply Log. 
 
 Convert Log. 
 to ordinary n 
 deduct unity 
 
 Log. of this is 
 
 (1) 
 Rby 
 
 (3) 
 umber 
 
 (4) 
 
 above 
 
 (4) 
 
 ■ (2) 
 
 R 
 
 Values. 
 
 Logs. 
 
 Loo- 
 
 RN 
 
 W—l^ 
 
 RN 
 
 -1 
 
 1- 
 
 
 
 RN-1 
 
 Log. Annuity 
 af/fZLog.RN_] 
 
 deduct Log. 7 
 
 A.y 
 
 RN-1 
 
 Ay(RN- 
 r 
 
 -1) 
 
 
 
 M 
 
 Rule 2. 
 
 (C) By Thoman's Table. M = Ky ("^L^) 
 per cent. years 
 
 Rules. 
 
 Log. Annuity 
 add Log. RN in 
 
 Table +10 (3) 
 
 deduct Log . a^ (8) 
 
 Ay 
 RN 
 
 AyRN 
 
 a^ 
 
 M 
 
 Required future amount, £ 
 
94 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Amount of One Pound per Annum. 
 
 Table III. Standard Form 3. 
 
 To find the number of years : 
 
 based on Calculation (XYIII) 4. 
 
 Given factors : 
 
 Annuity 
 
 Amount of annuity 
 Rate per cent. 
 
 Ratio ... 
 
 Interest of £1 
 Details of Method : 
 
 find 
 
 find, and deduct .. 
 
 R 
 
 r 
 
 Log. M 
 Log. Ay 
 
 Log. r 
 Log. RN- 
 
 RN 
 
 Log. RN 
 Log. R 
 
 7500-00 
 
 57468-48 
 
 300 
 
 1-03 
 
 0-03 
 
 57468-48 
 750000 
 
 -1 
 
 0-22987 
 1- 
 
 4-7594297 
 3-8750613 
 
 difference . . 
 find, and add 
 
 0-8843684 
 2-4771213 
 
 the sum is .. 
 
 i-3614897 
 
 find value of tliis log 
 add unity 
 
 
 whicb is tlie value of .. 
 find 
 
 1-22987 
 
 0-0898606 
 
 find 
 
 00128372 
 
 To find the number of years, divide the above log. of RN by 
 the above log. of R, as described in Chapter XXXII, 
 and the quotient is the number of years required, in 
 this case, 7 vears. 
 
 The Amount of One Pound per Annum 
 
 Table III. 
 To find the rate per cent : 
 
 Standard Form 3. 
 based on Calculation (XVIII) 7. 
 
 Given factors ; 
 
 
 
 
 Amount of annuity 
 
 M 
 
 1176-58 
 
 
 Annuity 
 
 A^/ 
 
 58-3715 
 
 
 Number of years . . . 
 
 
 16 
 
 
 Details of method : 
 
 
 
 
 find 
 
 Log. M 
 
 1176-58 
 
 30706241 
 
 find, and deduct ... 
 
 Log. A// 
 
 58-3715 
 
 1-7662008 
 
 diiference . . . 
 
 1-3044233 
 
 find value of this log. 
 
 
 20- 1569 
 
 
 which is the amount of an annuity of one pound for 16 years 
 at the required rate per cent. 
 
vSTANDARD CALCULATION FORMS 95 
 
 To ascertain tlie rate per cent., refer to Table III, giving 
 the amounts of one pound per annum, and find the nearest 
 value to the above amount of 20' 1569 in 16 years. If the 
 rate so found is not near enough, refer to Thoman's tables 
 and find the nearest log. to 1-3044233, which is ascertained 
 by deducting the log. of o" from the log. of E^, plus 10. 
 
 Eequired rate per annum, 3 per cent. 
 
 Note. In cases where the rate per cent, is not included in 
 the published tables of compound interest, or in Thoman's 
 tables, the above method will give only approximate results. 
 
 Standard Calculation Form, No. 3x. 
 
 Table III. To find the annual sinking fund instalment. 
 
 Chapter XIII. 
 This form has been used in the solution of problems of the 
 following nature : — 
 
 Calculation. 
 
 To find the annual sinking fund instalment, to be 
 set aside out of revenue or rate, and accumu- 
 lated at compound interest, to repay a stated 
 loan at the end of a prescribed period (^^) 1- 
 
 To find the annuity which Avill amount to a stated 
 sum in any number of years 
 
 To find the additional annual sinking fund 
 instalment required to provide the amount of 
 loan which will be unprovided for owing to a 
 deficiency in the amount in the fund (^^^I) 1- 
 
 To find the amount by which the original annual 
 sinking fund instalment may be reduced in 
 consequence of the withdrawal, during the 
 repayment period, of part of the loan from 
 the operation of the fund (XA III) 1. 
 
 To find the future annual increment to be added 
 to the fund, and accumiilated at compound 
 interest, to provide the balance of loan, which 
 will not be provided by the future accumula- 
 tion of the present investments representing 
 the fund (XYI)9. 
 
96 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Calculation Form, No. 3x, 
 Table III. To find the animal sinking fund instalment. 
 The following rules are explained at the head of Chapter XIII. 
 
 Here state the general nature of the problem. . Calculation 
 
 No. 
 Here state full details of the actual problem, 
 
 (A) By Formula Ai/ = M (jp^) Rulel. 
 
 Log . 
 W—1 
 
 ' Log. Eatio (1) 
 Multi'ply Log. E by 
 
 (3) 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 Log. of this is (4) 
 
 E 
 
 N 
 
 EN 
 
 Values. 
 
 Logs. 
 
 EN 
 
 -1 
 
 
 EN-1 
 
 Log. Amount of Loan 
 add Log. r (2) 
 
 M 
 
 r 
 
 deduct Log. (E^ _ \\ above 
 
 (4) 
 
 Mr 
 EN-1 
 
 Ay 
 
 
 Eequired annual instalment, £ 
 
 
 
 M 
 
 
 {^) 
 
 By Table III. Ai/ = Kn_i 
 
 r 
 
 Eule 2. 
 
 Log. Amount of Loan 
 Table III. years percent. 
 
 Amount of £1 per annum (5) 
 deduct Log. 
 
 M 
 EN-1 
 
 Av 
 
 Eequired annual instalment, £ 
 
 (C) By Thoman's Table. Ay ^ M ^^\ 
 per cent. years 
 
 Eule 3. 
 
 Log Amount of Loan 
 add Log. a« (8) 
 
 deduct Log. EN in 
 Table + 10 (3) 
 
 .AI 
 
 Ma" 
 EN 
 
 Ay 
 
 Required annual instalment, £ 
 
STANDARD CALCULATION FORMvS 
 
 97 
 
 The Sinking Fund Instalment. 
 Table III. Standard Form, 3x. 
 
 To find the number of years 
 
 based on Calculation (XV) 1. 
 
 Given factors : 
 
 Amount of loan ... M 
 
 Annual instalment Aiy 
 Rate per cent. 
 
 Ratio ^ 
 
 Interest of £1 ... ^ 
 
 26495 
 
 680-234 
 31 
 
 1035 
 0-035 
 
 Details of Method 
 
 find 
 
 find, and deduct ... 
 
 Log. M. 
 Log. Ay 
 
 Log. /■ 
 Log. RN-1 
 
 RN 
 
 . Log. RN 
 . Log. R 
 
 26495 
 680-234 
 
 1-36324 
 1- 
 
 4-4231639 
 2-8326581 
 
 difference . . 
 find, and add 
 
 1-5905058 
 2-5440680 
 
 the sum is .. 
 
 find value of tbis log 
 add unity 
 
 0-1345728 
 
 vvbicb is tbe value of .. 
 find ... ... .. 
 
 2-36324 
 
 0-3735087 
 
 find 
 
 00149403 
 
 To find tbe number of years, divide tbe above log. of R^ by 
 tbe above log. of R, as described in Cbapter XXXII, 
 and tbe quotient is tbe number of years required, in 
 tbis case, 25 years. 
 
gS REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Sinking Fund Instalment. 
 
 Table III. 
 To find the rate per cent 
 
 Given factors : 
 
 Standard Form, 3x. 
 based on Calculation (XY) 1. 
 
 Amount of loan ... 
 Annual instalment 
 Number of years ... 
 
 Details of method ; 
 
 find 
 
 find, and deduct ... 
 
 L 
 L 
 
 Ay 
 N 
 
 lo;. M. 
 Og-. Ay 
 
 26495 
 680-234 
 25 
 
 26495 
 680-234 
 
 38-94986 
 
 4-4231639 
 2-8326581 
 
 difference ... 
 find value of this log. 
 
 1-5905058 
 
 wliieli is tlie amount of loan which will be provided by an 
 annual instalment of one pound for 25 years at the 
 required rate per cent. 
 
 To ascertain the rate per cent., refer to Table III, giving 
 the amounts of one pound per annum, and find the nearest 
 value to the above amount of 38-94986 in 25 years. If 
 the rate so found is not near enough, refer to Thoman's 
 tables and find the nearest log. to 1-5905058 which is 
 ascertained by deducting the log. of (7» from the log. of E^, 
 plus 10. 
 
 Eequired rate per annum, 3| per cent. 
 
 Note. In cases where the rate per cent, is not included in 
 the published tables of compound interest, or in Thoman's 
 tables, the above method will give only approximate results. 
 
 Standard Calculation Form, No. 4. 
 
 Tablk IY. To find the present value of an annuity. 
 
 Chapter YII. 
 
 This form has been used in the solution of problems of the 
 following nature: — 
 
 Calculation. 
 
 To find the sum now payalde which is the equiva- 
 lent of tlie future annual sinking fund 
 instalments to be set aside to repay a given 
 loan at the end of a prescribed period of 
 years; and for which such annual instalments 
 might be redeemed Sec Chapter XXXII. 
 
STANDARD CALCULATION FORMS 
 
 99 
 
 Standard Calculation Form, No. 4. 
 
 Table IV. To find the present value of an annuity. 
 
 The following rules are explained at the head of Chapter VII. 
 
 Here state the general nature of the problem. Calculation 
 
 Here state full details of the actual problem. 
 
 (A) By Formula. 
 
 ^y {-w^) 
 
 Required present value, £ 
 
 No. 
 
 Rule 1. 
 
 
 ^Log. Ratio (1) 
 Multiply Log. R by 
 
 (3) 
 Convert Log. 
 
 to ordinary number 
 deduct Unity 
 
 Log. of this is (4) 
 
 Values. 
 R 
 
 N 
 
 Logs. 
 
 Log . 
 R^— 1 
 
 RN 
 
 RN 
 
 -1 
 
 
 RN-1 
 
 Log. Annuity 
 
 add Log. (RN-1) 
 above (4) 
 deduct Log. R^ above 
 
 (3) 
 
 Ay 
 RN_1 
 
 RN 
 
 deduct Log. r (2) 
 
 r 
 
 
 P 
 
 
 (B) By Table IV 
 
 /R^^-l\ 
 
 Table IV. years per cent. 
 Present Value £1 per 
 
 annum (6) 
 
 add Log. Annuity 
 
 RN-1 
 
 RNr 
 
 Ay 
 
 Required present value, £ 
 
 (C) By Thoman's Table. 
 
 per cent. years 
 
 a" 
 
 Required present value, £ 
 
 Rule 2. 
 
 Rule 3. 
 
 Log. Annuity 
 
 add 10 
 deduct Log. a" 
 
 («) 
 
 Ay 
 
 an 
 
 
 p 
 
REPAYMENT OF LOCAL AND OTHER I<OANS 
 
 The Present Value of One Pound per Annum. 
 Table IV. Standard Form. 4. 
 
 To find the number of years : 
 
 based on Calculation (XVIII) 14. 
 
 Given factors : 
 
 
 
 
 Annuity 
 
 Ky 
 
 40-215 
 
 
 Present value thereof 
 
 P 
 
 313118 
 
 
 Rate per cent. 
 
 
 3 
 
 
 Eatio 
 
 R 
 
 103 
 
 
 Interest of £1 
 
 r 
 
 003 
 
 
 Details of method : 
 
 
 
 
 find 
 
 Log.P 
 
 313-118 
 
 2-4957086 
 
 find, and deduct ... 
 
 Log. Ky 
 
 40-215 
 
 1-6043881 
 
 difference . . . 
 
 0-8913205 
 
 find value of this log. 
 
 
 7-7861 
 
 
 which is the present value of an annuity of one pound, at 
 3 per cent., for the required number of years. 
 
 To ascertain the number of years, refer to Table IV giving 
 the present values of one pound per annum, under 3 per 
 cent., and find the nearest value to the above amount of 
 7-7861. If the rate per cent, is not given in the tables, 
 refer to Thoman's tables, under the nearest rate per cent., 
 and find the nearest log. to 0-8913205, which is found by 
 deducting the log. of a" there given from 10. 
 
 Required period, 9 years. 
 
 Note. In cases where the rate per cent, is not included in 
 tbe published tables of compound interest, or in Thoman's 
 tables, the above method will give only approximate results. 
 
STANDARD CALCULATION FORMS 
 
 OP THE 
 
 UNIVERSJTV 
 
 OF 
 
 The Present Value of One Pound per Annum. 
 
 Table IV. 
 
 Standard Form. 4. 
 
 To find the rate per cent : 
 
 based on Calculation in Statement XXXII. D. 
 
 Given factors : 
 
 
 
 
 Annuity 
 
 A 
 
 1725-58 
 
 
 Present value thereof 
 
 P 
 
 28374-73 
 
 
 Number of years ... 
 
 N 
 
 23 
 
 
 Details of method : 
 
 
 
 
 find 
 
 Log.P 
 
 28374-73 
 
 4-4529318 
 
 find, and deduct ... 
 
 Log. Ay 
 
 1725-58 
 
 3-2369352 
 
 difference . . . 
 
 1-2159966 
 
 find value of this log. 
 
 
 16-4436 
 
 
 which is the present value of an annuity of one pound for 
 23 years, at the required rate per cent. 
 
 To ascertain the rate per cent, refer to Table IV, giving the 
 present values of one pound per annum, and find the 
 nearest value to the above present value of 16-4436 in 
 23 years. If the rate so found is not near enough, refer 
 to Thoman's tables and find the nearest log. to 1-2159966, 
 which is ascertained by deducting the log. of a" there 
 given from 10. 
 
 Eequired rate per annum, 3 per cent. 
 
 Note. In cases where the rate per cent, is not included in 
 the published tables of compound interest, or in Thoman's 
 tables, the above method will give only approximate results. 
 
102 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Standard Calculation Form, No. 5. 
 
 Table V. To find the annuity whicli a present sum will 
 puicLiase, or the annuity of which £1 is the 
 present value. Chapter YIII. 
 
 This form has been used in the solution of problems of the 
 following nature : — 
 
 Calculation, 
 
 To find the equal annual instalment of principal 
 and interest combined, to be paid to the 
 lender, in order to repay a stated loan in a 
 prescribed period (XII) 4. 
 
 To find the amount by which the original annual 
 sinking fund instalment may be reduced in 
 consequence of : — 
 
 (1) a surplus in the fund owing to an 
 excessive past accumulation of the fund (XYIII) 10. 
 
 (2) a surplus in the fund, due to the pay- 
 
 ment into the fund of any sum not 
 provided out of revenue or rate, 
 namely : — 
 
 (a) the proceeds of sale of part of 
 the assets representing the 
 security for the loan (XVII) 1. 
 
 or 
 
 (h) a realised profit upon the sale 
 of an investment representing 
 the fund. 
 
 To find the additional sinking fund instalment, to 
 be set aside, and added to the fund during 
 tlie unexpired portion of tlie repayment 
 period, to compensate for a deficiency in the 
 amount now in the fund (^^ ) ^' 
 
STANDARD CALCULATION FORMS 
 
 103 
 
 Table V. To find tlie annuity which a present sum will 
 purchase, or of which it is the present value. 
 To find the equal annual instalment of principal 
 and interest combined. 
 
 The following rules are explained at the head of Chapter VIII. 
 
 Here state the general nature of the problem. Calculation 
 
 No. 
 
 Here state full details of the actual problem. 
 
 (A) By Formula. At/ 
 
 / W r \ 
 
 Values. 
 
 Log 
 
 Log. Ratio (1) 
 
 Multiply Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 (4) 
 
 Log. Present Sum 
 add Log, RN above (3) 
 Log. r (2) 
 
 deduct Log. (R^ - 1) above 
 
 (4) 
 
 RN 
 
 RN 
 
 -1 
 
 RN-1 
 
 p 
 
 RN 
 
 r 
 
 RN-1 
 
 Ay 
 Required annuity, £ 
 
 Table V. years percent. 
 
 Annuity £1 will purchase (7) 
 add Log. Present Sum 
 
 RN 
 
 RN - 
 P 
 
 Ay 
 
 Required annuity, £ 
 
 Rule 1. 
 
 Logi 
 
 (B) By Table Y. ^V ^ ^^ (^ZTl) Rule 2. 
 
 (C) By Thoman's Table. Ai/ = P an 
 
 per cent. years 
 
 Required annuity, £ 
 
 Rule 3. 
 
 Log. Present Sum 
 add Log. a^ (8) 
 
 P 
 
 
 
 deduct 10 
 
 A?/ 
 
I04 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Annuity which One Pound will Purchase 
 
 and 
 
 The Equal Annual Instalment of Principal and Interest 
 
 Combined. 
 
 Table V. Standard Form, 5. 
 
 To find the number of years : 
 
 based upon Calculation (XV) 3. 
 
 Given factors : 
 
 
 
 ^ 
 
 Present sum 
 
 P 
 
 469-74 
 
 
 Annuity 
 
 Ky 
 
 45-594 
 
 
 Rate per cent. 
 
 
 31 
 
 
 Ratio 
 
 R 
 
 1-035 
 
 
 Interest of £1 
 
 r 
 
 0-035 
 
 
 Details of method : 
 
 
 
 
 find 
 
 Log. Ay 
 
 45-594 
 
 1-6589086 
 
 find, and deduct ... 
 
 Log.P 
 
 469-74 
 
 2-6718612 
 
 difference . . . 
 
 2'9870474 
 
 find value of tbis log. 
 
 
 0-097061 
 
 
 which is the annuity which one pound will purchase at 3^ per 
 cent, for the required number of years. 
 
 To ascertain the number of years refer to Table V, giving tbe 
 annuity which one pound will purchase, under 3^ per 
 cent., and find the nearest value to the above amount of 
 0- 097061. If the rate per cent, is not given in the tables, 
 refer to Thoman's tables, under the nearest rate per cent., 
 and find the nearest log. of a" to 2-9870474. This may 
 be ascertained by an inspection of the mantissa only. 
 
 Required period, 13 years. 
 
 Note. In cases where the rate per cent, is not included in 
 the published tables of compound interest, or in Thoman's 
 tables, the above method will give only approximate results. 
 
STANDARD CALCULATION FORMS 
 
 105 
 
 The Annuity which One Pound will Purchase 
 
 and 
 
 The Equal Annual Instalment of Principal and Interest 
 
 Combined. 
 Table V. Standard Form, 5. 
 
 To find the rate per cent : 
 
 based upon Calculation (XVIII) 10. 
 
 Given factors : 
 
 
 
 
 
 Present sum 
 
 Annuity 
 
 Number of years . . . 
 
 
 P 
 
 Ay 
 
 N 
 
 447-27 
 57-4446 
 9 
 
 
 Details of method : 
 
 
 
 
 
 find 
 
 find, and deduct . . . 
 
 L( 
 L( 
 
 Dg. Ay 
 
 57-4446 
 447-27 
 
 0-128434 
 
 1-7592493 
 2-6505698 
 
 difference . . . 
 find value of tbis log. 
 
 T-1086795 
 
 wbicb is tbe annuity wbicb one pound will purcbase for 9 years 
 at tbe required rate per cent. 
 
 To ascertain tbe rate per cent., refer to Table V, giving tbe 
 annuity wbicb one pound will purcbase, and find tbe 
 nearest value to tbe above annuity of 0' 128434 in 9 years. 
 If tbe rate so found is not near enougb, refer to Tboman's 
 tables and find tbe nearest log. of a" to 1-1086795. Tbis 
 may be ascertained by an inspection of tbe mantissa only. 
 
 Required rate per annum, 3 per cent. 
 
 Note. In cases wbere tbe rate per cent, is not included in 
 tbe published tables of compound interest, or in Tboman's 
 tables, tbe above metbod will give only approximate results. 
 
Section II. 
 
 The Methods of Repayment of the Loan Debt 
 
 of Local Authorities and Commercial 
 
 and Financial Undertakings. 
 
log 
 
 CHAPTER XI. 
 
 THE REPAYMENT OF THE LOAN DEBT OF LOCAL 
 AUTHORITIES AND COMMERCIAL AND FINAN- 
 CIAL UNDERTAKINGS. 
 
 Alternative methods allowed by the Public Health Act, 
 1875, AND other Acts. Comparison of methods as 
 regards the actual repayment to the lender, and the 
 annual charge against Revenue or Rate. 
 
 The Instalment Method. 
 
 BY AN EQUAL ANNUAL INSTALMENT OF PRINCIPAL TO BE REPAID 
 TO THE LENDER. No SiNKING FuND REQUIRED, BUT AN 
 EQUAL PERIODICAL REPAYMENT OF PRINCIPAL. AnNUAL 
 CHARGE AGAINST THE REVENUE OR RaTE AcCOUNT OF 
 
 successive years composed of an equal amount of 
 principal and a gradually decreasing amount of interest. 
 Statement showing the final repayment of the Loan. 
 
 Having obtained a series of rules and forinulse relating to 
 all problems involving compound interest, they will now be 
 applied to problems of actual finance, beginning witb the re- 
 payment of the loan debt of local authorities. This affords a 
 very good subject for treatment by the mathematical method, 
 not only on account of the variety of the problems occurring in 
 actual practice, but because the original conditions and regula- 
 tions are of a fairly uniform character. This uniformity is 
 much more pronounced than is the case with the loan debt of 
 commercial and financial undertakings, where not only much 
 more variable and elastic conditions exist, but also greater 
 facilities to alter the original arrangements between the borrower 
 and the lender. 
 
 Excluding for the present the provisions contained in early 
 Acts of Parliament, which vary considerably, the general 
 principles now in force, and governing the matter, are contained 
 in Section 234 of the Public Health Act of 1875. These 
 provisions may be accepted as the standard now adopted in all 
 
no REPAYMENT OF LOCAL AND OTHER LOANS 
 
 public general Acts, special Acts, and provisional orders of the 
 Local Government Board. There are of course small variations 
 in detail, but the principle remains in all cases substantially the 
 same, and, with the exception of the introduction of a new form 
 of " non-accumulating " sinking fund, there has not been any 
 change since 1875. These provisions relate to the repayment 
 of the debt, and will apply equally to commercial and financial 
 undertakings, as they are based upon general financial practice. 
 There is not anything new in the methods laid down in the Act, 
 but the merit of the section lies in the fact that for the first 
 time definite methods were prescribed in place of the very 
 varied practice previously followed. To state an extreme case, 
 borrowing powers will never be granted in future without any 
 obligation whatever as to redemption. This section specifies 
 three alternative methods, at the option of the local authority, 
 by which the loan debt may be repaid, and provides in effect 
 that : — 
 
 (a) The local authoritv shall repay the moneys so borrowed 
 by:~ ■ 
 
 (1) equal annual instalments of principal, or by 
 
 (2) equal annual instalments of principal and 
 
 interest combined ; 
 
 or (h) The local authority shall in every year set apart as a 
 sinking fund and accumulate in the way of compound 
 interest such a sum as will, with accumulations in the 
 way of compound interest, be sufficient to pay off the 
 moneys so borrowed within the period sanctioned. 
 
 The alternative methods of repayment are usually described 
 as : — 
 
 [a) (1) The instalment method. 
 (2) The annuity method. 
 
 (h) The sinking fund method. 
 
 The sinking fund method in the Act of 1875 is the same 
 as the accumulating sinking fund referred to in the 1893 
 clauses of the Local Government Board. 
 
 The sections of the Public Health Act, 1875, and other Acts 
 relating to the borrowing of money by local authorities lay 
 down five distinct principles, namely: — 
 
 The power to borrow is limited to works of a more or less 
 permanent nature. 
 
THE INSTALMENT METHOD iii 
 
 Tlie amount to be borrowed is limited. 
 
 The period of repayment must be fixed by tbe Local Govern- 
 ment Board, having regard to the relative permanency of 
 the works. 
 
 The Public General Acts contain a provision that the period 
 sanctioned by the Local Government Board for the 
 repayment of any loan shall not in any case exceed a 
 period prescribed in each Act. 
 
 The amount annually required to discharge the liability in 
 respect of interest upon the loan and the repayment of 
 the debt is chargeable against the rate or revenue account 
 of each year. 
 
 There are two main distinctions to be drawn between the 
 above methods (1) as regards the actual repayment of the loan, 
 and (2) as regards the charge upon the rate or revenue accounts 
 of the successive years of the repayment period. The instalment 
 and annuity methods both provide for the actual repayment to 
 the lender each year of a definite proportion of the loan or of 
 the loan and interest combined. The sinking fund method, on 
 the other hand, contemplates the provision annually of an 
 instalment of such amount as will, if set aside, invested, and 
 accumulated for the prescribed period, provide for the repay- 
 ment of the loan in one sum at the end of the period. Power is 
 given, however, under certain conditions to apply part of the 
 sinking fund in repayment of the loan during the prescribed 
 period of repayment. 
 
 Each of these methods will be considered in detail, taking 
 first : — 
 
 The Instalment Method, in which the loan is repaid to the 
 lender by equal annual instalments of principal only, and 
 interest is paid to him upon the balance of loan unpaid. This 
 me-thod applies mainly to advances made to local authorities by 
 the Public Works Loan Commissioners, and also to loans by 
 the larger insurance companies to the Metropolitan boroughs 
 and other local authorities. 
 
 This method is also commonly used by commercial and 
 financial undertakings, and is known as the deferred payment 
 system. The hire purchase system, on the other hand, is a 
 commercial form of the annuity method. The instalment 
 method is exceedingly simple in operation, seeing that it is 
 merely an arithmetical calculation, and does not involve any 
 question of compound interest whatever. 
 
112 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Generally the repayment period is 30 years or longer, but 
 in order to simplify the problem and to enable a comparison to 
 .be made with the annuity and sinking fund methods to be 
 hereafter described, in all cases the example will relate to the 
 repayment of a loan of £1,000, in ten years, with interest at 
 5 per cent., which rate will be assumed to be payable to the 
 lender, and will also be the rate of accumulation of the sinking 
 fund. 
 
 As will be seen by the following statement, the municipality 
 will repay to the lender at the end of the first year : — 
 
 ^/ 10 of the principal £100 
 
 Interest on £1,000 50 
 
 £150 
 
 and this amount will be charged to the rate or revenue account. 
 
 At the end of the second year the municipality will repay to 
 the lender : — 
 
 ^ 1 10 of the principal, as before ... £100 
 Interest on £900 45 
 
 £145 
 
 and so on each year until, at the end of the tenth year, they will 
 
 repay : — 
 
 ^ 1 10 of the principal, as before ... £100 
 Interest on £100 5 
 
 £105 
 
 the effect being, as regards the municipality, that the rate or 
 revenue account will be charged year by year with a gradually 
 decreasing amount, to the relief of the later generations of 
 ratepayers. 
 
 As regards the lender, he originally advanced to the 
 municipality a sum of £1,000, which is repaid to him at the 
 rate of £100 per annum, which will require to be invested each 
 year, with the result that his income will be constantly 
 fluctuating, and he will hold a number of small investments 
 instead of one large one. The following statement (XI. A.), 
 shows the operation of the repayment from year to year by 
 means of the constant instalment of £100 of principal, and also 
 shows the decreasing amount of interest annually paid to the 
 lender. As the principal and the interest are both charged to 
 the revenue or rate account, it shows the annually decreasing 
 loan charge. 
 
 Similar statements will be given relating to the annuity and 
 sinking fund methods; and, finally, a statomont will be pn^pared 
 
THE INSTALMENT METHOD 
 
 113 
 
 comparing tlie effect of the repayment by all three methods. 
 In the case of the instalment and annuity methods there is not 
 any accnmulating' sinking fund, and therefore there is not any 
 complication arising from the rate of accnmnlation. This will 
 be dealt with under the head of the sinking fund method. 
 The instalment method, so far as regards the actual repajanent 
 to the lender, is exactly similar to the ordinary repayment of 
 debt by commercial and finaneial undertakings, and there is not 
 any difference in principle if the instalments are not equal in 
 amount or are made at unequal intervals of time. The lender 
 receives interest each year, upon the actual balance owing to 
 him, since the last date to which interest has been paid. 
 
 Unlike the annuity and instalment methods, there is not 
 any variation in the calculation if the interest be paid half- 
 yearly or otherwise instead of yearly. It is a simple arith- 
 metical calculation not complicated in any degree by compound 
 interest. 
 
 STATEMENT XI. A. 
 
 The Repayment of Debt of Local Authorities. The Instalment 
 Method. 
 
 Showing the repayment of a Loan of £1,000 in 10 years, with 
 interest at 5 per cent, by equal annual instalments of 
 principal only. 
 
 Owing at 
 
 
 begin- 
 ning 
 of 
 year. 
 
 Interest 
 at 
 
 5% 
 
 Total 
 
 owing. 
 
 Repayments. 
 
 Balance 
 owing 
 at end 
 of year. 
 
 Charge to Ei 
 
 EVENUE. 
 
 
 ear 
 nd. 
 
 Princi- 
 pal. ] 
 
 [nterest 
 
 Total. 
 
 Princi- 
 pal. 
 
 Interes 
 
 . Total. 
 
 Yea 
 
 1 
 
 1000 
 
 50 
 
 1050 
 
 100 
 
 50 
 
 150 
 
 900 
 
 100 
 
 50 
 
 150 
 
 1 
 
 9 
 
 900 
 
 45 
 
 945 
 
 100 
 
 45 
 
 145 
 
 800 
 
 100 
 
 45 
 
 145 
 
 2 
 
 8 
 
 800 
 
 40 
 
 840 
 
 100 
 
 40 
 
 140 
 
 700 
 
 100 
 
 40 
 
 140 
 
 3 
 
 4 
 
 700 
 
 35 
 
 735 
 
 100 
 
 35 
 
 135 
 
 600 
 
 100 
 
 35 
 
 135 
 
 4 
 
 5 
 
 600 
 
 30 
 
 630 
 
 100 
 
 30 
 
 130 
 
 500 
 
 100 
 
 30 
 
 130 
 
 5 
 
 6 
 
 500 
 
 25 
 
 525 
 
 100 
 
 25 
 
 125 
 
 400 
 
 100 
 
 25 
 
 125 
 
 6 
 
 7 
 
 400 
 
 20 
 
 420 
 
 100 
 
 20 
 
 120 
 
 300 
 
 100 
 
 20 
 
 120 
 
 7 
 
 8 
 
 300 
 
 15 
 
 315 
 
 100 
 
 15 
 
 115 
 
 200 
 
 100 
 
 15 
 
 115 
 
 8 
 
 9 
 
 200 
 
 10 
 
 210 
 
 100 
 
 10 
 
 110 
 
 100 
 
 100 
 
 10 
 
 110 
 
 9 
 
 
 
 100 
 
 5 
 
 105 
 
 100 
 
 5 
 
 105 
 
 — 
 
 100 
 
 5 
 
 105 
 
 10 
 
 
 1000 
 
 275 
 
 — 
 
 1000 
 
 275 
 
 1275 
 
 — 
 
 1000 
 
 275 
 
 1275 
 
 
114 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XII. 
 
 THE REPAYMENT OF THE LOAN DEBT OF LOCAL 
 AUTHORITIES AND COMMERCIAL AND FINAN- 
 CIAL UNDERTAKINGS {Coiitinued). 
 
 The Annuity Method, 
 
 by an equal annual instalment of principal and 
 interest combined, to be repaid to the lender. 
 
 Methods of calculating the annual instalment by formula 
 AND tables; and the general rules based thereon. No 
 sinking fund required, but an equal periodical repay- 
 ment TO the lender of principal and interest combined. 
 The relation between such equal annual instalment, 
 the sinking fund instalment (Chapter XIII) and the 
 equal annual instalment of principal only (Chapter XI). 
 Statement showing the final repayment of the Loan. 
 
 Author's Standard Calculation Form, No. 5. 
 
 Formulae. 
 
 The whole of the formuhe at the head of Chapter VIII, 
 relating to Table V {the annuity which £1 will pwrhase) apply 
 to the present method. 
 
 General Rules deduced from the formulae relating to 
 Table V :— 
 
 To find the equal annual instalment of principal and interest 
 combined, to repay a given loan during a stated period. 
 
 Author's Standard Calculation Form, No. 5. 
 
 Rule 1. If the rate per cent, he not given in Table V, or in 
 Thoman's Tables : — 
 
 Proceed by the foriinda relating to Table Y . 
 
 Calculation {XII) 4 A. 
 
 Rule 2. If the rate per cent, be given in Table F : — 
 
 Multiply the annuity given in the table by the 
 amount of the loan. The product is the required 
 annual instalment. Calculation {XII) 4 B. 
 
THE ANNUITY METHOD 115 
 
 Rule 3. If the rate yet cent, he given in Thoman's Table : — 
 To the log. of the amount of the loan add the log. 
 of a", as given by Thoman : deduct 10 from the sum. 
 of the logs. The remainder is the log. of the 
 required annual instalment. Calculation {XII) 4 C . 
 
 Rule 4. Find the sinking fund instalment by any of the rules 
 given in the following chapter; add to the 
 instalment so found one year's interest upon the 
 loan. The rate pier cent, in both cases to be the rate 
 of interest to be paid to the lender. The sum is the 
 required annual instalment. Calculation {XII) 5. 
 
 To find the rate per cent, or number of years ^ proceed as 
 shown in the standard form for the purpose, relating to Table \ , 
 given m Chapter X. 
 
 The Annuity Method. Under this method there is, as in 
 the case of the instalment method, an actual repayment each 
 year to the lender, the whole of which is charged to the rate or 
 revenue account of each year. But in this case the lender 
 receives an equal amount each year, composed of principal and 
 interest combined. To the extent that it involves an equal 
 annual charge upon the rate or revenue account of the muni- 
 cipality during the whole of the repayment period, it is an 
 improvement upon the instalment method, but it still has not 
 any advantage to the lender. As will be seen by the detailed 
 statement, XII. A., following, and also by the comparative 
 statement in Chapter XIII, after the sinking fund method, the 
 annual amount repaid to the lender consists of an increasing 
 amount of principal and a decreasing amount of interest ; and, 
 further, if the lender be a trustee, or requires for any purpose to 
 allocate the repayment as between capital and income, he must 
 make a somewhat difficult calculation. The lender has to re- 
 invest each year a gradually increasing amount of principal, 
 unless he sets aside an equal annual proportion of the amount 
 paid to him, as a sinking fund, as will be explained later. 
 
 The formulae and tables will now be applied to ascertain the 
 equal annual instalment of principal and interest combined 
 required to repay a loan of £1,000 in 10 years at 5 per cent. 
 There are several ways of doing this, but the clearest, although 
 not the shortest, will be first described, being the one which 
 best illustrates the principles involved. Leaving the actual 
 calculation for the moment, the transaction will be divided into 
 
ii6 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 two parts, ignoring for the present the annual repayments to 
 the lender. The loan of £1,000, if not repaid, M-ill accumulate 
 at 5 per cent, compound interest with yearly breaks, and at 
 the end of the period will amount to £1628'9, as shown by 
 Calculation (XII) 1. The next step is to ascertain the amount, 
 at the end of 10 years at 5 per cent., of an equal annual instal- 
 ment of £1 per annum, or 12'5779, as shown by Calculation 
 (XII) 2. 
 
 It has now been ascertained that £1,000 in 10 years at 
 5 per cent, will amount to £1628-9 and that each £1 of equal 
 annual instalment, or annuity, will at the end of 10 years 
 amount to £12"57T9 ; and it is therefore obvious that the equal 
 annual instalment required will be, in sterling or other 
 currency, exactly the number of times that £125779 is con- 
 tained in £1628-9. By dividing £16289 by £12-5779, the 
 required equal annual instalment of principal and interest 
 combined is obtained, viz., £129-51, as shown by Calculation 
 (XII) 3. The actual details of the above Calculations (XII) 1 
 and (XII) 2 are given at the end of the chapter upon the 
 author's standard calculation forms, No. 1 and No. 3, both of 
 which are made by three methods : — 
 
 A. by formula. 
 
 B. by the published tables; 
 and C. by Thoman's tables. 
 
 The two factors referred to have now been ascertained: — 
 Calculation (XII) 1 shows that the original loan of £1.000 will 
 in 10 years, at 5 per cent., amount to £1628-90: and Calcula- 
 tion (XII) 2 shows that £1 per annum will in 10 years, at 
 5 per cent., amount to £12-5779; and the required annual 
 instalment of principal and interest combined is obtained by 
 dividing £162890 by £12-5779 by logarithms as follows. 
 
 CALCULATION (XII) 3. 
 
 To find the equal annual instalment of principal and interest 
 combined to repay a given loan. 
 
 Required the equal annual instalment of principal and 
 interest combined, to be repaid to the lender as and when 
 set aside, to repay £1,000 in 10 years at 5 per cent. 
 
 By Tables I and III and Logs. Based on CUilculations (XII) 1 
 and (XII) 2. 
 
THE ANNUITY METHOD 
 
 117 
 
 Table I, Calculation (XII) 1 : 
 
 Amount of £1,000 in 10 years at 
 
 5 per cent 
 
 deduct, log. 
 Table in, Calculation (XII) 2 : 
 
 Amount of £1 per annum for 10 years 
 at 5 per cent 
 
 1628-9 
 
 3-2118930 
 
 12-5779 10996079 
 
 2-1122851 
 
 which is the log. of the required equal annual instal- 
 ment of principal and interest combined, viz., £129-51 
 
 The principle involved in the above method of ascertaining 
 the equal annual instalment of principal and interest combined, 
 to be repaid to the lender as shown in Calculation (XII) 3, is, 
 that, on the one hand, there is the original loan of £1,000 
 quietly rolling up all by itself for the prescribed period ; and, 
 on the other hand, there is an equal annual instalment of 
 £12951 also rolling up at the same rate per cent, for the same 
 period. The annual instalment is of such amount that, at the 
 end of the period both accounts will amount to exactly the same 
 sum. Seeing that the rate of accumulation is the same in both 
 cases, it is obvious that the transfer from one account to the 
 other of an annual sum in repayment of principal and interest 
 combined, out of the accumulating credit, may be made without 
 in any way affecting the result arrived at by considering the 
 two factors as independent transactions. 
 
 The following table shows the methods of finding the equal 
 annual instalment of principal and interest combined in the 
 foregoing example, and also demonstrates the derivation of the 
 formula relating to Table Y from the formulae relating to 
 Tables I and III : — 
 
 Numerator : — 
 
 The amount of £1 in any 
 number of years 
 
 Denominator : — 
 
 The amount of £1 per annum, 
 in the same number of 
 years 
 
 By Published 
 
 Tables. 
 
 Actual 
 
 Values. 
 
 Table I 1628-90 
 
 Table III 125779 
 
 By 
 
 Formulae. 
 
 RN 
 
 KN- 
 
 Table V = 
 
 Table I 
 Table m 
 
 129-51 
 
 RN T 
 
ii8 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 In Chapter IX it was pointed out tliat the above formula 
 ^j^ — y is the equivalent of Thoman's factor (a"), both of which 
 
 denote the annuity which £1 will purchase for any number of 
 years. In Chapter VIII the same formula was arrived at, by 
 deduction, from Table IV, which gives the present values of an 
 annuity of £1. The present example proves that the formula 
 for Table \^ may be also found by deduction from Tables I and 
 III, and consequently that the equal annual instalment of 
 principal and interest combined may be found by Table V or by 
 Thoman's factor («") in a much more direct way than by using 
 Tables I and III as above. The calculation will therefore be 
 made by Table V, on the author's standard calculation form 
 No. 5, using the three methods therein contained, namely, by 
 formula, by the published table, and by Thoman's method. 
 See Calculation (XII) 4 at the end of this chapter. 
 
 The general rules relating to each method are given at the 
 head of this chapter. 
 
 The Relation between the Equal Annual Instalment of 
 Principal and Interest Combined, and the Sinking Fund 
 Instalment. On comparing the above Calculation (XII) 13, 
 relating to the annuity method, with Calculation (XIII) 1, in 
 the following chapter by which the sinking fund instalment is 
 ascertained, it will be seen that Calculation (XIII) 1 is the 
 simpler because it involves only one reference to the published 
 tables (No. III.). The present comparison is made in order 
 to compare the annual instalment of principal and interest 
 combined with the sinking fund instalment, ignoring the fact 
 that there is a more direct method of finding the equal annual 
 instalment of principal and interest combined by means of 
 Table V. It will be seen that the equal annual instalment of 
 [)rincipal and interest is greater than the sinking fund instal- 
 ment by £50, which is one year's interest upon the loan of 
 £1,000 at 5 per cent, per annum. 
 
 The equal annual instalment of principal and interest 
 combined, to be paid to the lender under the annuity method 
 may bo therefore ascertained in the following manner: — 
 
 First ascertain the sinhing fund instalment ivhich will 
 provide the loan at the end of the period, as in 
 Calculation [XIIT) 1 , taldng as the rate of arc^lVlulation 
 of the sinlnng fund the rate of interest to he paid' to the 
 lender under the annuity method. 
 
THE ANNUITY METHOD 119 
 
 Then add to the annual sinkiny fund instalment so found one 
 year's interest upon the loan at the same rate, which is 
 the rate of interest payable to the lender^ and the result 
 is the equMl annual instalment of principal and interest 
 combined under the annuity method, as follows : — 
 
 CALCULATION (XII) 5. 
 
 To find the equal annual instalment of principal and interest 
 
 combined to repay a given loan. 
 Required the equal annual instalment of principal and 
 
 interest combined, to be repaid to the lender as and when 
 
 set aside, to repay £1,000 in 10 years with interest at 
 
 5 per cent. 
 Based on the Sinking Fund Method. See Calculation (XIII) 1. 
 
 Amount of the annual instalment by the sinking 
 
 fund method as found by Calculation (XIII) 1, £ 79-51 
 Add one year's interest on £1000 at 5 per cent. 6000 
 
 £129-51 
 
 which is the required annual instalment of principal 
 and interest combined, as found by Calculations 
 (XII) 3 and (XII) 4. 
 
 It is important to bear in mind the requirements as to the 
 rate per cent. This method is of practical use only in finding 
 the equal annual instalment of principal and interest combined 
 under the annuity method. The sinking fund instalment is 
 more easily found by the direct method shown in Calculation 
 (XIII) 1, than by employing the method shown in Calculation 
 (XII) 3, to find the annual instalment of principal and interest 
 combined, and deducting therefrom the amount of interest upon 
 the loan. 
 
 The difference between the two instalments may be stated in 
 terms of the respective formulae by deducting the formula 
 relating to the sinking fund instalment from the formula 
 expressing the equal annual instalment of principal and interest 
 as follows : — 
 
I20 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 wliicli is tlie interest upon £1 for one year. One instalment 
 may also be expressed in terms of the other. If it be required 
 to find the equal annual instalment of principal and interest, 
 having found the sinking fund instalment, divide the formula 
 relating to the equal annual instalment of principal and interest 
 by the formula relating to the sinking fund instalment as 
 follows : — 
 
 thereby proving that the equal annual instalment, of principal 
 and interest combined, may be found by multiplying the 
 ascertained sinking fund instalment by the value of R^ as given 
 in' the published table (No. 1); or by logs., by adding to the 
 log. of the sinking fund instalment the log. of E^, as given in 
 Thoman's tables. The sum of the above logs, is the log. of the 
 equal annual instalment of principal and interest combined. 
 
 Applying this rule to the present example : — 
 
 Log. Sinking fund instalment = 795046 I'QOO 3921 
 
 add Log. EN 10 years, 5 per cent. l" 62889 0-211 8930 
 
 Log. equal annual instalment of principal 
 
 and interest combined = 2' 112 2851 
 
 which, as shown by Calculation (XII) 4, is £12951. 
 
 This method Avill rarely be used in practice, but is interest- 
 ing as furnishing a further example of the relation between the 
 above formulae. 
 
 The Eepayment of the Loan by the Annuity Method. 
 The following Statement XII. A. shows the repayment of the 
 loan, year by year; and should be compared Avith the similar 
 Statement XI. A., relating to the instalment method in the 
 previous chapter. It should also be compared with Statement 
 XIII. A., relating to the sinking fund method in the next 
 chapter, when it will be noticed that not only is the total 
 annual charge to the revenue or rate account uniform during 
 the whole of the repayment period under both the annuity and 
 sinking fund methods, but that the total annual charge is also 
 the same in amount in each case provided that the rate of 
 accumulation of the sinking fund is the same as the rate of 
 interest payable' upon the loan. The following Statement 
 XII. A. also shows the increasing amounts of principal and 
 
THE ANNUITY METHOD 121 
 
 tile couvsequent decreasing- amounts of interest contained in the 
 equal annual instalment repaid to the lender. 
 
 After considering the sinking fund method in the following 
 chapter the results under the three methods will be shown in 
 tabular form, both as regards the actual repayment to the 
 lender, and also the annual charges to the revenue or rate 
 account during the successive years of the repayment period, in 
 Statement XIII. B. 
 
 The two methods already discussed, namely, the instalment 
 method and the annuity method, involve the provision each 
 year, out of rate or revenue of part of the principal and interest, 
 and such annual provision is actually repaid to the lender as 
 and when set aside. They do not tlierefore in any way partake 
 of the nature of a sinking fund, which relates only to the 
 provision for the repayment of the principal in one amount at 
 the end of a definite period, and will be described in the 
 following chapter. 
 
REPAYMENT OF LOCAL AND OTHER LOANS 
 
 
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THE ANNUITY METHOD 
 
 123 
 
 Calculation (XII) 1. 
 
 Standard Calculation Form, A'o. 1. 
 
 To find the future amount of a present sum. 
 
 To find the amount which will be owing at the end of a stated 
 period in respect of a given loan if it be allowed to 
 accumulate at compound interest. Table I. 
 
 Required the amount of £1,000 at the end of 10 years at 5 per 
 cent, per annum, compound interest. 
 
 [A) By Formula, 
 
 A=PRN 
 
 Rule 1, Chapter lY. 
 
 Log r Log. Ratio 
 
 R^ ^ Mxilti'ply Log. R by 
 
 Log. Present Sum 
 add Loff. R^ above 
 
 R 
 
 N 
 
 105 
 10 
 
 00211893 
 10 
 
 RN 
 
 (1-05)10 
 
 0-2118930 
 
 P 
 
 RN 
 
 1000 
 
 3- 
 0-2118930 
 
 A 
 
 
 3-2118930 
 
 Required future amount, £1628-90. 
 
 (B) By Table I. 
 
 A=PRN 
 
 Rule 2, Chapter IV. 
 
 Table I. 10 vears, 5 per cent. 
 
 Amount of £1 j R^ 
 
 add Los;. Present Sum 
 
 A 
 
 1-628895 0-2118930 
 1000 3- 
 
 3-2118930 
 
 Required future amount, £1628-90. 
 
 (C) By Thoman's Table. A = P RN 
 5 per cent. 10 years. 
 
 Rule 3, Chapter lY 
 
 Log. Present Sum 
 aiZ^Log.RN 
 
 P 
 RN 
 
 A 
 
 1000 
 
 3- 
 0-2118930 
 
 3-2118930 
 
 Required future amount, £1628-90. 
 
124 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (XII) 2. 
 
 Standard Calculation Foriii, No. 3. 
 
 To find the amount of an aunuity in any number of years. 
 
 Table III. 
 Required the amount of £1 yier annum for 10 years at 5 per 
 cent, per annum, compound interest. 
 
 ^RN ]^^^ 
 
 (A) By Formula. M = Ay ( j Rule 1, Chapter VI. 
 
 
 ^Log. Ratio 
 
 Multiply Log. R by 
 
 Convert Log. 
 
 to ordinary numboi' 
 deduct unity 
 
 Log. of this is 
 
 Log. Annuity 
 add Log. RN - 1 above 
 
 deduct Log. r 
 
 11 
 
 N 
 
 105 
 10 
 
 00211893 
 10 
 
 Log 
 RN— 1 
 
 RN 
 
 (ro5)io 
 
 0-2118930 
 
 RN 
 
 -1 
 
 1-6289 
 1- 
 
 ^ 
 
 
 RN-1 
 
 0-6289 
 
 T-7985779 
 
 
 A^ 
 
 RN-1 
 
 1- 
 
 0- 
 
 1^7985779 
 
 
 A2/(RN- 
 r 
 
 -1) 
 
 1-7985779 
 2-6989700 
 
 
 M 
 
 
 1-0996079 
 
 Required amount, £125779. 
 
 (H) liy Table III. M = Ay f^^^^ Rule 2, Chapter VI. 
 
 Table 111. 10 years, 5 per cent. 
 Amount of £1 per annum 
 Add Log. Annuity 
 
 RN-l 12-5779 
 
 Ay 
 
 M 
 
 Required amount, £12-5779. This amount is given in Table III. 
 
 (C j By Thoman's Table . M = Ay ^ ^'^^ j Rule 3, Chapter VI . 
 5 per cent. 10 years. 
 
 Log. Annuity 
 add Log. RN iu 
 
 Table +10 
 
 deduct Log. a" 
 
 Ay 
 
 RN 
 
 0- 
 10-2118930 
 
 Ai/RN 
 
 10-2118930 
 9-1122851 
 
 M 
 
 1-0996079 
 
 Required amount, £12'5779. 
 
THE ANNUITY METHOD 125 
 
 Calculation (XII) 4. 
 
 Standard Calculation Form, No. 5. 
 
 To find the annuity wliicli a present sum will purchase for any 
 
 number of years. 
 To find the equal annual instalment of principal and interest 
 
 combined to repay a given loan. The Annuity Method. 
 
 Table Y. 
 Required the equal annual instalment of principal and interest 
 
 combined to l)e repaid the lender as and when set aside, to 
 
 repay £1,000 with interest in 10 years at 5 per cent. 
 
 A) By Formula. % = Pf^^J Rule 1, Chapter VIII 
 
 
 Log. Ratio 
 
 Multiply Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 1{ 
 
 N 
 
 
 1-05 
 10 
 
 0-0211893 
 10 
 
 Log . 
 RN_1 
 
 RN 
 
 
 (1-05)10 
 
 0-2118930 
 
 RN 
 
 -1 
 
 
 1-62889 
 1- 
 
 
 
 RN- 
 
 1 
 
 0-62889 
 
 T-7985779 
 
 Log. Present Sum 
 add Log. R^ above 
 Log. r 
 
 p 
 
 RN 
 
 r 
 
 
 1000 
 0-05 
 
 3- 
 0-2118930 
 
 2-6989700 
 
 deduct Log. (RN - 1) above 
 
 RN - 
 
 1 
 
 
 1-9108630 
 1-7985779 
 
 
 Aw 
 
 
 
 21122851 
 
 Required annuity, £129-5046. 
 
 (B) By Table V. Ay = P (^^^1^ ^ Rule 2, Chapter YIII. 
 
 Table V. 10 years, 5 per cent. 
 
 Annuity £1 will purchase 
 
 add Log-. Present Sum 
 
 RN 
 
 RN-1 
 
 p 
 
 1295 
 1000 
 
 1- 11 22851 
 3- 
 
 Ay 
 
 21122851 
 
 Required annuity, £129-5046. 
 
 (C) By Thoman's Table. At/^ P a" Rule 3, Chapter YIII. 
 5 per cent. 10 years. 
 
 Log. Present Sum 
 add Loo". a'^ 
 
 deduct 10 
 
 9-1122851 
 
 121122851 
 
 Ay 
 
 21122851 
 
 Required annuity, £129-5046. 
 
126 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XIII. 
 
 THE REPAYMENT OF THE LOAN DEBT OF LOCAL 
 AUTHORITIES AND COMMERCIAL AND FINAN- 
 CIAL UNDERTAKINGS {Continued). 
 
 The Sinking Fund Method. 
 
 BY SETTING ASIDE AND ACCUMULATING AN EQUAL ANNUAL 
 INSTALMENT IN OEDER TO PEOVIDE THE PRINCIPAL ONLY AT 
 THE END OF THE REDEMPTION PERIOD. 
 
 I, The Accumulating Sinking Fund. 
 Methods oe calculating the annual instalment by 
 
 FORMULA and TABLES AND THE GENERAL RULES BASED 
 
 THEREON. Description of the method and the calcula- 
 tion OF the annual instalment. Statement showing 
 
 THE final repayment OF THE LOAN. COMPARISON OF THE 
 instalment, ANNUITY AND SINKING FUND METHODS, ILLUS- 
 trated by a statement showing in each case the annual 
 charge to revenue or rate. 
 
 Author's Standard Calculation Form, No. 3x. 
 2. The Non-accumulating Sinking Fund. 
 
 The OBJECT of the fund and its relation to the METHODS 
 PRESCRIBED IN SeC. 234 OF THE PuBLIC HeALTH AcT, 1875. 
 
 Statement showing the final repayment of the loan 
 and the annual charges to revenue or rate. 
 
 Note. — Unless it is otherwise expressly stated, the term 
 
 " sinking FUND," WILL, THROUGHOUT THE BOOK, APPLY 
 ONLY TO AN ACCUMULATING SINKING FUND. 
 
 Formulae. 
 
 Variation of Table III, The aiuiuifij which uill amount to 
 
 £1 in any number of years, or ^ hT T^J'T 
 
THE SINKING FUND METHOD 127 
 
 A. To find the annuity which will amount to £1 in any 
 
 TMimber of years : — 
 
 (7) Formula, Ay= ( t^n _x j 
 
 by logs.: Log. (^required annuity) = Log. r — 
 Log. (EN-1) 
 
 (2) By Thoman's Method: — 
 
 Formula, Ay=^^^ 
 
 by logs.: Log. (^reqtiired annuity) — Log, a" — 
 {Log. EN +10) 
 
 B. To find the annual sinking fund instalment which tvill 
 
 amount to any given loan, in any number of years : — 
 
 (7) Formula, A.y = ^i( ^ J 
 
 by logs. : Log. (required instalvient)=Log. of Loan + 
 
 Log. r-Log. (R^-1) 
 
 (2) By Thoman's Method: — 
 
 /ft™ \ 
 Formula, Ay = M f t^^ 1 
 
 by logs.: Log. (required instalment) = Log. of 
 Loan + Log. a^-{Log. RN+IQ) 
 
 General Rules deduced from the above formulae. 
 
 To find the animal instalment to be set aside and accumulated 
 as a sinking fund to repay a given loan at the end of a prescribed 
 7iumber of years. Author's Standard Calculation Form, No. 3x. 
 
 Rule 1. If the rate per cent, be not given in Table III, or 
 in ThoTnan's Tables : — 
 
 Proceed by the formula derived from Table III, as 
 shoivn above. Calculation {XI II) 1 A . 
 
 fiule 2. If the rate per cent, be given in Table III : — 
 
 Divide the amount of the loan by the amount given 
 in the table. The qiiotient is the required annual 
 instalment. Calculation {XIII) 1 B. 
 
128 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Rule 3. If the rate per cent, he given in Th Oman's Table : — 
 To the log. of the loan, add the log. of a" as given by 
 Thoman. Deduct therefroiit the log. of W^ as given 
 by Thoman; also deduct 10. The remainder is the 
 log . of the reqtiired instalvient. 
 
 Calculation {XIII) IC. 
 
 To fnd the rate yer cent, or number of years ^ proceed as 
 shown in the standard form for the purpose, given inChapter X. 
 
 The Accumulating Sinking Fund. The sinking fund 
 metliod provides for the setting aside each year, and accumulat- 
 ing by way of compound interest, such a sum as will be sufficient 
 to pay off the money borrowed within the prescribed period. It 
 will be gathered from tbe above provision (which is laid down 
 in the Public Health Act, 1875, and is contained in principle 
 in all subsequent Acts) that this method differs from the instal- 
 ment and annuity methods in two particulars, viz. : — 
 
 1. It provides for the repayment of principal only, and is 
 
 quite apart from any question of interest on the loan. 
 
 2. The repayment of the principal money is not made by 
 
 instalments, but takes place at tlie end of the prescribed 
 period, with certain reservations which will be dealt with 
 later. 
 
 In both the instalment and annuity methods there is not 
 any question of the rate of accumulation, as the annual repay- 
 ments are made direct to the lender, and tbere is not therefore 
 any sinking fund set aside. 
 
 In the case of the annuity method as applied to the repayment 
 of the debt of a local authority, the lender may, or may not, be 
 able to reinvest the increasing proportion of principal included 
 in the annual instalment paid to him, at the calculated rate 
 which he receives upon his investment, but this does not enter 
 into the calculation in any way. So far as the local authority is 
 concerned they undertake to pay to the lender interest at the 
 agreed rate for such period only during which they have the 
 use of the money. As regards sinking funds relating to the loan 
 debt of commercial and financial undertakings, this is also 
 generally the case, but the purchaser of an annuity may require 
 that the annual instalment shall be fixed at such an amount as 
 will yield him a specified rate of interest upon his principal, and 
 at the same time enable him to reinvest the annual repayments 
 of principal at a lower rate than he receives as interest, in order 
 to ro]ilncc the capital. 
 
THE SINKING FUND METHOD 129 
 
 With regard to the provision in the Public Health Act that 
 the annual sum set apart shall be sufficient, after iKiying all 
 expenses, to pay oi! the money borrowed within the period 
 sanctioned, it is found in practice that the expenses, being of 
 uncertain amount, cannot be calculated actuarially. They are 
 therefore omitted from the calculation, and if small in amount 
 are charged direct to the rate or revenue account as and when 
 incurred. Where the expenses of raising the loan are large in 
 amount, as is the case when the loan is authorised by special 
 Act of Parliament, the Act generally provides that the cost of 
 obtaining the powers shall be repaid by means of a separate 
 sinking fund to mature in a short period, generally 5 to 10 years. 
 I The sinking fund method is the one now generally adopted 
 by all local authorities for the annual provision for redemption 
 of debt. It is called a sinking fund when it relates to loans, a 
 loans fund when it relates to the annual provision of principal 
 and the payment of dividends on stock, and a redemption fund 
 when it relates to stock issued under the stock regulations of the 
 Local Government Board. This is all very misleading and 
 confusing, but these are the statutory terms. The general term 
 sinking fund, with some distinguishing word added, would 
 better describe the nature of the fund which fulfils the same 
 purpose both in the case of loans and stock. The sinking fund 
 relates only to the ultimate repayment of principal by means 
 of an equal annual sum charged against the year's revenue or 
 rate, such annual sum being set aside and accumulated by 
 investment in outside securities. With regard to the interest 
 payable upon the loan, it is obvious, since no provision is made 
 for it in the sinking fund instalment, that during the whole of 
 the period of repayment the rate or revenue account of each year 
 will be charged with interest upon the full amount of the 
 original loan, and this notwithstanding the fact that part of the 
 sinking fund may have been applied in the redemption of part 
 of the loan before the expiration of the repayment period. 
 
 Since the interest paid upon the loan is quite outside the 
 question of the sinking fund, the rate of accumulation of the 
 fund may, and generally does, differ from the rate of interest 
 payable to the lender. Section 234 (5) of the Public Health 
 Act, 1875, provides that the local authority may apply the 
 whole or any part of the sinking fund in the repayment of the 
 debt, but if they do so they must pay into the sinking fund 
 annually a sum equivalent to the interest which would have 
 been produced by that part of the sinking fund so applied. 
 This provision, which is generally inserted in all general and 
 
I30 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 special Acts, is absolutely necessary. The sinking fund is 
 calculated to accumulate at a definite rate per cent., and if any 
 part of the fund be used to repay part of the debt the fund will 
 be deficient to that amount, and will lose the interest upon the 
 portion of the fund so applied. This provision is equal to 
 saying that any such application of the sinking fund shall be 
 treated as an investment of the fund as if it had been actually 
 invested in outside securities. 
 
 The section provides that the local authority shall pay into 
 the sinking fund a sum equivalent to the interest which would 
 have been produced by that part of the fund applied towards 
 the redemption of debt. But in practice it is usual to estimate 
 that the sinking fund will accumulate at a lower rate per cent, 
 than the interest paid upon the loan. This is in order to 
 provide for a fall in the rate of interest obtainable upon first- 
 class investments, and it results in a larger annual instalment 
 being set aside than would be the case if the sinking fund 
 were calculated to accumulate at the higher rate of interest 
 paid upon the loan. The general practice, when loans are 
 redeemed out of the sinking fund, is to pay into the fund the 
 actual amount of interest previously paid to the loan holders. 
 Any surplus thus arising helps to make up the deficiency caused 
 by the low rate of interest obtained Avhen part of the sinking 
 fund is in the bank awaiting investment, as often happens. 
 
 With regard to the investment of the sinking fund until it is 
 applied in the redemption of debt, it was until recently the 
 practice of Parliament and also of the Local Government Board 
 to require that it should be invested in outside securities, but 
 of late years Parliament has given power under special Acts to 
 invest the sinking funds in the stocks and loans of the same 
 local authority. The sinking fund, however, cannot be invested 
 in any other department of the same authority unless that 
 department has obtained statutory powers to borrow the amount, 
 and is therefore under a statutory obligation to set aside out of 
 revenue or rate a sinking fund for its redemption. 
 
 In the case of local authorities issuing stock at par which 
 afterwards commands a premium, the whole of the cost of any 
 part of the stock which is redeemed at a premium cannot be 
 taken out of the sinking fund, but only tlie par value of the 
 stock, the premium being charged to the rate or revenue account 
 at the time the stock is redeemed. If such purchases at a 
 premium are variable, both as to time and amount, they may 
 be dealt with by means of a su])plementary sinking fund relating 
 to the ]iremium only, in such a manner that the premium is 
 
THE SINKING FUND METHOD 131 
 
 spread equally over tlie unexpired period. If the premium is 
 fixed at the date of issue of the stock it should be included in 
 the original sinking fund calculation, but if the stock at any 
 time commands a premium beyond this amount the method of 
 providing for it in advance will be more difficult. 
 
 The Calculation of the Annual Instalment. The actual 
 calculation will now be considered. The instalment is required 
 to be set aside annually and accumulated at compound interest 
 in order to provide the principal sum only, and the question of 
 interest upon the loan does not enter into the calculation. 
 Under these conditions it would appear that the calculation is 
 much simpler than in the annuity method, using Tables I and 
 III, although not so if Table Y be used. The question to be 
 solved, therefore, is, taking as before a loan of £1,000 repayable 
 at the end of 10 years at 5 per cent., "what annuity accumulated 
 at 5 per cent, for 10 years will at the end of that period amount 
 to £1,000 "? This rate per cent, is the rate of accumulation of 
 the sinking fund and not the rate of interest payable upon the 
 loan. All questions involving the calculation of the amount of 
 an annuity are treated by the formula relating to Table III, 
 already referred to, namely, 
 
 M = A,(^l) 
 
 the actual values for £1 per annum being given in Table III. 
 
 The sinking fund calculation may be compared with that 
 made in the case of the annuity method, Calculation (XII) 3, 
 in which the instalment was required to provide a sum equal to 
 the " amount " of £1,000 accumulated at 5 per cent, compound 
 interest. 
 
 In this case the instalment has to provide only the capital 
 sum of £1,000 without interest. Consequently if the actual 
 loan be taken instead of the " amount " of the same sum at 
 the end of the period, as in Calculation (XII) 3, the required 
 annual instalment will be obtained for the reasons given in 
 ■discussing Calculation (XII) 1. The rule, therefore, to find the 
 sinking fund instalment is : — 
 
 " Divide the amount of the loan by the amount of £1 per 
 annum as given in Table III for the required number of 
 years at the stated rate per cent, and the quotient is the 
 required annual instalment.^' 
 
132 REPAYMENT OF LOCAL AXD OTHER LOANS 
 
 The problem resolves itself into the following : — 
 
 If £1 per annum in 10 years at 5 per cent, -will, at the end 
 
 of that period, amount to £12-5779, what annuity will, under 
 
 the same conditions, amount to £1,000? 
 
 The required formula is obtained by transposing the formula 
 relating to Table III as follows : — 
 
 A2/ =/ R^l j or A2/ = m(j^^^) 
 
 and the calculation will be made upon the author's standard 
 form, No. 3x, by the three methods previously referred to. 
 
 It may be interesting to point out that this calculation is an 
 example of how the use of a formula may lead to the discovery 
 of another method of making the same calculation. It will be 
 noticed in the above case that the numerator in the formula is 
 Mx7', (which means that £1,000 has been multiplied by 0'05) 
 and the result divided by (RN_i). But £1,000 x 005 = £50, 
 which is the interest upon £1,000 for one year at 5 per cent., 
 and therefore that an alternative rule may be stated as follows : 
 
 " To ascertain the sinking fund instalment, find the interest 
 upon the amount of the loan for one year at the sinking 
 fund rate of accumulation (not the rate of interest payable 
 upon the loan) and divide by (E^' — 1), which is the actual 
 value given in Table I, reduced by unity.'' 
 
 This rule is not of any practical advantage over those given 
 at the head of this chapter, and will not therefore be further 
 considered. 
 
 The Fixal REPAYiiExx of the Debt by the Operatiox of 
 THE Sinking Fund. The following statement shows the final 
 repayment of the loan by the operation of the sinking fund and 
 also the annual payment of interest upon the whole of the loan 
 until the end of the prescribed period when the accumulation of 
 the fund is equal to the amount of the loan which is then 
 repaid, the fund exhausted, and the annual contributions cease. 
 
 This statement shows that the fund is increased annually 
 by the instalment provided out of revenue or rate and by the 
 income received upon the investment of previous instalments. 
 This income from investments is the amount which the lender, 
 under the annuity method, would have obtained if he had taken 
 out of each annual instalment of £12951 paid to him the sum 
 of £50 bv wav of interest upon his loan, and invested the 
 
THE vSINKING FUND METHOD I33 
 
 reiiiaiuiug- £79-51 and the subsequent accumulations at 5 per 
 cent, annually to provide his capital at the end of the term. 
 He would by this means obtain a more regular income than by 
 treating as income the interest shown in the tables relating to 
 the annuity method, which decreases year by year. It will 
 further be noticed that the interest charged to the revenue or 
 rate account under the annuity method, as shown in the table 
 relating to that method, added to the income received from 
 investments, as shown in the table relating to the sinking fund 
 method, are together equal in each year to £50, which is the 
 interest paid to the lender annually under the sinking fund 
 method. See Statement XIII. A., page 139. 
 
 If, therefore, the lender, under the annuity method, requires 
 to equalise his annual income, he may do so by setting aside an 
 equal annual sum out of the instalment and accumulating it as 
 a sinking fund to provide his capital. 
 
 This mode of equalising the income might be adopted by 
 trustees and executors with the object of securing a fixed income 
 for a tenant for life, but will apply only to an annuity for a 
 fixed term. The above argument is, however, subject to the 
 reservation that the lender may not be able, year by year, to 
 reinvest the periodical repayments of principal to yield the rate 
 per cent, upon which the annual instalment Avas based. 
 
 Comparison of the Theee Methods. It is now possible to 
 compare the repayment of loans by instalment, annuity and 
 sinking fund methods, as above described, and this will be 
 done from the standpoints both of the lender and borrower by 
 means of the following statement (XIII. B., page 140. 
 
 In Chapter XI, the instalment method has been compared 
 with the annuity method, and it is interesting to compare the 
 annuity method with the sinking fund method. In each case 
 the annual instalment is ascertained by dividing a definite sum 
 by the same accumulated amount of an annuity of £1 for 10 
 years at 5 per cent., but in the case of the annuity method 
 the amount so divided is the amount of the principal sum 
 accumulated at compound interest, whilst in the sinking fund 
 method the amount so divided is the principal sum itself 
 without accumulations. This is owing to the fact that the 
 annual instalment in the case of the annuity method includes 
 interest, whereas the annual instalment in the case of the 
 sinking fund relates to the principal sum only. The annual 
 instalment in the sinking fund method, therefore, is smaller 
 than in the case of the annuity method. 
 
134 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 By tlie annuity method, Calculation (XII) 3, the 
 
 instalment of principal and interest is £129'51 
 
 By the sinking fund method the instalment of 
 
 principal only is £79"51 
 
 The difference being- one year's interest on £1000 at 
 
 5 per cent \ £5000 
 
 Under the sinking fund method, therefore, the total annual 
 charge to revenue or rate in respect of principal and interest is 
 exactly equal year by year to the total annual charge under the 
 annuity method, viz. £129'51 in each case. 
 
 This has already been referred to in discussing the annuity 
 method in Chapter XII. 
 
 With regard to the instalment method the total annual 
 charge to revenue or rate account in respect of principal and 
 interest is greater in the earlier years and is gradually reduced 
 from £150 to £105 in ten years. The relative merits of the 
 annuity method and the sinking fund method as regards the 
 annual incidence of local taxation are equal and are more 
 equitable than the instalment method. As regards the investor, 
 under the instalment method he receives a decreasing annual 
 payment made up of a constant amount of principal and a 
 decreasing amount of interest ; but he has definite knowledge of 
 how much is interest and how much is principal. Under the 
 annuity method he receives an equal annual payment made up 
 of an increasing amount of principal and a decreasing amount of 
 interest; but without an elaborate calculation he is unable to 
 apportion the amount paid to him between capital and income. 
 Under both the instalment and the annuity methods the 
 investor receives annual sums in respect of his capital which 
 he has to reinvest in small amounts. 
 
 Comparing the sinking fund method, on the one hand, with 
 the instalment and annuity methods on the other, from the 
 point of view of the investor, it will be seen that under the 
 sinking fund method he receives each year an equal amount by 
 way of interest upon his money, and has the further advantage 
 of a permanent investment of the whole of his capital for a 
 definite long; term. If he wishes to realise he has a definite 
 security to place upon the market either to be bought by some 
 other investor or to be redeemed by the local authority out of 
 the sinking fund. lender the sinking fund method he has to 
 run the risk of a fall in the market value in the case of a loan 
 
THE SINKING FUND METHOD 135 
 
 raised l>v tlie issue of stock; but, on tlie other hand, he may- 
 realise a profit. Summing up the respective merits of the various 
 methods of repayment of the debt of local authorities, it may 
 fairly be concluded that the accumulating sinking fund method 
 is by far the best. It bears equally upon the taxation or 
 revenue of each year of the repayment period; and as regards 
 the investor, it is at once more convenient and more equitable 
 than either of the other two methods. 
 
 The Non-Accumulating Sinking Fund. Up to this point 
 the enquiry has been limited to accumulating sinking funds 
 similar to the one prescribed in the Public Health Act, 1875. 
 The principal feature of such a fund is the provision out of 
 revenue or rate of an equal annual instalment to be set aside 
 and accumulated for a prescribed period at a rate per cent, to be 
 fixed in anticipation, with as near approach to accuracy as can 
 be obtained. In the case of loans with long repayment periods 
 this is very difficult, and it therefore becomes necessary to 
 compare the actual amount in the fund periodically with the 
 calculated amount which should be in the fund as shown by the 
 pro forma account. Any surplus or deficiency in an accumulat- 
 ing fund should be credited to, or charged against, the revenue 
 or rate account of each year, but this entails considerable 
 labour, and it is one of the objects of the non-accumulating 
 sinking fund to avoid this by providing an automatic accurate 
 accumulation of the fund irrespective of the rate of income 
 received on the investments representing the fund. The basis 
 of the method is the instalment system discussed in Chapter XI, 
 where each year a definite sum is charged to the revenue or rate 
 account and repaid to the lender. The annual instalment in 
 the case of the non-accumulating sinking fund is calculated 
 precisely as in the instalment method, namely, by dividing the 
 amount of the loan by the number of years in the repayment 
 period. But in this case the annual instalment of principal is 
 not repaid to the lender, but is invested by the local authority 
 in order to provide the amount of the loan at the end of the 
 period. Since an equal amount is added to the fund year by 
 year it requires merely an arithmetical calculation to ascertain 
 the amount which should be in the fund at any time. Seeing 
 that the total amount of the loan is provided by the actual equal 
 annual charges to revenue or rate, it is obvious that the income 
 arising from the investments rej^resenting the fund need not be 
 added to the fund. On comparing the actual instalments only. 
 
136 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 under tlie instalment metliod in Chapter XI, with tliose under 
 the accumulating sinking fund method in this chapter, it -will he 
 seen that the charge to revenue or rate under the instalment 
 method is greater than under the accumulating sinking fund 
 method, consequently in the case of a non-accumulating fund 
 tlie income to arise from the investments may be credited to the 
 rate or revenue account to which the annual instalment of 
 principal has been debited. The excess of the original annual 
 instalment in the non-accumulating fund over the instalment m 
 the accumulating fund Avill not be compensated by the reduction 
 therein due to the income received from the investment of the 
 fund, because such resulting income will be small during the 
 earlier years ; and an equality in the annual burden will not be 
 reached until the end of the fifth year out of ten. The first four 
 years will therefore bear an additional burden, and the last five 
 years will be relieved, as compared with the annual incidence 
 under the accumulating fund, in a similar manner to the instal- 
 ment method. As regards the ratepayer, the non-accumulating 
 fund will have all the disadvantages of the instalment method 
 already pointed out. The lender, on the contrary, will be in a 
 better position, since he obtains a permanent investment and is 
 relieved of tlie periodical reinvestment of small amounts of 
 capital. The actual method by which the local authority 
 provides the sinking fund has not any particular interest to him. 
 As between the instalment and annuity methods, on the one 
 hand, and the two sinking fund methods on the other, the only 
 difference is the date at which he shall be repaid, and he invests 
 in the particular loan which best meets his requirements. Under 
 the two periodical repayment methods the lender may be said to 
 keep his own sinking fund, whereas in both sinking fund 
 methods the local authority does this for him. There are not 
 any mathematical principles involved in the non-accumulating 
 fund, but it is merely an arithmetical one. The following table 
 shows the final repayment of the loan by the operation of the 
 fund. In order that it may be compared with the accumulating 
 sinking fund it has been assumed that the investments yield 
 6 per cent, per annum, and that no part of tlie fund is ap])lied in 
 I'edemptinn of debt during the period : — 
 
THE SINKING FUND METHOD 137 
 
 STATEMENT XIII. C. 
 
 The Repayment of the Debt of Local Altthoeities. 
 The Non-Accumulatixg Sinking Fund. 
 
 Showing the repajanent of a loan of £1,000, at the end of 
 10 years by an equal annual instalment of principal, to be 
 set aside and invested as a sinking fund, the annual income 
 upon the investments being applied in reduction of subse- 
 quent instalments. Interest at 5 per cent. 
 
 Net Total 
 
 Annual Deduct charge to charge to Amount 
 
 ^ear 
 end 
 
 Instal- 
 ment 
 
 Income 
 received 
 
 revenue 
 or rate 
 
 Interest 
 on loan 
 
 revenue 
 or rate 
 
 in 
 fund 
 
 Year 
 end 
 
 1 
 
 100 
 
 — 
 
 100 
 
 50 
 
 150 
 
 100 
 
 1 
 
 2 
 
 100 
 
 5 
 
 95 
 
 50 
 
 145 
 
 200 
 
 2 
 
 3 
 
 100 
 
 10 
 
 90 
 
 50 
 
 140 
 
 300 
 
 3 
 
 4 
 
 100 
 
 15 
 
 85 
 
 50 
 
 135 
 
 400 
 
 4 
 
 5 
 
 100 
 
 20 
 
 80 
 
 50 
 
 130 
 
 500 
 
 5 
 
 6 
 
 100 
 
 25 
 
 75 
 
 50 
 
 125 
 
 600 
 
 6 
 
 7 
 
 100 
 
 30 
 
 70 
 
 50 
 
 120 
 
 700 
 
 7 
 
 8 
 
 100 
 
 35 
 
 65 
 
 50 
 
 115 
 
 800 
 
 8 
 
 9 
 
 100 
 
 40 
 
 60 
 
 50 
 
 110 
 
 900 
 
 9 
 
 10 100 45 55 50 105 1000 10 
 
 The clauses authorising the non-accumulating sinking fund 
 contain the usual permission to apply part of the fund in 
 redemption of debt; but in this case there is not any necessity 
 to have regard to the interest which would have been received in 
 respect of the part of the fund so applied, because, although the 
 income received from the investments will be smaller in conse- 
 quence of such application, yet the interest payable upon the 
 loan will be correspondingly reduced, and there will not there- 
 fore be any alteration in the combined charge for interest and 
 redemption. There is another matter which may properly be 
 considered to the advantage of the non-accumulating fund, if 
 the greater burden imposed during the earlier years is not fatal 
 to its adoption. This affects the possible and very probable 
 variation in the rate of income to be received from the invest- 
 ments representing the fund. In the ease of the accumulating 
 fund, as already pointed out, it very rarely happens that the 
 
138 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 fund increases in accordance with the calculated amount, thus 
 rendering it necessary to make frequent adjustments through 
 the revenue or rate account. As will be shown in later chapters, 
 the variation in the rate per cent., whether of income from 
 investments or of accumulation, gives rise to many of the 
 problems which have to be dealt with. The non-accumulating 
 fund has this advantage, that any variation in the rate per cent, 
 is at once automatically adjusted, seeing that if from any cause 
 there is a fall in the rate of income from investments there is a 
 corresponding increase in the charge to revenue or rate due to 
 the decreased relief to subsequent annual instalments afforded 
 by the amount of income received from the investments. 
 
 In the case of a non-accumulating fund the annual instal- 
 ment of the full amount would be credited each year to the 
 sinking fund account and not debited direct to the revenue or 
 rate account, but to an intermediate account which might be 
 called the " non-accumulating sinking fund suspense account." 
 This suspense account Avould be credited with the income received 
 from the investments representing the fund, and with interest 
 allowed by the bank, if any, whether any part of the fund had 
 been applied in redemption of debt or not, and it would be 
 debited with the interest actually paid or accrued on the 
 outstanding loan. The balance remaining to the debit of this 
 suspense account would then be charged to the revenue or rate 
 account and would represent the total charge against the year in 
 respect of loan indebtedness. The accounts of an accumulating 
 sinking fund might be kept in a similar manner. A " sinking 
 fund interest suspense account " would be opened, and the 
 sinking fund account would be credited and the suspense interest 
 account debited with the actual amount of interest which should 
 yearly accrue to the fund as shown by the pro forma account. 
 The suspense interest account would be credited with the actual 
 income received from the investments, and bank interest; and 
 the balance, either debit or credit, but generally debit, would be 
 closed by transfer to the revenue or rate account. By this 
 means the sinking fund accounts Avould always stand at their 
 proper calculated amounts, any reduction in income would 
 immediately become apparent and there would be no possibility 
 of the gradual accumulation of many small deficiencies requiring 
 at some future time considerable correction by an increased 
 annual instalment. 
 
THE SINKING FUND METHOD 
 
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140 REPAYMENT OF LOCAL AND OTHER LOANS 
 
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THE SINKING FUND METHOD 
 
 141 
 
 Calculation (XIII) 1. 
 
 Standard Calculation Form, No. 3x. 
 
 To find the annual sinking fund instalment. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 as a sinking fund at 5 per cent, to provide £1,000 at the 
 end of 10 years. 
 
 A) By Formula. A(/= M ( j^y'l^ ) R^^le 1, Chapter XIII. 
 
 Log 
 R^— 1 
 
 'Log. Ratio 
 
 Multiply Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 R 
 
 N 
 
 Log of this is 
 og. Amount 
 add Loo^, r 
 
 105 
 10 
 
 00211893 
 10 
 
 Los". Amount of Loan 
 
 deduct Log. (R^ - 1) above 
 
 RN 
 
 (1-05)10 
 
 0-2118930 
 
 R^' 
 -1 
 
 1-6289 
 1- 
 
 
 RN-1 
 
 0-6289 
 
 1-7985779 
 
 M 
 
 r 
 
 1000 
 -05 
 
 3- 
 2-6989700 
 
 Mr 
 
 RN-1 
 
 
 1-6989700 
 1-7985779 
 
 A?/ 
 
 
 1-9003921 
 
 Required annual instalment, £79-5046. 
 M 
 
 (B) By Table III. A^ = RN-1 Rule 2, Chapter XIII. 
 
 Log. Amount of Loan M 
 Table III. 10 years, 5 per cent 
 Amount of £1 per annum 
 deduct Log. 
 
 1000 
 
 R^-1 12-5779 
 r 
 
 1-0996079 
 
 A?y 
 
 1-9003921 
 
 Required annual instalment, £79-5046. 
 
 (C j By Thoman's Table. A^ = ^^(^) ^^^^^ ^' Cliapter XIII. 
 5 per cent., 10 years. 
 
 Log. Amount of Loan 
 add Log. a" 
 
 M 
 
 a^ 
 
 1000 
 
 3- 
 9-1122851 
 
 deduct Log. R^ in 
 Table +10 
 
 Ma« 
 
 RN 
 
 
 121122851 
 10-2118930 
 
 
 A.v 
 
 
 1-9003921 
 
 Required annual instalment, £795046. 
 
Section III. 
 Sinking Fund Problems. 
 
 The Annual Instalment. 
 
vSINKING FUND PROBLEMS 145 
 
 CHAPTER XIY. 
 SINKING FUND PEOBLEMS. 
 
 RELATING TO : 
 
 (U The Amount in the Fund. 
 
 (2) The E ate per cent : — 
 
 (a) of income to be received upon the present invest- 
 ments representing the fund ; 
 (6) the future rate of accumulation. 
 
 (3) The Eedemption Period. 
 
 (4) The rate per cent, and the redemption period in 
 
 combination. 
 
 Definition of terms : 
 
 1. The present inve.stments. 
 
 2. The annual increment. 
 
 The Pro forma Account. Having discussed the several 
 alternative methods of repayment of loan debt by local authori- 
 ties laid doAvn by statute, and having described the methods 
 of finding the annual sums to be set aside for that purpose out 
 of revenue or rate, the subject M'ill now be considered in its 
 practical aspect. Most of these transactions extend over very 
 long periods and all trace of the original calculation is often 
 lost. It is therefore advisable in all cases involving a periodical 
 provision for repayment by means of a sinking fund to prepare 
 at the outset a pro forma account showing how the calculated 
 annual instalment should work out during the whole of the 
 period . 
 
 The Local Government Board auditors in many cases require 
 this to be done in respect of all loans coming under their 
 supervision, and it is a practice to be commended and followed. 
 Such a pro forma account enables a comparison to be made 
 annually between the actual and the calculated working out of 
 the fund so that any discrepancy may be immediately set right. 
 Especially does this apply to a deficiency caused by part of the 
 sinking fund lying uninvested in the bank and earning less than 
 
146 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 the calculated rate per cent, of accumulation or due to a general 
 decrease in such rate. Any such deficiency will be of small 
 amount in any one year, and may be charged against the 
 revenue or rate account of the particular year, so keeping the 
 sinking fund up to the proper amount. But cases have arisen 
 in which this has not been done, and from the above, and other 
 causes, the amounts in the sinking funds have been seriously 
 deficient. In such cases it becomes necessary to ascertain the 
 proper amount which would have been in the fund if the 
 original anticipations had been realised. This is a contingency 
 which may arise in the case of a local authority, and there are 
 other questions with regard to sinking funds which, although 
 not affecting local authorities, yet are very important in con- 
 nection with the sinking funds of commercial and financial 
 undertakings. 
 
 Nature of Problems. In dealing with all cases of adjust- 
 ment of a sinking fund it will be necessary to refer continually 
 to the present state of the fund as the basis upon which all such 
 adjustments are made, and later, when dealing with other 
 problems, it will be seen that the present position of the fund 
 plays an equally important part. Such questions will be 
 considered later, and will comprise : — 
 
 (1) A deficiency in the fund. Chapters XY, XYI. 
 
 (2) A surplus in the fund. Chapters XYII, XYIII. 
 
 (3) A variation in the rate per cent, at which the fund was 
 
 originally expected to accumulate. Chapter XIX, etc. 
 
 (4) A variation in the rate of income to be yielded by the 
 
 investments representing the fund. 
 
 Chapters XX, XXYII. 
 
 (5) A variation in the repayment period. Chapter XXIY. 
 
 (6) A variation in the repayment period accompanied by a 
 
 variation in the rate of accumulation. Chapter XXYI. 
 
 Any or all of the above contingencies may have to be taken 
 into account in an adjustment, and as they arise only after the 
 fund has been in operation for part of the original repayment 
 period, it is important to ascertain exactly the position of the 
 fund at the time the adjustment is required to be made. It is 
 generally the case with the sinking funds of local authorities 
 that the amount standing to the credit of the fund is required 
 
SINKING FUND PROBLEMS I47 
 
 to be invested in specific outside securities allocated to the fund, 
 or, wliicli is the same in effect, shall have been applied in part 
 repayment of the original loan. In the case of commercial and 
 financial undertakings it is usual to impose the obligation of 
 such outside investment in order to ensure that the original 
 purpose of the fund shall be carried out, and that the amount in 
 the fund shall be actually available for the repayment of the 
 debt at the end of the period. Any enquiry therefore into the 
 adequacy or otherwise of the amount in the fund at any time 
 will properly include, not only the value of the investments 
 representing the fund at the present time, but also an enquiry 
 as to the probable value at the end of the repayment period. 
 It will be necessary to ascertain whether they are yielding or 
 are likely to continue to yield a retvirn by way of income equal 
 to or differing from the calculated rate percent, of accumulation. 
 In the following chapters, treating of the above possible causes 
 of variation, as far as possible for purposes of convenience and 
 comparison, the position of an imaginary sinking fund will be 
 ascertained at the end of the 12th year of an original period 
 of 25 years, and the position of the fund will be shown at that 
 time, when the enquiry and any subsequent rectification is 
 made, in the following terms, viz. : — 
 
 (1) The. vahie of the present investments representing the 
 
 amount in the fund. 
 
 (2) The present annual increment at the time the enquiry is 
 
 made, and before the rectification to meet the new condi- 
 tions. 
 
 Present Investments. The term " present investments " 
 will be used to denote the value of the investments representing 
 the amount which actually stands to the credit of the fund and 
 not the amount which should so stand by calculation at the 
 original rate of accumulation as shown by the pro forma 
 account. In fixing the precise market value regard should be 
 had to the probability of the individual investments ultimately 
 yielding the original cost price, and if any fall in value has 
 occurred, or is likely to occur, it should as far as possible be 
 included in the adjustment. In dealing with a surplus or a 
 deficiency in the fund, any actual change in value should be 
 taken into account in calculating the amended annual instal- 
 ment ; but where the problem concerns the period of repayment 
 or the rate of accumulation, and especially if the fund has a 
 long unexpired period to run, it is hardly possible to make 
 
I4S REPAYMENT OF LOCAL AND OTHER LOANS 
 
 any exact forecast of tlie future value of the iuvestments, or of 
 the future rate of income to be receiTed therefrom, and this 
 should be provided for by making an allowance when deciding 
 upon the amended rate of accumulation, namely, by taking it at 
 a slightly lower rate than would otherwise be sufficient. In the 
 whole of the following examples, except a deficiency or a 
 surplus in the fund, it will be assumed that the fund stands at 
 the exact amount shown by the original calculation ; and, 
 further, that the various investments representing the fund are 
 each worth now the exact amount paid for them, and will be 
 so at the end of the period. This will sufiiciently explain 
 without further reference the meaning attached to the term 
 " present investments " in the following pages. 
 
 The Annual Increment. With regard to the annual 
 increment, it will be seen, on considering the sinking fund at 
 its inception, that there is then only one factor to deal with, 
 namely, the repayment of a definite loan (or the provision of a 
 definite sum) at the end of a stated number of years. This 
 term will be referred to in the following pages as the " period of 
 repayment or redemption," and in order to make the adjustment 
 it is necessary to fix an average rate per cent, at which the 
 future payments to the fund may reasonably be expected to 
 accumulate by subsequent investment. It is very difficult, if 
 not impossible, to do this correcth' in the case of a fund having 
 a long period of repayment, and the practice generally is to 
 assume a rate of accumulation slightly lower than the rate of 
 interest payable to the loauholders. This will alloAv for a fall 
 in the accumulation rate owing to fluctuations of the money 
 market or for a deficiency in the income of the fund caused by 
 delay in finding an investment which leaves money idle in the 
 bank, earning only a low rate of interest. If the annual 
 deficiency in the income of the fund or any annual surplus be 
 small it should be rectified, as and when it arises, by adjusting 
 it by means of the revenue or rate account, but if the annual 
 deficiency or surplus be large, it is better to adjust the annual 
 instalment immediately in the manner to be described later 
 under the head of variation in the rate of accumulation. Having 
 fixed the future estimated rate of accumulation, the calculation 
 is made in the manner shown in Calculation (XIII) 1, to 
 ascertain the annual instalment to be set aside each year to 
 accumulate at the estimated rate. This annual instalment thus 
 becomes the annual increment during the first year, but after 
 tl)e first instalment has been invested another factor is introduced 
 
SINKING FUND PROBLEMS i49 
 
 into the annual increment, namely, the income from the invest- 
 ments representing the fund. 
 
 It is not often that any question affecting the adequacy of 
 the amount in the fund arises during the earlier years of the 
 repayment period. Generally it is much later, and in the 
 following examples it has been taken as the 12th year of a 
 period of 25 years. By this time the fund will have amounted 
 to a large proportion of the total sum to be ultimately provided, 
 and the accruing annual income from investments will (with a 
 3i per cent, rate of accumulation) be about one-half of the 
 original annual instalment. Any adjustment of the fund at 
 the end of the 12th year will therefore depend largely upon the 
 future rate of income to be yielded by the present investments 
 representing the fund. And this adjustment may actually be 
 rendered necessary by a fall in the rate of income yielded by 
 the present investments, occurring at a time when the rate 
 yielded by other investments of all kinds is also falling. If the 
 original rate of accumulation be likely to be maintained in 
 spite of a fall in the income received from the present invest- 
 ments, there is not any need, as shown in Chapter XX (variation 
 B, in the rate per cent, of income) to make any adjustment by 
 calculation in the annual instalment. All that is required is to 
 take an additional annual sum out of revenue or rate, equal to 
 the amount of the reduction in the future annual income to be 
 received from the present investments, and the fund will 
 continue to accumulate as originally calculated. But where, as 
 in Chapter XXI (variation C in the rate per cent.) it is 
 necessary at the same time to provide for a reduction in the 
 rate of income from the present investments as well as a 
 reduction in the rate of accumulation, the problem becomes 
 more complicated because there are then two different rates per 
 cent, acting upon two diff'erent factors. The rate of income 
 upon the present investments has no relation to the annual 
 instalment provided out of revenue or rate which is acted upon 
 by the accumulation rate only. But the actual amount (if not 
 the rate per cent.) of the income from investments is also acted 
 upon by the accumulation rate, and it is possible to state 
 definitely the annual sum which will be received in respect of 
 such income. Consequently, the difficulty attending the two 
 rates per cent, may be avoided by treating the future income 
 from the present investments as an annuity certain which will 
 continue to be received during the whole of the unexpired 
 portion of the repayment period in exactly the same way as the 
 original annual instalment will continue to be set aside out of 
 
150 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 revenue or rate. These two annual factors tog-ether will be 
 considered as constituting the future annual increment to be 
 included as an asset in the adjustments, and to be supplemented, 
 as will be seen later, by any additional annual instalment (to be 
 provided out of revenue or rate) which may be found necessary 
 to make up for the decrease in the income from the present 
 investments, and also any further deficiency caused by a 
 reduction in the rate of accumulation. This supplemented 
 annual increment will be referred to later as the future or 
 amended annual increment, as defined in Chapter XXII. 
 Although in the examples which will be considered later a 
 reduction in the rate of income from investments will be 
 assumed, it is quite possible that there may be an increase in 
 the rate of income, which would have the effect of reducing the 
 original annual instalment instead of increasing it. It rarely 
 happens, however, that there is an increase in the rate of 
 accumulation. It is unwise to predict a change which will 
 have the effect of relieving the present revenue or rate account 
 to the possible detriment of future years, and if any surplus in 
 the fund arises in this way it is usually dealt with at the time. 
 The above remarks will explain the reason for the methods 
 adopted later of showing the position of the fund at the end 
 the 12th year when dealing with variations in the rates per cent, 
 of accumulation and income which dift'er from the methods 
 adopted to show the position at the end of a similar period when 
 dealing with a deficiency or a surplus in the amount in the 
 fund or with variations in the period of repayment without any 
 variation in the rate per cent, either of income or of accumula- 
 tion. In both the latter cases (see Statements XV. B. and 
 XXIV. A.), which do not involve any variation in the rate 
 of accumulation or in the rate of income, the assets of the fund 
 include the accumulated amount (using the term as in Table I) 
 of the value of the present investments at the end of the 
 respective re])ayment periods. This amount includes tlie 
 present value of the investments (£916-3 and £9932" 74) and 
 the accruing compound interest, because they both accumulate 
 at the same rate which is the same as the rate of income upon 
 the investments. 
 
 But m problems involving a variation in the rate per cent, 
 of accumulation, without any variation in the rate of income 
 from investments (as in Statement XIX. A.) it is necessary 
 to find the future amount of the present investments by two 
 calculations because whilst the present investments continue 
 to yield 3^ per cent, per annum, the income so yielded accumu- 
 
SINKING FUND PROBLEMS 151 
 
 lates at only 3 per cent. It is therefore requisite to include the 
 present value of the investments, viz., £993274 and to add 
 thereto the sum to which the annual income will accumulate 
 at the end of the period at the amended accumulation rate. As 
 above remarked, it is not necessary to consider the annual 
 increment in connection with problems involving a variation in 
 the rate of income from investments only, but later in Chapter 
 XXYI, when dealing- with problems, involving a variation in 
 the period of repayment complicated by a variation in the 
 accumulation rate, the annual increment again becomes an 
 important factor. The annual increment has been considered 
 in this exhaustive manner because it is a convenient way of 
 expressing the resulting correction required in consequence of 
 any of the above variations. 
 
 It is the adjusted annuity under the amended conditions 
 which is the equivalent of the original annuity under the 
 previous conditions. It may be divided, at both periods, into 
 its component parts of : — 
 
 (1) The income from the present investments received from 
 
 outside sources, and 
 
 (2) The annual instalment, to be provided out of revenue or 
 
 rate, which is the object of enquiry in all cases. 
 
 The term will be found very useful when dealing with all 
 actual adjustments, since by dividing the accretions to the fund, 
 as between income from outside investments and contributions 
 from internal revenue, a clearer insight is obtained into the 
 principles underlying the methods adopted. 
 
 Methods of Adjustment, based upon the Annual Increment. 
 (1) The Annual Increment {ratio) Method. It will be gathered 
 from the previous remarks that an adjustment in a sinking fund 
 due to any variation in the original conditions may be made in 
 terms of the annual increment, and that there is a definite 
 relation always existing between the annual increment before 
 adjustment (the present annual increment) and the annual 
 increment after the necessary adjustment has been made (the 
 future or amended annual increment). These terms are fully 
 defined at the head of Chapter X'XII, where the component 
 parts of each annual increment are exactly described. In both 
 cases the annual instalment may be found by deducting from 
 the annual increment the income from the present investments, 
 thereby eliminating from the calculation any variation in the 
 
152 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 rate per cent, of income from investments, and confininjs^ tlie 
 
 enquiry to tlie variation in tlie rate of accumulation only. Tlie 
 
 annual increment may be considered as a simple annuity to be 
 
 set aside for a number of years (N) and accumulated at a rate 
 
 per cent, per annum expressed by the factor (R) or ratio, and 
 
 the combination of-tbese factors, as regards an annuity or otber 
 
 R^ — 1 
 periodic sum is expressed by tbe formula '- tlie derivation 
 
 of whicb, from the simple formula A=P R^, is fully described 
 in Cbapter VI. There is an exact ratio always existing- between 
 a given annuity to be accumulated for a stated number of years 
 at a stated rate per cent., and. the equivalent annuity to be 
 accumulated for a varying number of years, at a varying rate 
 per cent., depending upon tbe respective values of X and E,. 
 This is the basis of the annual increment (ratio) method, 
 which is fully described in Chapter XXII, and which has been 
 used in many of the examples in the following chapters. 
 
 (2) The Annual Increment {balance of loan) Method. In 
 all problems involving an adjustment in a sinking fund there 
 are two fixed factors to be considered, namely : — 
 
 (1) The amount of loan to be ultimately repaid, and 
 
 (2) The amount now standing to the credit of the fund 
 
 represented by the present investments. 
 
 And in addition there are two variable factors, namely : — 
 
 (1) The future period of repayment (N years). 
 
 (2) The future rate of accumulation of the fund expressed by 
 
 the factor (R) or ratio. 
 
 Any variation in the future rate of income to be received upon 
 the present investments representing the fund has already been 
 eliminated by merging such annual income in the annual 
 increment. 
 
 In all problems involving a variation in the original 
 conditions governing a sinking fund the subject of inquiry is 
 the future amended annual obligation, and this may be ascer- 
 tained by reducing the present factors to a common basis, 
 namely, the balance of original loan which will be unprovided 
 if the amount now in the fund be immediately applied in 
 redeeming an equivalent part of the loan ultimately repayable. 
 The balance of loan, thereby unprovided for, represents the 
 accumulated amount of an annuity equal to the future or 
 amended annual increment to be set aside for the unexpired or 
 substituted repayment period and accumulated at the original 
 
SINKING FUND PROBLEMS 153 
 
 or varied rate of accumulation. This balance of loan may be 
 ascertained by deducting from the amount of loan ultimately 
 repayable the amount now in the fund as represented by the 
 present investments ; and the future annual obligation, which is 
 the future annual increment, may be ascertained by calculating, 
 on standard form, No. 3x, the sinking fund instalment required 
 to provide that amount under the altered conditions, both as 
 regards the period of repayment and the rate per cent, of 
 accumulation. The amended annual increment so ascertained 
 does not, however, represent the amount to be charged annually 
 against the revenue or rate account of the local authority. 
 The conditions governing a sinking fund, as laid down in 
 section 234 (5) of the Public Health Act, 1875, provide that if 
 at any time during the operation of a sinking fund any part of 
 such fund be applied in redemption of debt, the local authority 
 shall, out of its annual rate, pay into the sinking fund a sum at 
 least equal to the amount of interest Avhich would have accrued 
 to the fund if such amount had not been so applied. Conse- 
 quently the future amended annual instalment is found by 
 deducting from the future or amended annual increment, ascer- 
 tained in the above manner, the annual income to be received 
 upon the present investments which have been considered as 
 having been immediately applied in the redemption of an 
 equivalent part of the loan, whether the rate of income upon 
 such investments remains unaltered or is varied. 
 
 This is the basis of the annual increment (balance of loan) 
 method, which is fully described in Chapter XXII, and which 
 has been used in many of the examples in the following chapters. 
 
154 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTEE XY. 
 
 SINKING FUND PE(JBLE:\1S RELATING TO 
 THE AMOUNT IN THE FUND. 
 
 A deficiency in the fund ; how it may arise and how it 
 may be adjusted. 
 
 Peelimixaey calculation of a typical sinking fund to be 
 used to illusteate the problems to be discussed in the 
 following chapters. methods of ascertaining the 
 position of a sinking fund at any time. a deficiency in 
 the fund and the various ways in which it may be 
 corrected. general summary of methods of adjustment. 
 
 Before considering- in detail the various problems arising in 
 connection with a sinking fund it should be stated that there 
 are in each case several methods of making the required adjust- 
 ment, all of which depend upon the present position of the 
 fund, and the future variation in the original conditions. The 
 subsequent enquiry will include variations in all the funda- 
 mental factors relating to such a fund, namely, the amount of 
 the fund, the period of repayment of the loan, the rate of 
 accumulation of the fund, and the future rate of income to be 
 received upon the present investments representing the fund. 
 All these factors have each their own effect upon the ultimate 
 function of the fund, namely, the repayment of the loan, but 
 in addition they act and react one upon the other. 
 
 For the purpose of comparison, therefore, each of the possible 
 variations will be considered in relation to one and the same 
 fund, and it will be necessary to treat all the problems on, as 
 far as possible, parallel lines, with the result that in the first 
 instance the most direct method of making the adjustment will 
 not be discussed, although it will be afterwards fully described. 
 The first subject of enquiry will relate to the simple problem of 
 a deficiency in the amount in the fund without any further 
 complication, and the adjustment of such a deficiency will be 
 made by the deductive method, to be followed later when 
 dealing with other matters affecting the fund. 
 
 The following is a summary of the general rules as to the 
 adjustment of a deficiency in a sinking fund where the amount 
 in the fund only is in question, and the period of repayment, 
 
A DEFICIENCY IN THE FUND I55 
 
 the future rate of income upon the present investments, and the 
 future rate of accumulation all remain unaltered. In this 
 chapter a deficiency in the fund has been treated in a very- 
 exhaustive manner, perhaps more so than is due to its relative 
 importance. This course has been purposely adopted in order 
 to demonstrate the practical relation between the various 
 formulae and the tables deduced therefrom. 
 
 Summary of the methods of adjustment. 
 
 Variation I {^Deficiency), in which the adjustment is made 
 by an additional annual instalment to he set aside during 
 the whole of the unexpired portion of the original repayment 
 period . 
 
 Method I. The deductive method, based upon all the factors 
 governing the fund. Statement XV. B. 
 
 [1} Calculate the amount ichich should stand to the credit of 
 the fujid; being the accumulation, at the calculated rate, 
 of the annual instalments which should have been set 
 aside. Calculation {XV) 2. £9932-744. 
 
 [2) Ascertain the value of the present investments representing 
 the fund, including in the case of a local authority, the 
 loan repaid by means of the sinking fund. £9463-00. 
 
 {3} The difference between the above amounts so found will 
 be the deficiency or surplus in the amount of the fund at 
 the time of making the enquiry. £469-744. 
 
 (4) Calculate the amount to which the value of the present 
 investments {as in 2) will accumulate at the end of the 
 original repayment period. 
 
 Calculation {XV) 4. £14799-71. 
 
 {5) Calculate the amount of the remaining original annual 
 sinking fund instalments at the end of the same period. 
 
 Calculation {XV) 5. £10960-62. 
 
 {6) Deduct the sum of the two amounts so obtained {£25760-33) 
 from the amount of the original loan. 
 
 (7) The difference represents the amount of loan which will 
 be unprovided for in the case of a deficiency, or provided 
 for in excess, in the case of a surplus, at the end of the 
 original repayment period {actually £7346o9). £734-67. 
 
156 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 [8) Calculate the additional annual sinking fund instalment 
 
 required to inovide this sum at the end of tJic jcpayme7it 
 period. Calculations {XV) 3 and {XVI) 1. £4o-594. 
 
 (9) Adjust the original sinJcing fund instalment hy adding to 
 
 it the annual instalment so obtained in (8) in the case of 
 a deficiency or by deducting it in the case of a surplus. 
 
 [10) Prepare a statement sJwiving the final repayment of the 
 loan by the operation of the sinJdng fund under the 
 amended conditions. Statement -YT'7. A. 
 
 {11) Prepare a pro forma account shoiving the amount icliich 
 should be in the fund at the end of each year of the 
 unexpired portion of the repayment period for future 
 reference. Pro forma Account, To. 2, Chapter XVI . 
 
 Method II. In which the original instalment does not enter 
 into the calculation. Statement XVI. A. 
 
 {1) Calculate the amount to which the sum which should be 
 in the fund, as found by Calculation {XV) 2, will 
 accumulate at the end of the repayment period. 
 
 Calculation (XVII) 2. £loo34-37o. 
 
 {2) Calculate, and deduct from the sum so found, the amount 
 to lohich the value of the present investments (£9,463) 
 will accumulate at the end of the repayment period. 
 
 Calculation {XV) 4. £14799-71. 
 
 {3) The difference will be the amount of original loan which 
 will be unprovided for in the case of a deficiency or 
 provided for in excess in the case of a surplus {as found 
 in No. 7, Method 1). {actually £7 34' 669.) £734 66-5. 
 
 {4) Adjust the or-iginal instalment, as in Xos. 8 and 9 in 
 Method 1, above. 
 
 {5) Prepare a statement showing the final repayment of the 
 loan by the operation of tJte sinJxing fund under the 
 amended conditions. Statement XVI. A. 
 
 {6) Prepare a pro forma account, as mentioned above. 
 
 Xo. 2, Chapter XVI. 
 
 Method III. The dieect method, based entirely upon the 
 present position of the fund. Statement X^ I. A. 
 
 {1) Calculate the amount ivhich sliould stand to the credit of 
 the fund, being the accumulation at the calculated rate, 
 of the annual instalments ulilch should have been set 
 aside. Calculation {XV) 2. £9932-74. 
 
A DEFICIENCY IN THE FUND 157 
 
 (2) Ascertain the value of the -present investments represent' 
 
 the fund, including, in the case of a local authority the 
 loan repaid by means of the sinking fund. £946S-00. 
 
 (3) The difference between the above amounts so found, will 
 
 be the deficiency or surplus in the amount of the fund at 
 the time of making the enquiry. Deficiency. £469-744. 
 
 (4) Calculate the annuity or annual instalment of which this 
 
 sum is the p^'csent value, depending upon the period over 
 which the correction shall extend. 
 
 Calculation {XV) 3. £45-594. 
 
 (J) Adjust the original sinJying fund instalment by adding 
 to it the instalment so obtained in the case of a deficiency 
 or by deducting it in the case of a surplus. 
 
 (6) Prepare a statement showing the final repayment of the 
 
 loan by the operation of the fund under the amended 
 conditions. Statement XVI. A. 
 
 (7) Prepare a pro forma account, as mentioned above. 
 
 ^ ^ No. 2, Chapter XVI. 
 
 Method IV. The anxial increment (balance or loan) method, 
 
 based upon the future annual increment and the present 
 
 position of th e fund . Statement XV I. ^ B. 
 
 This method will be fully discussed in Chapters XVI 
 
 and XXII . 
 
 {!) Ascertain the value of the present investments represent- 
 ing the fund, including in the case of a local authority, 
 the loan repaid by means of the fund, as already 
 described. £9463-00. 
 
 {2) Deduct the value so obtained from the amount of original 
 loan repayable at the end of the prescribed period. 
 
 £2649500. 
 
 (3) The remainder represents the balance of original loan to 
 be provided by the accumulation of the future or amended 
 annual increment, xi/uo^ uu. 
 
 {4) Calculate the annuity, or annual increment required to 
 provide the remainder so found, at the end of the pre- 
 scribed period at the future rate of accumulation. 
 
 Calculation [XVI) 9. £1057-033. 
 
158 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (<5) Deduct therefrom the future annual income to he received 
 from the i}resent investments. £331' 205. 
 
 (6) 'The remainder will he the am,ended annual instalment to 
 
 he provided out of revenue or rate instead of the original 
 instalment. £725'828. 
 
 The difference hetiveen the two instalments will he the 
 additional annual instalment found hy either of the fjre- 
 ceding methods. £4o'594 
 
 (7) Prepare a statement showing the final repayment of the 
 
 loan hy the operation of the sinking fund under the 
 amended conditions. Statement XVI. A. 
 
 (5) Prepare a pro forma account, as mentioned ahove. 
 
 No. 2, Chapter XVl. 
 
 Note, Calculations {XV) 1 and {XV) 2 are given in full 
 at the end of this Chapter. The remainder are given, in an 
 ahhreviated. form, in the Appendix. 
 
 Calculation of a Typical Sinking Fund. The previous 
 chapter contains a brief summary of the nature of the problems 
 likely to arise with regard to sinking funds of all kinds both in 
 connection with local authorities and commercial or financial 
 undertakings. There may be at times a combination of the 
 several variations, but in the first instance each problem will 
 be considered alone, deferring the examination of more compli- 
 cated cases. In order to do this in a consecutive manner, an 
 imaginary sinking fund Mill be adopted which will be used to 
 illustrate the whole of the examples to be afterwards considered, 
 because by this means only is it possible to apply the results 
 obtained in considering the simpler problems, to those of a more 
 complex nature. It will be assumed that the sinking fund is in 
 respect of a loan of £26,495, payable at the end of a period of 
 25 years, and that the instalments will be set aside annually, 
 and will accumulate by investment at 3| per cent, per annum. 
 The first step is to ascertain the annual instalment, and the 
 calculation will bo made upon standard calculation form No. 3x, 
 by the three methods described at the head of Cliapter XIII. 
 See Calculation (XV) 1. In this case, as in all others, the 
 method by formula is shown because although the methods by 
 table, including Thoman's, are much shorter, yet all the 
 published tables contain only a limited number of rates percent. 
 
A DEFICIENCY IN THE FUND 159 
 
 Furtlier, the tables are not of mucli assistance when it is 
 necessary to ascertain the rate per cent, or the number of years 
 with accuracy, which can only be done by the method by 
 formula, and then sometimes only approximately. Anyone 
 depending upon the published tables alone without a knowledge 
 of the method by formula is at a great disadvantage when the 
 book of published tables is absent. An acquaintance with the 
 methods by formula and a table of logs, is all that is required, 
 and a very small memorandum book will contain the whole of 
 the f ormulse mentioned in this work, which will be found at the 
 head of the various chapters, and also in Chapter X dealing 
 with the standard calculation forms prepared by the author. 
 There is a further advantage gained by a knowledge of the 
 formula and how they are arrived at, namely, a clear under- 
 standing of the principles underlying the theory of compound 
 interest which renders it an easy matter to make all calculations 
 by one or more alternative methods and thereby prove the 
 accuracy of the results obtained. In making a calculation 
 similar to the foregoing in which it is necessary to multiply or 
 divide a large principal sum by a figure containing 5 places of 
 decimals it is important to be extremely careful to obtain the 
 exact logs, or antilogs. by means of the tables of proportional 
 parts which will be found in the margin of the log. tables. 
 In the above instance, and in all other cases where it is required 
 to find the log of U^ , the log. of R should be carefully ascer- 
 tained, especially as to the last 3 or 4 figures. In order to 
 obtain the Nth power of R, the log. of E is multiplied by 25, 
 and any error in the last two figures will have a material effect 
 upon tiie result so found by multiplication. For this reason, 
 in Table Y. (A.), in Chapter Y, containing the values of (R) 
 for various rates per cent, the corresponding logs, of (E) are 
 given to eight places instead of seven as in the usual log. tables. 
 These logs, may be multiplied by the number of years and the 
 seventh figure adjusted, leaving out the eighth figure. The 
 logs, of RN are given in Thonian's tables for many rates per 
 cent., and even in cases where the method by formula is used it 
 may be taken direct from Thoman's tables with a saving of 
 time. The logs, of (RN-1) cannot be found from the tables, 
 but only by calculation, although the actual values of R^-l 
 may be found by deducting unity from the actual values given 
 in Table I. 
 
 Calculation (XY) 1 shows that an annual instalment of 
 £680-234 is required, and the pro forma account No. 1 at the 
 end of this chapter shows the normal accumulation of the fund. 
 
i6o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Method of Asceetaixixg the Position of a Sinking Fund 
 AT ANY Time. Having ascertained tliat an annual sinking fund 
 instalment of £680"234 is required to be set aside and accumu- 
 lated at 3i per cent, per annum for 25 years to repay a loan of 
 £26,495, at the end of tliat period, this information will now be 
 applied to an enquiry into the position of the fund at the end 
 of the 12th year. In an investigation of this nature occurring 
 in actual practice the annual instalment would of course be the 
 basis of the enquiry as it would have been in operation for a 
 period of 12 years. The first stage of the actual enquiry is to 
 ascertain the amount which should now stand to the credit of 
 the fund, on the assumption that the annual instalment has 
 been regularly set aside and has been promptly invested at the 
 end of each year, to yield 3| per cent. This amount, as shown 
 by Calculation (XY) 2, should be £9932-744. The next step 
 is to ascertain the actual amount standing to the credit of the 
 fund in the books of the local authority or private undertaking 
 and then to compare this amount with the actual value of the 
 investments representing the fund, including in the case of a 
 local authority the loans redeemed by means of the fund. In 
 the case of a commercial or financial undertaking there may not 
 be any obligation to invest the fund in specific outside securities, 
 and the amounts to be charged annually against the profits of 
 the concern may be allowed to remain uninvested and go to 
 swell either the floating or fixed assets. In such a case it may, 
 and will most probably, happen that the book-keeping has been 
 correct, and that the profit and loss account of the undertaking 
 has been each year charged with the proper annual instalment 
 and also with the proper annual interest upon the increasing 
 balances to the credit of the fund. Under such conditions 
 there will rarely be any necessity for enquiry seeing that the 
 fund will always stand in the books at the correct amount, and 
 any deficiency of assets representing the fund will not be 
 apparent, but will be merged in the general state of the assets 
 of the concern. But in the case of commercial and financial 
 undertakings, where there is an obligation to take the amount 
 of the annual instalments out of the floating assets of the 
 concern and invest the same in specific outside securities, the 
 case is exactly similar to the conditions imposed by Parliament 
 upon all local authorities, and may be treated on precisely 
 similar lines. The deficiency in both cases may arise in twO' 
 ways, even if the annual instalments have been regularly set 
 aside and the proper amount of money actually paid into the 
 sinking fund account. The first cause of the deficiency may be 
 
A DEFICIENCY IN THE FUND i6i 
 
 tiiat owing- to delay in investing the instalments, or owing to a 
 fall in tlie rate of income received from the investments, the 
 fund has not accumulated at the rate originally anticipated and 
 upon which the calculation of the original annual instalment 
 was based. The second cause of the deficiency may he that the 
 investments have depreciated in value and cannot now be 
 considered as representing the amount standing to the credit of 
 the fund, and there may have been in addition an actual loss 
 on realisation. But it is necessary to go further and ascertain 
 whether these investments will, or will not, as far as can be 
 judged, be of such a value at the end of the repayment period 
 that they will fulfil the original purpose of redeeming their 
 proportion of the loan. As already remarked in dealing with 
 the present investments in Chapter XIY, this is a very difficult 
 matter if the unexpired repayment period is a long one ; and it 
 is therefore the general practice to assume the future estimated 
 rate of accumulation on the low side, leaving any further 
 adjustment to be made at a later date when the conditions will 
 be better known. In the case of local authorities, as will be 
 seen by a perusal of Article 11 (2) of the County Stock Eegula- 
 tions of 1891, the Local Government Board are empowered to 
 take cognisance of such matters, and the same supervision may 
 be said to apply to the whole of the loans of local authorities. 
 In the case of commercial or financial undertakings the 
 adequacy or otherwise of these investments and of the fund 
 generally would be investigated by the auditors of the company 
 or by or on behalf of the loan holders. In the present chapter 
 it will be assumed that there is a deficiency in the sinking fund 
 of a definite amount arising from any of the above causes, but 
 for the present the problem will not be complicated in any way 
 by a variation in the period of repayment or in the future rates 
 per cent, of income or of accumulation. 
 
 The Yaeious Methods of Correcting a Deficiency in a 
 Sinking Fund. Having assumed that there is now an actual 
 ascertained deficiency in the sinking fund the various methods 
 will now be considered by Avhich it may be made good. In the 
 case of a local authority such a deficiency may often arise, but 
 generally it is of small amount due entirely to a reduction in 
 the rate of income on part of the fund uninvested and in the 
 bank. In practice this is met by charging any such deficiency 
 to the general revenite or rate account of each year. If the 
 deficiency in the case of a local authority is large, owing either 
 to serious omissions in previous years or to the accumulation of 
 
i62 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 many small annual deficiencies, the matter wonld be decided by 
 tlie Local Government Board or by Parliament when next 
 powers are soug'ht by special Act. This need not now be 
 discussed in detail because all the available methods will be 
 fully described later. Taking actual fig'ures, it will be assumed 
 that the above imaginary sinking fund (requiring an annual 
 •nstalment of £680-234 to repay £26,495 in 25 years at an 
 accumulation rate of 3| per cent.) amounts at the end of 
 
 12 years to £9463000 
 
 instead of the correct amount shown by Calculation 
 
 XY. (2) ' £9932-744 
 
 or a deficiency of ... £469-744 
 
 and that the conditions governing- the fund require that this 
 deficiency should be made good in some manner out of rate or 
 revenue, or out of profits in the case of a commercial or financial 
 undertaking. 
 
 There are several ways in wliich such a deficiency may be 
 corrected, namely: — 
 
 (a) By an immediate payment of the deficiency of £469'744 
 into the fund, which need not, however, be considered, 
 because, although the soundest financially, it has no 
 bearing upon the subject under review. 
 (6) By an additional annual sinking fund instalment to be 
 spread over the whole of the unexpired 13 years of the 
 original repayment period, in augmentation of the 
 original annual instalment of £680-234. 
 
 (Variation I, Chaptei XYI.) 
 
 (r) By an additional annual sinking fund instalment to be 
 
 spread over a shorter period than the full unexpired term 
 
 of 13 years. (Variation II, Chapter XVI.) 
 
 Having dismissed the correction by an immediate payment 
 
 into the fund, the last two alternatives will be applied to the 
 
 imaginary deficiency in order to ascertain the corrected annual 
 
 instalment consequent thereon. The above deficiency of £469-744 
 
 represents an amount of money ])ayable now, being the amount 
 
 (in the sense in Avhich it is used in Table III) of past annual 
 
 omissions accumulated at 3| per cent. It does not represent an 
 
 equivalent amount of the original loan, as shown later by 
 
 Calculation (XV) 6. Stated in terms of the original loan, it is 
 
 the present value at 3^ per cent, per annum of £734*659, part 
 
 of that loan, repayable in 13 years from the present time, the 
 
A DEFICIENCY IN THE FUND 163 
 
 repayment of winch lias not in the past been provided for as it 
 should have been. 
 
 The several methods of adjusting the deficiency given in 
 summary form at the head of this chapter will now be 
 described in detail, commencing with the direct method, III, 
 which is the simplest, after which Method I will be considered, 
 followed by Method II, leaving Method lY to be dealt with in 
 the following chapter. 
 
 Method III, The ^^resent deficiency of £469" 744, if not 
 complicated by other varying factors of time or rate per cent., 
 may be regarded in its simplest form as the present value of an 
 additional future annual instalment required to be set aside and 
 accumulated during the unexpired portion of the original re- 
 payment period in augmentation of the original instalment ; 
 and in the summary of methods at the head of this chapter this 
 is described as the direct method No, III. The additional annual 
 instalment is found by Calculation (X.\) 3, which shows that the 
 deficiency of £469 "744 is the present value of an additional 
 annual instalment of £45'594 to be set aside and accumulated 
 at 3| per cent, during the unexpired 13 years of the original 
 repayment period. The same result is obtained by Calculation 
 (XYI) 1 in the following chapter, which shows that the annual 
 instalment which will amount to £734' 659 of original loan at 
 the end of the period is also £45"594. The above amount 
 (£734'659) of original loan (by Calculation (XY) 6 in this 
 chapter) is shown to be the accumulated amount of the present 
 deficiency of £469' 744. 
 
 Method I. The investigation will now be continued on the 
 lines set out in Method I, at the head of this chapter. The 
 present position of the fund may be stated in terms of the 
 present value of each of the component parts of the fund, 
 namely, the present investments, the deficiency, and the 
 remaining original annual instalments. Seeing, however, that 
 the object of the fund is to repay the loan, and that other causes 
 of adjustment all affect the ultimate amount of the loan, the 
 effect will be more clearly shown by reducing the whole of the 
 factors in all cases to terms of loan, repayable at the end of the 
 prescribed period. This will require three calculations, as 
 follows: — (1) Ascertain the sum to which the present invest- 
 ments (£9,463) will accumulate at the end of the unexpired 
 period of 13 years at 3^ per cent. See Calculation (XY) 4. 
 (2) Add to this amount the sum to which the remaining original 
 
i64 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 annual instalments of £680'234 will amount at the end of the 
 same period, also accumulated at 3| per cent. See Calculation 
 
 (XV) 5. The sum of these two factors will represent the 
 reduced portion only of original loan which would be provided 
 if the present deficiency were not corrected. This total added 
 to the amount of £734'659 to which the present deficiency of 
 £469'744 would accumulate in 13 years at 3^ per cent, [see 
 Calculation (XY) 6] will make up the total amount of the 
 original loan. This last factor is the measure of the deficiency 
 expressed in terms of original loan, and may be treated in the 
 same way as the full amount of the loan, in Calculation (XV) 1, 
 to find the original annual instalment. The required annual 
 instalment so found, namely, £45'594, represents the additional 
 annual sum to be set aside and accumulated in augmentation 
 of the original annual instalment of £680'234. See Calculation 
 
 (XVI) 1. 
 
 The three calculations to show the equivalent amounts of 
 original loan will be made as before by formula and logs., and 
 also by Table III and Thoman's tables. There is really not 
 any necessity to prove the result by further calculation because 
 the above results, added together, should be equal to the total 
 amount of original loan to be provided at the end of the 
 prescribed period. 
 
 Method II. As already stated in the summary at the head 
 of this chapter, the sum of £T34"659, being the amount of loan 
 which will remain unprovided if the present deficiency be not 
 corrected, may also be ascertained by leaving out of account the 
 future original annual instalments (which, yer sc, are unaffected 
 by any present deficiency in the fund), and comparing the 
 ultimate accumulated amount, at the end of the period, of the 
 present investments of £9463' 00 with the accumulated amount 
 of the sum of £9932'744 which, as shown by Calculation 
 
 (XVII) 2, should have been in the fund if the original anticipa- 
 tions had been realised, as follows: — 
 
 The ultimate amount of the present iuA-estments of 
 <£9463-00, as shown by Calculation (XV) 4, 
 will be £14799-710 
 
 and, in Calculation (XVII) 2, it is shown that the 
 above sum of £9932' 744 will in 13 years at 
 ^ per cent, amount to £15534-375 
 
 a difference of (actually £734-659) £734-665 
 
A DEFICIENCY IN THE FUND 165 
 
 wliicli, as proved by Calculation (XY) 6, is the iiltiiuate amovmt 
 of loan represented by the present deficiency of £469" 744. The 
 following summary Avill make the matter clear: — 
 
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i66 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (XV) 3 sliows tliat the above deficiency of 
 £469' 744 is the equivalent present value of an annual instalment 
 of £45o94, which, accumulated for 13 years at 3^ per cent., 
 will, as shown by Calculation (XYI) 1 in the following chapter, 
 provide the above portion namely £734" 659, of the original loan. 
 
 The following Statement XY. B. shows the present position 
 of the fund, and also the amount of loan which will be provided 
 at the end of the unexpired portion of the repayment period, 
 namely, 13 years, by the accumulation of the amount now 
 standing to the credit of the fund to be increased by the 
 remaining original annual instalments, but without any 
 correction being made to adjust the present deficiency of 
 £469" 744. The final repayment of the loan after correcting the 
 present deficiency by an additional annual instalment will be 
 shown in Statement XYI. A. in the following chapter. 
 
A DEFICIENCY IN THE FUND 167 
 
 A Deficiency in the Fund, Statement XV. B. 
 
 The Deductive Method. No. i. 
 
 Showing the position of the fund at the end of the 12th year, 
 and the amount of loan which will be unprovided at the 
 end of the repayment period if the present deficiency be 
 allowed to accumulate, instead of being immediately 
 corrected by an additional annual instalment. 
 
 Present investments (at end of 12th year) £946300 
 
 • Amount thereof, accumulated for 13 years at 
 
 3i per cent. Calculation (XV) 4 £14799-71 
 
 Original annual instalment : — 
 
 Amount of £680'234 per annum, for 13 years at 
 
 3i per cent. Calculation (XV) 5 £10960-62 
 
 Provision already made will repay loan of ... ... £25760-33 
 
 Deficiency, being the balance of loan unprovided 
 for, represented by the present deficiency of 
 £469-744, accumulated for 13 years at 3^ per 
 cent, (actually £734-659) Calculation (XV) 6 734-67 
 
 Amount of original loan £2649500 
 
 Additional annual instalment required, 
 
 Calculations (XV) 3 and (XVI) 1 £45-594 
 
 Amended annual instalment, 
 
 Original annual instalment £680-234 
 
 Additional annual instalment 45-594 
 
 £725-828 
 
 The final repayment of the loan by the operation of the 
 sinking fund after making the above adjustment in the annual 
 instalment is shown in Statement XVI. A., and by the pro 
 forma account, No. 2, Chapter XVI. 
 
i68 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Pro forma Sinking Fund Account, No. 1. 
 
 Loan of £26,495, re payable at the end of 2-5 years. 
 
 Annual Instalment. Calculation (XY) 1. £680-234. 
 Eate of Accumulation, 3| per cent. 
 
 Sliowino; the normal accumulation of tlie fund. 
 
 Year. 
 
 Amount in 
 
 tlie fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received from 
 
 investments 
 
 3^- per cen£. 
 
 .\nnual 
 sinkiaig fund 
 Instalment. 
 
 A)i;ount in 
 the fund 
 
 at 
 end of year. 
 
 Year 
 
 1 
 
 Nil 
 
 Nil 
 
 680-234 
 
 680-234 
 
 1 
 
 2 
 
 680-234 
 
 23-808 
 
 680-234 
 
 1384-276 
 
 • 2 
 
 3 
 
 1384-276 
 
 48-450 
 
 680-234 
 
 2112-960 
 
 3 
 
 4 
 
 2112-960 
 
 73-954 
 
 680-234 
 
 2867-148 
 
 4 
 
 5 
 
 2867-148 
 
 100-350 
 
 680-234 
 
 3647-732 
 
 5 
 
 6 
 
 3647-732 
 
 127-671 
 
 680-234 
 
 4455-637 
 
 6 
 
 
 4455-637 
 
 155-947 
 
 680-234 
 
 5291-818 
 
 7 
 
 8 
 
 5291-818 
 
 185-213 
 
 680-234 
 
 6157-265 
 
 8 
 
 9 
 
 6157-265 
 
 215-504 
 
 680-234 
 
 7053-003 
 
 9 
 
 10 
 
 7053-003 
 
 246-853 
 
 680-234 
 
 7980-090 
 
 10 
 
 11 
 
 7980-090 
 
 279-302 
 
 680-234 
 
 8939-626 
 
 11 
 
 12 
 
 8939-626 
 
 312-884 
 
 680-234 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 347-648 
 
 680-234 
 
 10960-626 
 
 13 
 
 14 
 
 10960-626 
 
 383-622 
 
 680-234 
 
 12024-482 
 
 14 
 
 15 
 
 12024-482 
 
 420-857 
 
 680-234 
 
 13125-573 
 
 15 
 
 16 
 
 13125-573 
 
 459-395 
 
 680-234 
 
 14265-202 
 
 16 
 
 17 
 
 14265-202 
 
 499-282 
 
 680-234 
 
 15444-720 
 
 17 
 
 18 
 
 15444-720 
 
 540-565 
 
 680-234 
 
 16665-519 
 
 18 
 
 19 
 
 16665-519 
 
 583-293 
 
 680-234 
 
 17929-046 
 
 19 
 
 20 
 
 17929-046 
 
 627-517 
 
 680-234 
 
 19236-797 
 
 20 
 
 21 
 
 19236-797 
 
 673-268 
 
 680-234 
 
 20590-299 
 
 21 
 
 22 
 
 20590-299 
 
 720-661 
 
 680-234 
 
 21991-194 
 
 22 
 
 23 
 
 21991-194 
 
 769-692 
 
 620-234 
 
 23441-120 
 
 23 
 
 24 
 
 23441-120 
 
 820-439 
 
 680-2:54 
 
 24941-793 
 
 24 
 
 25 
 
 24941-793 
 
 872-973 
 
 680-234 
 
 26495000 
 
 25 
 
A DEFICIENCY IN THE FUND 
 
 169 
 
 Calculation (XV) 1. 
 
 Standard Calculation Form, No. 3x. 
 
 To find the annual sinking fund instalment to be provided out 
 of revenue or rate to repay the loan under the original 
 conditions laid down at the time of borrowing. 
 
 Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 as a sinking fund at 3^ per cent, per annum to provide 
 £26,495 in 25 years. _____^ 
 
 (A) By Formula. Ky = ^i (^ ^p^-^^Rule 1, Chapter XIII. 
 Lost. Ratio 
 
 Log. 
 RN-1 
 
 Multiply Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 Log. of this is 
 
 Log. Amount of Loan 
 add Log. r 
 
 deduct Log. (R^ - 1) above 
 
 R 
 
 N 
 
 1-035 
 25 
 
 0-0149403 
 25 
 
 RN 
 
 (l-035)-^5 
 
 0-3735087 
 
 RN 
 
 -1 
 
 2-36324 
 1- 
 
 
 RN-1 
 
 1-36324 
 
 0-1345738 
 
 M 
 
 T 
 
 26,495 
 0035 
 
 4-4231639 
 2-5440680 
 
 Mr 
 
 RN-1 
 
 
 2-9672319 
 0-1345738 
 
 A.V 
 
 
 2-8326581 
 
 Required annual ii)stalment, £680-2336 
 
 M 
 
 (B) By Table III. Ai/- RN - 1 Rule 2, Chapter XIII. 
 
 Log . Amount of Loan 
 Table III. 25 years, 3^ per cent. 
 Amount of £1 per annum 
 ded^ict Log. 
 
 M 26,495 
 
 RN-1 38-94986 
 r 
 
 4-4231639 
 1-5905058 
 
 Ay 
 
 2-8326581 
 
 Required annual instalment, £680-2336 
 
 (C) By Thoman's Table. Ay = M (^ \ Rule 3, Chapter XIII. 
 3^ per cent., 25 years. ^^ 
 
 Log. Amount of Loan 
 add Loff. a^ 
 
 deduct Log. RN in 
 Table + 10 
 
 M 
 
 26,495 
 
 4-4231639 
 8-7830029 
 
 M a" 
 
 RN 
 
 13-2061668 
 10-3735087 
 
 A.v 
 
 2-8326581 
 
 Required annual instalment, £680-2336 
 
lyo REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Calculation (XV) 2. 
 
 Standard Calculation Form, -A o. 3. 
 
 To find the amount whicli sliould stand to the credit of a 
 sinking- fund at any time. 
 
 Required the amount which shoukl stand to the credit of a 
 sinking- fund representing- the accumulation of an annual 
 instalment of £680"2o4 for 12 years at 3^ per cent. 
 
 (A) By Formula. M = A2/(^^~^) Rule 1, Chapter YI. 
 
 Log 
 
 RN-1 
 
 Log. Ratio 
 
 Ji/uUiph/ Log. R by 
 
 Convert Log. 
 
 to ordinary number 
 deduct unity 
 
 Log. of this is 
 
 Log. Annuity 
 add Loff. R^ — 1 above 
 
 deduct Log, r 
 
 1035 
 12 
 
 00149403 
 12 
 
 RN 
 
 (1-035)12 0-1792842 
 
 RN 
 
 -1 
 
 1-51107 
 1- 
 
 RN-1 0-51107 
 
 1-7084792 
 
 RN-1 
 
 680-234 
 
 ^-8326581 
 1-7084792 
 
 AyCR^-1] 
 
 0-035 
 
 2^-5411373 
 2-5440680 
 
 M 
 
 3-9970693 
 
 Required amount, £9932- 744. 
 
 (B.l 
 
 By Table III. M = Ay ( — ^_ ^ ) Rule 2, Chapter YI. 
 
 Table III. 12 years, 3| per cent. 
 Amount of £1 per annum 
 add Log. Annuity 
 
 RN-1 14-60196 
 A^ 680-234 
 
 1-1644112 
 2-8326581 
 
 M 
 
 3-9970693 
 
 Required amount, £9932-744. 
 
 (Cj By Thoman's Table. M = A?/('^ ^ Rule 3, Chapter YI. 
 3i per cent., 12 years. 
 
 Log. Annuity 
 add Log. RN in 
 
 Table +10 
 
 deduct Log". «" 
 
 Ay 
 
 RN 
 
 680 
 
 234 
 
 2-8326581 
 10-1792842 
 
 AyRN 
 
 
 
 130119423 
 9-0148730 
 
 M 
 
 
 
 3-9970693 
 
 Required amount, £9932-744. 
 
THE CORRECTION OF A DEFICIENCY 171 
 
 CHAPTEE XYI. 
 
 SINKING FUND PROBLEMS RELATING TO 
 THE AMOUNT IN THE FUND. 
 
 The correction of a deficiency in the fund. 
 
 Variation I. 
 
 By an additional annual instalment to be set aside 
 during the whole of the unexpired portion of the 
 repayment period. statement xyi. a. 
 
 Variation II. 
 
 By an ADDITIONAL ANNUAL INSTALMENT TO BE SET ASIDE 
 
 during the earlier part only of the unexpired portion 
 of the repayment period. statement xvi. c. 
 
 Summary of the methods of adjusting a deficiency. The 
 several methods described. tlie annual increment 
 
 (balance of loan) METHOD. STATEMENT SHOWING THE 
 
 FINAL REPAYMENT OF THE LOAN BY THE OPERATION OF THE 
 
 AMENDED ANNUAL INSTALMENT, IN EACH OF THE ABOVE 
 VARIATIONS. 
 
 Summary of the methods of adjustment. 
 
 Variation I (Deficiency), in which the adjustment is made 
 by an additional annual instalment to he set aside during the 
 whole of the U7iexi)ired portion of the original repayment period. 
 
 Statement XVI. A. 
 
 (-7) Ascertain the amount of the present deficiency and 
 calculate the equivalent amount of original loan by one 
 of the methods described in Chapter XV . 
 
 Calculation {XV) 6. £734-659. 
 
 (2) Calctilate the additional annual sinking fund instalTuent 
 
 to be set aside and accuTnulated for the whole of the 
 
 unexpired portion of the original repayment period to 
 
 provide the above equivalent amount of original Loan. 
 
 Calculations {XV) 3 and (XVT) 1. £45-594. 
 
172 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (J) The additional annual instalment so ascertained added to 
 the original annual instalment will give the augviented 
 annual instalinent to he set aside during the whole of 
 the unex'pired 'portion of the repayment period. 
 
 (4) Prepare a statement showing the final repayment of the 
 
 loan by the operation of the fund under the amended 
 conditions. Statement XVI. A. 
 
 id) Prepare a pro forma account showing the amount which 
 should he in the fund at the end of each year of the 
 unexpired portion of the repayment period for after 
 reference . Pro forma Account, No. 2. 
 
 Variation II (Deficiency), in whicli the adjustment is made 
 hy an additional annual instalment (£104'039) to he set aside 
 during part only (-5 years) of the unexpired portion [13 years) 
 of the original repayTnent period {25 years). 
 
 Statement XYI. C. 
 
 Note. In order to Tnake tJie following summary perfectly 
 clear it contains (^in hracl-ets) the results ascertained in the 
 example afterwards tcorked out in detail. The anriual incre- 
 ment (ratio) method, previously referred to in Chapter XIV, 
 cannot he applied to cases in loliich the amended instalment is 
 not spread equally over the whole of the period. 
 
 (i) Ascertain the a'mount of the present deficiency {£469'744) 
 and calculate the equivalent amoxint {£734'659) of 
 original loan, as described in Chapter XV . 
 
 Calculation {XV) 6. 
 
 (2) Divide the unexpired portion {13 years) of the original 
 repayment period {25 years) into two parts, as follows : — 
 
 1st portion {5 years), during udiich the additional 
 annual instalment is required to he set aside. 
 
 2nd portion {8 years), during uliich the additional 
 annual instalment is not required, to he set aside, hut only 
 the annual instalment as originally ascertained. 
 
 (5) Calculate the present value {£557'908) of the above 
 
 equivalent amount {£734'659) of the original loan, as if 
 it were due at the end of a number of years {8) equal to 
 the second portion of the unexpired repayment period 
 {13 years). Calculation {XVI) 3. 
 
THE CORRECTION OF A DEFICIENCY 173 
 
 (4) Calculate the additional annual instalment (£104'039) to 
 
 be set aside and accumulated for a numher of years (5) in 
 the first 'portion of the unexpired repayment period' [13 
 years) to provide the present value {£5o7'90S) so found, as 
 above. Calculation {XVI) 4. 
 
 (5) The additional annual instalment so found {£104'039) 
 
 added to the original annual instalment {£680'234) icill 
 give the -augmented annual instalment [£784'273) to be 
 set aside during the first portion (5 years) of theunea-pircd 
 repayincnt period (13 years). 
 
 (6) The original annual instalment [£680'234) tvill continue 
 
 to be set aside and accumulated during the second portion 
 [8 years) of the unea-pired repayment period' (13 years). 
 
 (7) Prepare a statement shoicing the final repayment of the 
 
 loan by the operation of the fund binder the amended 
 conditions. Statement XVI. C. 
 
 (8) Prepare a pro forma account showing the amount which 
 
 should be in the fund at the end of each year of the 
 unexpired period for reference in after years. 
 
 Pro forma Account, Xo. 3. 
 
 Xote. The calculations in this and subsequent chapters itill 
 be found in the Appendix, but each calculation will be shown 
 by only one of the three methods given in the stcmdard forms. 
 
 Variation I. The correction of a deficiency in tlie fund by an 
 additional annual instalment to be set aside during- tlie 
 whole of the unexpired portion of the repayment period. 
 
 Statement XAI. A. 
 
 In Chapter XY, the factors relating to a deficiency in a 
 sinking fund have been fully discussed, and several methods 
 described by which to ascertain the resulting additional annual 
 instalment to be spread equally over the whole of the unexpired 
 portion of the repayment period. Two alternative methods 
 have been pointed out by which the deficiency may be corrected, 
 both of which agree in providing an additional annual instal- 
 ment, but differ as to the number of years over which such 
 increased contributions shall be spread. Sound finance demands 
 that the error should be put right by an immediate payment 
 of the deficiency into the fund, or that the increased annual 
 contribution should be spread over a shorter term than the full 
 unexpired portion of the original repayment period, but th:-- 
 
174 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 circumstances of individual cases may render it more equitable, 
 or perhaps more convenient, tliat the adjustment should be 
 spread over the longest possible period. 
 
 The present deficiency of £469' 744 if immediately paid into 
 the fund and accumulated until the end of the period, will then 
 provide £734"659 of original loan which would otherwise have 
 been unprovided for. The additional annual instalment of 
 £45"594, to be set aside and added to the fund during the whole 
 of the unexpired period, has already been ascertained by 
 Calculation (XY) 3, and it will now be proved by a further 
 Calculation (XYI) 1 upon the authors standard calculation 
 form No. 3x, based upon Table III, which is the usual method 
 of finding the sinking fund instalment. This and other 
 calculations subsequently referred to will l)e found in the 
 appendix. 
 
 Having ascertained the required additional annual instal- 
 ment, it is now possible to review the operation of the fund so 
 amended in order to show the final repayment of the loan by the 
 following Statement XYI, A., which will apply equally to all 
 similar cases of adjustment due to a deficiency in the fund, 
 irrespective of the method by which the additional annua] 
 instalment is ascertained provided that such additional annual 
 instalment be spread equally over the whole of the unexpired 
 portion of the repayment period. 
 
 The following Statement XYI. A. also shows that the 
 present investments of £9,463 will, if accumulated at 3^ per 
 cent, until the end of the period, then provide for the repayment 
 of £14799-71 of original loan. Before making the above 
 correction the balance of the loan unprovided for was repre- 
 sented by : — 
 
 The remaining original annual instalments of 
 £680'234 and their accumulations 
 
 Calculation (XY) 5 £10960-62 
 The deficiency at the end of the 12th year £469-74 
 and the loss of accumulated interest 
 
 ca used therebv 26493 
 
 . £734-67 
 
 Balance of Loan £11695-29 
 
 After making tlio above adjustment this amount will be provided 
 by the accumulation of the augmented annual instalment of 
 £725-828, as shown by Calcuhition (XYI) 2. 
 
THE CORRECTION OF A DEFICIENCY 175 
 
 The Annual Increment (balance of loan) Method. The 
 annual increment has been fully described in Chapter XIV, 
 where it is shown that it may be used to simplify the majority 
 of the adjustments in a sinking fund, rendered necessary by any 
 variation from the original conditions as to the repayment of 
 the loan. There is, however, one limitation, namely, that any 
 variation in the rate of income to be received upon the present 
 investments, or in the rate of accumulation, must apply equally 
 to the whole of the future period of repayment, which, however, 
 may be increased or reduced. It is also necessary that any 
 increased or reduced annual instalment, consequent upon any 
 such variation in the original conditions, shall be spread equally 
 over the whole of the unexpired portion of the repayment period. 
 For this reason, therefore, the method has been applied in 
 Statement XVI. B. to the foregoing example (Variation I) in 
 which the deficiency of £469' 744 is made good by an additional 
 annual instalment of £45"594, to be spread equally over the 
 whole of the unexpired portion of the repayment period, but the 
 method will not apply to the example following, namely, 
 Variation II, in which the additional annual instalment is 
 required to be spread over the earlier years only of such un- 
 expired term. If this method be applied to the latter example 
 the result would be only the equated annual instalment, which, 
 however interesting from a theoretical point of view, would not 
 be of any practical use under the actual conditions. An 
 example of an equated annuity is given and fully described in 
 Chapter XXVII. 
 
 This method of making the adjustment of a sinking fund by 
 means of the annual increment is practically the same as that 
 adopted in the case of local authorities, where the whole of the 
 annual instalments, as and when set aside, are immediately 
 applied in the actual repayment of debt. Section 234 (5) of the 
 Public Health Act, 1875, provides that where any part of the 
 fund is so applied there shall be paid into the fund and charged 
 to the rate account the interest which would have been earned by 
 the part of the fund so applied. If it be assumed that the 
 whole of the fund is so applied in repayment of the debt, and the 
 rate of interest payable upon the loan is the same as the rate of 
 accumulation of the fund, the amount charged annually to rate 
 account in respect of interest and redemption charges, is the 
 annual increment of the fund, using the term in the sense here 
 applied to it. 
 
176 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Deficiency in the Fund, Statement XVI, A. 
 
 Showing the final repayment of the Loan, by the operation 
 of the sinking fund, after making the adjustment in the 
 annual instalment, consequent ujion a deficiency in the 
 amount which should stand to the credit of the fund. 
 
 Variation I (Deficiency), in which the additional annual 
 instalment is set aside during the whole of the unexpired 
 portion of the repayment period. 
 
 Present investments (at end of 12thyear),£946300 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Amount thereof, accumulated for 13 years at 
 
 3i per cent. Calculation (XV) 4 £14799-71 
 
 Amended annual instalment : — 
 
 Original annual instalment £680'234 
 
 Additional annual instalment 
 
 Calculation (XVI) 1 45-594 
 
 £725-828 
 
 Amount thereof in 13 years at 3| per cent. 
 
 Calculation (XVI) 2 £11695-29 
 
 Amount of original loan £26495-00 
 
THE CORRECTION OF A DEFICIENCY 177 
 
 A Deficiency in the Fund. Statement XVI, B. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find the amended annual sinking fund instalment consequent 
 upon a deficiency in the amount which should stand to 
 the credit of the fund. 
 
 Variation I (Deficiency), in which the additional annual 
 instalment is set aside during the whole of the unexpired 
 portion of the repayment period. 
 
 Amount of originalloan (25 years) £2649500 
 
 deduct amount in the fund at the end of the 
 
 12th year £9463-00 
 
 Balance of loan £17032-00 
 
 Amended annual increment to be added to the fund 
 and accumulated at 3^ per cent., to provide 
 this amount at the end of 13 years. 
 
 Calculation (XYI) 9 £1057-033 
 
 deduct income to be received from the present 
 
 investments, £9,463, at 3^ per cent. £331-205 
 
 Amended annual instalment, feemg' £725-828 
 
 Original annual instalment ... £680-234 
 
 Additional annual instalment £45-594 
 
 £725-828 
 
 The rule relating to this method is stated at the head of 
 Chapter XXII. 
 
178 
 
 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Pro forma Sinking Fund Account, No. 2. 
 
 A Deficiency in the Fund. (Variation I.) 
 Loan of £26,495, repayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by tlie operation of 
 tlie amended annual instalment of £725'828, to be set aside 
 during the whole of the unexpired period of repayment. 
 
 Statement XVI. A. E-ate of accumulation, 3^ per cent. 
 
 Year. 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received from 
 
 investments 
 
 3i per cent. 
 
 Annual 
 sinking fund 
 instalment. 
 
 Amount in 
 
 the fund 
 
 at end 
 
 of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 3 
 
 4 
 
 The amount in the 
 
 fund at the end of the 
 
 4 
 
 5 
 
 12th year, £9,463, is an assumed amount, 
 
 5 
 
 6 
 
 and is 
 
 equivalent 
 
 to setting 
 
 aside an 
 
 6 
 
 7 
 
 annual instalment of £648'064, 
 
 as shown 
 
 7 
 
 8 
 
 by Calculation (XVI) 10, instead of the 
 
 8 
 
 9 
 
 correct annual instalment of £680"234. 
 
 9 
 
 10 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 9463000 
 
 12 
 
 13 
 
 9463000 
 
 331-205 
 
 725-828 
 
 10520-033 
 
 13 
 
 14 
 
 10520033 
 
 368-201 
 
 725-828 
 
 11614-062 
 
 14 
 
 15 
 
 11614062 
 
 406-492 
 
 725-828 
 
 12746-382 
 
 15 
 
 16 
 
 12746-382 
 
 446-123 
 
 725-828 
 
 13918-333 
 
 16 
 
 17 
 
 13918-333 
 
 487-142 
 
 725-828 
 
 15131-303 
 
 17 
 
 18 
 
 15131-303 
 
 529-596 
 
 725-828 
 
 16386-727 
 
 18 
 
 19 
 
 16386-727 
 
 573-535 
 
 725-828 
 
 17686090 
 
 19 
 
 20 
 
 17686090 
 
 619-013 
 
 725-828 
 
 19030-931 
 
 20 
 
 21 
 
 19030-931 
 
 666-083 
 
 725-828 
 
 20442-842 
 
 21 
 
 22 
 
 20442-842 
 
 714-799 
 
 725-828 
 
 21863-469 
 
 22 
 
 23 
 
 21863-469 
 
 765-221 
 
 725-828 
 
 23354-518 
 
 23 
 
 24 
 
 23354-518 
 
 817-408 
 
 725-828 
 
 24897-754 
 
 24 
 
 25 
 
 24897-754 
 
 871-418 
 
 725-828 
 
 26495-000 
 
 25 
 
THE CORRECTION OF A DEFICIENCY 179 
 
 Yariation II. The correction of a deficiency in the fund by 
 an additional annual instalment, to be set aside during 
 the earlier years only of the unexpired portion of the 
 repayment period. Statement XYI. C. 
 
 The correction of the deficiency in this manner is more 
 complicated than by spreading the additional annual instalment 
 equally over the whole of the unexpired portion of the repayment 
 period, but is not at all difficult. The factors immediately 
 concerned are (1) the present deficiency of £469"T44; (2) the 
 amount of original loan £734'659, represented by such deficiency, 
 and (3) the original annual instalment of £680'234. 
 
 In the present example it will be assumed that the additional 
 annual instalment is required to be of such increased amount (as 
 compared with the additional annual instalment of £45"594 to 
 be spread over the whole of the unexpired period) that it will 
 be sufficient to make up the present deficiency if set aside for 
 5 years only, instead of for 13 years. Under this alternative 
 method the unexpired period of 13 years is divided into two 
 parts. During the first five years the additional annual 
 instalment will be set aside and accumulated at 3^ per cent, in 
 augmentation of the original annual instalment. At the end 
 of the five years this additional annual instalment will cease, 
 and will then have amounted to a sum which will continue to 
 accumulate at compound interest for a further eight years. 
 The accumulated amount of the additional annual instalment at 
 the end of five years, should, at the end of the remaining eight 
 years, amount to the balance (£734'659) of loan not otherwise 
 provided for. The adjustment may be made by direct calcula- 
 tion, and may also be made by steps. A similar method by 
 step has been adopted Avhen dealing with a variation in the 
 future rate of income to be received upon the present invest- 
 ments when it is known in advance that such a variation will 
 take effect at a definite future date during the unexpired portion 
 of the redemption period, as explained in Chapter XXYII. In 
 order to determine the additional annual instalment to be set 
 aside and accumulated for the first period of five years, it is 
 first necessary to ascertain the sum to which it is required to 
 accumulate at the end of five years, which latter sum will in its 
 turn accumulate without further addition for a further period 
 of eight years. At the end of the unexpired period of 13 years 
 it is necessary to provide £734" 659, and the first step is to 
 ascertain the sum which, if accumulated at 3^ per cent, for 
 eight years, will amount to £734' 659 ; in other words, to find the 
 present value of £734"659 under the above conditions, namely, 
 
iSo REPAYMENT OF LOCAL AND OTHER LOANS 
 
 £557-908. Cak'ulatiou (XYI)3. The next step is to ascertain 
 the annual instalment which will amount to £557'908 if set 
 aside and accumulated at 3| per cent, for five years. 
 
 This is a similar problem to the previous one dealing with 
 the present deficiency of £469744, where it was required to find 
 the annual instalment to amount to £734"659, Calculation (XYI) 
 1, and also similar to Calculation (XY) 1 required to find the 
 original annual instalment of £680'2:')4. Calculation (XYI) 4 
 shows that the equal annual instalment to provide £557 '908 at 
 the end of five years at 3^ per cent, is £104-039. The method 
 of complying with the above conditions has noAv been ascertained. 
 It has been found by Calculation (XYI) 4 that an annual 
 instalment of £104-039 set aside for five years and accumulated 
 at 3^ per cent, will at the end of that time amount to £557-908, 
 and it has been found by Calculation (XYI) 3 that this sum of 
 £557-908, accumulated at 3^ per cent, for eight years, will 
 amount to £734-659, which is the portion of the original loan 
 not otherwise provided for, owing to the present deficiency of 
 £469-744. 
 
 The sinking fund, as amended by the results of the foregoing 
 calculations will now consist of : — 
 
 A present credit to the fund, represented by invest- 
 ments valued at ^ £9463-000 
 
 An augmented annual instalment for 5 years made 
 up as follows : 
 
 Original instalment £680-234 
 
 Additional instalment for 5 years 104-039 
 
 ■ £784-273 
 
 The original annual instalment to be continued 
 
 for a further 8 vears of £680-234 
 
 And the above provision accumulated at 3^ per cent., as 
 originally caktriated, will at the end of the prescribed period of 
 repayment, namely, 25 years, be sufficient to provide the full 
 amount of the original loan of £26,495. 
 
 In order to complete the argument it is necessary to show 
 the position of the fund at the end of the 17th year when the 
 additional annual instalment of £104039 will cease and to 
 continue the accumulation of the fund from that time until the 
 end of the original term of 25 years. During the second period 
 of eight years, as previously mentioned, the original instalment 
 
THE CORRECTION OF A DEFICIENCY i8i 
 
 uf £680'234 only will continue to be set aside and added to the 
 fund. The following Statement XYI. C. shows the final 
 repayment of the loan by the operation of the fund after making 
 the above adjustment. 
 
 In the foregoing statement a break has been made at the end 
 of the 17th year, being the end of the five years during which 
 the corrective instalment of £104089 is required to be set aside. 
 The calculation might have been simplified by ascertaining, 
 in the direct manner shown in Statement XYI. D.l, the 
 amount of loan Avhich will be provided by the accumulation at 
 the end of 13 years of the instalment of £104'0-j9 to be set aside 
 for five years only. This direct method by step is f ulh' explained 
 in Chapter XXYII, Statement C, where it is applied to find 
 the amount of loan which Mnll be provided by the accumulation 
 of the income from the present investments, such income being 
 at varying known rates per cent, during the, unexpired period. 
 (The calculation might also have been made in terms of the 
 amended annual instalment of c£T84'2T3). In conclusion, a 
 further Statement XYI. D".2, has been prepared, showing the 
 final repayment of the loan, which should be compared with 
 Statement XYI. C, in order to show the simplification of the 
 proof by the method by stej?. 
 
i82 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Deficiency in the Fund. Statement XVI. C. 
 
 Showing the final repayment of the loan, by the operation of 
 the sinking fund after making the adjustment in the 
 annual instalment consequent upon a deficiency in the 
 amount which should stand to the credit of the fund. 
 
 Variation II (Deficiency), in which the additional annual 
 instalment is set aside during the earlier part only of the 
 unexpired portion of the repayment period. 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments (at end of 12th year), £946300 
 
 Amount thereof, accumulated for 5 years at 
 
 31 per cent. Calculation (XYI) 5 £11239-07 
 
 Amended annual instalment : — 
 
 Original £680234 
 
 Additional 104039 
 
 £784-273 
 
 Amount thereof, accumulated for 5 years at 
 
 3i per cent. Calculation (XYI) 6 £4205-64 
 
 Amount in the fund, at end of 17th year £15444-71 
 
 Amount thereof, accumulated for 8 years at 
 
 31 per cent. Calculation (XVI) 7 £20337-74 
 
 Original annual instalment (resumed) :— 
 
 Amount of £680234 per annum, accumulated 
 for 8 years at 3| per cent. 
 
 Calculation (XVI) 8 £6157-26 
 
 Amount of original loan £26495-00 
 
THE CORRECTION OF A DEFICIENCY 183 
 
 A Deficiency in the Fund. Statement XVI. D(l). 
 
 The Amount of {the Amourit of £1 fer Annum) Method by Step, 
 by Thoinan's Tables. 
 
 To find the accumulated amount of an additional annual instal- 
 ment, or other annuity, to be set aside and added to the 
 sinking fund for a limited period of years ; and at the end of 
 that period the accumulated amount thereof to continue to 
 accumulate for a further specified period. The rate of 
 accumulation in both periods may be the same, or be at 
 different rates per cent. 
 
 Eequired the amount of an additional annual instalment of 
 £104-039, to be set aside for a period of 5 years, and 
 accumulated at 3^ per cent. At the end of 5 years the 
 annual instalment ceases, but the sum to which it has then 
 amounted continues to accumulate for a further period of 
 8 years, also at 3^ per cent. 
 
 First period, 5 years. Second period, 8 years. 
 
 Log. instalment A^/ 104-039 20171984 
 
 acZfZ: Log. RN3i per cent. 5 years EN 00747017 
 
 Log. RN 3i per cent. 8 years RN 0-1195228 
 
 2-2114229 
 
 add 10 to the log. 12-2114229 
 
 ^efZwrt : Log-fl*^, 3^percent. 5 years a^ 93453372 
 
 M 2-8660857 
 
 which is the log. of the required future amount, 
 
 namely ^'34-659 
 
 Note. This method may be inverted to find the additional 
 annual instalment in the first instance instead of as described 
 in the text. See Statement XXXIV. G. 
 
iS4 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Deficiency in the Fund. Statement XVI. D (2). 
 
 Showing the final repayment of the loan, by tlie operation of 
 tlie sinking fund after making the adjustment in tlie annual 
 instalment, consequent upon a deficiency in the amount 
 which should stand to the credit of the fund. 
 
 Variation II (Deficiency), in which the additional annual 
 instalment is set aside during the earlier part only of the 
 unexpired portion of the repayment period. 
 
 An alternative method to Statement XVI. C, based upon the 
 method by step. 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments (at end of 12th year),£946300 
 
 Amount thereof, accumulated for 13 years at 
 
 3i per cent. Calculation (XT) 4 £14799-71 
 
 Original annual instalment £680234 
 
 Amount thereof, accumulated for 13 years at 
 
 ^ per cent. Calculation (XY) 5 £10960-62 
 
 Additional annual instalment £104039 
 
 to be set aside for 5 years only, and accumu- 
 lated for a further 8 years at 3| per cent 
 " Method by step " Calculation (XVI) D.l £734'67 
 
 Amount of original loan £2649500 
 
THE CORRECTION OF A DEFICIENCY 
 
 185 
 
 Pro forma Sinking Fund Account, No. 3, 
 
 A Deficiency in tlie Fund. (Variation II.) 
 
 Loan of £26,496, reijayahJc at the end of 25 years. 
 
 Showing the final eepayment of the loan, by tlie operation of 
 the amended annual instalment of £784-273, to be set aside 
 during tbe first 5 years only of the unexpired period of 
 repayment. 
 
 Statement XYI. C. 
 
 Rate of accumulation, 3| per cent. 
 
 Year. 
 1 
 
 Amount in 
 the fund 
 
 at beginning 
 of year. 
 
 Income 
 
 received from 
 
 investments 
 
 3J per cent. 
 
 Amount in 
 Annual tlie fund 
 sinking fund at end 
 instalment. of year. 
 
 Year. 
 1 
 
 2 
 
 
 
 
 2 
 
 3 
 
 
 
 
 3 
 
 4 
 
 The amount in the i 
 
 :und at the end of the 
 
 4 
 
 5 
 
 12th year, £9,463, is an assumed amount, 
 
 5 
 
 6 
 
 and is 
 
 equivalent 
 
 to setting aside an 
 
 6 
 
 7 
 
 annual instalment c 
 
 )f £648-064, as shown 
 
 7 
 
 8 
 
 by Calculation (XVI) 10, instead of the 
 
 8 
 
 9 
 
 correct 
 
 annual inst; 
 
 alnient of £680-234. 
 
 9 
 
 10 
 
 
 
 
 10 
 
 11 
 
 
 
 
 11 
 
 12 
 
 
 
 9463-000 
 
 12 
 
 13 
 
 9463-000 
 
 331-205 
 
 784-273 10578-478 
 
 13 
 
 14 
 
 10578-478 
 
 370-247 
 
 784-273 11732-998 
 
 14 
 
 15 
 
 11732-998 
 
 410-655 
 
 784-273 12927-926 
 
 15 
 
 16 
 
 12927-926 
 
 452-477 
 
 784-273 14164-676 
 
 16 
 
 17 
 
 14164-676 
 
 495-761 
 
 784-273 15444-710 
 
 17 
 
 18 
 
 15444-710 
 
 540-567 
 
 680-234 16665-511 
 
 18 
 
 19 
 
 16665-511 
 
 583-293 
 
 680-234 17929-038 
 
 19 
 
 20 
 
 17929-038 
 
 627-516 
 
 680-234 19236-788 
 
 20 
 
 21 
 
 19236-788 
 
 673-288 
 
 680-234 20590-310 
 
 21 
 
 22 
 
 20590-310 
 
 720-661 
 
 680-234 21991-205 
 
 22 
 
 23 
 
 21991-205 
 
 769-692 
 
 680-234 23441-131 
 
 23 
 
 24 
 
 23441131 
 
 820-440 
 
 680-234 24941-805 
 
 24 
 
 25 
 
 24941-805 
 
 872-961 
 
 6S0-234 26495-000 
 
 25 
 
i86 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XVII. 
 
 SINKING FUND PROBLEMS RELATING TO 
 THE AMOUNT IN THE FUND. 
 
 A surplus in the fund ; how it may arise, and how it 
 may be adjusted. 
 
 Variation I. 
 
 Arising in consequence of an excessive past accumula- 
 tion OF the fund. 
 
 Variation IL 
 
 Arising in consequence of the payment into the fund of 
 the proceeds of sale of part of the assets representing 
 the security for the loan, or a realised profit upon 
 
 the SALE OF AN INVESTMENT REPRESENTING THE FUND. 
 
 Statement XYII. A. 
 
 Summary of the methods of adjustment. The various causes 
 leading to a surplus in the fund. Difference in 
 conditions and practice as between local authorities 
 and commercial and financial undertakings. comparison 
 
 OF THE VARIOUS METHODS OF DEALING WITH A SURPLUS. ThE 
 AN-NUAL INCREMENT (BALANCE OF LOAN) METHOD. STATEMENT 
 SHOWING THE FINAL REPAYMENT OF THE LOAN BY THE OPERA- 
 TION OF THE AMENDED ANNUAL INSTALMENT. 
 
 Summary of the methods of adjustment. 
 
 Variation I (Surplus), arising in consequence of an exces- 
 sive 'past accumulation of the fund. 
 
 (7) Ascertain the actual present surplus, as described in 
 Chapter XV . 
 
 {2) Calculate the annuity or annual instalment of lohich this 
 sum is the present value for the unexpired portion of the 
 repayment period. Similar to Calculation (XV) 3. 
 
A SURPLUS IN THE FUND 187 
 
 (t3) 2'he anmial instalment so ascertained, deducted jrora the 
 original anmial instalment loill give the reduced annual 
 instahne7it to he set aside during the tvhole of the un- 
 expired portion of the repayment period. 
 
 (4) Prepare a state merit shoudng the firuil repayment of the 
 
 loan by the operation of the sinking fund under the 
 amended conditions. Similar to Statement XVI. A., 
 with the necessary modifications relating to a surplus 
 instead of to a deficiency . 
 
 (5) Prepare a pro forma amount shoiving the amount which 
 
 should be in the fund at the end of each year of the 
 unexpired repayment period. 
 
 Note. The above method so closely resembles the one 
 adopted in the case of a deficiency and the folloioing method 
 relating to Variation II, that no further amplification is 
 required. Unlike a d&ficiency, however, a surplus should be 
 spread equally over the whole of the unexpired repa/yment 
 period, and consequently Method II [Deficiency) ivill rarely 
 apply. 
 
 Variation II (Surplus), arising in consequence of the 
 payment into the fmid of the proceeds of sale of part of the assets 
 representing the security for the loan, or a realised profit upon 
 the sale of an investment representing the fund. 
 
 Statement XVII. A. 
 
 (7) Ascertain in the manner described in Chapter XV, 
 whether there is a surplus or a deficiency in the fund 
 apart from the proceeds of realisation now under con- 
 sideration; and if so, calculate the corrective annual 
 sinking fund instalment required. Calculation {XV) 3. 
 
 (2) Calculate the annuity which may be purchased for the 
 
 unexpired portion of the repayment period, with the 
 amount now paid into the fund. Calculation [XVII) 1. 
 [Here refer to memo, after (^).] 
 
 (3) Deduct the annuity so ascertained from the original 
 
 annual instalment, and adjust the latter also, if required, 
 by the above corrective instalment, referred to in [1). 
 
 (4) The remainder will be the future reduced annual sinking 
 
 fund instalment, to be set aside and accumulated durmg 
 the whole of the unexpired portion of the repoAfment 
 period. 
 
i88 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (<5) Prepare a statement shoiciny the final repayment of the 
 loan hrj the operation of the sinliing fund under the 
 amended conditions. Statement XVII. A. 
 
 (6) Prepare a pro forma account showing the amount which 
 sJiould he in the fund at the end of each year of the 
 uned'pired repayment period. Pro forma Account, Xo. 4. 
 
 Memo. If the original annu(d instalment he a prescribed 
 suvi instead of being found by calculation in the ordinary way 
 {see [XVI) 1), proceed by the method described under T ariation 
 IV [Surplus) in Chapter XVIII, substituting for operation (7) 
 in that method the above operation [2). 
 
 A SuRPLrs IX A Sinking Fund and how it may arise. 
 
 Although it does not fall within the province of a work of 
 this character to mention all the varions causes which may 
 lead to the existence of a surplus in a sinking- fund, vet it is 
 very advisable to give a brief outline of the princi})al ways in 
 which this may happen, and which may be divided into the 
 following classes, any two or more of which may operate 
 simultaneously : — 
 
 (1) An excess in the amount of the annual instalments pre- 
 
 viously paid into the fund or an increase in the rate of 
 accumulation in excess of the rate assumed in calculating 
 the original instalment. Variation I. 
 
 (2) The payment into the fund of a realised profit upon the 
 
 sale of an investment representing the fund or the 
 proceeds of sale of part of tlie assets representing the 
 security for the loan. Variation II. 
 
 (3) In the case of commercial or financial undertakings, there 
 
 may be a change in the character of part of the original 
 loan, whereby the original obligation to set aside a 
 sinking fund is modified owing to the withdrawal of part 
 of the loan from the operation of the fund. 
 
 Tliis will be fully discussed in the following chapter, where 
 it will be shown that the precise metliod of making the adjust- 
 ment depends upon the nature of tlu' original annual instal- 
 
A SURPLUS IN THE FUND 189 
 
 ment, nnd the problem will be divided into two parts as 
 f olloAvs : — 
 
 A. In wliicli the original annual instalment was found by 
 
 calculation based upon a specified period of repayment 
 and rate of accumulation. Variation III. 
 
 B. In which the original annual instalment is a stated sum 
 
 and is not based, except in a general way, upon any 
 period of repayment or rate of accumulation. 
 
 Variation IV. 
 
 Variation I (Surplus), arising in consequence of an excessive 
 past accumulation of the fund. 
 
 This will be of rare occurrence if the pro forma account 
 already recommended has been made out showing the operation 
 of the fund until maturity, and any such minor instances may 
 be adjusted as and when they arise by transfers to the current 
 year's rate or revenue account. In the case of larger amounts 
 they may be treated in the manner mentioned in Chapters XV 
 and XVI, referring to a deficiency in the fund, but of course by 
 reducing the future annual instalment. 
 
 Variation II (Surplus), arising in consequence of the payment 
 into the fund of the proceeds of sale of part of the assets 
 representing the security for the loan or a realised profit 
 upon the sale of an investment representing the fund. 
 
 Statement XVII. A. 
 
 This chapter will deal fully with those cases in which the 
 sinking fund obligations are modified by the payment into the 
 fund of the proceeds of sale of part of the security for the loan 
 to be ultimately repaid, and attention will be directed to the 
 difference in practice as between local authorities and com- 
 mercial and financial undertakings. In the case of a local 
 authority the sinking fund instalment is set aside to repay the 
 loan at the end of the period allowed under the general or 
 special Act. These loans are invariably expended upon works 
 of a capital nature, and it sometimes happens that part of the 
 property representing the security for the loan is sold. The 
 practice generally followed in the case of local authorities is to 
 pay such proceeds into the fund and apply the same in the 
 redemption or repayment of part of the original loan. This is 
 as it shoiild be, and is the practice adopted in commercial and 
 financial undertakings, but it has an important effect upon the 
 sinking fund instalment. The repayment, during the period 
 
igo REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 allowed, of part of tlie debt out of tlie proceeds of sale of part 
 of the security (instead of out of the sinking fund provided out 
 of current rates or profits) anticipates the natural effect of the 
 sinking fund, and by reducing the loan repayable at the end of 
 the period correspondingly reduces the necessity to set aside in 
 future the full original sinking fund instalment. 
 
 It is obvious therefore that the original annual sinking fund 
 instalment niay be reduced during the remainder of the term to 
 such an amount as will provide the balance of the debt not 
 repaid out of the proceeds of the sale of part of the assets. 
 
 This principle is followed in the case of commercial and 
 financial undertakings, but in the case of local authorities the 
 Local Government Board may require that the proceeds of 
 such sales shall be paid into the sinking fund, and that the 
 full amount of the original annual instalment shall continue to 
 be set aside. The effect of this is to shorten the original 
 period allowed for the redemption of the debt. There is not 
 any objection to this method except that the result is to relieve 
 the later generation of ratepayers at the expense of the present, 
 but in its favour is the fact that it is always sound finance to 
 repay debt as soon as possible. In the case of commercial and 
 financial undertakings the practice varies, depending in each 
 instance upon the conditions laid down in the deed relating 
 to the loan. Generally speaking, it may be considered equitable 
 in the case of such undertakings to reduce the sinking fund 
 instalment and so maintain the original period allowed for the 
 repayment of the debt. In the case of a debenture stock repay- 
 able on a fixed future date this would necessarily require to be 
 so unless j^art of the stock were redeemed by purchase upon the 
 open market. 
 
 The proceeds of sales of capital assets forming part of the 
 security would, failing actual redemption, be invested in 
 securities authorised by the deed, and the resulting income 
 would be added to the sinking fund during the unexpired 
 portion of the repayment period, and therefore the future 
 annual instalments to be provided out of the profits of the 
 undertaking would be correspondingly reduced. If, however, 
 in the case of a commercial or financial undertaking any such 
 proceeds arising from the sale of part of the security were 
 actually applied in redemption of part of the loan, instead of 
 being invested in outside securities, the profit and loss account 
 of the undertaking would be relieved to the extent of the annual 
 interest payable upon such redeemed debt, but the sinking fund 
 -would not then be increased by any income arising from the 
 
A SURPLUS IN THE FUND 191 
 
 investment. Another difference between the sinking funds of 
 commercial and financial undertakings and those of local 
 authorities arises from the fact that in the former the annual 
 instalment is not always charged against the profits of the 
 undertaking but may be taken out of the general assets of the 
 concern. 
 
 The method of adjustment will be illustrated by the follow- 
 ing example relating to a commercial or financial undertaking. 
 
 A sinking fund has been set aside and accumulated to 
 provide for the repayment of a loan of £26,495 at the end of 
 25 years — and in fixing the annual instalment the rate of 
 accumulation was taken at 3^ per cent. At the end of the 
 12th year the fund stands at the proper estimated amount 
 shown by the pro forma account, and as found by Calculation 
 (XV) 2, namely, £9932'T44. At that time a portion of the 
 assets (forming part of the security for the loan) is realised, and 
 produces, say, £4,560. 
 
 The trust deed provides that this amount shall be paid into 
 the sinking fund and invested, and accumulated until the loan 
 is repayable, namely, at the end of the 25th year, and that the 
 future annual sinking fund instalments may be correspondingly 
 reduced. In the present example there is not any question of 
 the rate of income on the present investments, or the future rate 
 of accumulation, being less than 3^ per cent., the rate originally 
 assumed in calculating the annual instalment. The effect of 
 the realisation of part of the security for the loan is that the 
 amount in the sinking fund is suddenly increased by the sum 
 of £4,560, which was not anticipated when the original annual 
 instalment was calculated. If therefore this amount be paid 
 into the sinking fund and accumulated, and the original 
 instalments continue to be set aside in future and paid into the 
 fund until the end of the 25th year, the sinking fund will at the 
 end of that period be in excess of the amount required to repay 
 the loan, and the excess will be the amount of the above sum 
 of £4,560 accumulated at 3| per cent., compound interest, for 
 13 years. 
 
 The method of ascertaining the amount by which the 
 original annual instalment may be reduced during the un- 
 expired portion of the repayment period is exactly similar in 
 principle to that adopted in the case of the deficiency of 
 £469'744 in Chapter XY. In that case the deficiency was 
 converted into terms of original loan and the annual instalment 
 to be set aside during the remaining 13 years to redeem the 
 portion of the loan not already provided for was ascertained. 
 
192 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 lu the present case the sinking; fund stands at tlie proper 
 calculated amount at the end of the 12th year ; and the original 
 annual instalments, alone, if continued for a further I'S j-ears, 
 Avill be amply sufficient to provide for the ultimate repayment 
 of the debt. In addition there is a sum in hand of £4,560, 
 which may now be applied in repaying part of the loan, and the 
 equivalent annuity for the remainder of the period may be 
 applied in reduction of the future annual instalments to be 
 added to the fund. The £4,560 may be regarded as a sum 
 which may now be invested in the purchase of an annuity for 
 l^j years on a 3| per cent, basis. This method is the more 
 preferable seeing that the £4,560 is actually in hand, whereas 
 the deficiency of £469"T44 represented the present value of a 
 sum due at a future period and was a definite amoimt only so 
 far as it represented a sum which should have been in actual 
 possession, but was not so in fact. 
 
 When discussing the adjustment of a sinking fund in the 
 case of a deficiency in Chapters XV and XYI several alternative 
 methods were pointed out depending upon the period allowed in 
 which to make good past deficiencies. In the case of the 
 surplus under review, there is not any alternative to that already 
 considered if the original date of repayment be adhered to, 
 because the sum in question is a definite one and is actually in 
 hand. The calculation will be made upon the author's standard 
 form Xo. 5, relating to the annuity which £1 will purchase. 
 It will be seen that the sum now paid into the fund will effect 
 a decrease in the original annual instalment of £442" 6008 per 
 annum. Calculation (XYII) 1. 
 
 The final repayment of the loan by the operation of the 
 amended instalment during the remaining 13 years of the 
 original repayment period is shown in the following Statement 
 XVII. A. 
 
 The above method should be carefully compared with the 
 correction of a surplus in a sinking fund, caused by the with- 
 drawal of part of the loan from the operation of the fund owing 
 to the conversion of such part of tlie loan into ordinary share 
 capital. The difference in the methods will be fully described 
 in Chapter XVIII. 
 
 The Annual Increment (balance of loan) Method. This 
 method will now be used for the purpose of ascertaining the 
 amended annual instalment, based upon the future annual 
 increment, a summary of which is given at the beginning of 
 
A SURPLUS IN THE FUND 193. 
 
 Chapter XY, aud is fully described in Chapters XVI and XXII. 
 As this method is based upon the same actual conditions as the 
 previous example, Statement XYII. A., showing the final 
 repayment of the loan, will also apply. This method is shown 
 in Statement XVII. B. 
 
 Comparison of the Methods of Dealing with a Surplus 
 AND A Deficiency in a Sinking Fund. 
 
 It is instructive to compare the above results with the 
 example worked out in the case of a deficiency in the fund 
 (Variation I), seeing that both funds relate to loans identical a& 
 to amount, period of repayment, and rates per cent, of income 
 and accumulation. 
 
 In each case also the adjustment is made at the end of the 
 12th year, and is spread over the full remaining term of 13 
 years. 
 
 In the case of the deficiency in the fund there was an 
 ascertained amount of £469" 744 by which the present invest- 
 ments, £9463-00, fell short of the amount of £9932-744 which 
 should have been in the fund in order to carry out the original 
 obligation. This deficiency was corrected by setting aside an 
 additional annual instalment, in augmentation of the original 
 annual instalment of £680-234 during the unexpired portion of 
 the repayment period. 
 
 This instalment of £45-594, which was found by Calculation 
 (XV) 3, represents the annuity which might have been 
 purchased with the above amount of £469-744. 
 
 The present surplus consists of an actual amount of cash^ 
 namely, £4,560, paid into the fund, Avhich is applied in 
 providing an instalment in reduction of the original annual 
 instalment of £680-234. This instalment, as found by Calcula- 
 tion (XVII) 1, based on Table V, is £442-601, and represents 
 the annuity which might be purchased with the above amount 
 of £4,560 paid into the fund. 
 
 In both cases the amount, which should, as shown by 
 Calculation (XA^) 2, have been to the credit of the fund, is 
 £9932-744, which amount, if accumulated for 13 years at 3^ per 
 cent., would at the end of the period, as shown by Calculation 
 (XVII) 2, have amounted to £15534-375 of original loan. 
 
 In the case of the surplus caused by a payment into the fund 
 now under consideration, part of the amount which should be 
 in the fund at the end of the prescribed repayment period of 
 25 vears is actually in hand at the end of the 12th year, and 
 
194 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 tlierefore the future annual instalment must be correspondingly 
 reduced owing to the future accumulation of the sum of £4,560 
 paid into the fund. 
 
 In the two cases the amount which should have been to the 
 credit of the fund at the end of the 12th year was represented 
 as follows : — 
 
 In the case of a Deficiency in the Fund. 
 
 Amount at enfl Equivalent 
 
 of year. amount of loan. 
 
 Actual amount in the fund £946.3- £14799-71 
 
 Deficiency, involving an additional 
 
 annual instalment of £46-594 ... 469-744 734-67 
 
 £9932-744 £15534-38 
 
 In the case of a Surplus in the Fund. 
 
 Amount at end Equivalent 
 
 of year. amount of loan. 
 
 Actual amount in the fund £9932-744 £15534-38 
 
 Deficiency. Nil. — — 
 
 £9932-744 £15534-38 
 
 In both cases the amount of original loan to be provided by 
 the accumulation of the future annual instalments for 13 years 
 is the same, namely, £1096062, being the total of the original 
 loan, £26,495, after deducting the above amount of £15534*38 
 already provided for. 
 
 The manner in which this remaining portion of original loan 
 is dealt with in the two cases is shown in the following table : — • 
 
 In the case of a Deficiency in the Fund. 
 
 Equivalent 
 amount of loan. 
 
 Future original annual instalment of £680-234, to 
 be set aside and accumulated for 13 years at 
 ^ per cent. Calculation (XY) 5 £1096062 
 
A SURPLUS IN THE FUND 
 
 195 
 
 In the case of a Surplus in the Fund. 
 
 Cash in hand, being proceeds of security 
 sold and added to the fund. 
 
 Calculation (XVII) 3 £456000 £7131-64 
 
 Future reduced annual instalment of £237'633, 
 to be set aside and accumulated for 13 years 
 at 3i per cent. Calculation (XYII) 5 £3828-98 
 
 £10960-62 
 
 The above future reduced annual instalment of £237-633 is 
 arrived at by deducting from the original annual instalment of 
 £680-234 the annual instalment of £442601, which is the 
 future equivalent of the capital sum of £4,560 paid into the 
 fund. 
 
 The above tabulated summary shows the intimate relation 
 between " present value " and " future amount " at the 
 beginning and end of the same period and at the same rate per 
 •cent., and further demonstrates the connection between the 
 formulae. 
 
 There are here three expressions of the value of one and the 
 :same thing at the same date, namely : — 
 
 1, A sum in hand of 
 
 £4560-00 
 
 2. A future annuity for 13 years at 3i per cent, of £442-601 
 
 3. A sum due at the end of that time also at 
 
 3i per cent £7131-64 
 
196 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Surplus in the Fund. Statement XVII. A. 
 
 Showing the final repayment of the loan, by the operation o£ 
 the sinking fund after making the adjustment in the 
 annual instalment consequent upon a surplus over the 
 amount which should stand to the credit of the fund. 
 
 Variation II (Surplus), arising in consequence of the payment 
 into the fund of the proceeds of sale of part of the assets 
 representing the security for the loan, etc. 
 
 Equivalent 
 amount of 
 original loan. 
 
 Present investments (at end of 12th year), £9932" 74 
 
 Amount thereof, accumulated for 13 years at 
 
 3i per cent. Calculation (XYII) 2 £15534-38 
 
 Amount paid into the fund £456000 
 
 Amount thereof, accumulated for 13 years at 
 
 3i per cent. Calculation (XVII) 3 £7131-64 
 
 Amended annual instalment : — 
 
 Original instalment £680-234 
 
 reduced by, (XYII) 4, 442601 
 
 £237-633 
 
 Amount thereof, accumulated for 13 years at 
 
 31 per cent. Calculation (XYII) 5 £3828-98 
 
 Amount of original loan £26495-00 
 
A vSURPLUS IN THE FUND 197 
 
 A Surplus in the Fund. Statement XVII. B. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find the amended annual sinking fund instalment conse- 
 quent upon a surplus over the amount which should stand 
 to the credit of the fund. 
 
 Variation II (Surplus), arising in consequence of the payment 
 into the fund of the proceeds of sale of part of the assets 
 representing the security for the loan. 
 
 Amount of original loan (25 years) ..£2649500 
 
 deduct amount in the fund at the 
 
 end of the 12th year ... £9932-74 
 
 proceeds of sale paid into 
 
 the fund £4560-00 
 
 £14492-74 
 
 Balance of loan £12002-26 
 
 Amended annual increment to be added to the fund 
 and accumulated at 3^ per cent, to provide 
 this amount at the end of 13 years. 
 
 Calculation (XYII) 6 £744-879 
 
 deduct income to be received from the present 
 
 investments, £14492" 74 at 3^ percent. £507-246 
 
 Amended annual instalment, being :— £237-633 
 
 Original annual instalment £680-234 
 
 reduced by £442601 
 
 237-633 
 
 The rule relating to this method is stated at the head of 
 Chapter XXII. 
 
198 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Pro forma Sinking Fund Account, No. 4, 
 
 A Surplus in tlie Fund. (Variation II.) 
 
 Loan of £26,495, repayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by tlie operation of 
 the reduced annual instalment of £237"633. 
 
 Statement XYII. A. 
 
 Rate of accumulation, 3| per cent. 
 
 Year. 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received from 
 
 investments 
 
 3i per cent. 
 
 Proceeds of 
 Annual sale of assets 
 sinking fund paid into 
 instalment. the fund. 
 
 Amount in 
 
 the fund 
 
 at end 
 
 of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 
 3 
 
 4 
 
 The i 
 
 imount in 
 
 the fund at 
 
 the ( 
 
 3nd of 
 
 4 
 
 5 
 
 the 12th year, £9932744, is 
 
 the ( 
 
 correct 
 
 5 
 
 6 
 
 calculated amount, as shown 
 
 byO 
 
 alcula- 
 
 6 
 
 7 
 
 tion 
 
 (XY) 2, and by the 
 
 pro 
 
 forma 
 
 7 
 
 8 
 
 9 
 
 10 
 
 account, No. 1, 
 
 Chapter XY 
 
 
 
 8 
 
 9 
 
 10 
 
 
 
 
 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 456000C 
 
 ) 9932-744 
 
 12 
 
 13 
 
 14492-744 
 
 507-246 
 
 237-633 
 
 — 
 
 15237-623 
 
 13 
 
 14 
 
 15237-623 
 
 533-317 
 
 237-633 
 
 — 
 
 16008-573 
 
 14 
 
 15 
 
 16008-573 
 
 560-300 
 
 237-633 
 
 — 
 
 16806-506 
 
 15 
 
 16 
 
 16806-506 
 
 588-228 
 
 237-633 
 
 — 
 
 17632-367 
 
 16 
 
 IT 
 
 17632-367 
 
 617133 
 
 237-633 
 
 — 
 
 18487-133 
 
 17 
 
 18 
 
 18487-133 
 
 647-050 
 
 237-633 
 
 — 
 
 19371-816 
 
 18 
 
 19 
 
 19371-816 
 
 678-014 
 
 237-633 
 
 — 
 
 20287-463 
 
 19 
 
 20 
 
 20287-463 
 
 710-061 
 
 237-633 
 
 — 
 
 21235-157 
 
 20 
 
 21 
 
 21235-157 
 
 743-230 
 
 237-633 
 
 — 
 
 22216020 
 
 21 
 
 22 
 
 22216-020 
 
 777-561 
 
 237-633 
 
 — 
 
 23231-214 
 
 22 
 
 23 
 
 23231-214 
 
 813-092 
 
 237-633 
 
 — 
 
 24281-939 
 
 23 
 
 24 
 
 24281-939 
 
 849-868 
 
 237-633 
 
 — 
 
 25369-440 
 
 24 
 
 25 
 
 25369-440 
 
 887-927 
 
 237-633 
 
 — 
 
 26495-000 
 
 25 
 
A SURPLUS IN THE FUND 199 
 
 CHAPTER XYIII. 
 
 SINKING FUND PROBLEMS, RELATING TO 
 THE AMOUNT IN THE FUND. 
 
 A surplus in the fund, of a commercial or fixaxcial 
 
 UNDERTAKING ARISING ON THE WITHDRAWAL OF PART OF THE 
 LOAN FROM THE OPERATION OF THE FUND, OWING TO THE 
 CONVERSION OF SUCH PART OF THE LOAN INTO ORDINARY 
 SHARE CAPITAL OR STOCK OF THE UNDERTAKING. 
 
 Variation III, ix which the original annual instal- 
 ment WAS FOUND BY CALCULATION BASED UPON A SPECIFIED 
 PERIOD OF REPAYMENT AND RATE OF ACCUMULATION. 
 
 Statement XVIII. A. 
 
 Variation IV, in which the original annual instal- 
 ment is a stated sum and is not based, except in a 
 general way, upon any period of repayment or rate of 
 accumulation. Statement XYIII. D. 
 
 Summary of the methods of adjustment. Remarks as to the 
 sinking funds of commercial and financial undertak- 
 INGS. The ANTiTUAL INCREMENT (BALANCE OF LOAN) METHOD. 
 
 Statement showing the final repayment of the loan by 
 the operation of the amended annual instalment. 
 
 Summary of the methods of adjustment. 
 
 Variation III (Surplus), arising on the ivithdrawal of lyart 
 of the loan from the operation of the sinhing fund of a 
 commercial or financial undertaking owing to the conversion of 
 such part of the loan into ordinary share capital or stocJx of the 
 undertaking : — 
 
 in ivhich the original annual instalment was found by 
 calculation based upon a specified period of repayment and 
 rate of acctimulation. Statonent XVIII. A. 
 
200 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (1) Ascertain, in the manner described in Chapter XV^ 
 whether there is a surplus or a deficiency in the fund 
 apart from the speciM circumstances now und-er reojiew, 
 and if so, calculate the corrective annual sinldng fund 
 instalment required, by one of the methods there 
 described. Calculations {X.V) 3 or {X.VI) 1. 
 
 (2) Calculate the animal sinking fund instalment, whicli, 
 
 if set aside for the whole of the unexpired portion of the 
 repayment period, icill provide the part of the loan 
 converted into ordinary share capital, and thereby with- 
 drawn from the operation of the fund. 
 
 Calculation (XVIII) 1. 
 
 (3) Deduct the annual instalment so ascertained from the 
 
 original annual instalment, and adjust the latter if 
 required, by the above corrective instahnent. 
 
 Calculation {XYI) 1. 
 
 (4) The remainder icill be the future reduced annual iiistal- 
 
 ment, to be set aside and accumulated during the whole 
 of the unexpired portion of the repayment period. 
 
 Calculation (XVUI) 2. 
 
 (0) Prepare a statement showing the final repayment of the 
 
 loan by the operation of the sinking fund under the 
 amended conditions. Statement X] III. B. 
 
 (6) Prepare a pro forma account showing the amount which 
 should be in the fund at the end of ea-ch year of the 
 unexpired repayment period. Pro forma Account, .A o. 6. 
 
 Yariation TV (SuRPLi'S), arising on the withdrawal of part 
 of the loan from the operation of the sinhing fund of a 
 commercial or financial undertaking oiring to the conversion of 
 such part of the loan into ordinary share capital or stock of the 
 undertaking : — 
 
 in wlvich the original annual instalment is a stated sum, 
 and is 7iot based, except in a general way, upon any period 
 of repayment or rate of accumulation. 
 
 Statement XYIU. D. 
 
 (1) Ascertain from the actual records the value of the present 
 
 investments representing the fund. Ascertain also the 
 rate of income yielded on such value, and upon this and 
 other considerations, as elsewhere described, base the 
 future rate of accumulation of the fund. 
 
A SURPLUS IN THE FUND 201 
 
 (2) Ascertain by inspection of Table III, the ayfroximate 
 
 number of years in which the stated annual instalment 
 ivill accumulate to the amount of the origiiial loan at the 
 rate of accumulation fixed as in (1). Adopt the nearest 
 integral number of years so found as the approximate 
 period of repayment of the original loan, at the rate 
 of accumulation, ascertained as above, 
 
 (3) Calculate the annual sinking fund instalment required to 
 
 repay the full amount of the loan at the end of the 
 approximate period of repayment found in {2) at the rate 
 of accumidation fixed as in (-7). 
 
 Calculation (XVIII) 5. £7441-63. 
 
 (4) Calculate the amount which would be in the fund if the 
 
 annual instalment {£7 441' 63) so found [3) had been set 
 aside and accitmulated at the rate per cent, fixed in (1) 
 from the date of issue of the loan until the date of 
 con/version of part of the loan. 
 
 Calculation (XVIII) 9. £57021-21. 
 
 (5) Ascertain the apparent surplus or deficiency in the fund 
 
 by comparing the value of the present investments repre- 
 senting the fund (1) with the amount found in (4). 
 
 Surplus, £447-27. 
 
 (6) Calculate the corrective instalment, being the annuity 
 
 ivhich might now be purchased with the amount found 
 in [5), for the unexpired portion of the approximate 
 repayment period (2) at the rate of accumulation (1). 
 
 Calculation (XVIII) 10. £57-45. 
 
 (7) Calculate the annual sinldng fund instalment which, if 
 
 set aside for the unexpired portion of the approximate 
 repayment period (2), would p>rovide the portion of the 
 loan converted into ordinary share cap<ital or stock. 
 
 Calculation (XVIII) S. £4429-52. 
 
 [Here refer to the memo, after (12).'] 
 
 (8) Deduct from the annual sinking fund instalment 
 
 (£7441-63) found as in (3), but not from the fixed instal- 
 ment (£7,500) originally specified in the trust deed and 
 actually set aside, the annual instalment (£4429-52) 
 found in (7). 
 
202 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (9) The remainder [£3012'11), after adjustment in resyect of 
 the corrective instalment [£o7'45) fo^l.nd In (6) will be 
 the future reduced annual sinking fund instalment, to be 
 set aside and acciimulated during the whole of the 
 unexpired portion of the approximate repayment period 
 found in {2). ' £2954-66. 
 
 (^10) Prepare a statement showing the final repayment of the 
 loan by the operatio7i of the fund based (not tipon the 
 annual instalment (£7,500) originally fxed), but upon 
 the annual instalment (£7441'63) found in (3), as subse- 
 quently reduced by the annual instalment (£4429'52) 
 found in (7) and adjusted by the annual instalment 
 {£57-45) fotmd in (6). Statement XVIII. D. 
 
 (11) If the anrmal instalment (£7441-63) as found in (3) 
 
 differs considerably from the annual instalment (£7,500) 
 originally specified in the trust deed, an adjustment may 
 be made which icill have the effect of slightly increasing 
 or reducing the annual instalment found in (9), as here- 
 after described. 
 
 (12) Prepare the pro forma account mentioned in the previous 
 
 methods. Pro forma Account, No. 6. 
 
 Memo. If the above method be used to adjust a surplus 
 in the fimd arising in consequence of the payment into the 
 fund of the proceeds of realisation of part of the assets, as 
 described in Variation II (Surplus) in Chapter X\ II, biit in 
 whicJi the aiinual instalment is a stated sum not found by 
 calcAilation, substitute operation (2) in that method for the above 
 operation (7), iiamely : — 
 
 (2) Calculate the annuity which may be purchased for the 
 unexpired portion of the repayment period with the 
 amount now paid into the fund. Calculation (XVII) 1. 
 
 General Eemarks as to the Sinking Funds of Commercial 
 AND Financial Undertakings. 
 
 In the case of local authorities the method by which loans 
 are required to bo repaid by means of a sinking fund is well 
 defined, but in the case of commercial or financial undertakings 
 the conditions are much more variable, and the trust deed may 
 stipulate that it shall be provided either by : — 
 
A SURPLUS IN THE FUND 203 
 
 (1) An equal annual instalment to be calculated on tlie basis 
 
 of a given repayment period and a prescribed rate of 
 accumulation, similar to those of local authorities or 
 
 (2) A stated sum to be set aside each year. 
 
 Both methods must be considered. In order to attract investors 
 a commercial or financial undertaking, when inviting subscrip- 
 tions for bonds, debentures, debenture stock, or loan capital of 
 any other nature, may give the investor the option at a future 
 date, which may be specified or not, of converting the loan into 
 share capital or stock of the undertaking, on what may then be 
 very advantageous terms if the concern be making good profits. 
 In the meantime a sinking fund is required by the trust deed to 
 be set aside out of profits in order to repay the total loan on a 
 given date, the fund to accumulate at a rate per cent., which 
 may be specified or not, by investment in outside securities. 
 
 During the earlier years, if profits are low, the provision of 
 the annual instalment will have the effect of reducing the 
 dividends which may be paid to the ordinary shareholders, and 
 there will not therefore be any inducement to the loan creditors 
 to give up their security. But a time may come when the 
 position of the undertaking has been materially improved, and 
 if the profits have been good and are likely to continue so, some 
 of the loan creditors may be induced to convert their holding 
 into ordinary share capital or stock. The amount of loan so 
 converted will, of course, correspondingly reduce the amount to 
 be finally provided by means of the sinking fund, and, seeing" 
 that the period of repayment of the balance of the loan will 
 remain unchanged, the effect of the partial conversion will be 
 seen solely in a reduction in the future annual instalment to 
 be set aside out of profits during the unexpired portion of the 
 original repayment period. This reduction in the future annual 
 instalment arises in consequence of two factors, namely, the 
 amount of loan withdrawn from the operation of the fund by 
 reason of its conversion into ordinary share capital or stock; 
 and, further, from the fact that the amount now in the fund 
 represents the accumulation of past instalments set aside to 
 provide the whole of the loan. Stated in terms of the balance 
 of loan still unconverted, there is a present surplus in the fund 
 due to setting aside in the past Avhat will in future be excessive 
 annual instalments, and there is also an excessive future annual 
 instalment, both of which factors have been dealt with 
 individually in previous chapters. They are here combined: 
 and the problem is further complicated by the nature of the 
 
204 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 animal instalment. If it be a stated sum, it is very probable 
 tliat it was fixed originally AA-itli only an approximate regard to 
 the period of repayment, and it is therefore necessary to 
 ascertain not only the future period of repayment but also the 
 future rate of accumulation. This future rate of accumulation 
 may be based upon the rate of income now yielded by the 
 present investments of the fund, and therefrom it is possible 
 io calculate the period of repayment, both of which are 
 governing factors in the adjustment to be made. The problem 
 may be further complicated by other variations in the period 
 or rate per cent., or by a combination of both, but attention 
 will be directed only to the above factors. 
 
 Taeiation III (Surplus), arising on the withdrawal of part of 
 the loan from the operation of the sinking fund of a 
 commercial or financial undertaking owing to the conversion 
 of such part of the loan into ordinary share capital or stock 
 of the undertaking : — 
 
 in which the original an/iual instahneiif wa.'i found by 
 calculation, based upon a specified period of repayment 
 and rate of acciiinulation. 
 
 Statement XVIII. A. 
 
 The above variation will be illustrated by the information 
 previously obtained with regard to the imaginary sinking fund 
 already discussed, namely, the same amount of original loan, 
 £26,495, repayment period 25 years, rate of accumulation 3^ per 
 cent., amount in the fund £9,463, or a deficiency of £469"T4. 
 The assumed conversion of part of the loan takes place at the 
 end of the 12th year and affects £5,000 of the loan. The 
 ■original annual instalment, £680234 was arrived at bv 
 Calculation (XY) 1. 
 
 Statement XVIII, A, following, shows the successive steps 
 in the adjustment of the annual instalment, and Statement 
 XVIII, B, shows the ultimate repayment of the loan by the 
 ■operation of the fund after making such adjustment. 
 
 The above deficiency of £469"T4 has been purposely 
 introduced into this example in order to demonstrate that the 
 method adopted Avill apply to a combination of factors requiring 
 the adjustment. It will illustrate the remark made in a 
 previous chapter that it is not absolutely necessary to ascertain 
 the exact amount of the deficiency at the time of making the 
 adjustment seeing that the calculation is based upon the actual 
 amount now in the fund and the acruniulation thereof at the 
 
A SURPLUS IN THE FUND 205 
 
 future rate per ceut. This is clearly showu by the examples- 
 worked out iu this and other chapters by the annual increment 
 (balance of loan) method in which no mention is made, or 
 account taken, of any surplus or dehciency in the amount in 
 the fund as compared with the amount Avhich should be in the- 
 fund at the time of making the adjustment. 
 
 The reduction of £310'308 in the original annual instalment 
 is the sole effect of the withdrawal of the £5,000 of loan from 
 the operation of the fund, since there is not in this instance 
 any increase in the income to be added to the fund, as found 
 in Chapter XVII was the elfect of the payment into the fund 
 of tbe sum of £4,560 arising out of the proceeds of sale of part 
 of the security for the loan. See Statement XYII. A. 
 
 This method of adjusting a surplus in a sinking fund, owing 
 to the withdrawal of part of the loan from the operation of the 
 fund, should therefore be carefully compared with the method 
 found necessary in the case of a surplus arising from a cash 
 payment into the fund from proceeds of assets realised. This- 
 will be fully considered at the end of this chapter. 
 
 The Annual Inckemext (ualaxce of loan) Method. The 
 method of arriving; at the amended annual instalment based 
 upon the future annual increment is summarised at the 
 beginning of Chapter XV, and is fully described in Chapter 
 XVI. As the method about to be discussed is based upon the 
 same conditions as in the previous example, Statement X^ III. 
 B., showing the final repayment of the loan will again apply. 
 This method is shown in Statement XVIII. C following. 
 
2o6 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Surplus in the Fund. Statement XVIIL A. 
 
 The Deductive Method. 
 
 Showing the method of adjusting the annual instalment in 
 consequence of a surplus in the fund, arising on the with- 
 drawal of part of the loan from the operation of the fund 
 owing to the conversion of such part of the loan into 
 ordinary share capital or stock. 
 
 Yariatiox III (SuEPLus), in which the original annual instal- 
 ment was found bj' calculation, based upon a specified 
 period of repayment and rate of accumulation. 
 
 Calculation (XY) 1. 
 
 Equivalent 
 Annual amount 
 
 instalment. of original loan. 
 
 Amount of originalloan £26495 00 
 
 Original annual instalment. 
 
 Calculation (XY) 1 £680-234 
 
 Additional annual instalment to provide 
 
 present deficiency in the fund. 
 
 Calculation (XYj 3 £45-594 
 
 £725-828 
 
 Present investments (at end of 12th year) 
 
 £9463-00 
 
 Amount thereof, accumulated for 13 
 years at 3^ per cent. 
 
 Calculation (XY) 4 £14799-71 
 
 £725-828 £11695-29 
 
 Loan withdrawn from the operation of 
 
 the fund at the end of the 12th year £5000-00 
 
 equivalent to a reduction in the 
 annual instalment of 
 
 Calculation (XYIII) 1 £310308 
 
 £415-520 £6695-29 
 
 Amended annual instalment of £415-520 
 will provide £6695-29 in 13 years 
 at3l per cent. Calculation (XYIII) 2 £415520 £6695-29 
 
A SURPLUS IN THE FUND 207 
 
 A Surplus in the Fund. Statement XVIII. B. 
 
 Showing the final repayment of the loan, by the operation of 
 the sinking fund after making the adjustment in the annual 
 instalment consequent upon a surplus in the fund, arising 
 on the withdrawal of part of the loan from the operation 
 of the fund owing to the conversion of such part of the loan 
 into ordinary share capital or stock. 
 
 Variation III (Surplus), in which the original annual instal- 
 ment was found by calculation, based upon a specified 
 period of repayment and rate of accumulation. 
 
 Calculation (XV) 1. 
 
 Present investments (at end of 12th year) 
 
 £9463-00 
 
 Equivalent 
 Annual amount 
 
 instalment of original loan. 
 
 Amount thereof, accumulated for 13 
 years at 3^ per cent. 
 
 Calculation (XV) 4 £14799-71 
 
 Amended annual instalment : — 
 
 Original annual instalment £680'234 
 
 Additional annual instalment to 
 provide the present deficiency of 
 £469-744. Calculation (XVI) 1 £45-594 
 
 £725-828 
 
 Reduced annual instalment due to 
 withdrawal of £5,000 of loan. 
 
 Calculation (XVIII) 1 £310-308 
 
 £415-520 
 
 Amount thereof, in 13 years at 3^ per cent. 
 
 Calculation (XVIII) 2 £6695-29 
 
 Amount in the fund at the end of 25 years £2149500 
 
 heing amount of original loan ... £26495-00 
 less the amount converted as above £5000- 00 
 
 £21495-00 
 
2oS REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Surplus in the Fund. Statement XVIII. C. 
 
 The Annual Increment (balance of loan ) Method. 
 
 To find the amended annual sinking fund instalment consequent 
 upon a surplus in the fund, arising on the withdrawal of 
 part of the loan from the operation of the fund, owing to 
 the conversion of such part of the loan into ordinary share 
 capital or stock. 
 
 Variation III (Surplus), in which the original annual instal- 
 ment was found by calculation, based upon a specified 
 period of repayment and rate of accumulation. 
 
 Calculation (XV) 1. 
 
 Amount of original loan (25 years) .. . £2649500 
 
 deduct portion thereof converted into ordinary 
 share capital or stock and withdrawn 
 from the operation of the fund at the 
 end of the 12th year £5000-00 
 
 £21495-00 
 
 deduct amount in the fund at the end of the 
 
 12th vear £9463-00 
 
 Balance of loan £1203200 
 
 Amended annual increment to be added to the 
 fund, and accumulated at 3-| per cent, to 
 provide this amount at the end of 13 years 
 
 Calculation (XVIII) 3 £746-725 
 deduct income to be received from the present 
 
 investments (£9,463) at 3^ per cent. £331-205 
 
 Amended annual instalment, being £415-520 
 
 Original annual instalment £680234 
 
 increased by £45-594 
 
 £725-828 
 and reduced by £310-308 
 
 £415-520 
 
 The rule relating to tliis method is stated at tlie head of 
 Chapter XXII. 
 
A SURPLUS IN THE FUND 
 
 209 
 
 Pro forma Sinking Fund Account, No. 5. 
 
 A Surplus in the Fund. (Variation III.) 
 
 Loan of £26,495, repayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by the operation of 
 the reduced annual instalment of £415'520. 
 
 Statement XVIII. B. 
 
 Hate of accumulation, 3| per cent. 
 
 Year. 
 
 Amount in 
 
 the fund at 
 
 beginning of 
 
 year. 
 
 Income 
 received from 
 investments 
 3J per cent. 
 
 Annual 
 sinking fund 
 instalment. 
 
 Amount in 
 tlie fund 
 at end 
 of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 3 
 
 4 
 
 The amount in the f 
 
 und at the ( 
 
 ?nd of the 
 
 4 
 
 5 
 
 12th year, £9,463, is 
 
 1 an assume! 
 
 d amount, 
 
 5 
 
 6 
 
 and is 
 
 equivalent 
 
 to setting 
 
 aside an 
 
 6 
 
 7 
 
 annual 
 
 instalment of £648-064, 
 
 as shown 
 
 7 
 
 8 
 
 by Calculation (XVI) 10, instead of the 
 
 8 
 
 9 
 
 correct annual insta 
 
 Iment of £680-234. 
 
 9 
 
 10 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 9463-000 
 
 12 
 
 13 
 
 9463-000 
 
 331-205 
 
 415-520 
 
 10209-725 
 
 13 
 
 14 
 
 10209-725 
 
 357-340 
 
 415-520 
 
 10982-585 
 
 14 
 
 15 
 
 10982-585 
 
 384-390 
 
 435-520 
 
 11782-495 
 
 15 
 
 16 
 
 11782-495 
 
 412-387 
 
 415-520 
 
 12610-402 
 
 16 
 
 17 
 
 12610-402 
 
 441-364 
 
 415-520 
 
 13467-286 
 
 17 
 
 18 
 
 13467-286 
 
 471-355 
 
 415-520 
 
 14354-161 
 
 18 
 
 19 
 
 14354161 
 
 502-396 
 
 415-520 
 
 15272-077 
 
 19 
 
 20 
 
 15272077 
 
 534-523 
 
 415-520 
 
 16222-120 
 
 20 
 
 21 
 
 16222120 
 
 567-774 
 
 415-520 
 
 17205-414 
 
 21 
 
 22 
 
 17205-414 
 
 602-189 
 
 415-520 
 
 18223123 
 
 22 
 
 23 
 
 18223-123 
 
 637-809 
 
 415-520 
 
 19276-452 
 
 23 
 
 24 
 
 19276-452 
 
 674-676 
 
 415-520 
 
 20366-648 
 
 24 
 
 25 
 
 20366-648 
 
 712-832 
 
 415-520 
 
 21495000 
 
 25 
 
210 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Variation IY. (SuEPLrs), arising on the withdrawal of part of 
 the loan from the operation of the sinking fund of a 
 commercial or financial undertaking owing to the con- 
 version of such part of the loan into ordinary share capital 
 or stock of the undertaking : — ■ 
 
 in ivhicli the original annual instalment is a stated sum 
 and is not based except in a general way upon any 
 period of repayment or rate of accumulation. 
 
 Statement XVIII. D. 
 
 This variation is similar in principle to the one last 
 discussed, but requires different treatment owing to the fact 
 that the original annual instalment is a stated sum arrived at 
 in a somewhat empirical manner, without any calculation 
 similar to (XV) 1, which is based upon a prescribed period of 
 repayment and rate of accumulation. In the following 
 example, used to illustrate the variation under review new data 
 have been adopted, and the question is dealt with solely as 
 regards the loan debt of commercial or financial undertakings 
 without making any comparison with the methods to be adopted 
 in the case of a local authority. In the early days of municipal 
 finance the annual instalment to be set aside was often a fixed 
 amount, being generally a definite percentage of the amount 
 of the loan, but the conditions then imposed upon such 
 authorities were vague and indefinite both as regards the 
 accumulation of the fund and the gradual repayment of the debt, 
 and left entirely out of account the life or duration of utility 
 of the asset created out of the loan. 
 
 Example to Illustrate Variation IV (Surplus). The 
 sinking fund under review relates to the repayment of a loan 
 of £150,000, and an annual instalment of £7,500 is required 
 to be set aside for this purpose out of the profits of the undertak- 
 ing and invested in outside securities. Under the trust deed the 
 loan creditors have the option of converting their holding into 
 ordinary share capital or stock at any time within seven years 
 from the date of issue. The price payable, on conversion, for 
 the ordinary share capital is immaterial for the present purpose, 
 as is also the rate of interest payable upon the loan; but any 
 premium payable to the loan-holders upon conversion or 
 redemption should be taken into account. At the end of the 
 seventh year the holders of £45,000 of loan elect to exercise 
 the above option, and convert their loan holding into ordinary 
 share capital or stock. Seeing that no specified period is 
 
A SURPLUvS IN THE FUND 211 
 
 prescribed within which the loan shall be repaid by means of 
 the sinking fund and that the annnal sum to be set aside is 
 fixed at £7,500, there was not any necessity, at the date of issue 
 of the loan, to make any calculation of the annual instalment 
 as in the case of the sinking funds of local authorities. This 
 annual instalment of £7,500, it will be assumed, has been 
 regularly set aside and invested each year, and at the end of 
 the seventh year, when £45,000 of original loan is converted 
 into ordinary share capital or stock, it will have amounted to 
 £5746848, having earned an average accumulation rate of 
 8 per cent., as shown by Calculation (XYIII) 4. In actual 
 practice, of course, this amount would be obtained from the 
 actual records or books of account. 
 
 The position at the end of the seventh year will therefore be 
 as follows : — 
 
 1. Loan outstanding and unconverted £105,000 
 
 2. Amount in the sinking fund, invested and 
 
 yielding o per cent, per annum £57468'48 
 
 3. Present annual instalment £7500* 
 
 This annual instalment will be reduced in future years owing 
 to the withdrawal of £45,000 of loan from the operation of the 
 fund. 
 
 The next step in the adjustment is to ascertain the annual 
 amount by which this instalment may be reduced and yet fulfil 
 the original obligation to repay the unconverted portion of the 
 loan under the original conditions. There are several ways of 
 doing this, as may be gathered from previous examples. But 
 it is in any case first essential that the estimated future rate of 
 accumulation shall be fixed. In this case past experience is 
 available, and, for convenience, 3 per cent, will be taken, being 
 the rate of income already yielded by the present investments 
 representing the fund. Any variation in this rate per cent, 
 and in the future accumulation rate may be treated as 
 explained in Chapter XXI (variation in the rates per cent.). 
 Having decided upon the future estimated rate of accumulation, 
 it is next necessary to fix the period of redemption in order to 
 ascertain the reduction in the annual instalment of £7,500. 
 This would not be necessary but for the amount at present in 
 the fund. The instalment then would be lo^^^^^ths of the 
 original instalment of £7,500, or £5,250 per annum, but this 
 
212 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 will be reduced by au annual amount depending upon tlie 
 money now in tlie fund. There are therefore two factors to be 
 taken into account — (1) the accumulation of the £57468'48 
 now in the fund, and (2) the accumulation of the future 
 reduced annual instalment which it is required to ascertain. 
 It is not possible to combine, in one calculation, factors 
 involving the amount of £1, and also of £1 per annum 
 without reducing both to a common denomination, and there- 
 fore it is better as the simplest method to revert to the original 
 conditions at the date of the issue of the loan. 
 
 The first step is to ascertain the approximate number of 
 years in which an annual instalment of £7,500 will amount to, 
 and repay, a loan of, £150,000 if accumulated at -3 per cent. 
 per annum. This is the rate of income which has been yielded 
 by the present investments representing the fund and which it 
 is assumed will continue to be yielded by any future invest- 
 ments. The number of years may be ascertained approximately 
 by an inspection of Table III, seeing that if £7,500 per annum 
 will, at 3 per cent., amount to £150,000, £1 per annum will, at 
 the same rate and in the same time, amount to £20. Table III 
 gives the following figures : — 
 
 £1 per annum will, at 3 per cent., amount to — 
 
 in 15 years £1859891, 
 
 in 16 years 20-15688, 
 
 in 17 years 21"76159, 
 
 and an even 16 years is therefore adopted as the approximate 
 period of repayment, which will be slightly in excess of the 
 actual period required, and the calculated annual instalment 
 will therefore be less than £7,500. In order to make the 
 calculation in such a manner that the result may be proved as 
 in other cases, it is necessary to first ascertain the exact annual 
 instalment, to be accumulated at 3 per cent., to repay £150,000 
 in exactly 16 years. The annual instalment is £7441"63, as 
 shown by Calculation (XVIII) 5. 
 
 This annual instalment of £7441-63 is less than the stated 
 annual instalment of £7,500, as will be gathered from the above 
 extracts from Table III, which show that £1 per annum will 
 in 16 years, at 3 per cent., amount to £2015688; consequently 
 the prescribed annual instalment of £7,500 will amount to : — 
 
 (£20-15688 X £7,500) or to £151,17659 
 
 in 16 years at 3 per cent., as shown by Calculation (XYIII) 6. 
 
A SURPLUS IN THE FUND 213 
 
 Expressed in terms of the above difference it will be seen tliat : 
 
 £7,500 -£7441-63 per annum or £58-37 
 
 will in 16 years, at 3 per cent., amount to 
 
 £151,176-59 -£150,000 or £1176-58 
 
 as shown by Calculation (XYIII) 7. 
 
 By adopting the above annual instalment of £7441-63 
 instead of £7,500, an intentional error of £58-37 per annum is 
 introduced, relating- to the repayment of a loan of £150,000 in 
 16 years. But the reduced annual instalment which is required, 
 and which will be based upon the above annual instalment of 
 £7441-63, will relate to a loan of £105,000 repayable in 9 years 
 only. This intentional error may be corrected if thought 
 desirable or required in the manner to be afterwards explained. 
 The following data have now been ascertained: — A loan 
 of £105,000 is repayable in a period of 9 years, and towards 
 this there is in the fund an amount of £57468-48, which, it is 
 estimated, will accumulate at 3 per cent. There is an annual 
 instalment of £7441-63 to be set aside for 9 years, and an 
 intentional error of £58-37 per annum has been introduced into 
 the problem. 
 
 It is required to find the annual amount by which the above 
 instalment of £7441-63 may be reduced, consequent upon the 
 withdrawal of £45,000 of loan from the operation of the fund. 
 The problem differs somewhat from the surplus of £4,560, 
 already considered in Chapter XVII (Statement XVII. A.). 
 In that case the £4,560 was paid into the fund in consequence 
 of the realisation of assets forming part of the security for the 
 loan, and was applied in repaying part of the loan or remained 
 to swell the assets of the fund. 
 
 In the present instance the conversion of £45,000 of loan 
 into share capital or stock may be looked upon as an entirely 
 separate transaction, and may be regarded as so much cash 
 received in consequence of the issue of new share capital, and 
 applied in reduction of the loan debt. The undertaking, 
 except as afterwards mentioned, does not derive any benefit 
 from the substitution of its obligation to the new shareholders 
 for its obligation to the previous loanholders. Indeed, it may 
 happen that the inducement to the loanholders to convert their 
 secured debt into ordinary share capital or stock is the expecta- 
 tion of a higher rate of interest upon their investment. The 
 only benefit to the undertaking is that the capital is firmly 
 invested in the concern; and the saving in the sinking fund 
 instalment, if previously taken out of profits, will help to 
 provide any increased return payable by way of dividend to the 
 
214 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 origiual ioaniioldeis iu respect of the loau capital so converted. 
 It IS here necessary to depart from the method adopted in 
 dealing with the sum of £4,560 paid 'into the fund in the 
 example considered in Chapter XVII. The £4,560 was an 
 amount actually in hand, and Calculation (XVII) 1 shows the 
 method of finding- the future annuity which it would purchase. 
 The £45,000 in tlie present case, on the contrary, is an amount 
 due at a future time, namely, at the end of the sinking fund 
 period, and the problem therefore becomes inverted, and instead 
 of calculating the annuity which £45,000 will purchase, it is 
 required to ascertain the annuity or annual sinking fund 
 instalment which in the remaining unexpired period of 9 years 
 will amount to that sum. This is shown by Calculation 
 (XVIII) 8, which may be usefully compared with Calculation 
 (XVII) 1. The annual instalment so found is £4429-52, by 
 which amount the original annual instalment of £7441'63 may 
 be reduced, making the amended annual instalment £301211. 
 
 There is, however, a further slight correction to be made. 
 In order to simplify the calculation an even period of 16 years 
 has been adopted, which is in excess of the actual period of 
 repayment and requires a reduced annual instalment of 
 £7441-63 instead of £7,500. Until the conversion of part of 
 the loan into ordinary share capital or stock, the undertaking 
 had been setting aside an annual instalment of £7,500, so that 
 there is now an apparent surplus in the amount in the fund as 
 compared with what would have been in the fund if £7441-63 
 only had been annually set aside. To ascertain the amount of this 
 surplus it is requisite to ascertain the amount to which an annual 
 instalment of £7441-63 will accumulate in 7 years at 3 percent. 
 
 This, as shown by Calculation (XVIII ) 9, is £5702121 
 
 and on comparing this sum with the amount 
 
 actually in the fund, being the accumulation 
 
 of the stated instalment of £7,500 
 
 Calculation (XVIII) 4, viz. : £57468-48 
 
 the apparent present surplus is found to be £447*27 
 
 whioli amouiii, being now in the fund, will accumulate for 
 9 years at 3 per cent., and is the present value of an annuity, 
 as'shown by Calculation (XVIII) 10, of £574446 which may be 
 applied in further reduction of the annual instalment of 
 £7441-63 in the same way that the annual instalment of 
 £442001 was applied in the case of the surplus of £4,560 in 
 Chapter XVII, Statement XVII, A. 
 
A SURPLUS IN THE FUND 215 
 
 In the event of the calculated insstalnient found as above 
 exceeding the prescribed instalment of £7,500 there would be 
 an apparent deficiency in the fund, instead of a surplus, which 
 would alter the method, but not the principle, of the minor 
 adjustment under consideration. 
 
 The A'arious stages of the adjustment have already been so 
 fully described that it is not requisite to prepare a statement 
 similar to XYIII, A, in the j)i"evious example. 
 
 A statement has, however, been prepared, similar to 
 XVIII, B, showing the final repayment of the loan by the 
 operation of the fund after making the above adjustment in the 
 annual instalment. See Statement XVIII, D, and the pro 
 forma account No. 6 following. 
 
 Correction of the Intentional Error. There now 
 only remains the correction of the above intentional error of 
 £58'37 in taking the annual instalment at the calculated 
 amount of £7441'63 instead of £7,500, which lengthened the 
 period of repayment by part of a year. 
 
 The £7,500, or any other similarly prescribed annual 
 instalment, is generally fixed in an empirical manner with only 
 a rough approximation to the actual requirements based upon 
 the conditions in each case. It may therefore be concluded 
 that the instalment ascertained in the above manner will meet 
 any practical need likely to arise in such a case. If, however, 
 there is at any time a necessity for greater accuracy it may be 
 ascertained, approximately, by the following method : — • 
 The intentional annual error introduced was £58'37 
 
 This caused an apparent surplus in 7 years of £447'27 
 
 Equal to an annual instalment spread over 9 years of £57'45 
 
 This error related to a loan of £150,000 
 
 The correction will relate to a loan of only £105,000 
 
 Therefore the correction may be taken as ^''^/isoths 
 
 of £57-45. or £40-215 
 
 which would in 9 years amount, at 3 per cent., to 
 
 Calculation (XVIII) 13 £408-549 
 
 The present value of this annual sum in 9 years, at 
 
 3 per cent., is Calculation (XVIII) 14 £313-118 
 
2i6 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 aud the currectiou may be made by increasing each of the 
 reduced annual instalments of £2954' 66 by £40"215, or by 
 paying into the fund at the present time the above present 
 value thereof, namely, £313"118. If no such correction be 
 made, only £408'549 of original loan will remain unprovided for 
 at the end of the period (which is slightly less than 16 years) in 
 which the original instalment of £7,500 would have repaid 
 the loan. 
 
 The Axxual Inckemext (balance of loan) Method. The 
 method of arriving at the amended annual instalment based 
 upon the future annual increment is summarised at the 
 beginning of Chapter XY, and is fully described in Chapter 
 XVI. As the method about to be discussed is based upon the 
 same data as in the example previously used, the following 
 Statement XYIII, D, showing the final repayment of the loan 
 will still apply. For the reasons already given the calculation 
 cannot be made in terms of the stated instalment, but must 
 be made in terms of the approximate amount of £T441'63 found 
 by Calculation (XYIII) 5. This method is shown in 
 Statement XYIII, E, following. 
 
 A Surplus in the Fund. Statement XVIII, D. 
 
 Showing the final repayment of the loan, by the operation of 
 the sinking fund, after making the adjustment in the 
 annual instalment, consequent upon a surplus in the fund, 
 arising on the withdrawal of part of the loan from the 
 operation of the fund owing to the conversion of such part 
 of the loan into ordinary share capital or stock. 
 
 Yariation IY (Surplus), in which the original annual instal- 
 ment is a stated sum, and is not based, except in a general 
 way, upon any period of repayment or rate of accumulation. 
 
 Equivalent 
 Annual amount of 
 
 instalment. original loan. 
 
 Present investments fat end of 7 years), 
 Calculation (XVIII) 4 £57468-48 
 
 Amount thereof, accumulated for 
 9 years at 3 per cent. 
 
 Calculation (XYIII) 11 £74983-30 
 
A SURPLUS IN THE FUND 217 
 
 Amended annual instalment ;— 
 
 Urigiual annual instalment, as 
 
 provided by trust deed ... ^''''^OQ^O 
 
 Substituted annual instalment as 
 adopted in Calculation (XVIII) 5, 
 based upon a rate of accumulation 
 of 3 per cent, and a repayment 
 period of 16 years £7441"63 
 
 This will be reduced by the annual 
 instalment required to repay the 
 £45,000 of loan withdrawn, in 9 
 years at 3 per cent. 
 
 Calculation (XVIIIj 8 £442952 
 
 £301211 
 
 And will be further reduced by the 
 annual instalment to proAade 
 £447-27, being the surplus which 
 will be in the fund at the end of 
 16 years, due to taking an even 
 period of 16 years 
 
 Calculation (XVIII) 10 £57-45 
 
 £2954-66 
 
 Amount thereof, accumulated for 
 9 years at 3 per cent. 
 
 Calculation (XVIII) 12 £3001670 
 
 Amount in the fund, at the end of 16 years 
 
 hemg the original loan £15000000 
 
 reduced by the amount of 
 loan withdrawn from the 
 operation of the fund ... £4500000 
 
 £10500000 
 
 £10500000 
 
 This statement shows the method of making the correction 
 in the annual instalment, and also the final repayment of the 
 loan. The amended annual instalment may also be found by 
 the annual increment (balance of loan) method, as shown 
 in Statement XVIII. E. 
 
2lS 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A Surplus in the Fund. 
 
 Statement XVIII. E. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find the ameuded auimal sinking- fund instalment, couse- 
 qnent upon a surplus in the fund, arising on the withdrawal 
 of part of the loan from the operation of the fund owing 
 to the conversion of such part of the loan into ordinary 
 share capital or stock. 
 
 Variation IV (Surplus), in which the original annual instal- 
 ment is a stated sum, and is not based, except in a general 
 way, upon any period of repayment or rate of accumulation. 
 
 Amount of original loan (16 years) £15000000 
 
 deduct, portion thereof converted into ordinary 
 share capital and withdrawn from the 
 operation of the fund at the end of 
 the 7th year £45000-00 
 
 £10500000 
 deduct amount in tlie fund at the end of the 
 
 7th year £57468-48 
 
 Balance of loan 
 
 ... £475:U-52 
 
 Amended annual increment to be added to the 
 fund, and accumulated at 3 per cent, to provide 
 this amount at the end of 9 years 
 
 Calculation (XVIII) 15 
 
 deduct income to be received from the present 
 investments (£57468-48) at 3 per cent. 
 
 £4678-71 
 
 £172405 
 
 Amended annual instalment, being : — 
 
 Calculated instalment 
 
 reduced by •• 
 
 £7441-63 
 
 £4480-97 
 
 £2954-66 
 
 £295406 
 
 The final repayment of tlie h)an by tlie operation of the 
 sinking fund, after making the above adjustment in the annual 
 instalment is shown in Statement XVIII. D. 
 
A SURPLUS IN THE FUND 
 
 219 
 
 Pro forma Sinking Fund Account, No. 6, 
 
 A Surplus in the Fund. (Variation IV.) 
 
 Loan of Hr5(),()l)0, repayahJc bi/ a statcil annual insfahncnf of 
 
 £7,600. 
 
 Showing the final repayment of tlie balance of unconverted 
 loan, by the operation of the reduced annual instalment of 
 £2954-660. 
 
 Statement XVIII. D. 
 
 Rate of accumulation, 3 per cent. 
 
 Year. 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 received from 
 investments 
 3 per cent. 
 
 Annual 
 sinking fund 
 instalment. 
 
 Amount in 
 tlie fund 
 at end 
 of year. 
 
 Year. 
 
 1 
 
 Nil 
 
 Nil 
 
 7500 000 
 
 7500000 
 
 1 
 
 2 
 
 7500000 
 
 225000 
 
 7500000 
 
 15225-000 
 
 2 
 
 3 
 
 15525000 
 
 456-750 
 
 7500000 
 
 23181-750 
 
 3 
 
 4 
 
 23181-750 
 
 695-453 
 
 7500000 
 
 31377-203 
 
 4 
 
 5 
 
 31377-203 
 
 941-316 
 
 7500-000 
 
 39818-519 
 
 5 
 
 6 
 
 39818-519 
 
 1194-556 
 
 7500()00 
 
 48513-075 
 
 6 
 
 7 
 
 48513-075 
 
 1455-495 
 
 7500000 
 
 57468-480 
 
 
 8 
 
 57468-480 
 
 1724-054 
 
 2954-660 
 
 62147-194 
 
 8 
 
 9 
 
 62147-194 
 
 1864-416 
 
 2954-660 
 
 66966-270 
 
 9 
 
 10 
 
 66966-270 
 
 2008-988 
 
 2954-660 
 
 71929 918 
 
 10 
 
 11 
 
 71929-918 
 
 2157-898 
 
 2954 660 
 
 77042-476 
 
 11 
 
 12 
 
 77042-476 
 
 2311-274 
 
 2954-660 
 
 82308-410 
 
 12 
 
 13 
 
 82308-410 
 
 2469-252 
 
 2954-660 
 
 87732-322 
 
 13 
 
 14 
 
 87732-322 
 
 2631-970 
 
 2954-660 
 
 93318-952 
 
 14 
 
 15 
 
 93318-952 
 
 2799-569 
 
 2954-660 
 
 99073181 
 
 15 
 
 16 
 
 99073-181 
 
 J. 
 
 2972-159 
 \.mount of loan 
 
 2954-660 
 converted 
 
 105000000 
 
 16 
 
 
 45000000 
 
 
 Comparison of Methods Previously Discussed. This 
 concludes the examination of the various methods of adjusting 
 a sinking fund to compensate for a difference between the actual 
 amount in the fund and the amount which should be in the 
 fund at any time in order to carry out the original obligation ; 
 and the results may be summarised as follows. 
 
220 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 In the case of the adjustment (Yariation I) caused by a 
 deficiency in the fund, as shown by Statement XYI, A, in 
 Chapter XYI, the deficiency was corrected by an additional 
 sinking fund instalment to be set aside during the whole of the 
 unexpired portion of the repayment period, as shown by 
 Calculation (XYI) 1. 
 
 In Yariation II, described in Chapter XYII, Statement 
 XYII, A, relating to a surplus of £4,560, being the proceeds 
 of sale of assets, paid into the fund, a different method was 
 adopted. In that case there was an actual increase in the cash 
 assets of the fund which operated in two ways, (1) by increasing 
 the future income of the fund, in consequence of which the 
 present sum of £4,560 will ultimately repay, by accumulation, 
 £7131"64 of original loan, and (2) by reducing the amount of 
 loan ultimately repayable by £7131"64, it relieved the future 
 years of the amount of the sinking fund instalment (£442'60) 
 equivalent to that amount for the unexpired portion of the 
 repayment period of 13 years, which is the annuity which 
 might now be purchased with the sum of £4,560. In the two 
 Yariations III and lY which have just been considered there is 
 a surplus in the fund caused by the withdrawal of part of the 
 loan from the operation of the fund. There is here no actual 
 addition to the assets of the fund, as in the case of the payment 
 into the fund of the proceeds of sale of part of the security for 
 the loan. The surplus may in effect be considered as a 
 lightening of the burden previously borne by the undertaking 
 measured by the reduction in the amount of loan to be 
 ultimately provided. Consequently the surplus operates in one 
 direction only, namely, by reducing the original annual instal- 
 ment to be set aside, whether that instalment was arrived at 
 by calculation in the ordinary manner or was a round sum 
 specified in the trust or other deed under which the fund was 
 instituted. 
 
 Further Problems. There are other problems which may 
 arise in connection witli the sinking funds of commercial or 
 financial undertakings, but which have not been treated in an 
 exhaustive manner, because they may be solved by one or other 
 of the methods elsewhere described. Thev are as follows :■ — 
 
 Redemption uy Drawings : 
 
 // annxud, they nuiy be considered on tlie lines of the 
 instalment method of local authorities. (Chapter XI.) 
 
A SURPLUS IN THE FUND 221 
 
 7/ at fie nods of years, a sinking fund may te set aside 
 during each period to provide tlie proportion of tlie loan 
 repayable at the end of each period. 
 
 If at 2>cnods of years, in a series, a sinking fund may be 
 provided by setting aside and accumulating equal annual 
 amounts during the whole period in order to provide the 
 amounts repayable at the end of each period. This will 
 apply to the simultaneous provision out of profits of loans 
 repayable in certain priorities. 
 
 Redemption of Loans (Issued as Stock) at a Premium : 
 
 If the premium be stated, the sinking fund instalment 
 should be calculated to provide that amount in addition 
 to the par value, and there is not any change in the 
 method described. 
 
 If the premium depends upon the price at the date of 
 redemption, and cannot be accurately estimated, the 
 annual instalment should be based upon the par value of 
 the stock, and the premium provided for, as and when it 
 arises, by charging it to revenue account, or by making 
 prudent provision in anticipation. 
 
 Redemption of Loan in Part. 
 
 The trust deed may provide that if any part of the loan be 
 redeemed out of the fund, the interest previously paid upon 
 such redeemed loan shall be added to the fund, although the 
 rate of interest payable to the loanholder be higher than the 
 calculated rate of accumulation of the fund. This will cause 
 a surplus in the fund over the calculated amount, which will 
 liave the eifect of anticipating the final matvirity of the fund, 
 whether the loan is repayable on a specified date or by the 
 accumulation of a stated instalment. The possibility of 
 making any provision for such an event when calculating the 
 original instalment in the case of an ordinary sinking fund 
 will depend upon the circumstances of each individual case. 
 
 Cessation of Annual Contributions. Instead of making 
 the adjustment by spreading any surplus, however arising, 
 equally over the unexpired portion of the repayment period, it 
 may be provided that the amount in the fund shall continue 
 to accumulate, and the original instalments be annually paid 
 in, until such time as the fund is of such an amount that the 
 
222 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 present investments and the accumulations of the annual income 
 to be received therefrom in future will be sufficient, without 
 any further instalments, to provide the amount of loan re- 
 payable. (See Article 11 (2) County Stock Reo-ulations, 1891.) 
 
 CoxTixuATiox OF INSTALMENTS, It may be provided that 
 the original instalment shall continue to be set aside and added 
 to the fund until the loan is ultimately repaid, notwithstanding : 
 
 (1) The withdrawal of any part of the loan from the operation 
 
 of the sinking fund by reason of its being converted into 
 ordinary share capital or stock. 
 
 (2) The sale of any part of the assets forming part of the 
 
 security for the loan, and the payment of the proceeds 
 into the fund. 
 
 (3) Any other cause operating to produce a surplus in the 
 
 fund or to accelerate the date of maturity of the fund. 
 
 In such cases it may be necessary to determine the reduced 
 period of redemption which may be ascertained by one or other 
 of tlie methods described. 
 
THE RATE OF ACCUMULATION 223 
 
 CHAPTER XIX. 
 
 SINKING FUND PIIOBLEMS, RELATING TO 
 THE BATE PER CENT., 
 
 OF INCOME UPON THE PRESENT INVESTMENTS REPRESENTING 
 THE AMOUNT IN THE FUND; AND ALSO THE FUTURE RATE OF 
 ACCUMULATION OF THE FUND. 
 
 Variation A, in which there is a variation in the 
 
 RATE OF accumulation WITHOUT ANY VARIATION IN THE 
 
 rate of income upon the present investments, or in 
 the period of repayment. statement xix. b. 
 
 Summary of the methods of adjustment. General con- 
 siderations AS to variations in the rate per cent, to be 
 treated in detail in the following chapters. The 
 deductive method. Statement showing the final re- 
 payment OF THE loan by THE OPERATION OF THE AMENDED 
 ANNUAL INSTALMENT. 
 
 Summary of the methods of adjustment. 
 
 (7) The dedaicti've method, as summarised helotv, is of wider 
 application than the variation in the rate of accumulation only, 
 and has been so uwrded that it may he treated as the standard 
 method relating to all variations. Statement XIX. A. 
 
 {II) The direct method, without calculation, as summarised 
 at the head of Chaper XX, will not apply to this variation. 
 
 {Ill) The annual increment {balance of loan) method, as 
 summarised at the head of Chapter XXII, may be used, but 
 ivill not be applied to the example under revieiv. The method 
 of finding the amended annual increment is shown in Calcula- 
 tion {XIX) 5. 
 
 {IV) The annual increment (ratio) method, as summarised 
 at the head of Chapter XXIII. Statement XXII. C. 
 
 Note. The terms used in the following summary are fully 
 explained at the head of Chapter XXII. In all the above 
 methods, it is imperative that the rate of aecumulation and of 
 income from investments be uniform during the whole of the 
 une.vpired or substituted portion of the repayment period. 
 
224 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Summary of the deductive method, of ascertaining the 
 amended annual sinhinc) fund instalment due to a variation in 
 the rate yer cent, of accumulation, accompanied by, or without, 
 any variation in the rate of income to he received upon the 
 present investments repi-esenting the fund, and also due to any 
 variation in the period of repayment, or any combination of the 
 above factors. Statement XIX. A. 
 
 (1) Ascertain the value of the present investments in the 
 
 manner already described, and also the amount of the 
 present annual incowe yielded by such investments, up 
 to the time of malting the adjustment. 
 
 (2) To the present annual income, so ascertained, add the 
 
 present or original annual instalmeyit which has been set 
 aside and added to the sinl^ing fund up to the time of 
 making the adjustment . 
 
 [3] The total so obtained is the present annual increment of 
 the fund. 
 
 {4) Ascertain, or estimate, the rate per cent, at which the 
 fund will accumulate in future (the future rate). 
 
 (5) Calculate {in one sum or separately) the amount of the 
 
 present annual increment found, as in (3), for the number 
 of years in the ^inexpired or substittited period of repay- 
 ment, at the future rate of accumulation fxed in (4). 
 
 Calculations {XIX) 1 and 2. 
 
 (6) The amount or amourits, so ascertained, will represent the 
 
 portion of original loan which will be provided at the 
 end of the original or varied period of repayment . 
 
 (7) To this amount add the value of the present invest- 
 
 ments, as ascertained in (1), and deduct the sum from the 
 amount of the origimd loan. 
 
 (S) The remainder represents the portion of original loan 
 which is now ^inprovided for by the present investments 
 and the future accumulation of the present annual incre- 
 inent found in {3). 
 
 (9) Calculate the additional annual sinMng fund instalment 
 wliich, at the future rate of accumulation^ estimated as 
 in [4), will amount to the balance of loan found in (8) 
 at the end of the unexpired or substituted period of 
 repayment. Calculation [XIX) 3. 
 
THE RATE OF ACCUMULATION 225 
 
 (10) This additional annual instahnent, added to the ^present 
 
 annual increment found in {3) gives the same future or 
 amended annual increTnent, which is found by direct 
 calculation hy the annual increTnent {ratio) Tnethod. 
 
 (11) From the future, or amended annual increment, so ascer- 
 
 tained, deduct the future annual income from the present 
 investments ; and the remainder is the future or amended 
 annual instalment to be charged to revenue or rate in 
 substitution for the present or original annual instalnnent. 
 
 (12) Prepare a, statement showing the final repayment of the 
 
 loan by the operation of the fund under the amended 
 conditions . Statement XIX. B. 
 
 (13) Prepare a pro forma account sltowing the amount which 
 
 should be in the fund at the end of each year of the 
 unexpired or substituted period of repayment. 
 
 Pro forma Account, No. 7. 
 
 Memo. The above method is worded to apply to a reduction 
 in the rate of accumulation or other factor, hut it unll apply 
 equally to an increase in suck factors with very little modi f ca- 
 tion. It should be compared with the deductive method 
 summarised at the head of Chapter XXIV . 
 
 General Considerations as to the Rate per cent. 
 
 Having described the various methods of dealing with 
 problems arising out of a deficiency or a surplus in the sinking 
 fund, further questions will now be considered in connection 
 with the rate per cent., beginning Avith cases in which it is 
 anticipated that the original estimated rate of accumulation 
 will not be realised in future. This is mainly due to a 
 fluctuation in the money market of a more or less permanent 
 character affecting the future return on all investments. 
 Questions will also arise in consequence of a reduction in the 
 rate of income to be received in future on investments already 
 made, as was the case in 1888, when, under Mr. Goschen's 
 Finance Act, the rate of Consols was reduced from 3 per cent, 
 to 2f per cent, for 15 years, after which a further reduction to 
 2i per cent, took place. Other causes may operate in a similar 
 manner, especially in the case of commercial and financial 
 undertakings. 
 
 The problem will differ according as the variation in the 
 original conditions affects: — 
 
226 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (1) The rate of accumulation anticipated to be realised on 
 
 tlie investment of future accretions to the fund. 
 
 Variation A. 
 
 (2) The rate of income to be received on the present invest- 
 
 ments representing the fund. Variation B. 
 
 (3) Both the above rates in combination. Variation C. 
 
 In making the adjustments it will at times be difficult to 
 forecast accurately the future rate of income to be received on 
 the present investments. In such cases it is wise to form a 
 conservative estimate of the future rate and fix it on the low 
 side; or to take a slightly lower rate of accumulation and 
 therebv increase the annual instalment to be charged to revenue 
 or rate account. In discussing the following variations it will 
 be assumed that although the future rate of income to be 
 received upon the present investments will change, yet it will 
 be uniform during the whole of the unexpired repayment 
 period. But cases may arise in which this will not be so, but 
 in which the rate of income will again vary, during the term, 
 in a definite manner laid down in advance, as in the case of 
 Consols previously referred to. A variation of this nature, 
 occurring during the unexpired portion of the repayment period, 
 will be deferred to Chapter XXYII. When considering 
 Variation B (rate of income only) in Chapter XX, it will be 
 found that the future rate of accumulation is the most 
 important factor in the adjustment, although it may not be the 
 greater as regards the actual amount of money involved. 
 
 The following discussion will be confined to a reduction 
 only in both the above rates per cent., but it should be borne 
 in mind that the method to be adopted and described will apply 
 equally to an increase in both rates or to an increase in one and 
 a decrease in the other. This will be better appreciated after 
 considering the methods of making the adjustment by the 
 annual increment (ratio") method. 
 
 Any deficiencv in the fund at the time of making the 
 enquiry, and arising out of a reduction in the rate of income 
 received from investments previously made, or from other 
 causes, will not affect the present method of colrulation. Any 
 such deficiencv may or may not be discovered on ascertaining 
 the present position of the fund as described in the previous 
 chapter. The following method differs from the one there 
 described, in that, in the present example, the basis of the 
 adjustment is the value of the present investments, and not the 
 omonnt to which thev Avill accumulate at the end of the term. 
 
THE RATE OF ACCUMULATION 227 
 
 In dealing- with a deficiency, it was assumed that there 
 would not be any variation in the rate of accumulation, whereas 
 in the present example the reduction in the rate of accumula- 
 tion is the cause of the rectification under discussion. 
 
 In an actual enquiry of this nature, the amount in the fund 
 at the end of the 12tli year, as shown by the records, would most 
 probably be compared with the calculated amount which should 
 be in the fund according to the pro forma account, and the 
 deficiency or surplus thereby ascertained, but it is not absolutely 
 necessary to do this. The important factors are, the value of 
 the present investments, the future income they may be 
 expected to produce, and the rate of accumulation which will be 
 yielded by the investment of the future accretions to the fund. 
 In this connection Chapter XIY, dealing generally with the 
 present investments and the annual increment should be 
 consulted, especially as to the meaning of the term " present 
 investments." The deductive method will apply to the rectifi- 
 cation of a present deficiency or surplus combined with a 
 variation in the future rates of income or accumulation, because 
 in this case the enquiry is based upon the value of the invest- 
 ments now representing the fund ; and the method of approach- 
 ing the problem is not altered because that value is greater or 
 less than the amount which should be in the fund according 
 to the original calculation, and as shown by the pro forma 
 account. The method about to be described will show the 
 amended annual instalment to be charged to revenue or rate, 
 based upon the present state of the fund, but if it be required 
 to allocate this as between a present deficiency or surplus and 
 the future reduction in the rates of income or accumulation, it 
 will be necessary to make, first, the calculation as to the 
 deficiency or surplus, as already described, followed by the 
 enquiry as to the increased annual instalment due solely to the 
 fall in the rate or rates per cent. 
 
 Details of the Sinking Fund. The sinking fund which 
 will be used to illustrate all problems relating to a variation in 
 the rate per cent, will apply to a loan of £26,495, repayable at 
 the end of a period of 25 years, requiring an annual instalment 
 of £680234 to be set aside and accumulated at 3| per cent. 
 [Calculation (XV) 1], and it will in all cases be assumed, as 
 when considering the rectification of a surplus, that at the end 
 of the 12th year the fund stands at the proper calculated 
 amount of £9932-74, as found by Calculation (XY) 2. This 
 sum is represented by investments worth that amount which 
 
228 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 have up to the present yielded an annual income at the rate of 
 3^ per cent, per annum, being the original estimated rate of 
 accumulation upon which the above instalment was based. 
 This sum of £9932" 74, if accumulated at the above rate of 
 3| per cent., will provide for the repayment of £15534"38 of 
 original loan at the end of 25 years, as found by Calculation 
 (XVII) 2. 
 
 Variations in the Rate Per Cext., to be considered in 
 Detail. In order to illustrate the problems to be discussed in 
 this and following chapters three variations from the original 
 conditions as regards the rate of accumulation of 3^ per cent, will 
 be considered. In fixing this rate percent, in the first instance it 
 was assumed that it would continue to be received upon the 
 whole of the accumulations of the fund during the whole of the 
 repayment period of 25 years. If this anticipation had been 
 realised the rate per cent, of income upon investments and the 
 rate per cent, of accumulation would have been the same in all 
 cases, namely, 3^ per cent., and the fund would have pursued 
 its calculated course until maturity. 
 
 In the three examples about to be considered a gradual 
 decrease in the rate of income from investments, as well as in 
 the rate of accumulation, will be assumed to occur between each 
 set of conditions; but when comparing the several results in a 
 later chapter they will be considered only as regards the altera- 
 tion in the rate of accumulation as follows : — 
 
 Future rate of 
 
 income Future 
 
 on present rate of 
 
 Chapter. Variation. Compared witli investments. accumulation. 
 
 XIX A Original conditions unaltered reduced 
 
 XX B Variation A reduced unaltered 
 
 XXI C Variation A reduced reduced 
 
 The paramount importance of the rate of accumulation in 
 such problems has already been referred to, and it will be 
 noticed from the above table that Variations (A) and (C) alone 
 contain any variation in that rate. The following details as 
 to each variation are given for convenience of reference and 
 comparison : — 
 
 Chapter XIX. Variation {A) in the rate of accunndation onh/. 
 Compared with the conditions at the time the 
 
 original calculation was made. 
 In this example the rate of accumulation is 
 reduced from 3^ to 3 per cent., but the rate 
 of income upon the present investments 
 remains at 3^ per cent. 
 
THE KATE OF ACCUMULATION 229 
 
 Chapter XX. Variation {B) in the rate of income upon tlte 
 present investTnents only. 
 Compared with the couditions iu Variation (A). 
 In this example the rate of accumulation is 
 unaltered, and remains at 3 per cent., but 
 the rate of income upon the present invest- 
 ments is redviced from 3^ to 3 per cent. 
 
 Chapter XXI. Variation (C) in the rate of accumulation, as 
 well as in the rate of income upon the 
 present investments. 
 
 Compared with the conditions in Variation (A) . 
 
 In this example the rate of accumulation is 
 reduced from 3 to 2| per cent., and the rate 
 of income upon the present investments is 
 reduced from 3| to 3 per cent. 
 
 These variations will now be examined, and will be treated 
 as independent problems instead of variations of the same fund. 
 This procedure involves a certain amount of repetition, but is 
 adopted in order to emphasize the principles involved, with the 
 view of finding a shorter method of making the adjustments. 
 There is also a further advantage, namely, that each problem 
 may be studied separately so that any cases occurring in actual 
 practice may be referred to a similar example completely 
 worked out in detail. 
 
 It will be noticed on referring to the above details and to the 
 summary of results given in Chapter XXI, Statement XXI, C, 
 that the above variations are not isolated cases without any 
 connection. They are intimately related by design, and form a 
 series commencing with the original conditions and leading by 
 successive stages to Variation C (rate of income and accumula- 
 tion). When considering the derivation of a rule and formula 
 relating to the adjustment of a sinking fund in consequence of a 
 simultaneous variation in the rates per cent, of accumulation 
 and income on investments these variations will be combined, 
 and in one instance (Calculation XXII, C), Variation A will be 
 inverted to serve as an example of an increase in the rate per 
 cent, of accumulation. 
 
 Any decrease in the rate of income yielded by the present 
 investments or by the future investments of the annual 
 accretions to the fund will have the effect of reducing the sum 
 to Avhich the fund will amount at the end of the repayment 
 period. The amount of such deficiency will depend upon the 
 actual rates to be received in future as compared with the 
 
230 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 original rate of accumulation, namely, 13^ j)er cent. It is 
 necessary so to adjust the sinking fund that the deficiency due 
 to a fall in the rate either of income or of accumulation shall 
 not only be made good, hut be spread equally over the remain- 
 ing 13 years, by increasing the original sinking fund instalment 
 by such an annual amount as will be sufficient for the purpose. 
 
 The Deductive Method. In order to ascertain the amount 
 by which the annual instalment should be increased, the present 
 sinking fund factors may be reduced, either to terms of present 
 value or to equivalent amounts of original loan repayable at 
 the end of the 25 years, but, as in the former example, it is 
 preferable to deal with the figures representing equivalent 
 amounts of loan. 
 
 In each of the above variations the common factors are : — 
 
 (1) A sum of £9932' 74 standing to the credit of the fund at 
 
 the end of the 12th year, which is invested and expected 
 to realise that sum at the end of the repayment period. 
 
 (2) The income arising from the above present investments. 
 
 (3) The original annual instalment of £680234 to be set aside 
 
 for the unexpired term of 13 years, and which will also be 
 invested each year. 
 
 (4) The income to be received annually from (2) and (3) when 
 
 invested. 
 Items (2) and (3) constitute the present annual increment of the 
 fund, as described in Chapter XIV and in Chapter XXII. 
 
 In each case the original annual instalment of £680'234 will 
 be supplemented by an additional annual instalment to be ascer- 
 tained, and which, added to the present annual increment, Avill 
 give the future or amended annual increment of the fund. The 
 method of approaching the solution of the problem is the same 
 in each variation. 
 
 A statement will be prepared similar to XIX, A, showing 
 the position of the fund at the end of the 12th year, when the 
 assumed necessity arises to make the adjustment due to a 
 change in the rate per cent, either of income or accumulation 
 or both. This statement will commence with the amount now 
 in the fund, which will be included at its present value without 
 accumulation. This is equivalent to deducting that amount 
 from the original loan, leaving the balance to be provided by 
 the accumulation of the future or amended annual increment 
 which is composed of the future income from the present 
 investments and the amended annual instalment. 
 
THE RATE OF ACCUMULATION 231 
 
 This is a departure from the procedure followed previously 
 in dealing with a surplus or a deficiency in the fund, in which 
 cases there was not any change in either of the rates per cent. 
 
 The above Statement XIX, A, will next include the present 
 annual increment consisting of the income from the present 
 investments prior to the variation occurring, and also the 
 original annual instalment. Both these annual sums will be 
 converted, by calculation at the future accumulation rate, into 
 equivalent amounts of original loan repayable at the end of the 
 unexpired period. The balance will represent the amount of 
 original loan for which further provision has to be made caused 
 by the decrease in the rates of income or of accumulation, and 
 from this balance of loan the required additional annual 
 instalment may be ascertained on standard calculation form. 
 No. 3x. There is a diierence in the method of treating the 
 income from investments in Statements XX, A, and XXI, A, 
 as compared with Statement XIX, A, but they may all be 
 treated by the deductive method summarised at the head of this 
 chapter. A further statement similar to XIX, B, is then 
 prepared in each case showing how the fund will ultimately 
 work out to repay the full amount of the loan at the end of the 
 original repayment period. 
 
 Having ascertained the future or amended annual instalment 
 in each case by the deductive method, the results will 
 afterwards be used to derive therefrom a simple rule and 
 formula by which to make the calculation by direct reference 
 to the published tables or formulae. It will then be found that 
 by taking the present annual increment as the prime factor 
 instead of the annual instalment, all such variations may be 
 divided into two classes depending entirely upon the rate of 
 accumulation. In variations similar to A and C, in which the 
 rate of accumulation is reduced or increased, a calculation must 
 be made by means of the tables or formula, but in variations 
 similar to B, where there is a variation in the rate of income 
 only, the rate of accumulation remaining unaltered, the 
 amended annual instalment may be ascertained without 
 calculation. This method is shown in Statement XX. C. called 
 " the direct method," and it may appear superfluous to include 
 the longer deductive method shown in Statement XX, A. 
 
 It is necessary, however, to state that in all cases the income 
 from investments has been treated as being received annually, 
 whereas in all probability it would be received half-yearly. 
 The difference between an annual and a semi-annual accumula- 
 tion has been pointed out at the end of Chapter V, giving also 
 
232 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 the methods to be adopted in the case of any periodic accumula- 
 tion other than annual. 
 
 The Annual Increment (Ratio) Method. It has already- 
 been stated that instead of using the above deductive method 
 the same result may be obtained by direct calculation by means 
 of a formula and rule. This will be fully described in 
 Chapter XXII (Calculation XXII, C), which shows the future 
 or amended annual increment of £1060474 as found by the 
 above deductive method. From the amended annual increment 
 so found the future or amended annual instalment niay be 
 obtained by deducting therefrom the future annual income 
 from the present investments. This ratio method by direct 
 calculation will also apply to Variation C, where there is also a 
 change in the rate of accumulation, but in the case of 
 Variation B, in which the rate of accumulation remains un- 
 altered, the amended annual instalment can be ascertained by a 
 much more direct method without calculation, as explained 
 above. 
 
 The method of finding the amended annual instalment is 
 shown in the following Statement XIX, A, and the final 
 repayment of the loan thereby is shown in Statement XIX, B, 
 and in the pro forma account 'No. 7. 
 
 The Rate per cent. Statement XIX. A. 
 
 The Deductive Method. 
 
 Variation A, rate of accumulation only. 
 
 Showing the method of adjusting the annual instalment^ in 
 consequence of a variation in the rate of accumulation 
 without any variation in the rate of income from the 
 present investments, or in the period of repayment. 
 
 This example is compared with the conditions at the 
 time the original calculation was made. 
 
 Conditions before adjustment (at end of 12th year) 
 
 Amount of loan repayable in 25 years £26,495 
 
 Amount in the fund (at end of 12th year) £9932-74 
 
 Present annual income (previously) received there- 
 from, at 3^ per cent, per annum £347*648 
 
 Present annual instalment, to be accumulated for 
 
 13 years at 3| per cent £6(S0"234 
 
 Present annual increment £1027'882 
 
THE RATE OF ACCUMULATION 233 
 
 Variation from the above conditions : — 
 
 The rate of accumulation of the fund is reduced from 3| to 
 3 per cent. 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments (at end of 12th year), 
 
 representing the amount now in the fund ... £9932"74 
 
 Present annual income from investments ;— 
 
 Amount of an annuity of £347648 
 
 accumuhited for 13 years, at 3 per cent. 
 
 Calculation (XIX) 1 £5429-49 
 
 Original annual instalment : — 
 
 Amount of an annuity of £680"234 
 
 accumulated for 13 years, at 3 per cent. 
 
 Calculation (XIX) 2 £10623-75 
 
 Present annual increment £1027882 
 
 Provision already made will repay loan of £25985-98 
 
 Additional annual instalment :— 
 
 Balance, heing amount of original loan un- 
 provided for owing to the above decrease in 
 the rate of accumulation requiring an 
 additional annual instalment, to he set aside 
 and accumulated for 13 years at 3 per cent. £509-02 
 Calculation (XIX) 3 £32-592 
 
 Amount of original loan £26495-00 
 
 Amended annual increment, being : — 
 
 Income from investments ... £347-648 
 Amended annual instalment ... 712-826 
 
 £1060-474 
 
234 
 
 REPAYMENT OF LOCAL AND OTHER. LOANS 
 
 The Rate per cent. 
 
 Statement XIX. B. 
 
 Variation A, rate of accumulation only. 
 
 Showing the final repayment of the loan, by tlie operation of 
 the sinking fund after making the adjustment in the 
 annual instalment, consequent upon a variation in the rate 
 of accumulation, without any variation in the rate of 
 income upon the present investments, or in the period of 
 repayment. 
 
 Present investments (at end of 12th vear) 
 
 Equivalent 
 
 amount r{ 
 
 original loan. 
 
 £9932-74 
 
 Amended annual increment :— 
 
 Original annual instalment ... 
 Additional annual instalment 
 
 £680-234 
 32-592 
 
 Total out of revenue £712'826 
 Income from investments 347-648 
 
 £1060-474 
 
 Amount thereof, accumulated for 13 years at 
 
 3 per cent. Calculation (XIX) 4 £16562-26 
 
 Amount of original loan 
 
 £26495-00 
 
THE RATE OF ACCUMULATION 
 
 235 
 
 Pro forma Sinking Fund Account, No. 7. 
 
 A Variation in tlie Hate of Accumulatiou only. 
 
 Loan of £26,495 re'payahle at the end of 25 years. 
 
 Showing the final repayment of the loan, by the operation of 
 the increased annual instalment of £712'826. 
 
 Statement XIX. B. 
 
 Rate of accumulation, 3 per cent. 
 
 Year. 
 
 Amount in 
 the fund 
 
 at beginning 
 of year. 
 
 Income 
 received from 
 investments 
 31 per cent. 
 
 Annual i 
 sinking fund 
 instalment. 
 
 received from Amount in 
 nvestments made the fund 
 after l-2th year at end 
 3 per cent. of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 
 3 
 
 4 
 
 The 
 
 amount in the fun( 
 
 i at the end of 
 
 4 
 
 5 
 
 the 12th year, 
 
 £9932-744, is the correct 
 
 5 
 
 6 
 
 calculated amo 
 
 unt, as sill 
 
 own by Ca 
 
 Icula- 
 
 6 
 
 7 
 
 tion 
 
 (XV) 2, 
 
 and by 
 
 the- pro 
 
 forma 
 
 
 8 
 
 account, No. 1 
 
 , Chapter 
 
 XV. 
 
 
 8 
 
 9 
 
 
 
 
 
 
 9 
 
 10 
 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 347-648 
 
 712-826 
 
 — 
 
 10993-218 
 
 13 
 
 14 
 
 10993-218 
 
 347-648 
 
 712-826 
 
 31-814 
 
 12085-506 
 
 14 
 
 15 
 
 12085-506 
 
 347-648 
 
 712-826 
 
 64-583 
 
 13210-563 
 
 15 
 
 16 
 
 13210-563 
 
 347-648 
 
 712-826 
 
 98-335 
 
 14369-372 
 
 16 
 
 17 
 
 14369-372 
 
 347-648 
 
 712-826 
 
 133-099 
 
 15562-945 
 
 17 
 
 18 
 
 15562-945 
 
 347-648 
 
 712-826 
 
 168-906 
 
 16792-325 
 
 18 
 
 19 
 
 16792-325 
 
 347-648 
 
 712-826 
 
 205-787 
 
 18058-586 
 
 19 
 
 20 
 
 18058-586 
 
 347-648 
 
 712-826 
 
 243-775 
 
 19362-835 
 
 20 
 
 21 
 
 19362-835 
 
 347-648 
 
 712-826 
 
 282-903 
 
 20706-212 
 
 21 
 
 22 
 
 20706-212 
 
 347-648 
 
 712-826 
 
 323-204 
 
 22089-890 
 
 22 
 
 23 
 
 22089-890 
 
 347-648 
 
 712-826 
 
 364-714 
 
 23515-078 
 
 23 
 
 24 
 
 23515-078 
 
 347-648 
 
 712-826 
 
 407-470 
 
 24983-022 
 
 24 
 
 25 
 
 24983-022 
 
 347-648 
 
 712-826 
 
 451-504 
 
 26495000 
 
 25 
 
236 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XX. 
 
 SINKING FUND PROBLEMS, RELATING TO THE 
 RATES PER CENT. OF INCOME AND ACCUMULA- 
 TION {Co7itinued) . 
 
 Variation 13, in which there is a variation in the rate 
 OF income upon the present investments without any 
 
 VARIATION IN THE RATE OF ACCUMULATION OR PERIOD OF 
 
 Repayment. Statement XX. A. 
 
 Summary of the methods of adjustment. The deductive 
 METHOD. The direct method without calculation. 
 The annual increment (balance of loan) method. 
 Statement showing the final repayment of the loan by 
 
 THE operation OF THE AMENDED ANNUAL INSTALMENT. 
 
 Summary of the methods of adjustment. 
 
 (/) The deductive tnethod, as summarised at the head of 
 Chapter XIX, ivill not ayfly, since there is not any variation 
 in the rate of accumulation. The following adjustment by the 
 deductive method is only of academic interest and has hardly 
 any practical value. Statement XX. A. 
 
 (IT) The direct method, without calculation^ as summarised 
 below, should always he used in acttial practice. 
 
 Statement XX, C. 
 
 {Ill) The annual increment [balance of loan) method, as 
 summarised at the head of Chapter XXll, may be used. 
 
 Statement XX. D. 
 
 {IV) The annual increment [ratio) method, as summarised 
 at the head of Chapters XXIII, XXV ^ and XXVI, will not 
 apply to tit is variation, as there is not any change in the rate 
 of accumulation. 
 
 Note. The terms used in the following summary are fully 
 explained at the head of Chapter XXII. If it he l-nown or 
 anticipated that the rate of income to be yielded in future by 
 the present investments representing the fund, toill not be 
 uniform during the whole of the unexpired portion of the 
 repayment period the above methods will not apply, and the 
 
THE RATE PER CENT. OF INCOME 237 
 
 adjustment miist he viade by the metlind fully described in 
 Chapter XXV 17. 
 
 vSuMMARY OF THE DIRECT METHOD (tvithout Calculation), 
 of ascertaining the amended annual sinTcing fund instalment 
 due to a variation in the rate of income yielded by the 'present 
 investments without any variation in the rate of accumulation 
 or in the period of repayment. StatCTnent XX. C. 
 
 (i) Having ascertained the value of the present investTnents 
 in the manner already described, 
 
 (2) Calculate the annual income previously received there- 
 
 from during the expired portion of the original repay- 
 ment period {the present annual incoTne). 
 
 (3) Calculate the annual income expected to be received 
 
 therefrom during the unexpired portion of the original 
 repayment period at the future rate per cent, of income 
 {the future annual income). 
 
 (4) Ascertain the decrease or increase in such future annual 
 
 income as compared ivith the annual income previously 
 
 received. 
 {6) Add to, or deduct from, the original annual instalment 
 
 the anmial decrease or increase of income so ascertained. 
 {6) The result is the amended annual instalment to be set 
 
 aside out of revenue or rate during the unexpired portion 
 
 of the original repayment period. 
 
 (7) Prepare a statement showing the final repayment of the 
 
 loan by the operation of the sinking fund under the 
 amended conditions. Statement XX. B. 
 
 (8) Prepare a pro forma account showing the amount which 
 
 should be in the fund at the end of each year of the 
 unexpired repayment period. 
 
 Pro forma account, No. 8. 
 The amounts in the fund at the end of each year u'ill be 
 the same as in the original pro forma account since there 
 is not any variation in the rate of accumulation or period 
 of repayment, but the annual increment, although 
 unaltered will have a different origin. Pro forma 
 account A^o. 1, Chapter XV ^ will not apply in this case, 
 the rate of acciimulation being 3\ per cent. 
 
 The Deductive Method. After discussing the deductive 
 method of finding the amended annual instalment due to a 
 change in the rate per cent, of accumulation (Variation A) in 
 
238 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Chapter XIX, a summary of tlie successive stages of tlie adjust- 
 ment has been prepared and placed at the head of that chapter, 
 and it is therefore only necessary to refer to that summary. 
 Attention has already been drawn to the general considerations 
 to be borne in mind in the rectification of a sinking fund in 
 consequence of any variation in the rates per cent, of income or 
 accumulation. The variation about to be considered is based 
 upon the same imaginary sinking fund as Yariation A (rate of 
 accumulation), details of which are given in the previous 
 chapter. At the end of the 12th year the sinking fund stands 
 at the proper calculated amount of i69932'74, as found by 
 Calculation (XV) 2. But whereas the conditions in Variation A 
 (rate of accumulation) were compared with the original condi- 
 tions, the present Variation B (rate of income), will be compared 
 with the conditions in Variation A (rate of accumulation). 
 
 The rate of income is reduced from 3^ to 3 per cent., but 
 the rate of accumulation is unaltered, and remains at 3 per cent. 
 
 It has been stated in the previous chapter that the future 
 rate of accumulation is the most important factor in the 
 adjustment. That conclusion was based, in advance, upon the 
 results of the discussion of the present variation, because, 
 although the same deductive method will be used which has 
 been applied to Variation A (rate of accumulation), this method 
 is quite unnecessary in practice, although it is instructive as 
 illustrating the predominant effect of the variation in the rate 
 of accumulation. 
 
 It will be found that when the variation in the rate per cent, 
 applies only to the rate of income from the present investments 
 there is not any necessity to make .any calculation whatever 
 beyond adding to the original annual sinking fund instalment 
 an amount equal to the annual loss of income caused by the 
 reduced yield per cent, of the present investments, or by 
 deducting therefrom any increase in such annual income. The 
 remarks in tbe previous chapter, as to the three variations being 
 derived by successive stages from the original conditions should 
 be carefully remembered, and will be further emphasised. 
 
 The original and varied conditions are given in the following 
 Statement, XX. A., and attention is again drawn to the fact 
 that in this case also the income from investments is treated as 
 being received annually, instead of semi-annually. Two state- 
 ments will be prepared exactly similar in principle to those in 
 the previous chapter, dealing Avith Variation A (rate of 
 accumulation), showing in XX. A. the deductive method of 
 ascertaining the amended annual instalment, and in XX. B. 
 
THE RATE PER CENT. OF INCOME 239 
 
 the final repayment of the loan by the operation of the sinking 
 fund, under the altered conditions. For the purpose of the 
 comparison to be made later, this variation will also be 
 compared with the original conditions. (See Statements XX. A. 
 and XX. B, at end of chapter.) 
 
 The Direct Method (without calculation). It has been 
 pointed out in the previous chapter dealing with a variation 
 in the rate of accumulation only that instead of making use 
 of the deductive method, there described, for the purpose of 
 ascertainino- the amended annual instalment, the same resiilt 
 may be obtained by direct calculation by means of a rule and 
 formula, which will be fully described in Chapter XXIII, 
 namely, the annual increment (ratio) method. This remark 
 applied to Variation A as compared with the original conditions 
 in which there is a reduction in the rate of accumulation, but 
 without any variation in the rate of income from investments. 
 In the present case, Variation B, as compared with the condi- 
 tions in Variation A (rate of accumulation) there is a reduction 
 in the rate of income upon the present investments, without 
 any variation in the rate of accumulation, and the deductive 
 method will again be used. On comparing the two results, it 
 is found that in both cases the future or amended annual 
 increment is £1060-474, although the amended annual instal- 
 ment is increased, namely, from £712'826 in Variation A to 
 £762"490 in Variation B. The difference between the two 
 amended annual instalments is £49' 664, which is the amount by 
 which the future annual income in Variation A is reduced 
 owing to the fall of | per cent, in the rate of income to be 
 yielded by the present investments nnder the altered conditions 
 of Variation B, namelv, from £84T'648 in Variation A to 
 £297-984 in Variation B. 
 
 This proves that when the rate of accumulation remains 
 unaltered, there is not any alteration in the annual increment, 
 and, further, that the amended annual instalment may be 
 ascertained without any calculation whatever, by merelv 
 adding to tlie present annual instalment the amount of the 
 decrease in the annual income to be received from the present 
 investments under the altered conditions, and the same applies 
 equally to an increase in the rate per cent, yielded by the 
 present investments. 
 
 The following Statement XX. C. illustrates the ndjustment 
 by the direct method, without calculation. 
 
 Althoufrh the direct method of finding the amended annual 
 
240 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 instalment will be sufficient in all cases where there is not any 
 variation in the rate of accumulation, it should be proved by 
 preparing a statement similar to No. XX. B. showing the 
 position of the fund and the final repayment of the loan after 
 making the adjustment. The rule and formula to be described 
 later in Chapter XXIII (the annual increment (ratio) method), 
 by which the future or amended annual increment under the 
 altered conditions may be found by direct calculation from the 
 present annual increment under the previous conditions, cannot 
 obviously be applied to cases in which there is not any variation 
 in the annual increment, which depends entirely upon the 
 rate of accumulation. 
 
 In the previous chapter the deductive method is employed 
 to ascertain the amended annual instalment, consequent upon 
 a variation in the rate of accumulation only. In the following 
 Chapter (XXI), in discussing Variation C, it will be seen 
 that this deductive method is also available for ascertaining the 
 amended annual instalment consequent upon a variation in 
 the rate of accumulation, accompanied by a variation in the rate 
 of income from the present investments. But in the case of a 
 variation in the rate of income only, the deductive method may 
 be replaced by one much simpler. At the head of this chapter, 
 therefore, although reference is made to the deductive method 
 as summarised in Chapter XIX, the direct method without 
 calculation has been treated as the standard method to be 
 adopted in practice, and has been stated in summary form. 
 
 In Chapter XIX, the conditions in Variation A (rate of 
 accumulation) are compared with the original conditions, and 
 it has been found that an additional annual instalment of 
 £'32'592 is required to compensate for the decrease in the rate 
 of accumulation. Proceeding to Variation B, it has been found 
 that although the rate of accumulation remains unaltered, the 
 rate of income from investments is reduced. This reduction in 
 income requires a further increase in the annual instalment of 
 £49'664. It is now possible to compare the amended annual 
 instalment in Variation B, with the annual instalment under 
 the original conditions as follows: — 
 
 The original annual instalment was £680"234 
 
 Additionnl instalment due to the reduction in the 
 
 rate of accunuilation. Variation A. 32*592 
 
 Additional instalment due to the reduction in the 
 
 rate of income from investments. Variation B 49'664 
 
 Amended annual instalment. Variation B ,£762'490 
 
THE RATE PER CENT. OF INCOME 241 
 
 or au increase of £82*256, but on comparing tlie annual 
 increment in A'ariation B (rate of income), with tlie annual 
 increment under the original conditions, it is increased by only 
 £32-592, namely, from £102T-8cS2 to £1060-474. This further 
 proves that so long as the rate of accumulation remains 
 unaltered the annual increment does not require to be amended, 
 but if the portion of the annual increment derived from outside 
 investments is reduced, oM-ing to a fall in the rate of income 
 yielded by the present investments, the burden must be borne 
 by the other partner, namely, the revenue or rate account which 
 provides the annual instalment. 
 
 Statement XX. D. shows the method of making the adjust- 
 ment by the annual increment (balance of loan) method, which 
 will be fully described and summarised in Chapter XXII. 
 
 The Rate per cent. Statement XX. A. 
 
 The Deductive Method. 
 
 Variation B, rate of income only. 
 
 vShowing the method of adjusting the annual instalment in 
 consequence of a variation in the rate of income upon the 
 present investments without any variation in the rate of 
 accumulation or in the period of repayment. 
 
 This example is compared with the original conditions 
 as modified by Variation A. 
 
 Conditions before adjustment (at end of 12th year), 
 
 Amount of loan repayable in 25 years £26,495 
 
 Amount in the fund (at end of 12th year) £9932-74 
 
 Present annual income (previously) received there- 
 from, at 3| per cent, per annum £347'648 
 
 Present annual instalment, to be accumulated for 
 
 13 years at 3 per cent £71-^'8^6 
 
 Present annual increment £10604(4 
 
 Variation from the above conditions : — 
 
 The rate of income yielded by the present investments is 
 reduced from 3^ to 3 per cent. 
 
 Future annual income £297'984 
 
 Eeduction in annual income 49*664 
 
 Increased annual instalment 49-664 
 
 Future annual increment 1060-474 
 
242 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments (at end of 12th year), 
 
 representing the amovmt now in the fund ... £9932'74 
 
 Future annual income from present investments : — 
 
 Amount of an annuity of £297 '984 
 
 accumulated for 13 years, at 3 per cent. 
 
 Calculation (XX) 1 £4653-85 
 
 Original annual instalment : — 
 
 Amount of an annuity of £680"234 
 
 accumulated for 13 years, at 3 per cent. 
 
 Calculation (XIX) 2 £10623-75 
 
 Additional annual instalment (Variation A): — 
 Amount of an annuity of £32592 
 
 accumulated for 13 years, at 3 per cent. 
 
 Calculation (XIX) 3 £509-02 
 
 Provision already made will repay loan of £25719-36 
 
 Additional annual instalment required : — 
 
 Balance, being amount of original loan un- 
 provided for owing to the above decrease in 
 the rate of income from investments requir- 
 ing an additional annual instalment, to be 
 set aside and accumulated for 13 years at 
 3 per cent £775-64 
 
 Additional annual instalment 
 
 Calculation (XX) 2 £49-664 
 
 Amount of original loan £26495"00 
 
 Amended annual increment, hriiig : — 
 
 Income from investments £297-984 
 
 Amended annual instalment 762-490 
 
 £1060-474 
 
THE RATE PER CENT. OF INCOME 
 
 243 
 
 The Rate per cent. 
 
 Statement XX. B. 
 
 Variation B, rate of income only. 
 
 Showing the final eepayment of the loan, by the operation of 
 the sinking fund after making the adjustment in the 
 annual instalment, consequent upon a variation in the rate 
 of income upon the present investments without any 
 variation in the rate of accumulation, or in the period of 
 repayment. 
 
 Present investments (at end of 12th year) 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 £9932-74 
 
 Amended annual increment : - 
 
 Original annual instalment £680'234 
 
 Additional. Variation A 32-592 
 
 ditto. Yariation B 49-664 
 
 Total out of revenue 
 Income from investments 
 
 £762-490 
 297-984 
 
 £1060-474 
 
 Amount thereof, accumulated for 13 years at 
 
 3 per cent. Calculation (XX) 3 £1656226 
 
 Amoiint of original loan 
 
 £26495-00 
 
 Amended annual instalment 
 
 £762-490 
 
244 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. Statement XX. C. 
 
 The Direct Method (without calculation). 
 
 Variation B, rate of income only. 
 
 Showing the method of adjusting the annual instalment in 
 consequence of a variation in the rate of income upon the 
 present investments without any variation in the rate of 
 accumulation or in the period of repayment. 
 
 E-equired the amended annual instalment, to be set aside and 
 accumulated as a sinking fund to compensate for a 
 reduction, from 3^ to 3 per cent., in the rate of income 
 to be received from the present investments, valued at 
 £9932' 74. Rate of accumulation 3 per cent. 
 
 Annual sinking fund instalment, at date of adjust- 
 ment as calculated or as ascertained in 
 Variation A. Statement XIX. B. £712-826 
 
 Add decrease in annual income from investments 
 
 at 3i per cent £347-648 
 
 at 3 per cent 297984 
 
 £49-664 
 
 Amended annual instalment £762-490 
 
 Memo. In the case of an increase in the amount of the 
 future annua] income, such increased income should he deducted 
 from the oii<j-in:il anmin] instalment. 
 
THE RATE PER CENT. OF INCOME 245 
 
 The Rate per cent. Statement XX. D, 
 
 The Annual Increment (balance of loan) Method. 
 
 Variation B, rate of income only. 
 
 To find tlie amended annual sinking fund instalment consequent 
 upon a variation in the rate of income upon the present 
 investments, without any variation in the rate of 
 accumulation, or in the period of repayment. 
 
 Eate of income from investments reduced from ^ to 3 per cent. 
 
 Eate of accumulation, 3 per cent. 
 
 For Eule, see Chapter XXII. 
 
 Amount of original loan (25 years) £2649500 
 
 deduct amount in the fund at the end of the 
 
 12th year £9932-74 
 
 Balance of loan £1656226 
 
 Amended annual increment, to be added to the 
 fund, and accumulated at 3 per cent., to 
 provide this amount at the end of 13 years 
 
 Calculation (XX) 4 £1060-474 
 
 deduct income to he received from the present 
 
 investments (£9932-74) at 3 per cent. £297-984 
 
 Amended annual instalment £(62-490 
 
246 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Pro forma Sinking Fund Account, No. 8. 
 
 A Variation in the rate of Income upon tlie present Investments. 
 
 Loan of £26,495, repayable at the end of 25 yearn. 
 
 Showing the final repayment of the loan, by the operation of 
 the increased annual instalment of £762'490. 
 
 Statement XX. B, 
 
 Rate of accumulation, 3 per cent. 
 
 fear. 
 
 1 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received 
 
 from 
 
 investments. 
 
 Income 
 received from 
 Annual investments made 
 sinking fund after 12th year 
 instalment. 3 per cent. 
 
 Amount in 
 
 the fund 
 
 at end 
 
 of year. 
 
 Year. 
 1 
 
 2 
 
 1 
 2 
 
 
 
 
 
 ^ 
 
 3 
 
 
 
 
 
 
 3 
 
 4 
 
 The 
 
 amount in the fund at the en 
 
 d of 
 
 4 
 
 5 
 
 the 
 
 L2th year. 
 
 £9932-74^ 
 
 ;, is the correct 
 
 5 
 
 6 
 
 calci 
 
 dated amount, as shown by Calcula- 
 
 6 
 
 7 
 
 tion 
 
 (XY) 2, 
 
 and by 
 
 the pro f 
 
 arm a 
 
 7 
 
 8 
 
 account, No. ] 
 
 , Chapter 
 
 XV. 
 
 
 8 
 
 9 
 
 
 
 
 
 
 9 
 
 10 
 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 297-984 
 
 762-490 
 
 — 
 
 10993-218 
 
 13 
 
 14 
 
 10993-218 
 
 297-984 
 
 762-490 
 
 31-814 
 
 12085-506 
 
 14 
 
 15 
 
 12085-506 
 
 297-984 
 
 762-490 
 
 64-583 
 
 13210-563 
 
 15 
 
 16 
 
 13210-563 
 
 297-984 
 
 762-490 
 
 98-335 
 
 14369-372 
 
 16 
 
 17 
 
 14369-372 
 
 297-984 
 
 762-490 
 
 133-099 
 
 15562-945 
 
 17 
 
 18 
 
 15562-945 
 
 297-984 
 
 762-490 
 
 168-906 
 
 16792-325 
 
 18 
 
 19 
 
 16792-325 
 
 297-984 
 
 762-490 
 
 205-787 
 
 18058-586 
 
 19 
 
 20 
 
 18058-586 
 
 297-984 
 
 762-490 
 
 243-775 
 
 19362-835 
 
 20 
 
 21 
 
 19362-835 
 
 297-984 
 
 762-490 
 
 282-903 
 
 20706-212 
 
 21 
 
 22 
 
 20706-212 
 
 297-984 
 
 762-490 
 
 323-204 
 
 22089-890 
 
 22 
 
 23 
 
 22089-890 
 
 297-984 
 
 762-490 
 
 364-714 
 
 23515-078 
 
 23 
 
 24 
 
 23515-078 
 
 297-984 
 
 762-490 
 
 407-470 
 
 24983-022 
 
 24 
 
 25 
 
 24983022 
 
 297-984 
 
 762-490 
 
 451-504 
 
 26495-000 
 
 25 
 
THE RATES OF ACCUMULATION AND INCOME 247 
 
 CHAPTER XXI. 
 
 SINKING FUND PEOBLEMS EELATING TO THE 
 EATE8 PEE CENT. OF INCOME AND ACCUMULA- 
 TION {^Continued). 
 
 Variation C, in which there is a variation in the rate of 
 accumulation and also in the rate of income upon the 
 present investments, but without any variation in the 
 period of repayment. 
 
 Summary of the methods of adjustment. The deductive 
 METHOD. Comparison of results obtained in this and 
 
 PREVIOUS chapters IN ALL PROBLEMS INVOLVING A 
 VARIATION IN THE RATE PER CENT. STATEMENT SHOWING 
 
 THE FINAL REPAYMENT OF THE LOAN BY THE OPERATION OF 
 THE AMENDED ANNUAL INSTALMENT. 
 
 Summary of the methods of adjustment. 
 
 (/) The deductive method, as suniviarised at the head of 
 Chapter XIX, will apply, but has been slightly viodified in 
 Statement XXI. A. 
 
 [II) The direct method, ivithout calculation, as summarised 
 at the head of Chapter XX, will not apply to this variation. 
 
 {Ill) The annual increment {balance of loan) method, as 
 summarised at the head of Chapter XXII, may be used, but 
 will not be applied to the example under review. 
 
 {IV) The annual increment {ratio) method, as sumnnarised 
 at the head of Chapter XXIII , may be used, but will not be 
 applied to the example under review. 
 
 Note. The terms used in the summaries above mentioned 
 are fully explained at the head of Chapter XXJI . In all the 
 above methods it is imperative that the rates of accumulation 
 and of income from, investments be uniform during the whole 
 of the unexpired or substituted period, of repayment. 
 
248 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The enquiry into the methods of adjusting the annual 
 sinking fund instalment in consequence of any variation in 
 the rate per cent, is now almost completed. Variation A, which 
 alfected the rate of accumulation only, is fully discussed in 
 Chapter XIX. In the case of Variation B (Chapter XX) the 
 varying factor is the rate per cent, of income to be yielded 
 by the present investments representing the fund. The enquiry 
 will noAv be completed by examining Variation C, in which 
 there is a simultaneous change in both the above rates, and the 
 deductive method will again be used, as fully described in 
 Chapter XIX, and of which a summary of the various stages is 
 placed at the beginning of that chapter. The two preceding 
 chapters deal exhaustively with all general questions affecting 
 the enquiry, and they will apply equally to the present 
 variation. 
 
 The position of the fund at the time of making the adjust- 
 ment is fully set out in the following Statement XXI. A., which 
 is similar to those prepared to illustrate the variations already 
 considered. These conditions are based upon those obtaining 
 when the original calculation was made, and although the 
 present example Avill be compared with the conditions in 
 Variation A (rate of accumulation) they will also be compared 
 with those originally existing. As in previous instances it will 
 be assumed that all sums are added to the fund and accumulated 
 annually. In the following chapter (XXII) the whole of the 
 results of the enquiry into the rate per cent, will be compared 
 in order to show the general elfect of such rate upon the 
 accumulation of a sinking fund. The investigation will then 
 be extended in order to derive a rule and formula by means 
 of which the adjustments may be made by the more direct 
 annual increment (ratio) method. 
 
 Similar statements have been prepared as in the previous 
 variations, namely, XXI. A., showing the amended annual 
 instalment as ascertained by the deductive method ; and 
 XXI. B., showing the final repayment of the loan by the 
 operation of the amended annual instalment so ascertained, as 
 shown by the pro forma account, No. 9. 
 
 The results already obtained may now be briefly stated, 
 both with regard to the original conditions as well as Variations 
 A, B, and C, in order to show the progressive variations in the 
 examples which will be used later in discussing the derivation 
 of a rule and formula which may be ap])liod to any jiroblem 
 relating to the rate per cent. 
 
THE RATES OF ACCUMULATION AND INCOME 249 
 
 The siuking- fuud iustalment as origiually calculated, at a 
 rate of accumulatiou of '6j per ceut., was 
 
 Calculation (XV) 1 £680-2o4 
 
 In Variation A (rate of accumulation only), as 
 compared with the original conditions, the rate 
 of income from investments remained un- 
 altered, namely, 3^ per cent., but the rate of 
 accumulation was reduced from -i^ to o per 
 per cent., requiring an additional instalment, 
 as shown by Calculation (XIX) :J of £32-592 
 
 Amended instalment (A), Statement XIX. A. £T12'826 
 
 In \'ariation 13 (rate of income only) as compared 
 with Variation A, the rate of accumulation 
 remained at 3 per cent., but the rate of income 
 from investments was reduced from 3^ to 
 3 per cent., requiring an additional instalment 
 as shown by Calculation (XX) 2 of £49-664 
 
 Amended instalment (B), Statement XX. A. £762490 
 
 In Variation C, as compared with A'ariation A, the 
 rate of accumulation was further reduced from 
 3 to 2^ per cent., and the rate of income from 
 investments was also reduced from 3^ to 3 per 
 cent. This required an additional instalment, 
 as shown by Calculation (XXI) 4, of £83-099 
 but part of this was due to the reduc- 
 tion in the rate of income in 
 
 Variation B, as above £49-664 
 
 £33-435 
 
 Amended instalment (C.) Statement XXI. A. £795925 
 
 which amended annual instalment is required to be set 
 aside out of revenue or rate and accumulated in addition to 
 the income from the present investments, in order to provide 
 the loan repayable at the end of the prescribed period. 
 
 In Statement XXI. A. following, the additional annual 
 instalment is ascertained to be £8:5-099. This annual increase 
 is derived directly from the conditions in Variation A, and is 
 made up of the increased instalment due to the reduction in the 
 
250 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 rate oi income iu \ ariation B, viz., £49 664, and the above 
 aiuouut of £o6'-i6d due to tiie variation in the rate o± accumula- 
 tion. In the present example there is a variation in both the 
 rates of income and of accumulation, and in this respect it 
 combines the changes in Variations A and 13. In Variation 13, 
 where there is a change in the rate of income only, the annual 
 instalment is corrected by adding thereto the actual deficiency 
 in the future annual income. Statement XXI. A. shows by 
 the deductive method that the amount of original loan which 
 would be unprovided in consequence of the concurrent reduction 
 in the above rates is £1258" 15, requiring an additional annual 
 instalment of £8o"099, as shown by Calculation (.XXlj 4. This 
 additional annual instalment is the measure of the two annual 
 losses of interest, and it is possible to allocate to each rate the 
 proportions in which they contribute thereto. This will be seen 
 by referring to the following Statement XXI. C, in which 
 column (2) contains, in the case of Variation A, the original 
 and additional annual instalments, and also the income from 
 investments, making up the amended annual increment. This 
 agrees with Statement XIX. A. in total, but the income from 
 investments at 3^ per cent, has been divided as between o and 
 J per cent, iu order to compare this variation with Variation C. 
 The final column (8) shows the deficiency of original loan 
 caused by the accumulation of each of the component parts of 
 the annual increment at 2| per cent, in Variation C, instead 
 of at 3 per cent., as in Variation A. 
 
 This deficiency is arrived at by deducting the amount of 
 loan in column (7) from the amount in column (4), and is made 
 up as follows, expressed in terms of original loan : — 
 
 Deficiency due to the reduction in the rate of 
 accumulation, from 3 to 2^ per cent., of items 
 1 to 3 £482-51 
 
 Deficiency due to the reduction in the accumulation 
 
 of the decrease in income, item 4 £23'70 
 
 £506-21 
 
 Deficiency of annual income accumulated at 2i 
 
 percent £75194 
 
 £1258-15 
 
THE RATES OF ACCUMULATION AND INCOME 251 
 
 The deficiency of <£1258'15 of loan requires a total additional 
 annual instalment of i>8o'099, as previously ascertained, which 
 is made up of : — 
 
 the loss of income from investments £49'664 
 
 and the loss owing to the reduction in the rate of 
 
 accumulation £;ilj"4y5 
 
 £83-099 
 
 The above amount of <£33'435 includes the loss of accumula- 
 tion not only upon the remaining portion (£1010'810) of the 
 present annual increment in Variation A, as shown in column 2, 
 but also upon the reduction in the annual income, viz., £49664. 
 This proves that when, as in Variation C, the reduction in the 
 rate of income from investments is accompanied by a reduction 
 in the rate of accumulation, the additional annual instalment is 
 measured, not by the actual reduction in the annual income, as 
 in Variation B, but by the annual deficiency of income increased 
 in the ratio that the amount of £1 per annum at the past rate 
 bears to the amount of £1 per annum at the future rate, in each 
 case for the same number of years, being the unexpired portion 
 of the original repayment period. This will be referred to later 
 in Chapter XXII, when discussing Calculation (XXII) E. with 
 the object of arriving at a method of making the adjustment 
 by the more direct annual increment (ratio) method. In that 
 case the comparison will be made between Variation C and the 
 original conditions, but the same principles apply, and the 
 above table may be again referred to with advantage. 
 (Statement XXI. C. follows.) 
 
252 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 The Rate per cent. Statement XXI. A, 
 
 The Deductive Method. 
 Variation C, rates of accumulation and income combined. 
 
 Showing the method of adjusting the annual instalment in 
 consequence of a variation in the rate of accumulation 
 and also in the rate of income upon the present investments, 
 but without any variation in the period of repayment. 
 
 This example is compared with the original conditions 
 as modified by Variation A. 
 
 Conditions before adjustment fat end of 12th year) 
 
 Amount of loan repayable 'in 25 years £26,495 
 
 Amount in the fund (at the end of 12th year) ... £9932-74 
 
 Present annual income (previously) received there- 
 from, at 3| per cent, per annum £347'648 
 
 Present annual instalment, to be accumulated for 
 
 13 years at 3 per cent £712'826 
 
 Present annual increment £1060474 
 
 Variation from the above conditions : — 
 
 The rate of accumulation of the fund is reduced from 
 3 to 2| per cent. 
 
 The rate of income yielded by the present investments is 
 reduced from 3i to 3 per cent. 
 
 Future annual income £297984 
 
 The future rate of accumulation 2^ per cent. 
 
THE RATES OF ACCUMULATION AND INCOME 253 
 
 Present investments (at end of 12th year), 
 
 representing tlie amount now in the fund ... £9932' 74 
 
 Future annual income from present investments : — 
 
 Amount of an annuity of £29T'984 
 
 accumulated for 13 years, at 2| per cent. 
 
 Calculation (XXI) 1 £451101 
 
 Original annual instalment : — 
 
 Amount of an annuity of £680'234 
 
 accumulated for 13 years, at 2^ per cent. 
 
 Calculation (XXI) 2 £1029904 
 
 Additional annual instalment (Vdriafion A) : — 
 Amount of an annuity of £32'592 
 
 accumulated for 13 years, at 2^ per cent. 
 
 Calculation (XXI) 3 £493-46 
 
 Provision already made -will repay loan of £25236'85 
 
 Additional annual instalment required : — 
 
 Balance, being amount of original loan un- 
 provided for owing to tlie above decrease in the 
 rate of accumulation, and in the rate of income 
 from investments requiring an additional 
 annual instalment, to be set aside and 
 accumulated for 13 years at 2| per cent £125815 
 
 Additional annual instalment 
 
 Calculation (XXI) 4 £83-099 
 
 Amount of orio-inal loan £26495-00 
 
 Amended annual increment, J)ciuf/ : — 
 
 Income from investments £297-984 
 
 Amended annual instalment 795-925 
 
 £1093-909 
 
254 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. 
 
 Statement XXI. B. 
 
 Variation C, rates of accumulation and income 
 
 Showing the final repayment of the loan, by the operation of 
 the sinking fund after making the adjustment in the 
 annual instalment, consequent upon a variation in the rate 
 of accumulation, and also in the rate of income upon the 
 present investments, but without any variation in the 
 period of repayment. 
 
 Present investments (at end of 12th year) 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 £9932-74 
 
 Amended annual increment : — 
 
 Original annual instalment £680'234 
 
 Additional. Variation A 32-592 
 
 ditto. Variation C 83-099 
 
 Total out of revenue ... 
 Income from investments 
 
 . 795-925 
 . 297-984 
 
 £1093-909 
 
 Amount thereof, accumulated for 13 years at 
 
 2i per cent. Calculation (XXI) 5 £1656226 
 
 Amount of original loan 
 
 £2649500 
 
 Amended annual instalment ... £795925 
 
THE RATES OF ACCUMULATION AND INCOME 255 
 
 
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256 
 
 REPAYMENT OF LOCAL AND OTHETl LOANvS 
 
 Pro forma Sinking Fund Account, No. 9. 
 
 A Yariatiou in the rate of Accumulation, as well as in the 
 Eate of Income upon the present Investments. 
 
 Loan of £26,495, repayable at tJte end of 25 years. 
 
 Showing the final eepaymext of the loan, by the operation of 
 the increased annual instalment of £795" 925. 
 
 Statement XXI. B. 
 
 Eate of Accumulation, 2^ per cent. 
 
 Year. 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received 
 
 from 
 
 investments. 
 
 Annual 
 
 Sinking 
 
 Fund 
 
 instalment. 
 
 Income 
 
 received from 
 
 investments 
 
 '2i per cent. 
 
 Amount in 
 
 tlie fund 
 
 at end 
 
 of year. 
 
 Vea 
 
 1 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 
 2 
 
 3. 
 
 
 
 
 
 
 3 
 
 4 
 
 The 
 
 amount in the fund 
 
 at the end of 
 
 4 
 
 5 
 
 the 
 
 12th year, 
 
 £9932-744 
 
 , is the correct 
 
 5 
 
 6 
 
 calculated amount, as shown by Ca 
 
 Icula- 
 
 6 
 
 7 
 
 tion 
 
 (XY) 2, 
 
 and by 1 
 
 he pro 
 
 forma 
 
 7 
 
 8 
 
 account, No. ] 
 
 , Chapter 
 
 XT. 
 
 
 8 
 
 9 
 
 
 
 
 
 
 9 
 
 10 
 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 297-984 
 
 795-925 
 
 — 
 
 11026-653 
 
 13 
 
 14 
 
 11026-653 
 
 297-984 
 
 795-925 
 
 27-348 
 
 12147-910 
 
 14 
 
 15 
 
 12147-910 
 
 297-984 
 
 795-925 
 
 55-379 
 
 13297-198 
 
 15 
 
 16 
 
 13297-198 
 
 297-984 
 
 795-925 
 
 84-111 
 
 14475-218 
 
 16 
 
 17 
 
 14475-218 
 
 297-984 
 
 795-925 
 
 113-562 
 
 15682-689 
 
 17 
 
 18 
 
 15682-689 
 
 297-984 
 
 795-925 
 
 143-749 
 
 16920-347 
 
 18 
 
 19 
 
 16920-347 
 
 297-984 
 
 795-925 
 
 174-690 
 
 18188-946 
 
 19 
 
 20 
 
 18188-946 
 
 297-984 
 
 795-925 
 
 206-405 
 
 19489-260 
 
 20 
 
 21 
 
 19489-260 
 
 297-984 
 
 795-925 
 
 238-913 
 
 20822-082 
 
 21 
 
 22 
 
 20822-082 
 
 297-984 
 
 795-925 
 
 272-233 
 
 22188-224 
 
 22 
 
 23 
 
 22188-224 
 
 297-984 
 
 795-925 
 
 306-387 
 
 23588-520 
 
 23 
 
 24 
 
 23588-520 
 
 297-984 
 
 795-925 
 
 341-394 
 
 25023-823 
 
 24 
 
 25 
 
 25023-823 
 
 297-984 
 
 795-925 
 
 377-268 
 
 26495000 
 
 25 
 
Section IV. 
 Sinking Fund Problems. 
 
 The Annual Increment. 
 
259 
 
 CHAPTER XXII. 
 
 THE ANNUAL INCREMENT METHODS. 
 
 Definition of terms relating to the annual increment and 
 the methods of ascertaining the amended annual 
 
 INSTALMENT BASED THEREON. ThIS APPLIES TO ALL 
 
 VARIATIONS IN THE RATE OF ACCUMULATION AND THE PERIOD 
 OF REPAYMENT, WITH OR WITHOUT ANY VARIATION IN THE 
 RATE OF INCOME UPON THE PRESENT INVESTMENTS REPRE- 
 SENTING THE FUND. 
 
 SINKING FUND PROBLEMS RELATING TO THE 
 RATE PER CENT. OF ACCUMULATION. 
 
 Methods of ascertaining the amended annual instalment by 
 direct calculation in terms of the annual increment. 
 Comparison of the results already obtained in 
 Chapters XIX, XX, and XXI in terms of the annual 
 instalment with those obtained by means of the annual 
 increment and the varying rates of accumulation. 
 The annual increment (balance of loan) method. 
 
 Summary of the methods of adjustment. 
 
 (Z) The deductive method, as siimmarised at the head of 
 Chapter XIX, 
 
 as to the rate of acctnmdation, Statement XIX. A. 
 
 as to the rate of income and the rate of accumulation^ 
 
 Statement XXI. A. 
 
 (II) The direct method, tcithout calculation, as s^immarised 
 at the head of Chapter XX, will not apply to these variattons. 
 
 {Ill) The annual increment {balance of loan) method, as 
 summarised heloiv, is illustrated in the text. 
 
 (IV) I he annual increment (ratio) viethod, as summarised 
 at the head of ( 'liaptvr XXIII, State mcnt XXII. C. 
 
26o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Note. In all cases where the adjustment is made by the 
 annual increment methods it is imperative that the rates 
 per cent., both of accumulation and income from investments, 
 he uniform^ during the whole of the unexpired or substituted 
 period of repayment. 
 
 The Annual Increment Methods, Definition of Terms. 
 
 The present annual increment, at the time of making an 
 adjustment in the annual instalment, consequent upon a 
 variation in the rate of accumulation, or in the period of 
 repayinent, or in both these factors combined^ is composed of : 
 
 1. The present or original anmial instalment, tvhich has been 
 
 set aside and added to the sinking fund up to the time 
 of making the adjustment, and 
 
 2. The present anmial income from investments, representing 
 
 the fund, whicli has been received up to the date of 
 making the adjustment. 
 
 The future or amended annual increment, consequent upon 
 a variation in either or both of the factors of rate per cent, of 
 accumulation and period of repayment, is composed of: — 
 
 1. The future or amended anmial instalment^ required to 
 
 be set aside and added to the sinking fund in consequence 
 of the above variation or variations, and 
 
 2. The future annual income from investments, representing 
 
 the fund at the date of making the adjustment whether 
 the rate of income ^ipon such investments remains un- 
 altered, or is varied. 
 
 The annual increments, as above described^ are the primary 
 and final factors in all the adjustments by these methods. 
 
 The past rate denotes the rate of accumulation upon which is 
 based the present or original annual instalment included 
 in the present annual increment. 
 
 The future rate, denotes the rate of accumulation to be tised 
 instead of the past rate to calculate the future or amended 
 annual increment. It will he ihe same as the past rate 
 in problems involving a variation in the period of 
 repayment only, without any variation in the rate of 
 accuTnulation. 
 
THE RATE OF ACCUMULATION 261 
 
 The uriexpired ■period, denotes the unexpired portion, at the 
 time of making the adjustment^ of the original repay- 
 ment period upon irhich the present or original annual 
 instalment was based. 
 
 The substituted period, denotes the increased or reduced number 
 of years over ivJtich the future or amended annual 
 instalments shall be spread and at the end of which the 
 full amount of the loan will be repayable. It will be 
 the same as the unexpired period in problems involving 
 a variation in the rate of accumulatio7i only, without 
 a variation in the period of repayment. 
 
 The income from investments ^ representing the amount in the 
 fund does not enter into the actual calculation except as 
 a component part of the present and future or amended 
 annual increments, as above defined. 
 
 The future or amended annual instalment, is obtained in all 
 cases by deducting, from the ascertained amended annual 
 increment, the future annual income from the present 
 investments representing the fund, u-hether the rate of 
 income upon such investments remains unaltered or is 
 varied. 
 
 Note. The foregoing definitions will be referred to in 
 subsequent chapters, loithout any further explanation or 
 amplification. 
 
 Geneeal summaey of the annual increment (balance of 
 loan) method, of ascertaining the amended annual sinking 
 fund instalment due to a variation in either the rate of 
 accumulation, the period of repayment, the rate of income upon 
 the present investments representing the fund, or any of these 
 factors in combination. The terms used in the following 
 summary are fully explained above. 
 
 {!) Ascertain the value of the present investments in the 
 manner already described^ and deduct the value so 
 obtained from the amount of the original loan. 
 
 [2) The remainder represents the balance of loan to be 
 provided by the accumulation of the future or amended 
 annual increment, as previously defined, for the un- 
 expired or substituted repayment period at the future 
 rate of accumulation. 
 
262 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 (J) Calculate tJte annuifij, or annual increnicnt^ to be added 
 to the fund and accumulated for the period and at tJic 
 rate per cent, as in {2). 
 
 [4) From the amended anntuil increment ascertained as in (3), 
 deduct the future annual income to he received from the 
 present investments during the tvhole of the unexpired 
 or substituted repayment period. 
 
 (J) 'The remainder ivill represent tJie future or amended 
 annual instalment to be charged to revenue or rate 
 account, and added to the fund, during the whole of the 
 unexpired or substituted repayment period. 
 
 {6) Prepare a statement showing the final repayment of the 
 loan by the operation of the sinking fund under the 
 amended conditions. 
 
 (7) Prepare a pro forma account showing the amount lohich 
 should be in the fund at the end of each year of the 
 unexpired or substituted repayment period. 
 
 Memo. In tJie event of the income from investments not 
 being uniform over the tvhole of the repayment period^ proceed 
 by the method in Chapter XXYII. 
 
 Sinking Fund Problems, relating to the rate per cent. 
 OF accumulation. The Annual Increment. Iu previous 
 chapters dealing with the three variations in the rates per cent, 
 of accumuhition and income from the present investments, 
 the amended annual instalment has been ascertained by the 
 deductive method described in Chapter XIX. 
 
 This method is based upon (1) the value of the present 
 investments representing the fund as described in Chapter XI^ ; 
 (2) the annual income to be received therefrom, and (3) the 
 original annual instalment. All these factors have been 
 reduced to equivalent amounts of original loan ultimately 
 repayable, in order to ascertain the deficiency in the fund at 
 the end of the repayment period due to the reduction in the 
 rate per cent, of income or of accumulation. 
 
 This deficiency of original loan ultimately repayable has 
 been converted into an equal annual sinking fund instalment, 
 to be provided out of revenue or rate, in addition to the original 
 instalment. A statement has been prepared showing in each 
 case the final re})aymeni of the loan by the operation of the 
 amended annual instalment so ascertained at the end of the 
 12th year. 
 
THE RATE OF ACCUMULATION 263 
 
 In these statements tlie amount of the loan has been divided 
 into two parts; the first (£9932-74) being the value of the 
 present investments representing the fund, and the second 
 (£16562-26) being the amount of loan to be provided at the end 
 of the repayment period by the accumulation of the future or 
 amended annual increment, which consists of : — 
 
 1. Income from the present investments. 
 
 2. The original annual instalment. 
 
 3. The additional annual instalment ascertained in the above 
 
 manner, 
 thereby proving the accuracy of the results obtained by the 
 deductive method. But the original annual instalment is the 
 only constant factor, although it may in future accumulate at a 
 lower rate than was originally estimated. Consequently, in 
 arriving at the future or amended annual instalment two 
 variable factors have to be considered, namely, (1) the rate of 
 income upon the present investments, and (2) the rate of 
 accumulation. These two factors of rate per cent, are most 
 important in the after consideration of the problem because 
 they may vary in different directions and are not in any way 
 related. But any difficulty may be eliminated by treating the 
 amount of the future annual income to be received from the 
 present investments, at the future rate per cent, of income, as 
 an annuity certain in the same manner as the original annual 
 instalment. These two factors together constitute the future 
 or amended annual increment of the fund, which is acted upon 
 by the future rate of accumulation only, consequently the 
 problem has been reduced to an annuity certain for a definite 
 term at a given rate per cent. The annual income from the 
 present investments, included in the present annual increment 
 in all adjustments made by this method, is the annual amount 
 which has been received in the past and is not the future annual 
 income which will be yielded during the unexpired or 
 substituted period of repayment. This is one of the funda- 
 mental principles of the annual increment (ratio) method. The 
 enquiry is thereby transferred from the annual instalment to 
 the annual increment, and as this is an annuity of fixed amount 
 it is possible to arrive at a formula, and a rule based thereon. 
 The annual increment has been fully described in Chapter XIV. 
 In Chapters XIX, XX, and XXI, three variations in the 
 rate per cent, have been considered, and the amended annual 
 instalment in each case has been ascertained by the deductive 
 method. Up to this point the examples have been considered 
 only as individual problems, but they will now be treated in 
 
264 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 combination. In order, however, to avoid undue reference to 
 previous chapters, the following Statement XXII. A. has been 
 prepared containing the whole of the conditions in each case 
 and the actual results previously obtained. 
 
 A further classified Statement XXII. B. has been prepared 
 showing the initial conditions in each case and giving 
 references to methods and calculations by which the results 
 have been obtained. It should again be mentioned that, 
 although in each A-ariation a reduction has been assumed in the 
 rates per cent., as being more likely to occur in practice, yet 
 the same principles and methods will apply equally to an 
 increase in both rates per cent., or to an increase in one rate 
 and a reduction in the other. 
 
 Statement XXII. A. (page 265) contains full details of the 
 amended annual instalments found by the deductive method in 
 the three Variations A, B, and C, which are derived one from 
 the other and from the original conditions by gradual variations 
 in the rates of income and accumulation. There is therefore a 
 definite relation between the original annual increment of 
 <£1027'882 and the successive annual increments in Variations 
 A, B, and C, leading to the final annual increment of £1093909 
 in Variation C. This relation depends upon the respective 
 rates of accumulation in the four examples, and by this means 
 it is possible to derive the rule and formula required. State- 
 ment XXII. B. contains the annual increments only, and shows 
 the rates per cent, of income and accumulation in each case. 
 All these annual sums are derived from a common source, and 
 therefore may be treated as simple annuities for a term without 
 reference to any principal sum or other factor than the rate of 
 accumulation. In Statement XXII. B. the variations in the 
 rate per cent, are divided into two classes depending upon the 
 rate of accumulation. The first class contains the problems 
 in which the rate of accumulation remains unaltered, and there 
 is not therefore any necessity to sub-divide the class as regards 
 any variation in the rate of income on the investments, because, 
 as ascertained in considering Variation B, there is not any 
 question of compound interest involved. It is only necessary 
 to correct the original annual instalment by adding to or 
 deducting therefrom the difference between the annual amounts 
 of income yielded by the present investiuents at the past and 
 future rates respectively. The second class includes cases in 
 which there is a variation in the rate of accumulation, and this 
 class may be sub-divided according as the rate of income upon 
 the present investments is unaltered or is varied. Although it 
 
THE RATE OF ACCUMULATION 265 
 
 will be found that both sub-divisious of this class may be 
 treated by one and the same rule and formula, the present 
 distinction is useful in giving- emphasis to the fact. It will 
 also be seen, in dealing with problems in which there is a 
 change in the rate of income upon investments as well as in the 
 rate of accumulation, Chapter XXI, that the reason why the 
 rule applies is not so obvious as in the case of a simple variation 
 in the rate of accumulation only. 
 
 Class I. Variations in the rate of incoine from investments 
 
 only, the rate of accumulation remaining 
 
 unaltered. 
 
 Problems of this nature, in which the variation affects only 
 
 the rate of income on the present investments, but in which the 
 
 rate of accumulation remains the same, have been fully 
 
 described in Chapter XX, Variation B. The calculation of the 
 
 amended annual instalment in such cases may be made by the 
 
 deductive method, Statement XX. A., which applies equally to 
 
 all manner of variations in the rate per cent. But Statement 
 
 XX. C. shows that the amended annual instalment may be 
 
 arrived at by a simple direct method, without calculation, 
 
 although the deductive method may be used to prove the 
 
 accuracy of the conclusions. 
 
 Class II. Variations in the rate of accumulation. 
 
 This class has been sub-divided into two groups, as shown 
 in Statement XXII, B. as follows: — 
 
 {A) In which the rate of income upon the present invest- 
 ments is unaltered. 
 (B) In which the rate of income upon the present investments 
 is varied. 
 Each of these sub-divisions will be considered in detail, 
 taking as examples the figures given in Statement XXII. B. 
 
 The Rate per cent. Statement XXII, A. 
 
 Variation A. Hate of accumulation only. Chapter XIX. 
 
 Variation B. Rate of income only. Chapter XX, 
 
 Variation C. Rates of accumulation and income combined. Chapter XXI. 
 
 Showing, at the end of the 12th year, under the original conditions, and 
 under each variation : — 
 
 (1) The present, and future or amended annual increments. 
 
 (2) The additional annual instalment distinguishing between the 
 
 loss of income from the present investments, and the reduction 
 in the rate of accumulation. 
 
 (3) The provision of the future annual increment from internal 
 
 and external sources. 
 
266 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Loan ;^26,495. Amount in the fund at end of Original Variation Variation Variation 
 
 12th year, ;{;», 03-2 -74. Conditions A B C 
 
 Future rate of accumulation 3^ 3 3 2j 
 
 Future yield on present investments 3| 3| 3 3 
 \. Present annual increment : — 
 
 Original instalment 680234 080-234 680234 680-234 
 
 Income from present invest- 
 ments at end of 12tli year, 
 
 at above rates 347-648 347-648 297-984 297-984 
 
 Present annxial increment 
 wliicb will continue to be 
 accumulated at reduced rate 
 of accumulation 1027-882 1027-882 978-218 978-218 
 
 Additional annual instahnents 
 to make good tbe loss of in- 
 terest on present investments 
 and future accumulations, to 
 be added to tbe original 
 annual instalments and pro- 
 vided out of revenue or rate Nil 32-592 82-256 115-691 
 
 Future annual increment 1027-882 1060474 1060-474 1093909 
 
 II. Tbe above additional annual 
 instalments, as compared witli 
 tbe original conditions are 
 made up as follows : — 
 
 1, Decrease in income from tbe 
 
 present investments ... ]S'il Xil 49-664 49'664 
 
 2. Decrease in interest on 
 
 future accumulations due 
 
 to reduction in rate of 
 
 accumulation : — 
 
 Variation A — 32592 32-592 32-592 
 
 A'ariation C — — — 33-435 
 
 Nil 32-592 82-256 115-691 
 
 III. Future annual increment to he 
 provided as follows : — 
 A. To be taken out of revenue or 
 
 rate : — 
 Original annual instalment 680234 680-234 680-234 680-234 
 Deficiency in future income 
 
 from present investments — — 49-664 49664 
 
 Additional annual instalment 
 
 to compensate for decrease 
 
 in rate of accumulation ... — 32-592 32-592 66027 
 
 680-234 712-826 762490 795925 
 B. Income to be received in 
 future from present invest- 
 ments 347-648 347-648 297-984 297-984 
 
 Future annual increment 1027882 1060-474 1060-474 1093-909 
 
X 
 
 THE RATE OF ACCUMULATION 267 
 
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268 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Class II (A). In which the rate of aceuniulation is varied, hut 
 in ^vhich the rate of income upon the present 
 investments is unaltered. 
 
 TJie following examples in Statement XXII. B. fall under 
 this head : — 
 
 (Ij In Variation A, as compared with the original conditions, 
 the rate of income is in each case 3| per cent., but the 
 rate of accumulation is reduced from 3| to 3 per cent. 
 The effect as ascertained by the deductive method (State- 
 ment XIX. A.) is to increase the annual increment from 
 £1027-882 to £1060-474. See also Calculation XXII. C, 
 where the same result is obtained by the annual 
 increment (ratio) method. 
 
 (2) In Variation C, as compared with Variation B, the rate 
 of income is in each case 3 per cent., but the rate of 
 accumulation in Variation C is reduced from 3 to 2^ per 
 cent. The effect, as will be seen from Statement XXII. 
 B,, is to increase the annual increment from £1060474 
 to £1093-909. This adjustment is worked out in detail 
 in Calculation XXII. D. 
 
 Chapter XIX deals very fully with the process of finding, 
 by the deductive method, the amended annual instalment 
 consequent upon a variation in the rate of accumulation only, 
 taking as an example the original conditions as modified by 
 Variation A. In both cases the present annual increment 
 consists of : — - 
 
 Income from investments £347-648 
 
 Original annual instalment 680-234 
 
 £1027-882 
 
 but under the original conditions this annual increment 
 accumulated at 3| per cent., whereas in Variation A the rate of 
 accumulation Avas reduced to 3 per cent. This requires an 
 additional annual instabnent of £32"592 to be set aside out 
 of revenue or rate, as found by Calculation (XIX) 3. But it is 
 apparent that this represents the deficiency in the accumulation 
 of £1027-882 per annum at 3 per cent, instead of at 3^ per cent., 
 and is measured, not by the actual ratio between 3 and 3| per 
 cent., but by tlic ratio (ixisting between the respective amounts 
 of £1 per annum for 13 years at those rates. These amounts 
 
THE RATE OF ACCUMULATION 269 
 
 are given in Table III in the published tables of compound 
 
 interest. 
 
 The same principle applies to Variation C, as compared with 
 Variation B, as will be seen by the following Calculation 
 
 XXII. D. 
 
 The remarks upon Calculation XXII. C. relating to Varia- 
 tion A, as compared with the original conditions, apply equally 
 to this case. 
 
 Class II (B). In which the rate of accumulation and the rate 
 of income upon the present investments are both 
 varied. 
 The following examples in Statement XXII. B. fall under 
 
 this head : — 
 
 (1) In Variation C, as compared with the original conditions, 
 
 the rate of income is reduced from o^ to 3 per cent., 
 and the rate of accumulation is reduced from 3^ to 21 per 
 cent. The effect, as found by Calculation XXII. E., is 
 to increase the annual increment from £1027-882 to 
 £1093-909. 
 
 (2) In Variation C, as compared with Variation A, the rate 
 
 of income is reduced from 3i to 3 per cent., and the rate 
 of accumulation is reduced from 3 to 2^ per cent. The 
 effect, as will be seen from Statement XXII. B., is to 
 increase the annual increment from £1060-474 to 
 £1093-909. This calculation is not worked out in detail, 
 but follows from the premises as a matter of course. 
 (3) In Variation B, as compared with the original conditions, 
 the rate of income is reduced from 3i to 3 per cent., and 
 the rate of accumulation is reduced from ^ to 3 per cent. 
 The effect, as will be seen from Statement XXII. B., 
 is to increase the annual increment from £1027-882 to 
 £1060-474. This calculation, also, is not worked out m 
 detail. 
 
 In Calculation (XXII) E the original conditions are com- 
 pared with Variation C in which there is a reduction in the rate 
 of income from present investments from 3^ to 3 per cent., or 
 £49-664 per annum; and at the same time a reduction in the 
 accumulation rate from 3i to 2i per cent. Although it is not 
 so obvious as in those cases where there is a variation m the 
 rate of accumulation only, yet the same rule applies, as will be 
 seen by the following considerations. The method is a 
 
270 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 combination of those previously considered. It resembles 
 Variation A in that the rate of accumulation is reduced, and 
 it is therefore necessary to increase the original annual 
 increment in proportion to the respective amounts of £1 per 
 annum for 13 years at the past and future rates of accumulation, 
 as in Variation A., Calculation XXII. C. It resembles 
 Variation B only to the extent that the actual decrease in the 
 annual increment due to the reduced annual income of £49'664 
 must be added to the fund as an additional annual instalment to 
 be provided out of revenue or rate. But in Variation B there 
 is not any reduction in the rate of accumulation, as in this 
 case; and in Variation B, therefore, the annual loss of income 
 on the present investments is the actual measure of the deficiency 
 in tlie annual instalment. In this case there is a reduction 
 in the future income from the present investments accompanied 
 by, and acted upon by, a reduction in the rate of accumulation; 
 consequently if the original annual instalment be increased by 
 the loss of income only, as in Variation B, the fund will lose 
 the accumulation on that sum due to the reduction in the rate 
 of accumulation. It is clear, therefore, that the original annual 
 instalment must be increased, not by the actual loss of income, 
 of ^ per cent., or £49'664, which under the original conditions 
 accumulated at 3^ per cent., but by a larger annual amount 
 which, accumulated at 2^ per cent, only, will at the end of the 
 repayment period amount to the same sum. This question has 
 been very fully discussed in Chapter XXI, dealing with 
 Variation C, which contains a useful Statement XXI. C, 
 which may be consulted in this connection. Therefore, the 
 actual rate per cent, of income from the present investments 
 does not enter into the calculation of the annual increment, 
 which, as shown in Statement XXII. E., is exactly similar in 
 principle to Calculations XXII. C. and XXII. D. 
 
 On comparing the whole of the above Calculations XXII. 
 C'., D. and E., it will be seen from the formulae at the heading 
 of each that they all follow the same rule, although the 
 conditions in each are diiferent. It may appear s\;perfluous to 
 include them all, but they will be referred to again in order to 
 illustrate the variations in (1) the period of repayment, in 
 Chapter XXIV, and (2) the period of rcjiayment accompanied 
 bv a variation in the rate per cent, of accumulation, in Chapter 
 XXVI. 
 
 The Axnual Increment (balance of loan) Method. Three 
 adjustments have been made relating to variations in the rate 
 per cent, of accumulation as follows: — 
 
THE RATE OF ACCUMULATION 271 
 
 Class II (A). Variation A from the original conditions, 
 
 Calculation XXII. C. 
 ditto. Variation C from Variation B, 
 
 Calculation XXII. D. 
 
 Class II (B). Variation C from the original conditions, 
 
 Calculation XXII. E. 
 
 On comparing them it will be seen that they all follow the 
 same rule, and it will be further noticed that the numerator of 
 the fraction in each calculation (log. 4-2191205) is the same, 
 and represents the balance of original loan, £16562'26, to be 
 provided by the accumulation of the future annual increment 
 at the respective rates of accumulation in Calculation XXII. C. 
 (Variation A) at 3 per cent., and in Calculations XXII. D. 
 and XXII. E. (Variation C) at 2^ per cent. But in each 
 case the above numerator (£16562-26, balance of loan) is 
 divided by the amounts of £1 per annum for 13 years at 
 the above respective rates per cent, of accumulation, which, 
 as shown by Calculation (XV) 1, is the usual method by which 
 to obtain the sinking fund instalment, as shown in standard 
 calculation form. No. 3x. 
 
 The calculation is the same as if it had been assumed that 
 the present investments of £9932-74 had been applied in the 
 redemption of an equivalent amount of loan and an annual 
 sinking fund instalment set aside for the unexpired portion of 
 the repayment period of 13 years to repay the balance of 
 £16562-26 of original loan. 
 
 Calculated in this manner, the annual instalment would be, 
 in Variation C, Calculation XXII. C, the annual increment 
 of £1060-474, which is made up of: — 
 
 The annual sinking fund instalment of £712-826 
 
 l)lus the interest upon the loans repaid out of the 
 sinking fund which (as pointed out in con- 
 sidering the case of local authorities in 
 Chapter XIII), should be paid into the fund £347-648 
 
 £1060-474 
 
 The same remarks apply equally to Variation C, as shown 
 by Calculations XXII D. and XXII. E. On comparing these 
 calculations, both of which relate to Variation C, and referring 
 to Statement XXII. A., the onlv difference is found in XXII. E. 
 
272 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (Variation C, as compared with tlie original conditions) which 
 
 requires an orig^inal annual increment of £1027'882 
 
 whereas in XXII. D. it is compared with Variation 
 
 B., which requires an annual increment of £1060'474 
 
 a difference of £32-592 
 
 which is the increased annual instalment required in Varia- 
 tion A, as compared with the original conditions in consequence 
 of the reduction in the rate of accumulation from 3^ to 3 per 
 cent, in Variation A. From the above data, as shown by 
 Calculation XXII. C. and the deductive method previously 
 described in Chapter XIX relating to Variation A, it is possible 
 to deduce a further method of finding the future or amended 
 annual instalment consequent upon a variation in the rate of 
 accumulation accompanied or not by a change in the rate of 
 income to be received upon the present investments representing 
 the fund. This has been called the annual increment (balance 
 of loan) method, and the rule may be stated as follows : — 
 
 (1) From the amount of loan repayable at the end 
 
 of the original period of repayment (25 years) £26495' 00 
 
 (2) deduct the value of the present investments 
 
 representing the fund at the time the adjust- 
 ment is required to be made, namely, at the 
 end of the 12th year £9932-74 
 
 (3) and treat the balance of loan, viz. £16562-26 
 
 as an original amount to be provided at the end of the unexpired 
 portion (13 years) of the original repayment period by means 
 of an annual increment based upon the future rate of accumula- 
 tion. 
 
 (4) This, as shown by Calculation (XIX) 5, requires 
 an annual increment to be accumulaled at 
 3 per cent., of £1060-474 
 
THE RATE OF ACCUMULATION 273 
 
 (5) From this aunual sum deduct the future annual 
 income to be received from the present invest- 
 ments at the future rate per cent., whether 
 unaltered, increased or reduced (in this case) <£o47"648 
 
 (6) and the remainder £712'826 
 
 is the future or amended annual sinking fund instalment to be 
 set aside out of revenue or rate for the unexpired portion of the 
 original repayment period, as ascertained in XIX. A. and 
 
 XXII. c. 
 
 In a later chapter it will be found that the above rule, with 
 modifications in the wording only, may be applied equally to 
 variations in the period of repayment accompanied or not by 
 variations in the rates per cent, of income and accumulation. 
 
 This will be shown in Chapter XXIV dealing with a variation 
 in the period of repayment only, and in Chapter XXYI, dealing 
 with a concurrent variation in the period of repayment and the 
 rate of accumulation. For this reason the summary of the 
 method at the head of this chapter has been so worded that it 
 will apply to the whole of the problems above referred to. 
 
274 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. 
 
 Calculation XXII. C. 
 
 The Annual Increment (ratio) Method. 
 Class II. A. To find the amended annual increment (and 
 therefrom the additional annual instalment) in a sinking 
 fund in which the rate of accumulation is reduced, hut in 
 which the income from the present investments, and the 
 period of repayment, remain unaltered. 
 
 The original conditions compared with Variation A by 
 the deductive method. Statement XIX. A. 
 
 The rule relating to this method is stated at the head 
 of Chapter XXIII. 
 
 Required the annual increment to he accumulated for a period 
 of 13 years at 3 per cent., which is equivalent to an annual 
 increment of £1027'882, to he accumulated for the same 
 period at 3| per cent. 
 
 Income from investments, 3| per cent. 
 
 1027-882 -r- 
 
 r Amount of £1 per annum, 13 years, 3|% -\ 
 Amount of £1 per annum, 13 years, 3% -' ~ ' 
 
 or hy Table III, giving the amounts of £1 per annum : 
 1027-882 X 1611303 
 
 ,r.r,^o = 1060-474 
 
 15 61(8 
 
 Log. Present annual increment 
 
 add Log. Amount of £1 per annum 
 
 1027-882 3-0119434 
 
 Table III, 13 years, 3i per cent. 16-11303 1-2071771 
 
 deduct Log. Amount of £1 
 
 TqKIo TTT 1.'^ TTOC,T>C 
 
 jjog. ^mouni oi ^1 per annum 
 Table III, 13 years, 3 
 
 per cent. 
 
 16562-26 
 15-6178 
 
 4-2191205 
 1-1936196 
 
 Log . Am ended annual increment 
 
 Amended annual increment 
 
 To find the amended annual instalment: — - 
 
 deduct the income from investments, 3^ 
 per cent 
 
 Amended annua] instalment 
 
 being Original annual instalment ... 680234 
 Additional annual instalment ;')2-592 
 
 3-0255009 
 1060-474 
 
 347-648 
 
 712-826 
 
 712-826 
 
THE RATE OF ACCUMULATION 
 
 "21^ 
 
 The Rate per cent. 
 
 Calculation XXII. D. 
 
 The Annual Increment (ratio) Method. 
 
 Class IT . A. To find tlie amended annual increment (and 
 therefrom the additional annual instalment) in a sinking 
 fund in which the rate of accumulation is reduced, but in 
 which the income from the present investments, and the 
 period of repayment, remain unaltered. 
 Variation B compared with Variation C. 
 
 This calculation is exactly similar in principle to 
 XXII. c. 
 
 The rule relating to this method is stated at the head 
 of Chapter XXIII. 
 
 Required the annual increment to be accumulated for a period 
 of 13 years at 2^ per cent., which is equivalent to an annual 
 increment of £*1060'4T4, to be accumulated for the same 
 period at 3 per cent. 
 
 Income from investments, 3 per cent. 
 
 r Amount of £1 per annum, 13 years, 3% 1 
 
 1060-474 -r , „^/ ^ ^5T°r [=1093-909 
 
 I, Amount ot i/1 per annum, 13 years, i\/o ) 
 
 or by Table III, giving the amounts of £1 per annum : — 
 
 1060-474 X 15-6178 
 
 - = 1093-909 
 
 15-14044 
 Log. Present annual increment 
 add Log. Amount of £1 per annum 
 Table III, 13 years, 3 per cent. 
 
 deduct Log. Amount of £1 per annum 
 Table III, 13 years, 2\ per cent. 
 
 1060-474 
 
 15-6178 
 
 16562-26 
 
 1514044 
 
 3-0255009 
 1-1936196 
 
 4-2191205 
 1-1801386 
 
 Log. Amended annual increment 3-0389819 
 
 Amended annual increment 1093-909 
 
 To find the amended, annual instalment : — ■ 
 
 deduct the income from investments, 3 
 per cent 
 
 Amended annual instalment 
 
 being Original annual instalment ... 762-490 
 Additional annual instalment 33-435 
 
 297-984 
 
 795-925 
 
 795-925 
 
276 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. 
 
 Calculation XXII. E. 
 
 The Annual Increment (ratio) Method. 
 
 Class 11. B. To find the amended annual increment (and 
 therefrom the additional annual instalment) in a sinking 
 fund in which the rate of accumulation and the income 
 from the present investments are both reduced, but in 
 which the period of repayment remains unaltered. 
 
 The original conditions compared with Variation C. 
 
 The rule relating to this method is stated at the head 
 of Chapter XXIII. 
 Eequired the annual increment to be accumulated for a period 
 of 13 years at 2\ per cent., which is equivalent to an annual 
 increment of £1027-882, to be accumulated for the same 
 period at 3| per cent. 
 
 The rate of income from investments is reduced from 
 '3| to 3 per cent. 
 
 (Amount of £1 per annum, 13 years, 3^% 
 
 1027-882 
 
 = 1093-909 
 
 ] Amount of £1 per annum, 13 years, 2^% 
 or by Table III, giving the amounts of £1 per annum : — 
 1027-882 X 16-11303 
 
 15-14044 
 
 Log. Present annual increment 
 add Log. Amount of £1 per annum 
 Table III, 13 years, 3| per cent. 
 
 = 1093-909 
 .. 1027-882 
 1611303 
 
 30119434 
 
 1-2071771 
 
 deduct Log. Amount of £1 per annum 
 Table III, 13 3-ears, 2^ per cent. 
 
 Log. 
 
 16562-26 
 1514044 
 
 4-2191205 
 1-1801386 
 
 Amended annual increment 
 
 Amended annual increment 
 To find the amended annual instalment :- 
 
 deduct the income 
 per cent 
 
 from investments, 3 
 
 3-0389819 
 1093-909 
 
 297-984 
 
 Amended annual instalment 
 
 heiyuj Original annual instalment ... 680234 
 Additional :nniual instalment 115-691 
 
 795-925 
 
 795-925 
 
THE RATE OF ACCUMULATION 277 
 
 CHAPTER XXIII. 
 
 SINKING FUND PEOBLEMS, EELATING TO THE 
 EATE PEE CENT. UE ACCUMULATION {Contmued). 
 
 Hekivation of a rule and formula relating to a variation 
 in the rate per cent. of accumulation based upon the 
 
 FOREGOING RESULTS BY THE ANNUAL INCREMENT (RATIO) 
 METHOD. 
 
 The Annual Increment (ratio) Method. 
 
 The rule as to a variation in the rate of accumulation may 
 be stated as folloios, using the terms as ex plained at the head 
 of Chapter XXII. Statement XXII. C. 
 
 EuLE. 7'o find the amended annual instalment to be set aside, 
 and added to the existing sinking fund, 
 
 to be accumulated in future at a rate per cent, 
 greater or less than the rate at which the present 
 annual instalment teas calculated 
 
 {the future rate), 
 and to be set aside during the unexpired portion of the 
 original repayment pcrtod 
 
 iythe unexpired period). 
 
 Proceed as follows : — 
 
 [1) Ascertain the present annual increment of the fund, as 
 
 described in Chapter XXII. 
 
 (2) Multiply the anmial increment so found by the amount 
 
 of £1 per annum at the past rate for the unexpired 
 period. 
 
 {3) Divide the above product by the amount of £1 per annum 
 at the future rate for the same micxpired period. 
 
 [4) The amount so found will represent the future or 
 amended annual increment of the fund under the new 
 conditions. The amended annual sinkirig fund instal- 
 ment may be found by deducting therefrom the future 
 anmial income from the present investments representing 
 the fund. 
 
278 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (Jj Prepare a statement shoioing the jinal repayment of the 
 loan by the operation of the sinhiny fund under the 
 amended conditions . Statement XIX. B. 
 
 [6) Prepare the usual pro forma account previously recom- 
 mended. Pro forma Account 3 o. 7. 
 
 This rule icill not apply to cases in which the rate of income 
 on investments only is varied. Such problems may be solved 
 by the simjjle direct method, without calculation, described in 
 Chapter XX, Statement C. It is imperative, in using this 
 method, that the future rate of accumulation and, the rate of 
 income upon the present investments shall be uniform during 
 the whole of the unexpired portion of the period of repayment. 
 
 The Annual Inchemext (eatio) Method. Derivation of a 
 rule and formula relating to a variation in the rate per cent, of 
 accumulation. The previous chapters illustrate the various 
 methods of adjusting the annual sinking fund instalment in 
 consequence of all possible combinations of changes in the rates 
 per cent, of income and accumulation, with the result that the 
 variations have been divided into two broad groups depending 
 upon the future rate of accumulation, as shown in Statement 
 XXII. B. All variations relating to the rate of income and 
 accumulation may be adjusted by making the calculation by 
 the deductive method described in Chapter XIX, but where the 
 variation affects only the rate of income upon investments, and 
 the rate of accumulation remains unaltered, the deductive 
 method is superfluous and may be replaced by the more simple 
 direct method without calculation, as described in Chapter XX, 
 Statement XX. C. The variations affecting the rate of 
 accumulation have been divided into two sub-classes according 
 as the variation in the rate of accumulation is accompanied or 
 not by a change in the rate of income upon the present invest- 
 ments. 
 
 The effect of a variation in the rate of income upon the 
 present investra'ents has been eliminated by ascertaining the 
 actual amount of such income to be yielded annually in future, 
 and treating the same as an anunuity to be paid into the fund 
 and accumulated along with the amended annual instalment. 
 These two annual sums have been combined under the term 
 annual increment which is acted upon by the rate of accumula- 
 tion only, and the enquiry is therefore confined to the rate of 
 accumulation. 
 
THE RATE OF ACCUMULATION 279 
 
 By this method the original annual instalment, as such, 
 takes only a minor place in the calculation which is made in 
 terms of the annual increment. Having found the future or 
 amended annual increment required, under the new conditions, 
 to be paid into the fund and accumulated for the unexpired 
 portion of the original repayment period, the future annual 
 income from investments is deducted therefrom in order to 
 ascertain the future or amended annual instalment to be set 
 aside out of revenue or rate. The difference between this 
 amended instalment and the original instalment is the 
 additional annual charge to revenue or rate due to the variation 
 in the rates of both income and accumulation. 
 
 Having reduced all problems to terms of the present annual 
 increment at the date of making the adjustment, it is found 
 that this annual sum must be increased or reduced in a definite 
 ratio depending upon the original and amended rates of 
 accumulation. If it be required to ascertain the respective 
 amounts of principal which will provide a given annual sum 
 in perpetuity at two varying rates per cent., they will be 
 inversely proportional to the respective rates. But if it be 
 required to find, as in the problems now under discussion the 
 respective annuities which will amount to a given sum at the 
 end of a given term at varying rates per cent., the element of 
 accumulation enters into the calculation, although the resulting 
 annuities are still, in a sense, in inverse ratio to the rates per 
 cent. Very little consideration will show that the ratio, instead 
 of being expressed in terms of the actual rates per cent., must 
 be expressed in terms of the amounts of £1 per annum at the 
 respective rates per cent., both for a number of years equal to 
 the unexpired portion of the period of repayment. This latter 
 provision is important; it is not the factor (R) so often used 
 
 (which is £1 increased by interest for one year) but , 
 
 in which N represents the number of years in the unexpired 
 portion of the repayment period, and which expresses the 
 amount of an annuity of £1 in any number of years, as shown 
 in Chapter VI, dealing with Table III. In the previous 
 discussion of the subject in Chapter XXII this method has 
 been applied to three of the examples previously considered, 
 and results have been obtained identical with those found by 
 the deductive method. These results are shown in Calculations 
 XXII. C, D., and E. On referring to these calculations it 
 will be seen that in each case the actual working is prefaced 
 by a formula commencing with the present annual increment 
 
28o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 at the time the adjustnieut is required to be made, which annual 
 increment is multiplied by a fraction. In all cases the 
 numerator of this fraction is the amount of £1 per annum at 
 the past rate of accumulation governing the above annual 
 increment up to the time of making the adjustment. The 
 denominator of the fraction is, in each case, the amount of 
 £1 per annum at the future or substituted rate of accumulation 
 which will govern the futiire or amended annual increment 
 required. 
 
 The following table will make the matter clear and will be 
 useful for future reference when considering the question of a 
 variation in the rate per cent, of accumulation accompanied 
 by a variation in the period of repa^anent. It shows the 
 respective variations in the rate of accumulation in the 
 examples previously used to illustrate the derivation of a rule 
 and formula applying to all such variations, namely, the 
 annual increment (ratio) method: — 
 
 Rate of Accumulation Amount of £1 per annum, 
 
 reduced : — for 13 years. 
 
 Calculation. From To Numerator. Denominator. 
 
 XXII, C. oi per cent. 3 per cent. o^ per cent. 3 per cent. 
 
 XXII. D. ;}' „ 21 „ ;j „ 2i „ 
 
 XXII. E. :ji „ 21 „ 31 „ 21 „ 
 
 In the whole of the progressive examples used to illustrate 
 the consideration of the general question of variations in 
 the rates per cent, of income upon investments and of accumula- 
 tion, a gradual reduction in both rates has been assumed. It 
 has been frequently pointed out that the methods already 
 adopted will apply equally to an increase in such rates, and 
 an inspection of Statement XXII. B. will confirm this. It 
 will be seen later, in Chapter XXYI, when considering the 
 question of a variation in the rate of accumulation, complicated 
 by a variation in the period of repayment, that the sam.e rule 
 holds good, seeing that the numerator of the fraction is always 
 based upon the past rate of accumulation, and the denominator 
 upon the future rate. 
 
 A rule and formula may now be stated, based upon the 
 foregoing considerations and upon Calculations XXII. C, D., 
 and E., for finding by direct calculation from the present 
 annual increment (not the annual sinking fund instalment) the 
 future or amended annual increment due to a variation in the 
 rate of accumulation, whether accompanied or not by a variation 
 in the rate of income upon the investments representing the 
 
THE RATE OF ACCUMULATION 
 
 281 
 
 fund at the time of making the adjustment. In stating the rule 
 and formula relating to a variation in the rate of accumulation 
 in this chapter, as well as the rules relating to a variation in 
 the period of repayment in Chapter XXV, and a concurrent 
 variation in both period and rate of accumulation in Chapter 
 XXYI, the abbreviated terms which are given at the head of 
 Chapter XXII will be used, as follows : — 
 
 The Past Rate denotes the rate of accumulation upon which 
 was based the original annual instalment included in the 
 present annual increment. 
 
 The Future Rate denotes the rate of accumulation to be 
 used instead of the past rate, to calculate the amended annual 
 increment. It will be the same as the past rate in problems 
 involving a variation in the period of repayment only without 
 any variation in the rate of accumulation. 
 
 The Unexpired Period denotes the unexpired portion at the 
 time of making the adjustment of the original repayment period 
 upon which the present or original annual instalment Avas 
 based. 
 
 The Substituted Period denotes the increased or reduced 
 number of years over which the future or amended annual 
 instalment shall be spread, and at the end of which the full 
 amount of the loan will be repayable. It will be the same as 
 the unexpired period in problems involving a variation in the 
 rate of accumulation only, without any variation in the period 
 of repayment. 
 
 The rule as to a variation in the rate of accumulation only 
 (the annual increment (ratio) method) is stated in full at the 
 head of this chapter. 
 
 The above rule is sufficiently explicit, but as it will, in 
 Chapter XXVI, be combined with the rule relating to a varia- 
 tion in the period of repayment, it is expressed as a formula 
 as follows : — 
 
 Variation in the Rate of Accumulation. 
 
 The Annual Increment {ratio) Method. 
 
 ' Amount of £1 per annum 
 
 at past rate 
 
 for unexpired period. 
 
 Present 
 
 annual 
 
 increment. 
 
 Amount of £1 per annmn 
 
 at f^iture rate 
 
 for unexpired period. 
 
 Future 
 
 or 
 amended 
 
 annual 
 increment. 
 
282 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The amounts of £1 per aunum iu the above rule and formula 
 are at varying rates per cent, of accumulation, but are for the 
 same number of years. 
 
 Calculation XXII, C. will now be expressed in terms of the 
 above formula, but in this case the problem will be inverted 
 to apply to an increase in the rate of accumulation instead of a 
 decrease, as follows : — 
 
 /15-61779' 
 1060-474 X 1,,.^^-^. I = 1027-882. 
 
 /lO Di< ii}\ 
 
 60-474 X (i^.^TT^. I = 1027-^ 
 \lo-ii-jOo/ 
 
 In Chapter XXVI this calculation will be combined with the 
 similar calculation showm in Chapter XXY, but relating to a 
 variation in the period of repayment. 
 
 It Avill be noticed that the above rule, and the formula 
 expressing it, do not contain any reference to the future rate 
 of income to be yielded by the present investments representing 
 the fund, and that the sole governing factor is the varying 
 rate of accumulation. This rule and formula will apply 
 equally to an increase or decrease in the future rate of 
 accumulation, and it i.s important to remember that an increase 
 in the rate of accumulation will cause a reduction iu the annual 
 instalment to be charged to revenue or rate account in future 
 years; an increase in the repayment period will, on the other 
 hand, involve a decrease in the future annual instalment. 
 
 The object of expressing the above rule in formula form 
 will be seen later in Chapter XXV, when discussing the 
 adjustment of the annual instalment in consequence of a 
 variation in the period of repayment only, and also when 
 discussing, in Chapter XXVI, the adjustment in the annual 
 instalment due to a variation in the period of repayment 
 accompanied by a variation in the rate of accumulation. 
 
 In Chapter XXVI both the above formulae Avill be combined, 
 but in this case Calculation XXII. C. will be used in an 
 inverted form in order to obtain an example of an increase in 
 the rate of accumulation from 3 to '3| per cent, which will be 
 used as the basis of Calculation XXVI. C. 
 
 On comparing the above formula with the formula in 
 Chapter XXV, relating to a variation in the period of repay- 
 ment, it will be noticed that the denominator in the above 
 formula is the same as the numerator in the formiila in 
 Chapter XXV. 
 
THE REDEMPTION PERIOD 283 
 
 CHAPTER XXIV. 
 
 SINKING FUND PROBLEMS, RELATING TO THE 
 REDEMPTION PERIOD. 
 
 A VARIATION IN THE PERIOD OF REPAYMENT WITH OR WITHOUT 
 ANY VARIATION IN THE RATES PER CENT. OF INCOME OR 
 ACCUMULATION. SuMMARY OF METHODS. GENERAL CON- 
 SIDERATIONS AS TO THE REDEMPTION PERIOD. ThE DEDUCTIVE 
 METHOD. The annual INCREMENT (rATIO) METHOD, AND 
 
 the annual increment (balance of loan) method. 
 Statement showing the final repayment of the loan by 
 
 THE operation OF THE AMENDED ANNUAL INSTALMENT. 
 
 Summary of the methods of adjustment. 
 
 (/) The deductive method, as summarised below {see note). 
 
 Statement XXIV. A. 
 
 {II) The direct method, without calculation, as sunnmarised 
 at the head of Chapter XX, will not apply to this cariation. 
 
 {Ill) The annual increment [balance of loan) method^ as 
 suTnmnarised at the head of Chapter XXll. 
 
 Statement XXIY . D. 
 
 {IV) The annual increment {ratio) method, as summarised 
 at the head of Ch.apter XXV. Statement XXIV. C. 
 
 Note. The terms used in the following summary are fully 
 explained at the head of Chapter XXII. The deductive method 
 summarised below relates only to a variation in the period of 
 repoAfment, and is of limited application^ in that the rates of 
 accumulation and of income from investments are both the 
 same and remain unaltered. The metliod described in Chapter 
 XIX is more generally applicable, and should be folloived in 
 all cases. 
 
 Summary of the deductive method, of ascertaining the 
 amended annual sinking fund instalment due to a variation 
 in the period of repaymeiit only, without any variation in the 
 rates per cent, of acciirmilation or of income from, the present 
 investments representing the fund, both of which must be the 
 same. Statement XXIV . A. 
 
284 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (i) Ascertain the vahie of the 'present investments as 
 previously described. 
 
 [2) Calculate the amount thereof, if accumulated for tlie 
 substituted repayment period at the past unaltered rate 
 of accumulation . Calculation [XXIV) 1. 
 
 {3) Calculate the ainouiit of an annuity equal to the present 
 or oriyinal annual instalment for the substituted period 
 at the past unaltered rate of accumulation. 
 
 Calculation [XXI \') 2. 
 
 (4) The amount found in [2) added to the amount found in 
 
 (3) will represent the amount of oriyinal loan ivhich will 
 be provided thereby at the end of the substituted period 
 of repay inent. 
 
 (5) Deduct the sum found in (4) from the amount of oiiymal 
 
 loan, and the remainder represents the portion of 
 oriyinal loan which will be unprovided for by the 
 accumtilation of the present investments and the present 
 or oriyinal annual instalment at the past unaltered rate 
 of accurtiulation. 
 
 (6) Calculate the additional annual sinkiny fund instalment 
 
 which at the past unaltered rate of accmnulation will 
 amount to the balance of loan found in (J) at the end of 
 the substituted period of repayment. 
 
 Calculation [XXIV) 3. 
 
 (7) The additional annual instalment found in [6) added to 
 
 the oriyinal or present annual instalment, as in {3), yives 
 the future amended annual instalment to be set aside and 
 added to the fund duriny the substituted period of 
 repayment. 
 
 {8) Prepare a statement shoioiny the final repayment of the 
 loan by the operation of the fund under the amended 
 conditions. Statement XXJY . B. 
 
 (9) Prepare a pro forma account showing the amount which 
 should be in the fund at the end of each year of the 
 substituted repayment period. 
 
 Pro forma Account, No. JO. 
 
 Memo. The aborc method vill apply equally to an increase 
 or reduction in the period of repayment. 
 
THE REDEMPTION PERIOD 285 
 
 General Considerations. It very rarely happens that 
 there is any alteration in the period originally allowed 
 for the repayment of any individual loan of a local 
 authority. It may be taken as a general rule that 
 in the special or general Act, provisional order, or sanction 
 of the Local Government Board, authorising the expenditure 
 and the consequent borrowing, there is a specified period 
 imposed for the final repayment of the loan out of revenue 
 or rate, and this period is strictly adhered to. The Local 
 Government Board have power under the Local Government 
 Act, 1888, and the Public Health Acts Amendment Act, 
 1890, to extend or vary the periods within which loans may be 
 discharged, but this power is limited to the consolidation of 
 debt, and the exercise of such power is therefore confined to 
 the equation of the repayment periods of the several loans sa 
 consolidated. The discussion of this part of the subject will 
 be deferred to Chapter XXXII, where it will be fully con- 
 sidered. It is different with the sinking funds set aside to 
 repay the loan debt of commercial or financial undertakings. 
 In these cases the conditions are much more elastic than in the 
 case of local authorities, and almost every kind of variation is 
 met with in practice. These problems may arise at the time the 
 sinking fund is inaugurated in order to meet any special 
 obligations imposed at the time the loan is arranged, or to meet 
 any future contingency, which it is anticipated may arise 
 during the continuation of the fund. It may also happen 
 that events occur after the fund has been in operation for some 
 years which require that the period of repayment shall be 
 increased or reduced, and any alteration in the period may be, 
 and generally is, accompanied by a variation in the rate per 
 cent, of accumulation. Any variation in the rate of interest 
 payable to the loan holders rarely affects the sinking fund 
 instalment, and may generally be ignored, but in all questions 
 of this nature it is most important to ascertain the whole of 
 the conditions in order that the proper adjustment may be 
 made. 
 
 The Methods of Adjustment. The dedtictlve method. 
 Although a shorter method has been found of making the 
 adjustment in the annual instalment, in the present instance 
 the deductive method will again be first used, afterwards 
 making the same adjustment by the methods described as the 
 annual increment (ratio) method and the annual increment 
 (balance of loan) method. 
 
286 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 In tliis chapter tlie variation will be assumed to relate only 
 to the period of repayment without any complication arising in 
 consequence of a variation in the rate of accumulation or of 
 income upon the present investments. In the following 
 chapter (XXV) the annual increment (ratio) method will be 
 reduced to a rule and formula relating to the period of repay- 
 ment only, in a similar manner to that adopted in Chapter 
 XXIII, relating to the rate of accumulation. It will, however, 
 sometimes happen that an adjustment is required to be made 
 owing to a concurrent variation in the rate of income to be 
 received from the present investments and also from the invest- 
 ment of the future accretions to the fund, and these again may 
 be at different rates. All questions arising out of a variation 
 in the rate per cent, generally, have been considered in previous 
 •chapters, and the adjustment due to a simultaneous variation 
 in both period and rate per cent, will be deferred to Chapter 
 XXVI. 
 
 The present problem will be illustrated by the now familiar 
 example of the sinking fund already discussed, which relates 
 to the repayment of a loan of £26,495 at the end of 25 years, 
 requiring an annual instalment of £680'284, to be set aside and 
 accumulated at 3^ per cent., as found by Calculation (XV) 1. 
 
 Circumstances have arisen which impose upon the under- 
 taking the necessity to accelerate the final redemption of the 
 loan indebtedness by the operation of the fund. It is not 
 necessary to enquire into the special reason for such acceleration 
 because the principle is the same in any event. The adjustment 
 will again be based upon the position of the fund at the end 
 of the 12th year. The undertaking or company was originally 
 required to repay the loan of £26,495 at the end of the 25th 
 year, namely, in 13 years from the present time, and, towards 
 this, there is in the fund the proper calculated amount, which 
 is represented by investments valued at £9932*74, as found by 
 Calculation (XV) 2, yielding an assured future annual income, 
 at 3| per cent., of £347"64(S. The altered conditions demand 
 that the operation of the fund shall be accelerated and that the 
 original annual instalment shall be increased to such an amount 
 as will repay the loan in (S years from the present time instead 
 of at the end of 13 years thereby reducing the original repay- 
 ment period from 25 to 20 years. 
 
 This reduction in the period affects the future accumulation 
 of the annual instalment of £680"234, as originally calculated, 
 and also the future accumulation of the amount of £993274 
 now in the fund. In order to compare the resulting increased 
 
THE REDEMPTION PERIOD 287 
 
 annual instalment with the original instalment it will be 
 assumed that the original estimated rate of accumulation, 
 namely, 3^ per cent., will continue to be received during the 
 remaining 8 years, both as regards the income from the present 
 investments and the amended annual instalment. 
 
 All the present factors will be again reduced to equivalent 
 amounts of original loan which will be provided at the end of 
 the substituted period of 8 years by the accumulation of such 
 factors in order to ascertain, by the deductive method, as shown 
 in Statement XXIV. A., the portion of original loan which 
 remains to be provided by an additional annual instalment. 
 If the rate of accumulation remains unaltered the reduction in 
 the period of repayment will have the effect of increasing the 
 annual instalments as originally calculated. If, on the 
 contrary, the unexpired period of 13 years be extended instead 
 of reduced, there will be an apparent surplus in the fund which 
 will lead to a reduction in the annual instalment. 
 
 The additional annual instalment required, as shown in 
 Statement XXIY. A., by the deductive method, is £801-862. 
 The balance of loan, £7258-21, shown in Statement XXIY. A., 
 which will be unprovided for owing to the reduction in the 
 redemption period from 13 years to 8 years is made up as 
 follows : — 
 
 Present investments £9932-74 
 
 Amount thereof, accumulated for 
 13 years at 3^ per cent. 
 
 Calculation (XYII) 2 £15534-38 
 Amount thereof, accumulated for 
 8 years at 3^ per cent. 
 
 Calculation (XXIY) 1 £13079-53 
 
 £2454-85 
 
 Original annual instalment : — 
 
 Amount of £680-234 per annum, 
 accumulated for 13 years at 3^ 
 percent. Calculation (XY) 5 £10960-62 
 
 Amount of £680-234 per annum, 
 accumulated for 8 years at 3| 
 per cent. Calculation (XXIY) 2 £6157-26 
 
 £4803-36 
 
 Balance of loan unprovided for £7258-21 
 
288 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 It lias tlius been ascertained tliat tlie ultimate amount of 
 loan which will be unprovided at the end of the substituted 
 period in consequence of the reduction in the original redemp- 
 tion period is £7258"21, and this deficiency has been divided 
 between the accumulations of the present investments and of 
 the original annual instalment. The portion of the deficiency 
 due to the reduced accumulation of the present investments is 
 £2454'85, and has been expressed in terms of the capital value, 
 but it may also be expressed in terms of the annual income of 
 £347'648 to arise from the present investments, as follows : — 
 
 Amount of £347 648 per annum in 13 years at 
 
 31 per cent. Calculation (XXIY) 4 £5601-66 
 
 Amount of £347" 648 per annum in 8 years at 
 
 31 per cent. Calculation (XXIV) 5 £3146-81 
 
 £2454-85 
 
 Statement XXIV. A. shows that the reduction in the period 
 of repayment from 25 years to 20 years (but with the same rate 
 of accumulation) taking place at the end of the 12th year, 
 results in an increased annual burden of £801-862 chargeable 
 against the revenue of the undertaking. It only now remains 
 to review the operation of the fund under the altered conditions 
 in order to ascertain that the amended annual instalment of 
 £1482096 so found will carry out the purpose of the fund, 
 namely, to repay the loan of £26,495, but at the end of 20 
 instead of 25 years. This is shown in Statement XXIV. B., 
 and by the pro forma account, No. 10. 
 
 The Annual Increment (ratio) Method . In previous 
 chapters dealing with each of the variations in the rates per 
 cent, of income and accumulation, the additional annual 
 instalment was first ascertained by the deductive method, as 
 fully described in Chapter XIX. This method is based 
 essentially upon the ultimate separate accumulation at the 
 future rate of each of the present factors of the fund, namely, 
 the annual instalment as originally calculated, the value of the 
 present investments, and the future income to arise therefrom, 
 all of which were reduced to equivalent amounts of original 
 loan which they will indiAndually provide at the end of the 
 period of redemption. In Chapter XXII iho whole of these 
 adjustments were again made by direct calculations based upon 
 
THE REDEMPTION PERIOD 289 
 
 the auuiial increnieut of the fund as defined in Chapter XIY, 
 and it was found that by this means it was possible to simplify 
 tbe calculation and eliminate altogether the effect of any 
 variation in the rate of income to be received in future upon 
 the present investments representing the fund. It was found 
 that there is an exact ratio existing between the present and 
 future annual increments depending upon the respective 
 amounts of £1 per annum; and in Chapter XXIII, relating 
 solely to the rate of accumulation, this method of calculation 
 was reduced to a rule and formula, called the annual increment 
 
 (ratio) method. 
 
 For the purpose of the following adjustment the present 
 
 annual increment, which is the basis of the calculation, is made 
 
 up as follows : — 
 
 Original annual instalment £680"234 
 
 Income from present investments 34T"648 
 
 £1027-882 
 
 The above annual income from the present investments, as 
 in all adjustments made by this method, is tlie amount which 
 has been received in the past, and is not the amount which will 
 be yielded thereby during the substituted period of repayment. 
 
 This is one of the fundamental principles of the annual 
 increment (ratio) method, as fully explained in the opening 
 paragraphs of Chapter XXII. This method will now be applied 
 to a variation in the period of repayment, as shown in 
 Calculation XXIY. C. 
 
 The Annual Increment (balance of lo.\n) Method. 
 It has been found in Chapter XXII, dealing with a variation in 
 the rate of accumulation, that the future or amended annual 
 increment, and therefrom the future or amended annual 
 instalment, may be obtained by deducting the value of the 
 present investments representing the fund, from the total 
 amount of loan repayable at the end of the prescribed period, 
 and treating the balance as an original amount to be provided 
 by an annual sum to be accumulated during the unexpired 
 portion of the original repayment period at the future amended 
 rate of accuni illation. The annual sum so found is the 
 equivalent of the future or amended annual increment, and the 
 future or amended annual instalment under the new conditions 
 
290 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 is found by deducting tlierefrom the annual income to be 
 received in future upon tlie present investments representing 
 the fund, at any rate per cent, whether increased or reduced. 
 This is the annual increment (balance of loan) method, and 
 although its derivation is not described until Chapter XXII, it 
 has been used in previous chapters. Statement XXIV. D. 
 following gives details of the present example worked out by 
 this method. 
 
 The Redemption Period. Statement XXIV. A. 
 
 The Deductive Method, 
 
 Showing the method of adjusting the annual instalment 
 in consequence of a variation in the redemption period 
 without any variation in the rate per cent, of accumulation 
 or of income from the present investments, both of which 
 rates are the same. 
 
 If these rates are unequal or are varied proceed as in 
 Chapter XIX. A. 
 
 Conditions before adjustment, at end of 12th year : 
 
 Amount of original loan, repayable in 25 years £26,495 
 Amount in the fund, at end of 12th year ... £9932"74 
 Present annual income (previously) received 
 
 therefrom, at 3-| per cent., per annum ... £347"648 
 Present annual instalment, to be accumulated 
 
 for 13 years, at ^ per cent £680-234 
 
 Present annual increment £102T"882 
 
 Variation from the above conditions ;— 
 
 The period during which the loan shall be redeemed is 
 reduced from 13 to 8 years. 
 
 The substituted period of repayment 8 years. 
 
 Present investments (at end of 12th year) £9932"74 
 
 Equivalent 
 
 amount of 
 
 oriffinal loan. 
 
 Amount thereof, accumulated for 8 years at 
 
 3i per cent. Calculation (XXIV) 1 £13079-53 
 Original annual instalment £680234 
 
 Amount of £680-234 per annum, for 8 years 
 
 at 3i per cent. Calculation (XXIV) 2 £6157-26 
 
 Provision already made, will repay loan of £19236-79 
 
THE REDEMPTION PERIOD 
 
 291 
 
 Additional annual instalment required : — 
 
 ]ialauce, being- amount of original loan unpro- 
 vided for, owing to the above decrease in the 
 redemption period requiring an additional 
 annual instalment to be set aside and 
 accumulated for 8 years at 3^ per cent. 
 
 Additional annual instalment 
 
 Calculation (XXIY) 3 £801-862 
 
 Amount of original loan 
 
 Amended annual increment : — 
 
 Annual income from investments. . 
 Amended annual instalment 
 
 £347-648 
 £1482-096 
 
 £1829-744 
 
 £7258-21 
 
 £26495-00 
 
 The Redemption Period. 
 
 Statement XXIV. B. 
 
 Showing the final repayment of the loan, by the operation of 
 the sinking fund after making the adjustment in the 
 annual instalment consequent upon a reduction in the 
 period ol repayment, Avithout any variation in the rate per 
 cent, of accumulation, or of income from the present 
 investments. 
 
 Present investments (at end of 12tli year) ... . 
 
 Amended annual increment : — 
 
 Original annual instalment £680-234 
 
 Additional annual instalment ... 801*862 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 £9932-74 
 
 Total out of revenue £1482-096 
 
 Income from investments 347*648 
 
 Total £1829-744 
 
 Amount thereof, accumulated for 8 years at 
 
 3i per cent. Calculation (XXIV) 6 £16562-26 
 
 Amount of original loan £26495'00 
 
 Amended annual instalment £1482096 
 
292 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Redemption Period. Calculation XXIV. C, 
 
 The Annual Increment (ratio) Method. 
 
 To find the amended annual increment (and tlierefrom the 
 amended annual instalment) in a sinking fund in which 
 the original period of repayment is varied, accompanied or 
 not by any variation in the rates of accumulation or of 
 income from the present investments. 
 
 The rule relating to this method is stated at the head 
 of Chapter XXV. 
 
 Required the annual increment to be accumulated for a period 
 of 8 years, which is equivalent to an annual increment of 
 £1027-882, to be accumulated for a period of 13 years, the 
 rate of accumulation in both cases being 3^ per cent. 
 
 1027-882 ( ^^^o^^^^ o^ ^1 Pe^ annum, 13 years, 3|% | _.,^pc)-744 
 
 ) Amount of £1 per annum, 8 years, 3^% ( 
 or by Table III, giving the amounts of £1 per annum. 
 
 1027-882x1611303 
 9M05I68 
 
 1829-744 
 
 Log. Present annual increment ... 1027882 30119434 
 add Log. Amount of £1 per annum 
 
 Table III, 13 years, 3i percent. 16-11303 1-2071771 
 
 deduct Log. Amount of £1 per annum 
 Table III, 8 years, 3| per cent. 
 
 16562-26 
 9-05168 
 
 Log. Amended annual increment 
 
 Amended annual increment 
 
 To find the a iiiriidcd. anniuil insfal »)ent : — - 
 
 deduct the inc(mie from investments, 3^ 
 per cent. 
 
 Amended annual instalment 
 
 being PreseTit annmil instalment ... 680-2-)4 
 Additiomil annual instalment 801-862 
 
 4-2191205 
 0-9567296 
 
 3-2623909 
 
 1829-744 
 
 347-648 
 
 1482096 
 
 1482096 
 
THE REDEMPTION PERIOD 
 
 293 
 
 The Redemption Period, 
 
 Statement XXIV. D. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find the amended annual sinking fund instalment consequent 
 upon a variation in the period of repayment with or without 
 any variation in the rate of income to be received from 
 the present investments or in the rate of accumulation. 
 
 For Eule see Chapter XXII. 
 
 Amount of original loan (25 years) £26495- 00 
 
 deduct amount in the fund at the end of the 
 
 12th year £9932-74 
 
 Balance of loan 
 
 ... £16562-26 
 
 Amended annual increment, to be added to the 
 fund, and accumulated at 3| per cent., to 
 provide this amount at the end of 8 years. 
 
 Calculation XXIV. C. £1829-744 
 
 deduct therefrom income to be received from 
 the present investments, £9932-74, at 3| 
 per cent. 
 
 £347-648 
 
 Amended annual instalment 
 
 being Original annual instalment ... £680-234 
 Additional annual instalment 801-862 
 
 £1482096 
 
 £1482-096 
 
294 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Pro forma Sinking Fund Account, No. 10. 
 
 A A'^ariation in the Redemption Period. 
 
 Loan of £26,495, repayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by the operation 
 of the increased annual instalment of £1482'096. 
 
 Statement XXIV. B. 
 
 Rate of accumulation, 3^ per cent. 
 
 Year. 
 1 
 
 Amount in 
 
 the fund 
 
 at beginning 
 
 of year. 
 
 Income 
 
 received from 
 
 investments 
 
 3i per cent. 
 
 Annual 
 
 sinking fund 
 
 instalment. 
 
 Amount in 
 
 the fund 
 
 at end 
 
 of year. 
 
 Year 
 1 
 
 2 
 
 3 
 
 
 
 
 
 2 
 3 
 
 4 
 
 The amount in the 
 
 fund at the 
 
 end of 
 
 4 
 
 5 
 
 the 12th 
 
 year, £9932-744, is the 
 
 correct 
 
 5 
 
 6 
 
 calculated amount, 
 
 as shown hj Calcula- 
 
 6 
 
 7 
 
 tion (XV) 2, and 
 
 hy the pro 
 
 forma 
 
 7 
 
 8 
 
 account, 
 
 No. 1, Chapter XV. 
 
 
 8 
 
 9 
 
 
 
 
 
 9 
 
 10 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 347 
 
 648 
 
 1482-096 
 
 11762-488 
 
 13 
 
 14 
 
 11762-488 
 
 411 
 
 687 
 
 1482-096 
 
 13656-271 
 
 14 
 
 16 
 
 13656-271 
 
 477 
 
 969 
 
 1482-096 
 
 15616-336 
 
 15 
 
 16 
 
 15616-336 
 
 546 
 
 572 
 
 1482-096 
 
 17645-004 
 
 16 
 
 17 
 
 17645004 
 
 617 
 
 575 
 
 1482-096 
 
 19744-675 
 
 17 
 
 18 
 
 19744-675 
 
 691 
 
 063 
 
 1482-096 
 
 21917-834 
 
 18 
 
 19 
 
 21917-834 
 
 767 
 
 124 
 
 1482-096 
 
 24167-054 
 
 19 
 
 20 
 
 24167-054 
 
 845-850 
 
 1482-096 
 
 26495 000 
 
 20 
 
 21 
 
 
 
 
 
 21 
 
 22 
 
 
 
 
 
 22 
 
 23 
 
 
 
 
 
 23 
 
 24 
 
 
 
 
 
 24 
 
 25 
 
 
 
 
 
 
 25 
 
THE REDEMPTION PERIOD 295 
 
 CHAPTER XXV. 
 
 SINKING FUND PROBLEMS, RELATING TO THE 
 REDEMPTION PERIOD {Continued). 
 
 Derivation of a rule and formula relating to a variation 
 in the period of repayment based upon the foregoing 
 results by the annual increment (ratio) method. 
 
 The Annual Increment (ratio) Method. 
 
 The rule as to a variation in the period of repayment, may 
 he stated as follows^ using the terms explained at the head of 
 Chapter XXII. Statement XXIV. C. 
 
 Rule. To find the amended annual instalment to he set aside, 
 and added to the existing sinking fu7id, 
 
 to he accumulated in f^iture at the same rate per cent, 
 at tvhich the present annual instalment was calculated 
 
 (the future rate), 
 and to he set aside for a reduced or increased number 
 of years as com.pared loith the unexpired portion of 
 the original repayment period 
 
 (the suhstituted period). 
 Proceed as follows : — 
 
 (i) Ascertain the present animal increment of the fund, as 
 described in Chapter XXII. 
 
 (2) Multiply the an?iual increment so found by the amount 
 
 of £1 per annum at the future rate for the unexpired 
 period. 
 
 (3) Divide the above product by the amount of £1 per 
 
 annum, at the future rate for the suhstituted period. 
 
 (4) The amount so found will represent the future or 
 
 amended annual increment of the fund binder the new 
 conditions. The amended annual sinking fund instal- 
 ment may be found by deducting therefrom the future 
 annual income from the present investments representing 
 the fund. 
 
296 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (5) Prepare a statement showing the final repay inent of the 
 
 loan by the operation of the sinking fund under the 
 a/mended conditions. Statement XXIV . B. 
 
 (6) Prepare the usual pro forma account previously recom- 
 
 mended. Pro forma Account, No. 10, Chapter XXIV. 
 
 It is imperative, in using this method, that the future rate 
 of accumulation and the rate of income from the present invest- 
 ments, shall he uniform dur-ing the whole of the substituted 
 period of repayment. 
 
 The Annual Increment (eatio) Method. Derivation of 
 a rule and formula, relating to a variation in the period of 
 repayment. The subject of enquiry in Chapter XXIII is the 
 derivation of a rule and formula by which to ascertain the 
 future or amended annual increment, and therefrom the 
 amended annual instalment, due to a variation in the rate of 
 accumulation only. The present object is to find a similar 
 rule and formula Avhich will apply to a variation in the period 
 of repayment, and the method to be adopted will be the same in 
 principle. In discussing the effect of a variation in the rate 
 of accumulation upon the future or amended annual increment 
 in Chapter XIX, Variation A, the amended annual increment 
 was ascertained by the somewhat roundabout, although instruc- 
 tive, deductive method there described (Statement XIX. A.). 
 This method was used purposely in order to emphasise the 
 principles involved and to show the effect of the variation in 
 the rate of accumulation upon each of the actual factors of the 
 fund, namely, the present investments, the annual income to 
 arise therefrom, and the original annual sinking fund instal- 
 ment. This deductive method of enquiry was again adopted in 
 Chapter XX (Variation 13, rate of income upon investments), 
 and the amended annual increment was ascertained as shown in 
 Statement XX. A. In Chapter XXI, the same method was 
 applied to ascertain the amended annual increment due to a 
 dual variation in the rates per cent, of accumulation and of 
 income upon investments (Variation C), and the result is con- 
 tained in Statement XXI. A. 
 
 Chapter XXII contains a tabular summary fXXII. A.) of 
 the results obtained in all the above investigations into the 
 effect of variations in the rate per cent. This summar}- shows 
 that in each of llie above eases the original and amended annual 
 increments bear a certain definite ratio one to the other; and 
 
THE REDEMPTION PERIOD 397 
 
 fioiii this ratio it is possible to derive a rule and formula by 
 which, to derive the amended annual instalment directly from 
 the original annual instalment. 
 
 In the above examples the period of repayment remained 
 unaltered, and it has been ascertained that any variation in the 
 rate per cent, of accumulation has the effect of increasing or 
 reducing the present annual increment in proportion to the 
 ratio existing between the amounts of £1 per annum for the 
 unexpired portion of the original repayment period at the past 
 and future rates of accumulation respectively. A similar 
 method will now be applied to the derivation of a rule and 
 formula by which to find the future or amended annual 
 increment, and therefrom the amended annual instalment, due 
 to a variation in the period of repayment, the rate of 
 accumulation remaining the same, and it will be demonstrated 
 by means of the results obtained in the example just considered 
 in Chapter XXIV. In this instance there is a present annual 
 increment, receivable for 13 years, composed of : — 
 
 Original annual instalment £680"234 
 
 Income from present investments 347 "648 
 
 £1027-882 
 
 and this present annual increment, if accumulated at 3^ per 
 
 cent, for 13 years, is sufficient to provide a definite amount of 
 
 loan at the end of that time. The above annual income from 
 
 the present investments, as in all adjustments made by this 
 
 method, is the amount which has been received in the past, 
 
 and is not the amount of income which Avill be yielded thereby 
 
 during the unexpired or substituted period of repayment. This 
 
 is one of the fundamental principles of the annual increment 
 
 (ratio) method, as fully explained in the opening paragraphs of 
 
 Chapter XXII. For this purpose it is not necessary to know 
 
 the actual amount of the loan, but the above present annual 
 
 increment may be treated as a simple annuity certain, for a 
 
 period of 13 years, to be acciimulated at 3^ per cent. It is 
 
 required to ascertain the equivalent annuity or annual 
 
 increment accumulating at the same rate to repay the same 
 
 loan, but at the end of a term of 8 years instead of at the end of 
 
 13 years. It has already been ascertained, in Chapter XXIV, 
 
 that this equivalent annual increment is £1829" 744. In the 
 
 case of the previous Calculations XXII. C, D., and E., tlie 
 
29? REPAYMENT OF LOCAL AND OTHER LOANS 
 
 period of repaymeut remaiued the same, but the rate of 
 accumulation varied ; consequently the ratio was expressed in 
 terms of the amounts of £1 per annum at the respective rates 
 per cent., but for the same number of years. In the present 
 instance the rate of accumulation remains unaltered, but the 
 period of repayment is varied. Consequently the ratio is 
 expressed in terms of the amounts of £1 per annum for the 
 respective unexpired and substituted periods of repayment, but 
 at the same rate per cent, of accumulation. 
 
 In the formula in Chapter XXIII relating to a variation in 
 the rate of accumulation, the numerator is the amount of £1 
 per annum for the unexpired period at the past rate of 
 accumulation, and the denominator is the amount of £1 per 
 annum for the same unexpired period at the future rate of 
 accumulation, thus taking as the basis of the ratio the varying 
 rates of accumulation. But as the formula about to be 
 ascertained depends as to its ratio upon the varying periods of 
 repayment, and there is not any variation in the rate of 
 accumulation, the numerator becomes the amount of £1 per 
 annum at the rate of accumulation common to the two periods 
 for the unexpired period, and the denominator becomes the 
 amount of £1 per annum at the same rate of accumulation for 
 the substituted period. Substituting the above terms for those 
 in the previous formula, the amended formula is ascertained 
 for dealing with problems involving variations in the period of 
 repayment only, but not at the same time involving any 
 variation in the rate of accumulation. The rule and formula 
 as to a variation in the period of repayment only will be 
 expressed in the same abbreviated terms used in Chapter 
 XXIII, dealing with a variation in the rate of, accumulation, 
 and these abbreviated terms should be carefully considered. 
 They are fully explained at the head of Chapter XXII. In 
 this case there is not any variation in the rate of accumulation, 
 consequently the past and future rates are the same, and are, in 
 effect, the past rate. This is important when considering a 
 variation in the period of repayment only, or a concurrent 
 variation in the rate of accumulation as well as in the period of 
 repayment. 
 
 It is therefore necessary to use the term " future rate " in 
 the after consideration of this and the formula relating to the 
 dual variation in rate and period. 
 
 The rule as to a variation in the period of repayment only, 
 the annual increment (ratio) method, is stated in full at the 
 head of this chapter. 
 
THE REDEMPTION PERIOD 299 
 
 As stated in Chapter XXIII, the above rule is sufficiently 
 explicit, but as it will be necessary in the following chapter to 
 combine it with the previous rule relating to a variation in the 
 rate of accumulation it will be expressed as a formula, as 
 follows : — 
 
 Variation in thp: Period of Repayment. 
 The Aniiual Increment (ratio) Method. 
 
 \ 
 
 Present 
 
 annual 
 
 increment 
 
 'Amount of £1 per annum 
 
 at future rate 
 
 for unexpired period 
 
 Amount of £1 per annum 
 
 at future rate 
 
 for substituted period 
 
 Future 
 
 or 
 amended 
 
 annual 
 increment 
 
 The amounts of £1 per annum in the above rule and formula 
 are at the same rate per cent, of accumulation, but are for 
 varying numbers of years. In this case the future rate is the 
 same as the past rate. Calculation XXIV. C. may now be 
 stated in terms of the above formula, as follows : — 
 
 1027-882 X f^fi^^f ) =1829-744 
 V 9-05168 / 
 
 and in Chapter XXVI this calculation will be combined Avith 
 the similar calculation in Chapter XXIII. 
 
 The above rule and formula will apply equally to an 
 increase or reduction in the period of repayment, and it is 
 important to remember that an increase in the period will have 
 the effect of reducing the annual instalment to be charged to 
 revenue or rate account in future years. When considering 
 the rate of accumulation in Chapiter XXIII it was found that 
 an increase in the rate of accumulation will reduce the annual 
 instalment in future years. In the following chapter (XXVI) 
 the above formula, relating to a variation in the period of 
 repayment will be combined with the formula found in 
 Chapter XXIII, relating to a variation in the rate of accumula- 
 tion, for the purpo.se of deriving therefrom a formiila which 
 may be applied to a concurrent variation in the period of 
 repayment and the rate of accumulation. 
 
 It will be noticed that the numerator in the above formula, 
 relating to the period, is the same as the denominator in the 
 formula in Chapter XXIII, relating to the rate per cent. 
 
300 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XXVI. 
 
 SINKING FUND PKUBLEMS, RELATING TO THE 
 RATE PER CENT. OF ACCUMULATION AND THE 
 REDEMPTION PERIOD IN COMBINATION. 
 
 Summary of methods. General considerations. The 
 methods of ascertaining the amended annual instalment 
 due to a variation in both the above factors in 
 
 COMBINATION. ThE DEDUCTIVE METHOD, THE ANNUAL 
 
 INCREMENT (RATIO) METHOD, AND THE ANNUAL INCREMENT 
 (balance of loan) METHOD. STATEMENT SHOWING THE 
 
 final repayment of the loan by the operation of the 
 amended annual instalment. 
 
 Derivation of a rule and formula relating to a dual 
 variation of this nature based upon the foregoing 
 
 RESULTS, BY THE ANNUAL INCREMENT (rATIO) METHOD. 
 
 Summary of the methods of adjustment. 
 
 (/) The deductive method, as summarised at the head of 
 Chapter XXIV; ichich may he compared with the method 
 summarised at the head of Chapter XIX. 
 
 Statement XXTI. A. 
 
 ill) The direct method, ivithout calculation, as surmnarised 
 at the head of Chapter XX, will not apply to this variation. 
 
 {Ill) The annual increment {balance of loan) method, as 
 summarised at the head of Chapter XXII. 
 
 Statement XXVI . H . 
 
 {IV) The annual inclement {ratio) method, as summarised 
 below. Statement XXVI. C . 
 
 Note. The terms used in the following summary are fully 
 discussed at the head of Chapter XXII . It is imperative, in 
 using the above methods, that the future rate of accumulation 
 and the rate of income from the present investments shall be 
 uniform during the whole of the substituted period of repay- 
 Tnent. 
 
THE RATE PER CENT. AND PERIOD 301 
 
 The Annual Increment (ratio) Method. 
 
 The rule as to a conmirrent variation in the rate of accumu- 
 lation, as well as in the 'period of repay 7ne7it, inay he stated as 
 follows, using the terms explained at the head of Chapter 
 XXI L Statement XXV I. C. 
 
 Rule. To find the amended annual instalment to he set aside, 
 and added to the existing sinking fund, 
 
 to he accumulated in future at a rate per cent, 
 greater or less than the rate at which the present 
 annual instalTnent ivas calculated 
 
 [the future rate), 
 and to he set aside for a reduced or increased number 
 of years, as cornpured ivith the unexpired portion of 
 the original repayment period 
 
 {the s^ihstituted\ period). 
 Proceed as follows : — 
 
 {!) Ascertain the present annual increment of the fund, as 
 described in Chapter XXII. 
 
 (2) Multiply the annual incrcTnent so found by the amount 
 
 of £1 per anmim at the past rate for the unexpired 
 period. 
 
 (3) Divide the above product by the amount of £1 per 
 
 annum, at the future rate for the substituted period. 
 
 (4) The amount so found loHl represent the future or 
 
 amended annual increment of the fund under the new 
 conditions . The amended annual sinlcing fund instal- 
 ment may be found by deddicting therefrom the futtire 
 annual income from, the present investments representing 
 the fund. 
 
 [6) Prepare a statement showing the flrial repayment of the 
 loan by the operation of the fund under the amended 
 conditions . Statement XXVI . B. 
 
 (6) Prepare the usual pro forma account previously recom- 
 mended. Pro forma Account, No. 11. 
 
 General Considerations. The predommant factor in all 
 problems of tliis nature is the variation in the period of 
 repayment because its effect upon the amended annual instal- 
 ment is far greater than that due to the variation in the rate 
 
302 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 of accumulation which will generally lie within very narrow 
 limits. For the reasons given in Chapter XXIV, a variation 
 of this two-fold nature will rarely arise in connection with 
 any individual loan of a local authority, and if such a problem 
 arises in connection with the consolidation of several such loans 
 it will be complicated by other factors which will render 
 necessary a different mode of treatment, as will be explained 
 in Chapter XXXII, dealing generally with the equation of the 
 period of repayment. 
 
 The principal application of the methods to be discussed in 
 this chapter will relate to the sinking funds of commercial and 
 financial undertakings, and all the general considerations as to 
 a variation in the period of repayment only, stated in Chapter 
 XXIV, will apply to this example without further reference 
 or amplification. 
 
 In dealing with problems which may arise in connection 
 with the sinking funds of local authorities and commercial and 
 financial undertakings, the following important factors have 
 already been discussed, namely : — 
 
 1. The amount in the fund. 
 
 2. The rate per cent. — 
 
 (a) of income upon the present investments. 
 (6) of future accumulation. 
 
 3. The period of repayment. 
 
 In discussing the problems relating solely to a variation in 
 the rate per cent, or the period of repayment', in each case there 
 has been combined in one factor, " the annual increment," 
 (1) the original or amended annual instalment, and (2) the past 
 or future income arising from the present investments represent- 
 ing the fund at the time the variation occurs in the rate or 
 period. 
 
 This annual increment is fully discussed and described in 
 Chapters XIV and XXII. The majority of the examples used 
 to illustrate the above problems relate to a sinking fund to 
 repay a loan of £26,495 at the end of 25 years, and it has been 
 assumed that ihe variation, and the consequent necessity for 
 adjustment, occurs at the end of the 12tli year in each case. 
 As regards a variation in the rate per cent., it has been proved 
 that the problem may be confined, so far as the actuarial 
 calculation is concerned, to the rate of accumulation. It has 
 also been ascertained that there is a simple ratio existing 
 
THE RATE PER CENT. AND PERIOD 303 
 
 between the original and amended annual increments due to a 
 variation in both the rate and the period, and that this ratio 
 is based, not upon the respective rates per cent, of accumulation 
 or upon the number of years in the period of repayment, but 
 upon the respective amounts of £1 per annum as follows : — 
 
 1. In the case of a variation in the rate of accumulation, 
 upon the amounts of £1 per annum for the same period of 
 repayment, but at the respective rates per cent, of accumulation. 
 
 Chapter XXIII. 
 
 2. In the case of a variation in the period of repayment, 
 upon the amounts of £1 per annum at the same rate per cent, 
 of accumulation, but for the respective periods of repayment. 
 
 Chapter XXV. 
 
 In the case of variations in the rate per cent, the necessary 
 adjustment has been made, in the first instance, by the deductive 
 method, fully described in Chapter XIX, based upon the whole 
 of the factors governing the fund, after which the result so 
 obtained has been verified by the annual increment (ratio) 
 method based upon the annual increment, as described in 
 Chapter XXII. These results have been utilised to deduce a 
 rule, and a formula expressing the rule, which is fully described 
 in Chapter XXIII. 
 
 The enquiry was then extended in a similar manner to an 
 adjustment rendered necessary by a variation in the period of 
 repayment which was considered in Chapters XXIV and XXV, 
 and a similar rule and formula was deduced. In each case it 
 was found that the methods applied equally to an increase or a 
 reduction in the rate of accumulation or period of repayment. 
 
 The adjustment consequent upon a dual variation in the rate 
 of accumulation, as well as in the period of repayment, will be 
 fully considered in the present chapter, using the whole of the 
 methods already described, after which a rule and formula 
 relating to the adjustment will be deduced from the results so 
 obtained. 
 
 The Deductive Method. The present enquiry will also 
 be illustrated by a sinking fund to repay a loan of £26,495 at 
 the end of 25 years, but with a rate of accumulation of 3 per 
 cent., requiring an annual instalment of £712''826 to be set 
 aside for the remaining 13 years. This has been ascertained 
 in Chapter XIX, Statement XIX. A. 
 
 The necessity to make the adjustment arises at the end of 
 the 12th year, at which time the amount in the fund is 
 £993274, which is represented by investments valued at that 
 
304 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 amount, bringing in an annual income at 3^ per cent, per 
 annum of £347'648, and it will be assumed tbat this income is 
 assured for the whole of the unexpired portion of the original 
 repayment period. At the end of the 12th year, this period is 
 for some reason reduced from 13 years to 8 years, and the rate 
 of accumulation is increased from 3 to 3^ per cent., as in the 
 original conditions in Chapter XY. 
 
 The effect will be that the annual instalment of £712-826 
 will be increased in consequence of the reduction of the period 
 of repayment, but it will not be increased to such an amount as 
 it would have been if the rate of accumulation had remained at 
 3 per cent,, and had not been increased to 3| per cent. This 
 will be shown later in this chapter in detail, where the amended 
 annual instalment will be divided between these factors. As in 
 previous examples by the deductive method, all the present 
 factors of the fund will be reduced to equivalent amounts of 
 orit^inal loan which they will each provide by accumulation at 
 the future rate of 3^ per cent., at the end of the substituted 
 period of 8 years, in order to ascertain, by deduction, the 
 portion of original loan which will remain to be provided by the 
 future accumulation of an additional annual instalment to be 
 charged to revenue or rate or deducted from profits, and a final 
 calculation will be made to ascertain such additional instalment. 
 
 This is fully shown in Statement XXYI. A., which is similar 
 in principle to previous statements illustrating the deductive 
 method. This statement shows that the reduction in the original 
 period of repayment from 25 to 20 years (but with an increase 
 in the rate of accumulation) taking place at the end of the 
 12th year results in an increased annual burden of £769"'270 
 chargeable against the rate account or the revenue account of 
 the undertaking. 
 
 It is now possible to review the operation of the sinking fund 
 under the altered conditions in order to ascertain that the 
 amended annual instalment so found will carry out the purpose 
 of the fund, namely, to repay the loan of £26,495, at the end 
 of 20 instead of 25 years. This is shown in Statement XXYI. B., 
 which is exactly similar in principle to the previous statements 
 prepared to illustrate the accuracy of the amended annual 
 instalments found by the deductive and other methods. 
 
 The Annual Increment (ratio) Mettiod. (Tfule and 
 Formula.) In Chapter XXII (a variaiiou in the rate of 
 accumulation) as well as in Chapter XXIY (a variation in the 
 pciiod of repaymeni) the actual adjustment has been made bv 
 
THE RATE PER CENT. AND PERIOD 305 
 
 the annual increment (ratio) method there described, and from 
 the results so obtained the formula relating to the method has 
 been deduced. In both these variations the ratio is a simple 
 one, depending upon the respective amounts of £1 per annum at 
 the varying rates per cent, in one case and for the varying 
 periods in the other. As the rules and formulse relating to the 
 above variations have been already ascertained, it is only 
 necessary in the present instance to revert to those formulae in 
 order to deduce therefrom a modified formula relating to a 
 combination of the above causes of adjustment, and afterwards 
 to make the calculation in the manner shown in Chapters XXII. 
 and XXIV. It would appear from the above theoretical con- 
 siderations that the two formulae may be combined in order to 
 deduce therefrom a simple formula which will apply to all 
 problems involving a dual variation in the rate of accumulation 
 and the period of repayment. It is therefore necessary to 
 combine the formula relating to a variation in the rate of 
 accumulation given in Chapter XXIII with that relating to a 
 variation in the period of repayment in Chapter XXV. The 
 factors required are : — 
 
 1. The present annual increment. 
 
 2. The past and future rates of accumulation. 
 
 3. The unexpired and substituted periods of repayment. 
 
 For the purpose of the following adjustment the present 
 annual increment, which is the basis of the calculation, is made 
 up as follows : — 
 
 Original annual instalment (Statement XIX. A.) ... £T12"826 
 Income from present investments 347'648 
 
 £1060-474 
 
 The above annual income from the present investments, as in 
 all adjustments made by this method, is the amount which has 
 been received in the past, and is not the amount of income 
 which will be yielded thereby during the unexpired or 
 substituted period of repayment. This is one of the funda- 
 mental principles of the annual increment (ratio) method, as 
 fully explained in the opening paragraphs of Chapter XXII. 
 
 The method of making the adjustment by this method is 
 shown in Calculation XXVI. C, at the end of this chapter. 
 
 In each of the examples discussed in Chapters XXII and 
 XXIV the original annual increment was multiplied by the 
 
3o6 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 fractional ratio of £1 per annum. It is therefore obvious tliat 
 a combination of the above formulae to relate to the dual 
 variation under discussion must be made by multiplying the 
 present annual increment by each fractional ratio in succession. 
 As already pointed out, the numerator in the fractional ratio 
 relating to the period of repayment is the same as the 
 denominator in the fractional ratio relating to the rate of 
 accumulation, which will cancel out when the respective 
 formulae are multiplied together ; therefore the product of these 
 fractional ratios Avill consist of the numerator of the ratio 
 relating to the rate of accumulation and the denominator of the 
 ratio relating to the period of repayment as follows : — 
 
 Variation in the Rate of Accumulation and the 
 Period of Repayment. 
 
 The Annual Increment (^ratio) Method. 
 
 Variation in Bate. 
 Chapter XXJII. 
 
 Variation in Period^ 
 Chapter XXV . 
 
 \ 
 
 Present 
 
 annual 
 
 increment 
 
 ( Amount of £1 per ann. 
 at past rate, 
 for unexpired period 
 
 Amount of £1 per ann. 
 at future rate, 
 for unexpired period 
 
 Amount of £1 per ann. 
 at future rate, 
 for uncrpired period 
 
 Amount of £1 per ann. 
 
 at future rate, 
 for substituted period 
 
 Future 
 
 or 
 amended 
 annual 
 ^increment 
 
 Note. The factors in the above formulae which are printed 
 in italics are common to both and will cancel out in the multi- 
 plication. Calculation XXA^I. C. may now be stated in terms 
 of the above formula in a similar manner to that adopted in 
 Chapters XXIII and XXV: — 
 
 1000-474 X 
 
 L5-617 79( 
 ieill808( 
 
 Hill 
 
 9-05168 
 
 308) _ 
 
 4~ 
 
 = 1829-744 
 
 The result is the foHowing simplified formula relating to a 
 concurrent variation in tlie rale of accuniulaiion and the period 
 of repayment : — 
 
THE RATE PER CENT. AND PERIOD 
 
 307 
 
 Variatiox in the Eate of Accumulation and the Period 
 OF Repayment. 
 
 The Annual Increment (ratio) Mctliod. 
 [Amount of £1 per annum 
 
 Present 
 
 annual 
 
 increment 
 
 at past rate, 
 for unexpired period 
 
 Future 
 
 or 
 amended 
 
 annual 
 increment 
 
 Amount of £1 per annum 
 
 at future rate, 
 for substituted period 
 
 The amounts of £1 per annum in tlie above formula are at 
 varying rates per cent., and are also for different numbers of 
 years. 
 
 Calculation XXYI. C. will now be expressed in terms of tbe 
 above formula as follows : — 
 
 1060-474 X 
 
 ( l5-6l779 | 
 I 9-05168 j 
 
 1829-744 
 
 It is now possible to state a rule based upon tbe foregoing 
 formula, using tbe abbreviated terms set out in full at tbe 
 head of Chapter XXII, and explained in Chapter XXIII, 
 dealing with the rule relating to a variation in the rate of 
 accumulation. The same terms are used in Chapter XXY, 
 in the rule relating to a variation in the period of repayment. 
 The rule relating to the variation under review is stated in full 
 at the head of this chapter. 
 
 Proof of the aeove Method. The foregoing results which 
 have been obtained by taking both variations into account will 
 now be proved, and the effect of each variation will be shown 
 separately, beginning with the. variation in the period of 
 repayment. In Chapter XXIY, an adjustment Avas made in 
 the annual instalment consequent upon a reduction in the 
 period of repayment from 13 to 8 years, but without any 
 variation in the rate of accumulation. This reduction in the 
 period involved an ultimate deficiency of loan of £7258-21, 
 requiring an additional annual instalment of £801-862 to be set 
 aside for the substituted period of 8 years, as shown in Statement 
 XXIV. A. The accuracv of the calculation was proved by 
 dividing the deficiency in the amount of loan, £7258-21, 
 between the reduced accumulation of the present investments. 
 £2454-85, and of the original annual instalment, £4803-o6. 
 
3o8 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Tile future deficiency in tlie accumulation of the present invest- 
 ments, £2454*85, was also reduced to terms of the annual 
 income to arise therefrom. 
 
 Although the same method of proof may be applied to the 
 present example the problem will be reduced to terms of the 
 present annual increment of £1060474, and by deducting 
 therefrom the income from investments, £347'648, included 
 therein, it will be possible at the same time to express in figures 
 the effect upon the additional annual instalment of the 
 reduction in the period of repayment, as distinguished from the 
 effect of the increase in the rate of accumulation. 
 
 The calculation will be made by the annual increment 
 (ratio) method, which is the most convenient for the purpose. 
 The problem Avill be divided into two parts in order to ascertain 
 in the first place the amended annual increment due to the 
 reduction in the period of repayment only, on the assumption 
 that the rate of accumulation remained the same, namely, 
 3 per cent. This amended annual increment, as shown by the 
 following Statement XXYI. D., is £1862'532, requiring an 
 additional annual instalment of £802"058. 
 
 Although it will be necessary to consider the above 
 additional annual instalment of £802'058 later in this chapter, 
 the proof will be continued by taking up the above amended 
 annual increment of £1862'532 in order to ascertain the 
 reduction therein due to the increase in the rate of accumulation 
 from 3 per cent, to 3^ per cent. The calculation cannot be 
 made in terms of the above additional annual instalment of 
 £802-058 for the reasons given in Chapter XXII, Calculation 
 XXII. E., because the benefit of the accumulation of the 
 income from the present investments at the increased rate of 
 accumulation would be lost. The calculation might be made 
 in terms of each of the above factors, namely the annual 
 instalment and the income from investments composing the 
 annual increment of £1862'532, but this would involve only 
 increased labour without any corresponding advantage, seeing 
 that the accuracy of the calculation may be proved by 
 comparing the additional annual instalment to \v? obtained 
 with that found by the deductive method, and also by 
 comparing the amended annual increment with that foimd 
 previously by the annual increment (ratio) method, Calculation 
 XXVI. C. This method of proof shows the advantage of the 
 annual increment as a factor even in cases where there is not 
 any variation in the rate of income from the present invest- 
 
THE RATE PER CENT. AND PERIOD 309 
 
 ments. This is sliowu in Statement XXYI. E. at the end of 
 this chapter. 
 
 The results obtained by the foregoing calculations may now 
 be summarised in order to prove the accuracy of the previous 
 methods of adjustment. The additional annual instalment of 
 £769270 found in Statement XXVI. E. agrees with that found 
 by the deductive method (Statement XXYI. A.), and the 
 amended annual increment of £1829-744 in Statement XXVI. E. 
 agrees with that found by the annual increment (ratio) method 
 (Statement XXVI. C). Attention may now be drawn to the 
 additional annual instalment so found in order to ascertain the 
 relative effects thereon of the variation in each factor of period 
 and rate per cent. 
 
 The original annual instalment accumulated at 
 3 per cent, before these variations occurred, as 
 shown by Statement XIX. A., was £712'826 
 
 and the result of the reduction in the period of 
 repayment, taken by itself, is to increase the 
 original annual instalment by an additional 
 annual amount, as shown by Statement XX\ I. 
 D. of £802-058 
 
 but the effect of the increase in the rate of accumula- 
 tion is to reduce this annual amount bv £32'788 
 
 leaving a net increase in the original annual 
 instalment, as shown by Statement XXVI. E. 
 of £769-270 
 
 Proof of Method (continued). The present example has 
 already been compared with that used in Chapter XXIV, to 
 illustrate the effect of a variation in the period of repayment 
 only. It was found by Calculation XXIV. A. that the 
 additional annual instalment in that case was £801-862. In 
 the present case, which is complicated by a variation in the rate 
 of accumulation, an additional annual instalment of £802-058 
 is required. As in both cases the reduction in the period of 
 repayment is the same, namely, from 13 years to 8 years, the 
 two examples may be connected, bearing in mind, however, 
 that the rate of accumulation in the present instance is 3 per 
 cent., and that therefore the annual instalment at the time of 
 making the adjustment is £721-826 instead of £680-234, and 
 
3IO REPAYMENT OF LOCAL AND OTHER LOANS 
 
 a rate of accumulation of 3^ per cent., as in the previous 
 example. (See Statement XIX. A.) 
 
 In that case it was found by the deductive method (State- 
 ment XXIV. A) that the ultimate dehciency in the amount 
 of loan to be provided, in consequence of the reduction in the 
 period of repayment, was i/7258"21, Avhich requires, as shown by 
 Calculation (XXIV) 3, an additional annual instalment of 
 <£801'8G2 to be accumulated at o| per cent, for 8 years. 
 
 It is therefore necessary to ascertain the equivalent annual 
 instalment to jjrovide the same amount of loan, i;;7258'21, at 
 the end of 8 years, but to be accumulated at 3 per cent, instead 
 of 3;! per cent. This is shown to be £816"232 in Statement 
 XXVI. F. 
 
 Statement XXVI. G. shows that if the additional annual 
 instalment of £816'232, as found by Calculation XXVI. F. be 
 adopted, there will, at the end of the substituted period of 
 8 years, be in the fund an amount of i>12G'04 in excess of the 
 amount actually required to repay the loan ; and therefore that 
 the additional annual instalment of i>816"232 must be reduced 
 by an annual sum which, if accumulated at 3 per cent., will 
 amount to £126'04: at the end of 8 years, which is the annual 
 sinking fund instalment which will provide, or the annuity 
 which will amount to that sum, under the above conditions. 
 By Calculation (XXVIj 5 the annual sum is found to be 
 £14174. 
 
 The correct additional annual instalment required for the 
 purpose of showing the separate eifect of the variation in the 
 period therefore is : — 
 
 The above calculated instalment of £816232 
 
 reduced by the above annuity of ' £14'174 
 
 leaving the actual additional annual instalment of £802'058 
 
 which agrees Avitli the amount found by Statement XXVI, D. 
 by the annual increment (ratio) method. 
 
 The Annual Ixchkment (balance of loan) Method. In 
 previous chapters attention has been directed to the principles 
 
 underlying this meiliod. It I'esembles very closely the practice 
 adopted by such local authorities as are able to apply the whole 
 of the annual instalments towards the immediate actual 
 redemption of debt. In such cases the interest upon the debt 
 
THE RATE PER CENT. AND PERIOD 311 
 
 so redeemed, and the future annual instalments, constitute the 
 annual increment of this method provided the rate of interest 
 upon such redeemed debt is the same as the rate of accumula- 
 tion. In case there is any variation in these two rates per cent, 
 the annual difference may be transferred as and when it arises 
 to the debit or credit of the revenue or rate account. It is an 
 essential principle of this method that the resulting annual 
 instalment, the future rate of income from the present invest- 
 ments, and the rate of accumulation shall continue without 
 variation during the whole of the unexpired portion of the 
 repayment period. Any departure from uniformity in these 
 respects has already been pointed out. Chapter XYI, dealing 
 with the adjustment of a deficiency in the fund contains fvill 
 particulars of the method of finding the additional annual 
 instalment to be spread over a portion only of the unexpired 
 repayment period, and Chapter XXVII explains the method 
 of correcting the annual instalment in consequence of a 
 variation in the future rate of income to be received from the 
 present investments, which is expected to occur at a future date 
 during such unexpired period, and a similar method of adjust- 
 ment will apply to a variation in the rate of accumulation 
 occurring at such a future date. 
 
 In dealing with such a future variation in the rate per 
 cent, of income or of accumulation in Chapter XXVII, a dis- 
 tinction has been drawn between a reduction which, although 
 anticipated, is uncertain both as to rate and time, and one in 
 which both factors are definite, as was the case with the reduc- 
 tion in the dividend on consols under Mr. Goschen's Finance 
 Act, 1888. In Chapter XXVII attention is also directed to 
 the difference between the arithmetical and true mathematical 
 methods of arriving at the equated rate per cent., and it is there 
 pointed out that the same difference in such methods occurs in 
 the equation of the period of repayment, as will be fully 
 described in Chapter XXXII. Any variations in the above 
 factors of rate per cent, or period of repayment anticipated to 
 arise during the unexpired portion of the repayment period, 
 whether definite or estimated, are met by finding the amount of 
 an annuity by the method " by step," fully described in 
 Chapters XVI and XXVII, both of which contain a description 
 of the longer as well as the simplified method of such calcula- 
 tion. 
 
 The following Statement, XXVI. H., shows the method of 
 proving the previous results by the annvial increment (balance 
 of loan) method. 
 
312 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Redemption Period and Statement XXVI. A. 
 
 The Rate per cent. 
 
 The Deductive Method. 
 
 Showing tlie metliod of adjusting the annual instalment in 
 consequence of a variation in the period of repayment 
 accompanied by a variation in the rate of accumulation, 
 the rate of income from the present investments being 
 unaltered and being the same as the future rate of 
 accumulation. If these rates are unequal or are varied 
 proceed as in Chapter XIX. 
 
 Conditions before adjustment (at end of 12th year), 
 
 Amount of original loan repayable in 25 years ... £26,495 
 
 Amount in the fund (at end of 12th year) £9932-74 
 
 Present annual income (previously) received there- 
 from at 3| per cent, per annum £347'648 
 
 Present annual instalment, to be accumulated for 
 
 13 years, at 3 per cent £712"826 
 
 Present annual increment £1060474 
 
 Variation from the above conditions : — 
 
 The period during which the loan shall bo redeemed is 
 
 reduced from 13 to 8 years, and 
 the rate of accumulatioji of the fund is increased from 3 to 
 
 3^ per cent. 
 
 Tlie substituted jiciiod of repayment ... 8 years. 
 The future rate of accumulatioii 3^ per cent. 
 
THE RATK PER CENT. AND PERIOD 
 
 313 
 
 Present investments (at end of 12tli year) 
 
 £9932-74 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Amount thereof, accumulated for 8 years at 
 
 3| per cent. Calculation (XXIV) 1 £13079-53 
 
 Present annual instalment 
 
 £712-826 
 
 Amount thereof, accumulated for 8 years at 
 
 31 per cent. Calculation (XXVI) 1 £6452-28 
 
 Provision already made will, at the end of 8 years, 
 
 repay loan of £19531-81 
 
 Additional annual instalment required : - 
 
 Balance, being amount of original loan un- 
 provided for owing to the above reduction 
 in the period of repayment from 13 to 
 8 years, but reduced in consequence of the 
 increase in the rate of accumulation from 
 3 to 31 per cent., requiring an additional 
 annual instalment to be set aside and 
 accumulated for 8 years at 3^ per cent. ... £6963-19 
 
 Additional annual instalment 
 
 Calculation (XXVI) 2 £769-270 
 
 Amount of original loan ... 
 
 £2649500 
 
 Amended annual increment, being :— 
 
 Annual income from investments .. 
 Amended annual instalment 
 
 £347-648 
 1482-096 
 
 £1829-744 
 
314 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Redemption Period and Statement XXVI. B. 
 
 The Rate per cent. 
 
 Showing thp: final repayment of the loan, by the operation 
 of tlie sinking fund, after making the adjustment in the 
 annual instalment consequent upon a variation in the 
 period of repayment accompanied by a variation in the 
 rate of accumulation. 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments (at end of 12th vear) £99;j2'T4 
 
 Amended annual increment : — 
 
 Present annual instalment i^T12'826 
 
 Additional annual instalment ... 769270 
 
 Total out of revenue £1482-096 
 
 Income from investments 347'648 
 
 £1829-744 
 
 Amount thereof, accumulated for 8 years at 
 
 3i per cent. Calculation (XXIVj 6 £16562-26 
 
 Amount of orin-iual loan £26495-00 
 
 Amended annual instalment ... £1482096 
 
THE RATE PER CENT. AND PERIOD 
 
 315 
 
 The Redemption Period and 
 The Rate per cent. 
 
 Calculation XXVI. C. 
 
 The Annual Increment (ratio) Method. 
 
 To iiud the amended aunual increment (and therefrom the 
 amended annual instalment) in a sinking fund, in which 
 there is a variation in the period of repayment accompanied 
 by a variation in the rate of accumidation, with or without 
 any variation in the rate of income upon the present 
 investments. 
 
 liequired the annual increment to be accumulated for a period 
 of 8 years at o-| per cent., which is equivalent to an annual 
 increment of £1060-474, to be accumulated for a period of 
 lo years at o per cent. 
 
 ( Amount of £1 per ann., 13 years, 3% ) 
 
 1060-474 \ T^T-H^^^'^'"^^ 
 
 ) Amount of ,£1 per ann., 8 years, o-| % j 
 
 or by Table III, giving the amounts of £1 per annum 
 
 1060-474 X 15-61779 
 
 9-05168 
 
 Log. Present annual increment 
 add Log. Amount of £1 per annum 
 Table 111, lo years, o per cent. 
 
 = 1829-744 
 1060-474 
 15-61779 
 
 3-0255000 
 1-1936196 
 
 deduct Log. Amount of £1 per annum 
 Table III, 8 years, 3i per cent. 
 
 Loff. Amended annual increment 
 
 16562-26 
 905169 
 
 4-2191196 
 0-9567296 
 
 Amended annual increment ... 
 To find the ainendcd animal instalinent : — 
 deduct the income from investments, 
 per cent. 
 
 31 
 
 Amended annual instalment 
 
 being Present annual instalment... 712-826 
 Additional annual instalment 769-270 
 
 3-2623900 
 
 1829-744 
 
 347-648 
 
 1482-096 
 
 1482-096 
 
3i6 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 The Redemption Period and Statement XXVI. D. 
 
 The Rate per cent. 
 
 The Annual Increment (ratio) Method. 
 
 Enquired the annual increment to be accumulated for a period 
 of 8 years at 3 per cent,, whicli is equivalent to an annual 
 increment of £1060'4T4, to be accumulated for a period of 
 13 years, also at o per cent. 
 
 To show the separate eii'ect of the variation in the period. 
 
 Present annual increment 1060474 30255000 
 
 multiply by amount of £1 per 
 
 annum, 13 years, 3 per cent. 15"61779 11936196 
 
 16562-26 4-2191196 
 
 divide by amount of £1 per 
 
 annum, 8 years, 3 per cent. ... 8*89234 
 
 Log. Amended annual increment 
 
 Amended annual increment 
 
 beiny Income from investments ... 347*648 
 Present annual iiislalment ... 712'826 
 Additional annual instalment 802058 
 
 0-9490159 
 
 3-2701031 
 
 1862-532 
 
 1862-532 
 
THE RATE PER CENT. AND PERIOD 
 
 317 
 
 The Redemption Period and 
 The Rate per cent. 
 
 Statement XXVI. E. 
 
 The Annual Increment (ratio) Method. 
 
 Required the annual increment to be accumulated for a period 
 of 8 years at 3| per cent., which is equivalent to an annual 
 increment of £1862-532, to be accumulated for a like period 
 of 8 years, but at -3 per cent. 
 
 To show the separate effect of the variation in the rate per cent. 
 
 Present annual increment 1862'532 3"270103T 
 
 muUiphj htj amount of £1 per 
 annum, 8 years, 3 per cent. ... 8-89234 0-9490159 
 
 16562-26 4-2191196 
 
 divide hy amount of £1 per 
 
 annum, 8 years, 3^ per cent. ... 905169 
 
 0-9567296 
 
 Lop^, amended annual increment 3-2623900 
 
 Amended annual increment 1829-744 
 
 heinn Income from investments 
 
 mg 
 
 347-648 
 
 Present annual instalment... 712-826 
 Additional annual instalment 769270 
 
 1829-744 
 
3i8 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Redemption Period and Statement XXVI. F, 
 
 The Rate per cent. 
 
 The Annual Increment (ratio) Method. 
 
 Eequired tlie annual instalment to be accumulated for a period 
 of 8 years at 3 per cent., which is equivalent to an annual 
 instalment (as in XXIY. A.) of £801-862, to be accumulated 
 for a like period of 8 years, but at 3| per cent. 
 
 Present annual instalment 80r862 2-9040988 
 
 vniltipjy hij amount of £1 per 
 
 annum, 8 years, 3^ per cent. ... 9-05169 0-9567296 
 
 7258-21 3-8608284 
 
 divide hy amount of £1 per 
 
 annum, 8 years, 3 per cent. ... 8-89234 0-9490159 
 
 Loff. amended annual instalment 
 
 2-9118125 
 
 Amended annual inslalment 
 
 816-232 
 
THE RATE PER CENT. AND PERIOD 319 
 
 The Repayment Period and Statement XXVI, G, 
 
 The Rate per cent. 
 
 The Deductive Method. 
 
 Showing for purpose of proof only the surplus wliicli will arise 
 in the fund by adopting the additional annual instalment 
 of £816-232 found in Statement XXVI. F., instead of the 
 instalment of £802-058 in Statement XXYI. D. The 
 correction of this surplus is shown below. 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Present investments £9932-74 
 
 Present annual increment £1060-474 
 
 Amount thereof, accumulated for 8 years at 
 
 3 per cent. Calculation (XXYI) 3 £9430-09 
 
 Amended annual instalment, 
 
 as in XXVI. F. £816-232 
 
 Amount thereof, accumulated for 8 years at 
 
 3 per cent. Calculation (XXVI) 4 £7258-21 
 
 Amount in the fund at end of 8 years £26621-04 
 
 Amount of original loan 26495-00 
 
 Surplus £12604 
 
 Annual instalment to provide £12604 at the end 
 of 8 years at -\ per cent. 
 
 Calculation (XXVI) 5 £14-174 
 
 being the annual instalment, as 
 
 shown in Statement XXVI. F. . . . £816-232 
 
 less the annual instalment, as shown 
 
 in Statement XXVI. D £802-058 
 
 £14174 
 
320 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Repayment Period and 
 The Rate per cent. 
 
 Statement XXVI . H. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find tlie amended annual sinking fund instahnent consequent 
 upon a variation in tlie period of repayment, accompanied 
 by a variation in the rate of accumulation. 
 
 For Rule, see Chapter XXII. 
 
 Amount of original loan £2649500 
 
 deduct amount in the fund at the end of the 
 
 12th year £9932-74 
 
 Balance of loan 
 
 ... £16562-26 
 
 Amended annual increment, to be added to the 
 fund, and accumulated at 3^ per cent., to 
 provide this amount at the end of 8 years 
 
 Calculation (XXIY) 6 £1829-744 
 deduct income to be received from the present 
 
 investments, £9932-74, at 31 per cent. ... £347*648 
 
 Amended annual instalment £1482090 
 
 heing Present annual instalment ... £712-826 
 Additional annual instalment 769270 
 
 £1482-096 
 
THE RATE PER CENT. AND PERIOD 
 
 321 
 
 Pro forma Sinking Fund Account No. 11. 
 
 A Variation in the Redemption Period, and in the Rate of 
 
 Accumulation. 
 
 Loan of £26,495, repayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by the operation 
 of the increased annual instalment of £1482'096. 
 
 Statement XXYI. B. Rate of accumulation, 3^ per cent. 
 
 Year. 
 
 Amount I 
 in the fund 
 at beginning 
 of year. 
 
 nconie received 
 from 
 investments 
 Zk per cent. 
 
 Annual 
 sinking fund 
 instalments. 
 
 Amount 
 
 in the fund 
 
 at end 
 
 of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 3 
 
 4 
 
 The amount in the 
 
 fund at the 
 
 end of 
 
 4 
 
 5 
 
 the 12th 
 
 year, £9932- 744, is the 
 
 correct 
 
 5 
 
 6 
 
 calculated amount, ; 
 
 as shown by C 
 
 ^alcula- 
 
 6 
 
 7 
 
 tion (XY) 2, and 
 
 by the pro 
 
 forma 
 
 7 
 
 8 
 
 account, 
 
 No. 1, Chapter XV. 
 
 
 8 
 
 9 
 
 
 
 
 
 9 
 
 10 
 
 
 
 
 
 10 
 
 11 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 347-648 
 
 1482096 
 
 11762-488 
 
 13 
 
 14 
 
 11762-488 
 
 411-687 
 
 1482^096 
 
 13656-271 
 
 14 
 
 15 
 
 13656-271 
 
 477-969 
 
 1482-096 
 
 15616-336 
 
 15 
 
 16 
 
 15616-336 
 
 546-572 
 
 1482-096 
 
 17645-004 
 
 16 
 
 IT 
 
 17645-004 
 
 617-575 
 
 1482-096 
 
 19744-675 
 
 17 
 
 18 
 
 19744-675 
 
 691-063 
 
 1482-096 
 
 21917-834 
 
 18 
 
 19 
 
 21917-834 
 
 767-124 
 
 1482-096 
 
 24167-054 
 
 19 
 
 20 
 
 24167054 
 
 845-850 
 
 1482-096 
 
 26495-000 
 
 20 
 
 21 
 
 
 
 
 
 21 
 
 22 
 
 
 
 
 
 22 
 
 23 
 
 
 
 
 
 23 
 
 24 
 
 
 
 
 
 24 
 
 25 
 
 
 
 
 
 25 
 
322 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XXVII. 
 
 SINKING FUND PEOBLEMS, RELATING TO THE 
 
 RATE PER CENT. OF INCOME UPON THE 
 
 PRESENT INVESTMENTS REPRESENTING THE 
 FUND [in continuation of Chapter XX). 
 
 In which the rate of income yielded by such investments 
 
 IS NOT uniform during THE WHOLE OF THE UNEXPIRED 
 PORTION OF THE REPAYMENT PERIOD. 
 
 A. In WHICH THE FUTURE VARIATION IN THE RATE OF 
 
 INCOME IS KNOWN, AND IS DEFINITE, BOTH AS TO TIME 
 AND AMOUNT. STATEMENT XXVII. A. 
 
 B, In WHICH THE FUTURE VARIATION IN THE RATE OF 
 
 income is anticipated, but is uncertain, both as 
 to time and amount. statement xxvii. d. 
 
 Summary of methods. How the variation may arise and 
 
 THE general considerations APPLICABLE THERETO. ThE 
 DEDUCTIVE METHOD. ThE ANNUAL INCREMENT (BALANCE 
 
 OF loan) method. Statement showing the final repay- 
 ment OF THE loan by THE OPERATION OF THE AMENDED 
 ANNUAL INSTALMENT. COMPARISON WITH VARIATION A 
 
 (rate of income), in Chapter XX, where the rate is 
 
 UNIFORM DURING THE WHOLE PERIOD. CALCULATION OF THE 
 EQUATED ANNUAL INCOME BY THE ARITHMETICAL METHOD AND 
 DEMONSTRATION OF THE ERROR INVOLVED. 
 
 Summary of the methods of adjustment. 
 
 (7) The ded/iictive method, {A) as summarised helow. 
 
 Statement XXVII. A. 
 
 {II) The direct method, without calcidation, as summarised 
 at the head of Chapter XX, will not apply. 
 
 (Ill) The annual increment (balance of loan) mrfJiod, as 
 summarised at the head of Chapter XXII, will ap/dij after 
 finding the equated annual income by the drduetive 
 method (/?), suiiimaiised below. Statement XXVII. D. 
 
THE RATE OF INCOME 323 
 
 (/!') The annual increment [ratio) method^ as summarised 
 at the heads of Chapters XXIII, XXV, and XXVI, will not 
 apply, as there is not any variation in the rate of accumulation. 
 
 SuMMAUY OF THE DEDUCTIVE METHOD, (A) of ascertaining the 
 future or amended annual sinking fund, instalment due to a 
 variation in the rate of income to he received upon the present 
 investments representing the fund when it is Jcnoivn at the time 
 of making the adjustment that such future rate of income 
 will not he uniform during the unexpired period of repayment, 
 hut will he varied by a definite amount at a hno^vn futiire date. 
 In this case the pioblem is not complicated by any variation in 
 the rate of accumulation or the period of repayment. 
 
 Statement XXVII. A. 
 
 The terms used in the following stimmary ai'e fully cavplaincd 
 at the head of Chapter XXII . The unexpired period of repay- 
 ment is divided into two known parts, namely : — 
 
 The first period, during which the rate of income upon the 
 present investments tvill remain unaltered. 
 
 The second period, during which the rate of irucoine upon 
 the present investments will he varied by a knotvn 
 a, mount. 
 
 Memo. (A). If the rate of income he varied at the time of 
 making the adjustment, as ivell as at a known future date, 
 adjust the annual instalment, as described in Chapter XX, 
 before operation (6) following . 
 
 (i) Having ascertained the value of the present investments 
 in the manner already described, calculate the annual 
 amount of income (at each rate per cent.) to be received' 
 during the first and second periods respectively . 
 
 [2) Calculate the amount of an annuity, equal to the annual 
 
 income to he received during the first period at the 
 original rate of income, for the number of years in that 
 period, at the future rate of accumulation. 
 
 Calculation (XXVII) 1. 
 
 (3) Calculate the sum to which the amotint so found in (2) 
 
 will accumulate at the end of the number of years m the 
 second period at the future rate nf accumulation. 
 
 Calculation (XXVII) 2. 
 
324 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (4) Calculate the amount of an annuity, equal to the annual 
 income to he received during the second period at the 
 reduced rate of income, for the 7iumber of years in that 
 period at the future rate of accumulation. 
 
 Calculation {XXVII) 3. 
 
 (<5) The amount found in [3) added to the amoiint found in 
 (4) will give the accumulated amount of the income from 
 investments at the end of the unexpired period of repay- 
 ment, expressed in terms of original loan. 
 [Here refer to Memo. A, ahove.^ 
 
 (6) Calculate the accumulated amount of the original or 
 
 amended annual instalment for the total number of 
 years in the unexpired repayment period at the future 
 rate of accumulation, expressed in terms of original 
 loan. Calculation [XIX) 2. 
 
 (7) Add together the amounts foiind in (5) and (6) and the 
 
 value of the present investments found in {1), and deduct 
 the total from the amount of the original loan. 
 
 {8) The difference will he the amount of loan unpn^ovided for 
 in consequence of the above decrease in the rate of 
 income upon the present investments during the second 
 period. 
 
 {9) Calculate the annual instalment which will provide the 
 amount of loan found in [8) at the end of the total 
 unexpired portion of the repayment period at the future 
 rate of accumulation. Calculation (XXVII) 5. 
 
 [10) The annual iiistalment found in (9) added to the original 
 
 or amended annual instalment foxind in (6) will be the 
 future or amended annual instalment required. 
 
 [11) Prepare a statement showing the final repayment of the 
 
 loan by the operation of the fund under the amended 
 conditions. Statement XXVII . B. 
 
 [12) Prepare the usual pro forma account previously recom- 
 
 Tnended. Pro form Account, No. 12. 
 
 If the above problem be complicated by a variation in the 
 rate of accunfiulation, or the period of repayment, or both, it 
 may be solved by a combination of the methods previously 
 (Iciiionsf rated , hut irhuli need not. be specially described. 
 
THE RATE OF INCOME 325 
 
 SuMMAliY OF THE DeDUCTIVE MeTHOD (Bj of (ISCe rUilnDUJ 
 
 the future equated (Uinual income tiijon the yrese^it investments 
 representing the fund when it is antici'pated that the future rate 
 of income will not he uniform during the unexpired period of 
 repayment, hut the amount of the variation, and the date at 
 ivhich it will occur, are not knoivn at the time of maJdng the 
 adjustment. 
 
 In this case the problem is not complicated by any variation 
 in the rate of accumulation, or the period of repayment. 
 
 The terms used in the following summary are fully ex- 
 plained at the head of Chapter XXII. The unexpired period 
 of repayTnent is divided into two estimated parts, as follows : — 
 
 Hie first period, during which the present rate of income 
 will continue to be received. 
 
 The second period, during which the rate of income is 
 expected to be varied, but the exact amount of such 
 variation can only be estimated, 
 
 (i) Estimate, as accurately as possible, the period during 
 which the present investments will continue to yield the 
 rate of income now received] {The first period.) 
 
 [2) Deduct the number of years, so estimated, from the 
 
 unexpired portion of the original repayment period. 
 
 [The second period.) 
 
 [3) Estimate as accurately as possible, the rate of income 
 
 which will be yielded by the present investments during 
 the second period, as ascertained^ ni (2) . 
 {4) Ascertain the value of the present investments in the 
 manner already described. 
 
 (5) Calculate the annual amount of income to be received 
 
 during the first period, estimated as in {!), at the present 
 unaltered rate of income. 
 
 (6) Calculate the annual amount of income expected to be 
 
 received during the second period, as ascertained in [2), 
 at the rate per cent., estimated as in {3). 
 
 (7) Calculate the accumulated amount of an annuity equal 
 
 to the annual income to be received duririg the first 
 period as ascertained in {S) for the 7iumber of years rn 
 the first period as estimated in [l] at the present un- 
 altered rate of accumulation. Calculation {XXVII) I. 
 
326 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 (8) Calculate the sum to tvhich the amount found in (7) will 
 
 accumulate at the end of the second period found in [2) 
 at the inesent unaltered rate of accuvudation. 
 
 Calculation (XXVIl) 2. 
 
 (9) Calculate the accumulated amount of an annuity equal 
 
 to the annual income estimated to he received^ as found 
 in [6) during the second period found in (2) at the rate 
 per cent, of income estimated in (3) at the present un- 
 altered rate of accumulation. Calculation {XX] 11) 3. 
 
 [10) The amount found in (8) added to the amount found in 
 
 {9) will represent the amount of original loan which will 
 he provided at the end of the unexpired period of repay- 
 ment by the future accumtilation, at the unaltered rate, 
 of the annual amounts of income from the present invest- 
 ments found as above, 
 
 as to the first period, in (<5). 
 as to the second period, in [6). 
 
 [11) Calculate in the inanner already described, using the 
 
 author's standard calculation form, Xo. 3x^ Chapter X^ 
 the equal annual instalment or annuity which loill 
 amount to the total sum found in (10) at the end of the 
 unexpired repayment period, at the unaltered rate of 
 accumulation. Calculation (^Y^T //) 6. 
 
 (12) The annuity, or equal animal sum, found in [11) is the 
 equated annual income required, and may be treated as 
 part of the future or amended annual increment in all 
 problems involving a variation in the rate per cent, of 
 income upon the present investments accompanied by a 
 variation in the rate of accumulation. 
 
 Fro forma Account, Xo. 13. 
 
 General Considerations. Reference lias already been 
 made in previous chapters to the difficulty which arises, 
 especially in cases where the repayment of the loan is spread 
 over long periods, of fixing the future rate of accumulation of 
 the sinking fund, and a similar difficulty will also occur in 
 connection with the future rate of income to be received upon 
 the present investments representing the amount in the fund. 
 Tn adjustments similar to those under review the future rate of 
 accumulation will nearly always be a matter of speculation 
 and any uncertainty in the matter is met in practice by assum- 
 ing a rate of accumulation on the low side. The rate of income 
 
THE RATE OF INCOME 337 
 
 to be received iu future upon the investments representing the 
 fund at tlie time of making' tlie adjustment niay in some cases 
 be assured for the whole of the unexpired portion of the repay- 
 ment period, and in Chapters XIX, XX, and XXI, dealing 
 with Variations A, B, and C, it has been assumed that this will 
 be the case in order to simplify the calculation and to demons- 
 trate the principle. It has in fact been assumed that the 
 reduction in the rate of income on the present investments in 
 Variations B and C has been partly the cause of the rectification 
 of the annual instalment. If at any future time the rate of 
 income from the present investments should again be reduced 
 it would be necessary to repeat the adjustment. This reduction 
 in the rate of income yielded by the present investments may 
 be due to a decrease in the rate of interest upon a security 
 similar to a mortgage without any fall in the capital value of 
 the investment, or might be due to the realisation of part of 
 the security at a loss, in which case the additional annual 
 instalment would inckide the replacement of the deficiency of 
 capital, and a further amount due to the reduced income upon 
 such capital realised, although the actual rate per cent, yielded 
 on the re-investment might remain the same. But the rate of 
 income to be received from the present investments may be 
 reduced at the time of making the adjustment, and at the same 
 time it may also be provided that a further additional reduction 
 shall take place at a fixed future date. These are definite data 
 which may be made the subject of actual calculation. Such an 
 instance occurred in 1888, when, by Mr. Groschen's Finance Act, 
 the rate of interest on Consols was reduced from 3 per cent, to 
 2f per cent, for a period of 15 years until 190-3, and the Act 
 provided that the interest should be then further reduced to 2j 
 per cent., the present rate. If the typical Sinking Fund which 
 has been used to illustrate the previous examples had been, in 
 1888, invested in Consols and had then an unexpired period of 
 13 years to run, the method of calculation adopted in all the 
 variations already considered would have been accurate, and it 
 would have been quite correct to base the additional annual 
 instalment on an assured yield of 2| per cent. But if the fund 
 had, in 1888, been invested in Consols, and had then an 
 unexpired period of 20 years to run, the problem would have 
 been very different seeing that the present investments would 
 yield 2| pe-r cent, for 15 years and 2| per cent, for the remain- 
 ing 5 years. 
 
 A similar calculation " by step " has already been made 
 when dealing with the adjustment of a deficiency in the fund 
 
328 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 by means of au additional annual instalment to be spread over 
 the earlier years only of the unexpired repayment period (see 
 Variation II (Deficiency), Chapter XVIj, and a similar pro- 
 cedure will apply to the above conditions. At the end of this 
 chapter the method of ascertaining the amount of an annuity 
 in this way will be further esjjlained and illustrated by a 
 shorter mode of calculation. (Statement XXYII. C. 
 
 Illustration of the Method. The method of making the 
 adjustment will be illustrated by means of the results obtained 
 in Chapters XIX and XX. In Variation A, Chapter XIX, the 
 future rate of income upon the present investments is assumed 
 to be I3| per cent, for the whole of the unexpired period of 13 
 years, and in Variation B, Chapter XX, to be reduced to 3 per 
 cent, for the same unexpired term. This reduction in the rate 
 of income in Variation B is assumed to take place at the end of 
 the 12th year and to continue unaltered during the remaining 
 13 years, but a similar change in the rate of income to that in 
 the case of Consols already referred to might take place during 
 the unexpired term of 13 years in Variation A. The present 
 annual increment in Variation A includes income at 3^ per 
 cent, on investments valued at £9932"74, viz., £347"648 per 
 annum which, at the end of the unexpired period of 13 years, 
 will amount at 3 per cent., as shown by Calculation (XIX) 1, 
 to £5429-494. 
 
 In Variation B the present annual increment includes 
 income at 3 per cent, on the same investments, viz., £297984 
 per annum, and this at the end of the period of 13 years will 
 amount at 3 per cent., as shown by Calculation (XX) 1, to 
 £4653-85. 
 
 Both the above annual amounts of income are assumed to 
 accumulate at 3 per cent, for the 13 years so that the question 
 of the rate of accumulation does not affect the problem. But 
 if in Variation A there had been a reduction in the rate of 
 income taking place at the end of the 8th year of the unexpired 
 period of 13 years, the accumulated amount of the annual 
 income at tlie end of the 13 years would have been different. 
 
 Instead of £347-648 per annum at 3| per cent, for 13 years 
 there would have been : — 
 
 Income at 3| per cent., or £347-648 per annum for 8 years, 
 
 followed by 
 Income at 3 per cent., or £297*984 per onnum for 5 years, 
 
 both accumulating at 3 per cent, for the above periods; and in 
 
THE RATE OF INCOME 329 
 
 additiou the accumulation at 3 per cent, of a sum (to which 
 £347' 648 per auuum will amount at 3 per cent, at the end of 
 8 years) continued without further annual additiou for a period 
 of 5 years. 
 
 The amount of loan which will be provided at the end of 
 the period of 13 years by the accumulation of the above income 
 from investments may be ascertained by the following method 
 by " step." 
 
 Amount of £347 '648 per annum for 8 years 
 accumulated at 3 per cent. 
 
 Calculation (XXVII) 1 £3091-403 
 
 Amount of the above sum of £3091'403 in 5 years 
 accumulated at 3 per cent 
 
 Calculation (XXVII) 2 £3583-783 
 Amount of £297-984 per annum for 5 years 
 accumulated at 3 per cent 
 
 Calculation (XXVII) 3 £1582-037 
 
 Accumulated amount at the end of 13 years ... £5165-820 
 
 as compared with the following amounts already ascertained 
 on the assumption that tbe rate of income will be uniform 
 during the whole period of 13 years : — 
 
 at 3i per cent XIX. A. £5429-494 
 
 at 3 percent XX. A. £4653850 
 
 The above sum of £5165-82 represents the portion of 
 original loan which will be provided at the end of the period 
 of 13 years by the accumulation at 3 per cent, of the income 
 from investments (at 3^ per cent, for the first 8 years and at 
 3 per cent, for the remaining 5 years). 
 
 The above figures show the deficiency in the amount of 
 original loan to be provided by the accumulation of the annual 
 income from investments if such investments had yielded the 
 above definite although variable rates during the period of 13 
 years, as compared with a uniform rate of 3^ per cent, as 
 assumed in the calculation of the amended annual instalment 
 in Variation A, Chapter XIX. The following Statement, 
 XXVII. A, shows the deductive method of ascertaining the 
 amended annual instalment in consequence of a reduction in 
 the rate of income of the above character. 
 
330 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The original annual instalment as shown in State- 
 ment XIX. A. is £712-826 
 
 and the additional annual instalment due to the 
 variation in the rate of income from the present 
 investments now under review, as found by 
 Calculation (XXVII) 5, is £16-883 
 
 or an amended annual instalment as shown by 
 
 Statement XXYII. A., of ]. £729-709 
 
 The above reduction in the rate of income from the present 
 investments at the end of the 8th year involves a further 
 deficiency of £263-67 in the amount of loan which will be 
 provided at the end of the unexpired repayment period of 13 
 years, and is the difference between the future accumulation of 
 the income from investments shown in 
 
 Statement XIX. A £5429-49 
 
 and the amount ascertained as above £5165-82 
 
 £263-67 
 
 requiring a further additional annual instalment of £16-883 as 
 shown by Calculation (XXVII) 5. 
 
 The Future Equated Annual Income. The future or 
 amended annual increment will now be considered. Statement 
 XIX. A., Variation A, shows that the future or amended annual 
 increment to be accumulated for 13 years at 3 per cent, is 
 £1060-474. This is made up of : — 
 
 Amended annual instalment £712826 
 
 and future annual income from investments at 3^ 
 
 percent £347-648 
 
 £1060-474 
 
 In the case now under consideration the future annual 
 increment will still be £1060-474 seeing that there is not any 
 variation in the rate of accumulation, as proved by the results 
 obtained in Chapter XX, Variation B, but it will be an equated 
 and not an actual annual increment, ascertained as follows : — 
 
THE RATE OF INCOME 331 
 
 Present annual instalment, as above £712'826 
 
 Additional annual instalment due to the reduction 
 in tlie rate of income from the present invest- 
 ments from 3^ to 3 per cent, during- the last five 
 years of the period of repayment 
 
 Calculation (XXVII) 5 i>16-8S3 
 
 Amended annual instalment (Statement XXVII. A.) £729" 709 
 leaving to be provided, an equated annual amount of 
 
 income from the present investments £330'765 
 
 Amended annual increment, as above £1060474 
 
 Under the altered conditions, the actual annual income 
 from the present investments will be £347 '648 per annum for 8 
 years, followed by £297"984 per annum for 5 years, and these 
 annual sums accumulated at 3 per cent, will, at the end of the 
 13 years, amount together to £5165'82. If the calculation be 
 correct the above equated annual income (£330'765) should 
 represent an equated sinking fund instalment which will 
 provide £5165"82 at the end of 13 years if accumulated at 3 per 
 cent., and this is found to be the case by Calculation (XXVII) 6. 
 The above amount of £330'765 may therefore be described as 
 the true equated annual income, being the annual sum which, 
 accumulated for 13 years at 3 per cent., is equivalent to the 
 two annual amounts of income of £347'G48 and £297984 to be 
 accumulated at 3 per cent, for the successive periods of 8 and 5 
 years as above described. This equated annual amount of 
 income of £330765 does not take any part in the actual working 
 of the fund. It is merely the average annual equivalent (over 
 the whole period) of the known actual varying amounts which 
 will be received during the period and is used here merely to 
 demonstrate that the actual successive annual amounts of 
 income are the equivalents of the calculated equated annual 
 income. 
 
 Where the future variation in the rate of income during" 
 the unexpired portion of the repayment period is definite both 
 as to time and rate per cent, and is not an estimate the above 
 deductive method (A) should be adopted exclusively. This 
 method is summarised at the head of this chapter. 
 
 The Method by Step. In Chapter XVI, dealing with the 
 correction of a deficiency in a sinking fund, two methods of 
 
332 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 adjustmeut have been jsliowu depending upon the period during 
 wliicli the additional annual instalment is required to be set 
 aside and added to the fund. In Variation I, such additional 
 annual instalment is spread equally over the whole of the 
 unexpired portion of the repayment period of l-J years, and as 
 shown in that chapter the method of adjustment is a simple one. 
 In Variation II the conditions are more complicated because 
 it is required that the additional annual instalment shall be 
 set aside and added to the fund during the earlier years only 
 of the unexpired portion of the repayment period. This 
 involves an increased annual charge as compared with Varia- 
 tion I, as follows : — 
 
 In Variation I the additional annual instalment to 
 be spread equally over the whole of the unex- 
 pired repayment period of 13 years, as shown 
 by Statement XVI. A., is £45-594 
 
 In Variation II, the additional annual instalment to 
 be set aside during the first 5 years only of the 
 unexpired repayment period of 13 years, has 
 been obtained by the method " by step " there 
 described, and as shown in Statement XVI. C, is £104*039 
 
 proving that the increased annual burden is due solely to the 
 reduction in the period allowed for the adjustment of the 
 deficiency, the rate of accumulation being the same in both 
 cases. In this example there is a variation from the general 
 rule applicable to the accumulation of a given sum of money 
 now in hand and also of an annual or other periodic sum, or 
 annuity, which rule is based upon a steady and uninterrupted 
 accumulation, and does not provide for any break in any of the 
 factors of rate per cent, or period. 
 
 The present example relates to the correction of a sinking 
 fund in consequence of a variation in the future rate of income 
 to be received on the present investments representing the fund, 
 which occurs in the middle of the unexpired portion of the 
 repayment period, and therefore a similar method by " step " 
 may be adopted. 
 
 Having ascertained the additional annual instalment of 
 £729'709 in the above manner. Statement XXVII. B. has been 
 prepared showing the final repayment of the loan. This state- 
 ment shows the accumulated amount, £11106"10 of the annual 
 increment of £1077'357 at the end of the 13 years by the longer 
 method " by step " by two Calculations, (XXVII) 1 and 2, 
 
THE RATE OF INCOME 333 
 
 relating- to tlie annual income of £347'648. The same calcula- 
 tion may be made by tbe shorter m^ethod shown in Statement 
 XXVII. C, described later, which is similar to Calculation 
 XVI. D. 1. The method shown in Statement XXVII. A. is 
 the one which should be adopted in cases where the further 
 reduction in the rate of income from the present investments 
 is not an estimate, but is known and is definite as to the rate 
 per cent, as well as the period. The difference between the 
 arithmetical and true methods of arriving at the equated period 
 of repayment will be fully discussed in Chapter XXXII where 
 it will be shown that the same principle applies, and at the end 
 of this chapter the question of equation as applied to the rate 
 per cent, wall be briefly discussed. 
 
 Calculations (XXVII) 1 and 2 are made with the object of 
 finding the amount which will be provided at the end of 13 
 years by the accumulation at 3 per cent, of an annuity of 
 £347' 648 to be set aside for the first 8 years of that period when 
 the annuity ceases, but the sum to which it then amounts will 
 continue to accumulate at the same rate for the remaining 5 
 years. 
 
 Calculation (XXVII) 1 shows that at the end of 8 years the 
 annuity of £347-648 will amount to £3091-403, and Calculation 
 (XXVII) 2 shows that this sum accumulated for a further 5 
 years will amount to £3583-783. The amount of £3091-403 
 found by Calculation (XXVII) 1 becomes the basis of Calcula- 
 tion (XXVII) 2, but being only an intermediate factor is not 
 of any further interest in the problem. The two calculations 
 may therefore be combined with advantage as shown in State- 
 ment XXVII. C, using Thoman's method, as being the simpler, 
 pointing out, however, that if the calculation be required at any 
 rate per cent, not worked out by Thonian, the values of the 
 factors may be ascertained by means of the formulse already 
 referred to. All that is necessary is to remember that (a") of 
 Thoman may be found by means of the following factors 
 referred to in Chapters IX and X. 
 
 Log a" = Log RN + Log ^. _ Log ^ j^n _ i) 
 
 It is important to remember, how^ever, that the method 
 applies equally to cases in which the rate of accumulation is 
 not the same in both periods, and that the periods in each 
 factor EN may be different. In the following Calculation 
 XXVII. C. 10 has been added to the sum of the logs of the 
 annuity and of U^, for the reason fully explained in Chapter 
 IX, dealing with Thoman's Tables. 
 
334 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Arithmetical Method of Finding the Equated 
 Annual Income. Althougli the actual amount of the imme- 
 diate reduction in the future rate of income to be received upon 
 the present investments may be known at the time of making 
 the adjustment, it may be anticipated that there will be a 
 further reduction of an unknown amount at some future date 
 and it may be deemed advisable to make allowance therefor. 
 The above method may be used although in this case there is 
 one known and one estimated rate per cent, of income. 
 
 In such a case the amount of the further reduction is 
 problematical, and it is therefore preferable and permissible 
 to use the shorter and more direct arithmetical method of find- 
 ing the equated annual income to be received over the period 
 of 13 years, and this will now be shown as applied to the above 
 particulars in order to compare the result with the mathe- 
 matical method just described. 
 
 There is (1) an annual income for 8 years at 3| per 
 
 cent, of ". £347-648 
 
 and (2) an annual income for a further 5 years at 
 
 3 per cent, of ' £297-984 
 
 and it is required to find the equated annual income for 13 
 years, which is equivalent to the above. Proceed as follows : — 
 
 (1) 347-648 X 8 = £2781-184 
 
 (2) 297-984 x 5 - 1489-920 
 
 £4271104 
 
 this total, divided by 13, gives an annual sum of £328-546 
 
 which is the arithmetical equivalent of the above annual 
 amounts of £347-648 and £297-984 for the above respective 
 periods. 
 
 This calculated annual income of £328-546 upon £9932-74, 
 the value of the present investments, is equivalent to 3-31 per 
 cent., but the actual rate per cent, is immaterial. What is 
 important is the fact that the estimated annual income to be 
 received, calculated in this manner, is only £328546, as com- 
 pared with the true equated annual amount of £'):{0-765, ascer- 
 tained by the mathematical method as above, or a decrease of 
 £2-219 per annum. 
 
THE RATE OF INCOME 335 
 
 Seeing that tlie total future or amended annual increment 
 tnwst be £1060" 474 in order to provide the balance of loan 
 at the above rate of accumulation the estimated annual 
 deficiency of £2'219 must be added to the additional instalment 
 of £16-883 (found as above in Calculation (XXYII) 5), to be 
 charged against revenue or rate, with the result that at the end 
 of the period the fund will be in excess of the proper amount 
 by the accumulation of the larger actual amounts of the future 
 annual income owing to the fact that the actual income received 
 will be in excess of the equated amount assumed. Further, 
 the actual amounts in the fund at the end of each year will 
 exceed the amounts shown by the pro forma account already 
 referred to, by an increasing annual surplus, for the same 
 reason. The difference in this case is only small, but it is proof 
 that the arithmetical method of equation is incorrect, and the 
 extent of the error depends upon the actual rates per cent, of 
 income, the periods during which they operate, and the amount 
 in the fund. The same error will be found in the arithmetical 
 method generally adopted when considering the equation of the 
 period of repayment in Chapter XXXII, and the two results 
 should be carefully compared. 
 
 The Rate per cent. Statement XXVII. A. 
 
 The Deductive Method. 
 
 Showing the method of adjusting the annual instalment, in 
 consequence of a known variation in the rate of income 
 upon the present investments, to occur at a known future 
 date, without any variation in the rate of accumulation 
 or in the period of repayment. 
 
 The original conditions in this example are similar to 
 Variation B, in Chapter XX, in which case the reduction 
 in the rate of income, from 3^ to 3 per cent., took effect at 
 the end of the 12th year. In the present instance, how- 
 ever, the reduction in the rate of income, from 3^ to 3 per 
 cent., does not operate immediately, but occurs at the end 
 of 8 years. The future annual income from the present 
 investments will therefore be : — 
 
 for 8 years, at 3^ per cent., on £9932-74 . . . £347648 
 for 5 years at 3 per cent., on £9932-74 ... 297984 
 
336 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Present investments (at end of 12tli year) 
 
 Income from present investments :— 
 
 Amount of £'347"648 per annum, 
 for 8 years at 3 per cent. 
 
 Calculation (XXVII) 1 £3091-403 
 
 Amount of £3091-403, in 5 years at 
 3 per cent. 
 
 Calculation (XXVII) 2 £3583-783 
 Amount of £297-984 per annum, 
 for 5 years at 3 per cent. 
 
 Calculation (XXVII) 3 £1582-037 
 
 Present annual instalment (Variation A) : — 
 
 Amount of £712-826 per annum, 
 for 13 years at 3 per cent. 
 Calculation (XIX) 2 (£680-234) £10623-75 
 Calculation (XIX) 3 (£32-592) 509-02 
 
 Present annual instalment 
 
 £712-826 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 £9932-74 
 
 £5165-82 
 
 - £11132-77 
 
 Provision already made will repay loan of 
 
 Additional annual instalment required :— 
 
 Balance, being amount of original loan un- 
 provided for, owing to tlie above reduction 
 in the rate of income from tbe present 
 investments during the later years of the 
 unexpired portion of the repayment period, 
 requiring an additional annual instalment 
 to be set aside and accumulated for 13 years 
 at 3 per cent 
 
 Additional annual instalment 
 
 Calculation (XXVII) 5 £16-883 
 
 £26231-33 
 
 £263-67 
 
 Amount of original loan 
 
 Amended annual instalment £729-709 
 
 £2649500 
 
THE RATE OF INCOME 
 
 337 
 
 The Rate per cent. 
 
 Statement XXVII. B. 
 
 Showing the final repayment of the loan, by the operation 
 of the sinking fund, after making the adjustment in the 
 annual instalment, consequent upon a variation in the rate 
 of income upon the present investments to occur at a known 
 future date, without any variation in the rate of accumula- 
 tion or period of repayment. 
 
 Amended annual increment for 8 years : — 
 
 Present annual instalment ... £729709 
 
 Equivalent 
 
 amount of 
 
 original loan. 
 
 Income from investments, 3. 
 per cent 
 
 347-648 
 
 £1077-357 
 
 Amount thereof, accumulated for 8 years at 
 
 3 per cent. Calculation (XXVII) 7 £9580-22 
 
 Amount of this sum, accumulated for a further 
 
 5 years at 3 per cent. Calculation (XXVII) 8 £11106-10 
 
 Amended annual increment for 5 years : — 
 
 Annual instalment, as above... £729-709 
 Income from investments, 3 
 
 percent 297-984 
 
 £1027-693 
 
 Amount thereof, accumulated for 5 years at 
 
 3 per cent. Calculation (XXVII) 9 £5456-16 
 
 £16562-26 
 
 « 
 
 Present investments (at end of 12th year) £9932-74 
 
 Amount of original loan £26495-00 
 
338 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. 
 
 Statement XXVII. C. 
 
 The Amount of {the Avio^mt of £1 per annum). 
 Method by Step, by Thoman's Tables. 
 
 To find tlie accumulated amount of an annuity to be added 
 to the sinking fund for a limited period of years, and at 
 the end of that period the accumulated amount thereof 
 to continue to accumulate for a further specified period. 
 The rate of accumulation in both cases may be the same, 
 or be at different rates per cent. 
 
 Eequired the amount of an annuity of <£347"648 to be added to 
 the sinking fund for a period of 8 years, and accumulated 
 at 3 per cent. At the end of 8 years the annuity ceases, 
 but the sum to which it has then amounted continues to 
 accumulate for a further period of 5 years at 3 per cent. 
 
 First period, 8 years; second period, 5 years. 
 
 Log. annuity 
 
 add Log. W^, 3 per cent. 8 years 
 Log. RN, 3 per cent. 5 years 
 
 347-648 
 
 2-5411397 
 0-1026978 
 0-0641861 
 
 2-7080236 
 
 deduct Log. a", 3 per cent. 8 years 
 
 add 10 12-7080236 
 91536819 
 
 3-554341' 
 
 which is the log. of the required future amount 
 
 at the end of 13 years £3583-783 
 
 Note. This statement may be compared with Statements 
 XVI. D.l and XXXIV. G. 
 
THE RATE OF INCOME 
 
 339 
 
 Pro forma Sinking Fund Account, No, 12, 
 
 A Variation in tlie Hate of Income from Investments, which 
 is not uniform over the unexpired Hepayment Period. 
 
 Loan of £26,495, rejjayable at the end of 25 years. 
 
 Showing the final repayment of the loan, by the operation 
 of the increased annual instalment of £729709. 
 
 Statement XXVII. B. Rate of accumulation, 3 per cent. 
 
 Year. 
 1 
 
 2 
 
 Amount 
 in the fund 
 
 at 
 
 beginning of 
 
 year. 
 
 Income 
 received'fiom 
 investments 
 
 made up 
 to Vith year. 
 
 Annual 
 
 sinking 
 
 fund 
 
 instalment. 
 
 received from 
 investments 
 
 made after 
 l'2th year 
 
 3 per cent. 
 
 Amount 
 
 in the fund 
 
 at 
 
 end of 
 
 year. 
 
 Year. 
 1 
 
 
 
 
 
 
 JL 
 
 2 
 
 3 
 
 
 
 
 
 
 3 
 
 4 
 
 The 
 
 amount in the fun( 
 
 i at the end of 
 
 4 
 
 5 
 
 the : 
 
 I2th year. 
 
 £9932-744, is the correct 
 
 6 
 
 6 
 
 calculated amount, as sh 
 
 own by Calcula- 
 
 6 
 
 7 
 
 tion 
 
 (XV) 2, 
 
 and by 
 
 the pro 
 
 forma 
 
 7 
 
 8 
 
 9 
 
 10 
 
 account, No. 1, 
 
 Chapter XV. 
 
 
 8 
 
 9 
 
 10 
 
 
 
 
 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 347-648 
 
 729-709 
 
 Nil 
 
 11010-101 
 
 13 
 
 14 
 
 11010101 
 
 347-648 
 
 729-709 
 
 32-320 
 
 12119-778 
 
 14 
 
 15 
 
 12119-778 
 
 347-648 
 
 729-709 
 
 65-610 
 
 13262-745 
 
 15 
 
 16 
 
 13262-745 
 
 347-648 
 
 729-709 
 
 99-900 
 
 14440002 
 
 16 
 
 17 
 
 14440002 
 
 347-648 
 
 729-709 
 
 135-216 
 
 15652-575 
 
 17 
 
 18 
 
 15652-575 
 
 347-648 
 
 729-709 
 
 171-594 
 
 16901-526 
 
 18 
 
 19 
 
 16901-526 
 
 347-648 
 
 729-709 
 
 209070 
 
 18187-953 
 
 19 
 
 20 
 
 18187-953 
 
 347-648 
 
 729-709 
 
 247-654 
 
 19512-964 
 
 20 
 
 21 
 
 19512-964 
 
 297-984 
 
 729-709 
 
 287-407 
 
 20828-064 
 
 21 
 
 22 
 
 20828-064 
 
 297-984 
 
 729-709 
 
 326-860 
 
 22182-617 
 
 22 
 
 23 
 
 22182-617 
 
 297-984 
 
 729-709 
 
 367-494 
 
 23577-804 
 
 23 
 
 24 
 
 23577-804 
 
 297-984 
 
 729-709 
 
 409-350 
 
 25014-847 
 
 24 
 
 25 
 
 25014-847 
 
 297-984 
 
 729-709 
 
 452-460 
 
 26495000 
 
 25 
 
340 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The Rate per cent. Statement XXVII. D. 
 
 The Annual Increment (balance of loan) Method. 
 
 To find the amended annual sinking fund instalment, conse- 
 quent upon a known variation in the rate of income upon 
 the present investments to occur at a known future date, 
 based upon the equated annual income. 
 
 The rule relating to this method is stated at the head 
 of Chapter XXII. 
 
 Amount of original loan (25 years) £2649500 
 
 deduct amount in the fund at the end of the 
 
 12th year £9932-74 
 
 Balance of loan £16562-26 
 
 Annual increment, to be added to the fund, and 
 accumulated at 3 per cent., to provide this 
 amount at the end of 13 years 
 
 ' Calculation (XX) 4 £1060474 
 deduct the equated annual income to be 
 received from the present investments ascer- 
 tained as described in the text £330-765 
 
 Amended annual instalment £729-709 
 
 heing Present annual instalment ... £712-826 
 Additional instalment, as 
 found in 
 
 Statement XXVII. A. £16-883 
 
 £729-709 
 
 Note. This method will be of use where the variation in 
 the rate of income is of the above unequal nature, and is 
 combined with a variation in the rate of accumulation, as in 
 Chapter XXI (Variation C). 
 
THE RATE OF INCOME 
 
 341 
 
 Pro forma Sinking Fund Account, No. 13, 
 
 A Variation in the Rate of Income from Investments, which 
 is not uniform over the unexpired Repayment Period. 
 
 Loan of £26,495, repoi/ahle at the end of 25 yearx. 
 
 Showing the final kepayment of the loan, by the operation 
 of the increased annual instalment of £729709, and the 
 equated annual income of £3-30'765. 
 
 Statement XXYII. D. 
 
 Rate of accumulation, 3 per cent. 
 
 ^ear. 
 
 Amount 
 
 in the fund 
 
 at beginning 
 
 of year. 
 
 Equated 
 
 annual income 
 
 from 
 
 investments 
 
 after li'th year. 
 
 Annual 
 sinking fund 
 instalment. 
 
 Income to be 
 
 received from 
 
 investments 
 
 made after 
 
 12th year. 
 
 Amount 
 
 in the fund 
 
 at end 
 
 of year. 
 
 Year. 
 
 1 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 
 
 
 2 
 
 3 
 
 
 
 
 
 
 3 
 
 4 
 
 The 
 
 amount in the fund at the end of 
 
 4 
 
 6 
 
 the 
 
 12th year, 
 
 £9932-744, is the correct 
 
 5 
 
 6 
 
 calculated amount, as sh 
 
 own by Ca 
 
 Icula- 
 
 6 
 
 7 
 
 tiou 
 
 (XV) 2, 
 
 and by 
 
 the pro 
 
 forma 
 
 7 
 
 8 
 
 9 
 
 10 
 
 acco 
 
 unt, No. 1, 
 
 Chapter XV. 
 
 
 8 
 
 9 
 
 10 
 
 
 
 
 
 
 11 
 
 
 
 
 
 
 11 
 
 12 
 
 
 
 
 
 9932-744 
 
 12 
 
 13 
 
 9932-744 
 
 330-765 
 
 729-709 
 
 — 
 
 10993-218 
 
 13 
 
 14 
 
 10993-218 
 
 330-765 
 
 729-709 
 
 31-814 
 
 12085-506 
 
 14 
 
 15 
 
 12085-50G 
 
 330-765 
 
 729-709 
 
 64-583 
 
 13210-563 
 
 15 
 
 16 
 
 13210-563 
 
 330-765 
 
 729-709 
 
 98-335 
 
 14369-372 
 
 16 
 
 17 
 
 14369-372 
 
 330-765 
 
 729-709 
 
 133-099 
 
 15562-945 
 
 17 
 
 18 
 
 15562-945 
 
 330-765 
 
 729-709 
 
 168-906 
 
 16792-325 
 
 18 
 
 19 
 
 16792-325 
 
 330-765 
 
 729-709 
 
 205-787 
 
 18058-586 
 
 19 
 
 20 
 
 18058-586 
 
 330-765 
 
 729-709 
 
 243-775 
 
 19362-835 
 
 20 
 
 21 
 
 19362-835 
 
 330-765 
 
 729-709 
 
 282-903 
 
 20706-212 
 
 21 
 
 22 
 
 20706-212 
 
 330765 
 
 729-709 
 
 323-204 
 
 22089-890 
 
 22 
 
 23 
 
 22089-890 
 
 330-765 
 
 729-709 
 
 364-714 
 
 23515-078 
 
 23 
 
 24 
 
 23515078 
 
 330-765 
 
 729-709 
 
 407-470 
 
 24983-022 
 
 24 
 
 25 
 
 24983-022 
 
 330-765 
 
 729-709 
 
 451-504 
 
 26495000 
 
 25 
 
Section V. 
 
 Sinking Fund Problems. 
 
 The Date of Borrowing and the 
 
 Redemption Period. 
 
345 
 
 CHAPTER XXVIII. 
 
 SINKING FUND PROBLEMS RELATING TO THE 
 DATE OF BORROWING AND THE REDEMPTION 
 PERIOD 
 
 Without any complication as kegakds the life or duration 
 
 OF continuing utility of the ASiSET CHEATED OUT OF THE 
 LOAN. 
 
 Loan borkowed over several years, in one sum in each year, 
 
 EACH year's borrowings BEING REPAYABLE IN A PRESCRIBED 
 PERIOD FROM THE DATE OF BORROWING. 
 
 1. By MEANS OF ONE SINKING FUND ONLY. 
 
 2. By separate sinking funds for each year's 
 
 borrowings. 
 
 The foregoing chapters deal with the various problems 
 likely to arise in connection with the sinking funds of local 
 authorities and commercial and financial undertakings affecting 
 (1) the amount in the fund at any time; (2) the rate per cent, 
 of accumulation; (3) the rate per cent, of income upon the 
 present investments representing the fund; (4) the period of 
 repayment; and (5) various combinations of the above factors. 
 In the whole of the examples which have been used to illustrate 
 such problems it has been assumed, for the purpose of 
 calculating the original or amended annual instalment to be 
 set aside and accumulated as a sinking fund to provide a given 
 loan at the end of any period, that the loan was borrowed in 
 one year and on one date, namely, at the beginning of the 
 financial year, and that the first annual instalment was set 
 aside at the end of that year. This method of treating an 
 annuity or other periodic payment is the basis of all such 
 calculations and upon which the formulae and tables are 
 constructed. This ideal procedure may, it is true, be met with 
 occasionally; but as a matter of fact it very seldom occurs in 
 actual practice. It has been assumed in all cases that it has so 
 happened in order to simplify the conditions and to demonstrate 
 the actuarial principles underlying the repayment of debt in 
 this manner, without introducing any extraneous complications. 
 The time has now arrived when it is necessary to consider the 
 conditions occurring in actual practice. 
 
 The variations from the ideal method of borrowing are of a 
 twofold nature and arise when the borrowings are made at 
 
346 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 various dates in any one year or are spread over several years. 
 If the total loan be repayable on a given date both these 
 variations may necessitate an adjustment in the annual instal- 
 ment. In the case of a loan borrowed at various dates during 
 any one year the necessity for any adjustment depends upon 
 the magnitude of the loan and affects only the first and last 
 years of the term. Any neglect to make such adjustment 
 cannot prolong the period of repayment for more than part 
 of a year, but in the case of borrowings spread over several 
 years the matter becomes more important. There are other 
 factors which may further complicate the problem, depending 
 upon the nature of the outlay and the periods of repayment 
 allowed for each class. In some cases the power or sanction 
 specifies not only the total amount of loan authorised, but also 
 gives details of the component parts of such loan divided as 
 between the various classes of outlay, each having its own 
 period of repayment. In some cases, however, the total amount 
 authorised is stated without any such subdivision, and an 
 equated period is prescribed for the repayment of the total loan. 
 
 The after consideration of the subject will be divided into 
 two parts depending upon the character of the outlay, which 
 may be all of one nature having a similar life or period of 
 utility and a consequent equal period of repayment. On the 
 other hand, the outlay under one power or sanction may consist 
 of various classes for each of which a separate, and varying, 
 period of repayment is imposed. 
 
 It may be accepted as a general rule that the repayment 
 period now allowed by Parliament is fixed with regard to the 
 probable life or duration of continuing utility of the works. 
 This variation in the life of the asset imports special difficulties 
 into the problem, relating to the vexed question of the adequacy 
 or otherwise of the sinking fund instalment as a provision for 
 depreciation, obsolescence and supersession. 
 
 Dealing first with the actual borrowings, the subject will be 
 treated in the following order, namely : — 
 
 I. Loans authorised for outlay all of one nature having the 
 same period of repayment. 
 As regards the actual borrowing such loans may be divided 
 into three classes as follows : 
 
 (a) Loan borrowed over several years,, in one sum in each 
 year, repayable over a term of years in a prescribed 
 period from the several dates of borrowing. Such loans 
 will be described in this chapter. 
 
THE DATES OF BORROWING AND REPAYMENT 347 
 
 (6) Loan borrowed over several years, in one sum in each 
 year, repayable in one sum on a certain specified date. 
 
 Chapter XXIX. 
 
 (c) Loan borrowed in one or more years, in varying amounts 
 at varying dates in each year, repayable in one sum on 
 a certain specified date, where it is required that the 
 revenue or rate account of each year of borrowing shall 
 be charged with a proportionate part of the annual 
 sinking fund instalment. Chapter XXX. 
 
 II. Loans authorised for outlays of varying nature, each 
 having a different life or period of continuing utility, 
 the whole of the loan to be repaid on one uniform date. 
 
 Chapter XXXII. 
 
 It is not necessary to do more than point out that if at any 
 future time during the operation of the fund any question 
 should arise as to a variation in the repayment period, the rate 
 of accumulation, or the rate of income to be received from the 
 present investments representing the fund, the problem may 
 be solved by one or other of the methods described in previous 
 chapters. 
 
 The total amount of any loan sanctioned for purposes of large 
 public works is rarely required to be raised in one year. The 
 actual construction often occupies several years, and such an 
 amount only is borrowed in any one year as will be sufficient 
 to pay for the works actually constructed in that year. The 
 complication of the sinking fund owing to the loan being 
 borrowed over a period of years may be obviated by borrowing 
 one amount in advance; but the results of borrowing largely 
 in excess of the actual annual requirements are : — 
 
 (1) A loss of interest owing to the money borrowed lying 
 
 in the bank. 
 
 (2) The excessive sinking fund instalments which have to 
 
 be set aside, and provided out of revenue or rate, in 
 
 respect of the amount borrowed in excess of the actual 
 
 requirements. 
 
 The Act authorising the borrowing generally contains a 
 
 clause limiting the period of repayment either to a definite 
 
 number of years from the date of borrowing or to a specified 
 
 date, and the limitation applies to the amount borrowed in each 
 
 year. If the construction extends over, say, four years, this 
 
 will entail four separate calculations of the annual instalment; 
 
 and the generally recognised practice is to treat each year's 
 
348 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 borrowings a.s a separate loan repayable in the prescribed 
 ntimber of years from the actual date of borrowing and 
 requiring a separate sinking fund. In the case of a large 
 municipality this entails the keeping of a great number of 
 sinking fund accounts, and the annual provision of the instal- 
 ments becomes a very detailed process. Further, the final 
 repayment of the loans is spread over a term of years equal 
 to the extended period of borrowing. 
 
 In the case of loans raised by the issue of stock the whole 
 of such stock is made redeemable at the end of a specified 
 number of years, or, as is generally the case, on a definite 
 date. In this case there need only be one sinking fund with 
 four separate annual instalments set aside each year for 
 decreasing periods, but all calculated to mature at the same 
 date and at the same rate of accumulation. 
 
 The present example will be illustrated by the sinking fund 
 fully described in Chapter XY, relating to the repayment of a 
 loan of £26,495 in 25 years at an accumulation rate of 3^ per 
 cent. In this case it has been ascertained that if the whole of 
 the loan were borrowed in one year, namely, at the beginning 
 of the financial year, it required an annual instalment of 
 i:/680"234 to be set aside at the end of the first and 24 subse- 
 quent years in order to repay the original loan. If the loan, 
 instead of being borrowed in one year, were borrowed over a 
 period of four years, and the sanction or authorisation imposed 
 the period of redemption of 25 years in respect of each amount 
 borrowed, the conditions would havebeen considerably modified. 
 There is in this case the equivalent of four separate loans each 
 repayable in 25 years, but maturing at the end of four successive 
 years. It will be assumed that each amount of loan Avas 
 borrowed at the beginning of the financial year, or if borrowed 
 on several dates in that year that no necessity exists to equate 
 the borrowing at the various dates. It will be further assumed 
 that the above amount of £26,495 was borrowed in unequal 
 amounts in each of the four years, and, in order to avoid making 
 four separate additional calculations, that a definite proportion 
 of the loan was borrowed in each year. Seeing that the annual 
 instalment is based upon the amount of £1 per annum for 
 25 years, it is obvious that it is directly proportionate to the 
 amount of the loan and that the four annual instalments may 
 be found by dividing the original annual instalment of 
 £680'234 in the same proportions as the total loan is divided. 
 
 The following table shows the actual details of the loan 
 under consideration : — 
 
THE DATES OF BORROWING AND REPAYMENT 349 
 
 TABLE XXYIII. A. 
 
 Loan of £26,495, borrowed over four years. Eepayment spread 
 over a similar period. 
 
 Original annual instalments all calculated to mature in 25 years 
 but at the end of successive years. 
 
 Rate of accumulation 3| per cent. 
 
 Year 
 
 of 
 
 borrowing. 
 
 Redemption 
 period. 
 
 Proportion 
 borrowed 
 each year. 
 
 Amount 
 borrowed 
 each year. 
 
 Annual 
 instalment 
 on yearly 
 borrowing. 
 
 Annual 
 instalment 
 at end of 
 each year. 
 
 First 
 
 25 vears 
 
 ^/:4 
 
 3785- 
 
 97-176 
 
 97-176 
 
 Second 
 
 j» 
 
 'U. 
 
 5677-5 
 
 145-764 
 
 242-940 
 
 Third 
 
 j> 
 
 Vi. 
 
 7570- 
 
 194-353 
 
 437-293 
 
 Fourth 
 
 jj 
 
 'U. 
 
 9462-5 
 
 242-941 
 
 680-234 
 
 
 26495- 
 
 680-234 
 
 — 
 
 There are two alternative methods of keeping the sinking 
 fund accounts in such a case. One method is to keep one 
 sinking fund only, and to set aside an increasing instalment 
 during the first four years, a constant instalment during the 
 next 21 years, and a decreasing instalment during the final 
 three years of the total period of 28 years during which the 
 fund will run. If this method be applied to the foregoing 
 example, the annual instalments added to the fund will be as 
 follows : — 
 
 TABLE XXVIII.B. 
 
 Loan of £26,495, borrowed over four years. Repayment spread 
 over a similar period. 
 
 Annual instalments to be added to one sinking fund relating- to 
 the total loan. 
 
 To be set aside at end of 
 
 1st year 
 
 2nd year 
 
 3rd year 
 
 4th year 
 
 5th to 25th year = 21 years 
 
 26th year 
 
 27th year 
 
 28th year 
 
 Annual instalments. 
 
 97-176 
 242-940 
 437-293 
 680-234 
 680-234 
 583-058 
 437-294 
 242-941 
 
 Total. 
 
 97-176 
 242-940 
 437-293 
 680-234 
 14,284-914 
 583-058 
 437-294 
 242-941 
 
 being the equivalent of 25 annual instalments 
 
 pf £680-234 , £17,005-850 
 
350 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 There are several objections to keeping the sinking fund 
 accounts in this manner, all of which are practical. The first 
 is that unless a proper pro forma account be at once made out 
 showing the operation of the fund until maturity there will be 
 a liability to continue the full instalment of £680"234 beyond 
 the 25th year. A further error may possibly arise owing to 
 the application, during the period, of part of the fund in 
 redeeming part of the debt. As previously stated, if any part 
 of the fund be so applied it is requisite and obligatory to pay 
 into the fund, annually, interest upon the loan so redeemed at 
 a rate per cent, at least equal to the calculated rate of accumula- 
 tion. Although this obligation may be remembered and carried 
 out during the first 25 years it may then be overlooked that at 
 that time £3,785 of loan has been fully redeemed and that 
 interest upon this amount of loan repaid need no longer be 
 charged to the revenue or rate account and added to the fund. 
 The same factor of error may arise at the end of the 26th, 2Tth 
 and 28th years. Taking the above possible sources of error 
 into consideration, it is preferable to adopt a method which will 
 avoid them, although it may entail a little more clerical work. 
 The proper method in such cases is to keep a separate sinking 
 fund for each year's borrowings, and to prepare at the dates 
 of borrowing a pro forma account showing the operation of each 
 fund until maturity. 
 
 It cannot be too often repeated that this pro forma account 
 should be prepared in respect of every sinking fund. If the 
 method of separate sinking funds be adopfed it will ensure that 
 proper payments of interest in respect of debt redeemed out of 
 the fund are made to the fund each year and will also enable 
 arrangements to be made to repay each loan at the end of the 
 prescribed period. It will also, in the case of long repaj-ment 
 periods, avoid the necessity of referring to old ledgers or books 
 of account which may have been destroyed. 
 
 For this purpose it is an advantage to earmark each fund, 
 and also the corresponding instalment, in some such way as the 
 following : — 
 
 Gas Works Sinking Fund. Sanction 1900. 
 Loan of 1901 — 25 Years, 
 
 and to number each instalment. 
 
 The charge to tlie revenue or rate account at the end of the 
 third year would be made up as follows : — 
 
THE DATES OF BORROWING AND REPAYMENT 351 
 
 Sanction 1900 — 25 Years. 
 
 Loan of 1901. 3rd Instalment £97-176 
 
 1902. 2nd do 145-764 
 
 1903. 1st do 194-353 
 
 Totallnstalment £437-293 
 
 At the end of each of the last four years one of the original 
 year's borrowings will be repaid ; and the charge to revenue or 
 rate at the end of the 26th year (when the loan borrowed 
 during the first year has been entirely repaid) will be as 
 follows : — 
 
 Sanction 1900 — 25 Years. 
 
 Loan of 1901. Repaid nil 
 
 1902. 25th Instalment £145-764 
 
 1903. 24th do 194-353 
 
 1904. 23rd do 242-941 
 
 Totallnstalment £583058 
 
 Although the method of keeping separate sinking funds for 
 each year's borrowings under each sanction or authorisation is 
 here advocated, it must not be assumed that this method 
 requires that separate bank accounts should be kept for each 
 fund. This would become intolerable in practice even if the 
 bank would agree to do so. Neither is it necessary to keep a 
 separate investment account for each fund. One bank account 
 and one investment account for each department of the local 
 authority is quite sufficient because at the end of any year the 
 amount in the bank, the amount invested, and the loans repaid 
 out of sinking fund should together be equal to the amount 
 standing to the credit of the fund, or to the credit of the whole 
 of the funds of the particular department. In this connection 
 it is important to point out that loans repaid by means of the 
 sinking fund should be treated as an investment of so much of 
 the fund so applied and be debited to a special account instead 
 of being debited to the sinking fund account, in the same way 
 that investments in outside securities are kept in separate 
 accounts. The reason for doing this is to ensure that the 
 revenue or rate account is annually debited, and the sinking 
 fund credited, with the proper amount of interest in respect of 
 such part of the fund so applied in redemption of debt. If the 
 
352 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 accounts are kept in such a manner tlie sinking fund will, at the 
 end of each year, show the amount of loan provided for out of 
 revenue or rate, and will enable a comparison to be made with 
 the pro forma account already referred to and recommended. 
 By this means only can any variation from the calculated 
 amount which sliould be in the fvmd at any time be readily 
 ascertained and immediately adjusted. This applies to all 
 sinking funds. 
 
 It ought, however, to be pointed out that if only one bank 
 account and one investment account be kept it will be necessary 
 to apportion, as between the different sinking funds, the interest 
 allowed by the bankers and the income received from invest- 
 ments whether the investments be in outside securities or 
 consist of loans redeemed out of the sinking fund. In ordinary 
 cases there may be some difficulty in doing this because the 
 interest allowed by the bank upon balances in hand will almost 
 certainly be at a lower rate than the calculated rate of accumu- 
 lation of the fund. This difficulty is, however, removed by the 
 fact that there is in the case of each sinking fund a standard 
 to work to, namely, the pro forma account previously prepared 
 showing the amount which should stand to the credit of each 
 fund at the end of each year of the repayment period. If the 
 amount of income actually received from the investment of the 
 sinking fund in outside securities and in loans redeemed falls 
 short of the amount originally calculated to be received, such 
 deficiency should be made good each year by charging it to the 
 revenue or rate account and paying the deficiency into the 
 sinking fund bank account. The necessity to apportion the 
 interest allowed by the bank and the income received from 
 investments may be entirely removed by crediting the interest 
 allow^ed by the bank, as well as the income received from the 
 investments, to a sinking fund interest suspense account. The 
 suspense account should be debited with the total amount of 
 interest which ought, according to the pro forma accounts, to 
 be credited to the various sinking funds, and the balance 
 remaining to the debit of the suspense account, will show the 
 amount of the deficiency of interest to be debited to the revenue 
 or rate account. By this means not only will the amount 
 standing to the credit of the sinking fund agree each year with 
 the amount which should so stand according to the pro forma 
 account, but there will be the further advantage that eacli 
 year's revenue or rate account will bear its proper burden and 
 there will never arise any necessity to make provision for a large 
 deficiency in any sinking fund caused by an accumulation of 
 
THE DATEvS OF BORROWING AND REPAYMENT 353 
 
 many annual deficiencies in the income which ought to have 
 accrued to the fund. 
 
 This method of keeping separate sinking funds for eacli 
 year's borrowings is not required in the case of loans issued 
 by way of a stock redeemable at a fixed date, seeing that the 
 repayment of the loan is not spread over a number of years 
 equal to the number of years occupied by the borrowing. 
 
 A loan borrowed over a series of years repayable in one sum 
 at a fixed date will be considered in the next chapter, and, for 
 the sake of comparison, the figures used in this example will be 
 further utilised. 
 
354 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XXIX. 
 
 SINKING FUND PROBLEMS, RELATING TO THE 
 DATE OF BORR(JWING AND THE REDEMPTION 
 PERIOD 
 
 Without any complication as eegards the life or duration 
 OF continuing utility of the asset created out of the 
 LOAN [continued). 
 
 Loan borrowed over several years, in one sum in each 
 year, repayable in one sum on a certain specified date : 
 
 1. Where the date of repayment is known at the 
 
 time the money is borrowed. 
 
 2. Where the date of repayment is fixed after the 
 
 sinking fund has been in operation for a number 
 of years, and an adjustment of the fund is 
 required. 
 
 Summary of the methods of adjustment. 
 
 (1) Summary of the method of ascertaining, at the end of 
 tlie period of construction, the future or amended equal annual 
 instalment to he set aside and added to the amount now in 
 the fund which has been provided by the accumulation, during 
 the jjeriod of construction, of temporary instalments set aside 
 in respect of amounts borrowed over a series of years, all of 
 which were, and still are, repayable in one sum on a certain 
 specified date which was known at the time the money was 
 borrowed. The problem is not complicated by any variation in 
 the i^eriod of repayment due to the life of the asset. 
 
 (7) Ascertain from the actual records the a/nouuf standing 
 to the credit of the fund, at the time the adpistment is 
 required to be made. Statement XXIX B. 
 
 (2) Calculate the amount of loan which u-ill be provided at 
 
 the end of the unexpired repayment period^ by the 
 accumulation of this amoit/it noiv in the fiind^ at the 
 future rate of accumulation. 
 
 Standard Calculation Form, No. 1. 
 
THE DATES OF BORROWING AND REPAYMENT 355 
 
 (3) Deduct the am,ount so found ^ as in (2), from the amoiint 
 
 of the original loan. . 
 
 (4) The remainder will represent the balance of loan to he 
 
 j)rovided hy the accumulation, at the future rate, of the 
 required amended annual instalment to be added to the 
 fund during the tine j^jji red. repayment 'period. 
 
 {5) Calculate the annual instalme7it so required. 
 
 Standard Calculation Form, No. 3x. 
 
 (6) The annual instahnent so ascertained should be equal to 
 
 the sum of the several annual instalments already set 
 aside in respect of the amounts borroioed in each year 
 provided there is not any variation in the period of 
 repayment or rate of accumulation. Table XXIX A . 
 
 (7) Any variation between the annual instalment $0 ascer- 
 
 tained, as in (J), and the sum of tlie several annual 
 instalments already set aside will be due to an abnormal 
 past accumulation of the fund, and will result in a 
 surplus or deficiency in the amount of loan to be provided 
 at the end of the repayment period. 
 
 (8) Such surplus or deficiency {if any) in the amount of loan 
 
 to be ultimately provided should be corrected in the 
 manner already described under these heads in previous 
 chapters. 
 
 (.9) Prepare a statement showing the final repayment of the 
 loan by the operation of the sinking fund under the 
 amended conditions. Statement XXIX. B. 
 
 {10) Prepare the usual pro forma account. 
 
 (2) Summary of the method of ascertaining, at some 
 future time, the future or amended equal annual instalments 
 to be set aside and added to the amount now m the ftmd which 
 has been provided by the accumuhition of previous instalments 
 set aside in respect of loans borrowed over a series of years, 
 all of wliicli were orig^inally repayable at the end of successive 
 years, l)ut wliicli are now repayable in one sum on a certain 
 date now for tlie first time specified. The problem is not 
 complicated by any variation in the period of repayment due 
 to the life of the asset. 
 
 {1) Ascertain from the actual records the amount standing 
 to the credit of the fund at the time flic adjustment is 
 required to he mdde. Table XXIX D. 
 
356 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 (2) Calculate the amourit of loan which will he lyrovidcd at 
 the end of the unexinred repayment fjeriod hy the 
 accumuJation of this amount noiv in the fund at the 
 future rate of accumulation. 
 
 Standard Calculation Form, No. 1. 
 
 {3) Deduct the amount so found, as in (2), front the amount 
 of the original loan. 
 
 {4) The remainder ivill represent the balance of loan to be 
 provided by the accumulation at the future rate of the 
 required amended equal annual instalment to he added, 
 to the fund during the unexpired repayment period 
 in substitution for the annual instalment, as originally 
 set aside. 
 
 (5) Calculate the annual instalment so required. 
 
 Standard Calculation Form, No. 3x. 
 
 (6) Prepare a statement shoiving the fnal repayment of the 
 
 loan by the operation of the sinMng fund under the 
 amended conditions . 
 
 (7) Prepare the usual pro forma account. 
 
 The loans about to be considered differ from tlie preceding 
 example only in the fact tbat, altbougb the borrowings are 
 spread over a series of years, the loan is repayable in one sum 
 instead of at the end of successive years corresponding to the 
 number of years during which the money was borrowed. The 
 enquiry is still limited to loans in respect of outlay having a 
 uniform period of repayment. The date of repayment of the 
 loan is generally prescribed in the original sanction or 
 authorisation, and may be either (1) a specified date, (2) a 
 definite number of years from the date of the sanction, or from 
 the commencement of operations, or (3) a given number of years 
 from a date later than the sanction ; or, in other words, a 
 deferred sinking fund. On the other hand, the date of repay- 
 ment may be fixed by the local authority or by Parliament 
 some years after the loan has been borrowed and a sinking fund 
 or funds established. This may arise on the consolidation of 
 existing loans, and also under the folloM-ing or other similar 
 conditions. A local authority has obtained powers to construct 
 certain works and to borrow on loan, and the power or sanction 
 provides that the loan shall be repaid in 25 years from the 
 dates of borrowing. The actual construction of the works 
 extends over a period of three years, and such an amount only 
 
THE DATEvS OF BORROWING AND REPAYMENT 357 
 
 is borrowed iu each year avS will pay for the works actually 
 constructed in that year. At the end of three years the works 
 authorised are completed and the full amount of the loan has 
 been borrowed. During the period of construction the proper 
 instalments have been regularly set aside out of revenvie or 
 rate, to provide the amount of loan repayable at the end of each 
 of the prescribed periods of 25 years. The local authority then 
 decide to convert the loans into stock redeemable on a fixed 
 date. 
 
 This date may be specified under further powers granted, or 
 may be fixed by the local authority at the time of making the 
 adjustment. 
 
 The above instances may be divided into two classes requir- 
 ing different treatment, although the loan relates to outlay of 
 one character only, namely : — 
 
 Class 1. Loans in respect of which the date of repayment is 
 known at the time the money is borrowed, and 
 
 Class 2. Loans in respect of which the date of repayment is 
 fixed after the sinking fund has been in operation 
 for some years, and an adjustment becomes 
 necessary. 
 
 The method of making the adjustment, however, rather than 
 the cause of the adjustment, is the principal object of enquiry. 
 
 Class I. Loans in respect of one class of outlay only, borrowed 
 over a series of years, repayable in one sum on a specified 
 date, which date is known at the time the money is 
 borrowed. 
 
 The first example will relate to a loan raised by the issue of 
 stock repayable on a specified date. The actual borrowing is 
 spread over three years (the period of construction of the 
 works), and the period of repayment is 25 years from the 
 commencement of operations. It will be assumed for the 
 purpose of simplifying the conditions, that the local authority 
 has borroM-ed the money immediately prior to the beginning 
 of the financial year and that work has been commenced on 
 that date. Subsequent borrowings are made on the first day 
 of the two following financial years, and there is not therefore 
 any complication due to the loan being borrowed at various 
 dates in any one year. 
 
 The rate of accumulation of the sinking fund is 3| per cent. 
 
358 
 
 REPAYMENT OF LOCAE AND OTHER LOANvS 
 
 Altlioug-li it is requisite to keep only one sinking fund, separate 
 calculations must be made of the annual instalments to be 
 charged to revenue or rate, and added to the sinking fund, in 
 respect of each annual amount of loan borrowed. The complete 
 conditions are shown in the following table : — 
 
 TABLE XXIX. A. 
 
 Loan of £11,355, borrowed over three years, repayable in one 
 sum on a specified date. 
 
 Annual instalments calculated for varying periods, all to 
 mature on the same date. Rate of accumulation 3| per 
 cent. 
 
 Year of 
 borrowing. 
 
 Redemption 
 period. 
 
 Amount 
 borrowed 
 each year. 
 
 Annual 
 
 instalment 
 on yearly 
 borrowing. 
 
 Annual 
 
 instalment 
 
 at end of 
 
 each year. 
 
 First. 
 
 25 years 
 
 3,785 
 
 97-176 
 
 97-176 
 
 Second. 
 
 24 years 
 
 3,785 
 
 103-227 
 
 200-403 
 
 Third. 
 
 23 years 
 
 3,785 
 
 109-836 
 
 310-239 
 
 
 11,355 
 
 310-239 
 
 
 It will be noticed that the loan is borrowed in equal annual 
 amounts during the period of construction, and that the annual 
 instalments in respect of the several amounts borrowed are 
 gradually increased owing to the reduction in the period of 
 repayment. The instalment to be set aside at the end of the 
 first year is £97-176 only and increases until the end of the 
 third year when it attains the maximum of £310239, which 
 will be continued for a further 22 years when the amount in 
 the sinking fund should be £11,355, provided care has been 
 taken, at the end of each year, to see that the fund has accu- 
 mulated at the proper rate in accordance with the pro forma 
 account which should have been prepared. 
 
 In this instance there is not any decreasing instalment 
 during the later years of the repayment period as was the case 
 in the previous example, Table XXVIII. B, seeing that 
 altliougli the borrowing is spread over three years the instal- 
 ments are calculated on the basis that the whole of the loan 
 will matvire on the same date. The final repayment of the loan 
 is shown in the following statement: — 
 
THE DATEvS OF BORROWING AND REPAYMENT 359 
 
 STATEMENT XXIX. B. 
 
 Loau of £11,355, burrowed over three years, repayable in one 
 sum on a specified date. 
 
 Showing the final repayment of the loan by the operation of 
 the sinking fund and the annual instalments shown in 
 Table XXIX. A. 
 
 Amount in the fund : — 
 
 At end of first year, instalment 
 
 At end of second year : — 
 
 Interest, 3| per cent. ... £3401 
 Instalment £200403 
 
 £97176 
 
 £203-804 
 
 At end of third year : — 
 
 Interest, 3^ per cent. ... £10-534 
 Instalment £310-239 
 
 £300-980 
 
 £320-773 
 
 Amount in the fund at the end of the third year ... £621-753 
 
 At the end oi the 25th year the amount in the fund 
 will be as follows: — 
 
 Amount of £621-753 for 22 years at 3^ per cent. 
 
 per annum. Standard Calculation Form, No. 1 £1325-3 
 
 Amount of £310239 per annum for 22 years at 
 3^ per cent, per annum 
 
 Standard Calculation Form, No. 3 £10029-7 
 
 Total amount of loan ... 
 
 £113550 
 
 Class 2. Loans in respect of one class of outlay only, borrowed 
 over a series of years repayable in one sum on a specified 
 date, such date of repayment being fixed after the sinking 
 fund has been in operation for a number of years and an 
 adjustment is required. 
 
 The second class of loans borrowed over a series of years 
 will now be considered, namely, those in which the date of 
 repayment of the whole of the loan is fixed after the sinking 
 fund has been in operation for some years, prior to which time 
 
36o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 each year's borrowiugs were repayable at the end of successive 
 years. In such a case it is necessary to make an adjustment 
 in order to ascertain the future annual instalment to be added 
 to the sinking fund during the whole of the newly ascertained 
 redemption period, in substitution for the varying instalments, 
 as shown in Table XXYIII. B. This adjustment depends upon 
 two factors, namely, the amount now in the fund, and the 
 exact date fixed for the redemption of the AA^hole of the loan. 
 Seeing that the loan in Table XXA'III. B. was originally 
 repayable over a series of four years, and is now repayable on 
 one uniform date, it is advisable to adhere as closely as possible 
 to the original conditions as to repayment, by expediting the 
 repayment of part of the loan and delaying the repayment of an 
 equivalent part. In fact there is here a mild form of the 
 equation of the period of repajanent, and an average eqviation 
 of two years will be adopted since the present example is chosen 
 to illustrate the method of making the adjustment rather than 
 to demonstrate the proper mathematical method of finding the 
 equated period of repayment. This will be fully considered in 
 Chapter XXXII, where it Avill be shown that the ordinary 
 arithmetical method of finding the equated period is incorrect, 
 but not to such an extent as to make any appreciable difference 
 in two years seeing that in such calculations the nearest 
 whole number of years is adopted. 
 
 In the following example the original conditions as to the 
 amounts of loan borrowed, and the annual instalments required, 
 are the same as in Chapter XXYIII, the only difference being 
 that the loan is repayable in one sum instead of at the end of 
 four successive years. The following table contains the original 
 conditions in the example now" under consideration, and, as 
 regards the actual figures, is a copy of Table XXYIII. A. 
 
 TABLE XXIX. C. 
 
 Loan of £26,495, borrowed over four j^ears, repayable in one 
 sum on a specified date fixed after the fund has been in 
 operation for a number of years and an adjustment is 
 required. 
 
 Original anniuil instalments all calculated to mature in 25 
 years, but at the end of successive years. Rate of accu- 
 mulation 3^ per cent. 
 
 This table is a copy of Table XXYIII A. 
 
THE DATES OF BORROWING AND REPAYMENT 361 
 
 Year of 
 borrowing. 
 
 Redemption 
 period. 
 
 Proportion 
 borrowed 
 each year. 
 
 Amount 
 borrowed 
 each year. 
 
 Annual 
 instalment 
 on yearly 
 borrowing. 
 
 Annual 
 instalment 
 
 at end of 
 each year. 
 
 First. 
 
 25 years 
 
 ^/>4 
 
 3785- 
 
 97-176 
 
 97-176 
 
 Second . 
 
 25 years 
 
 'U. 
 
 5677-5 
 
 145-764 
 
 242-940 
 
 Third. 
 
 25 years 
 
 Vl4 
 
 7570- 
 
 194-353 
 
 437-293 
 
 Fourth. 
 
 25 years 
 
 'U. 
 
 9462-5 
 26495 
 
 242-941 
 
 680-234 
 
 
 680-234 
 
 
 Tables XXYIII. A, and XXVIII. B show the annual in- 
 stalments to be set aside to repay the above loans at the end 
 of the 25th, 26th, 27th and 28th years. 
 
 As stated in the preliminary remarks in this chapter, during 
 the 5th year circumstances arise which render it necessary to 
 provide for the repayment of the whole of the loan on one date, 
 instead of at the end of 4 successive years, and it will be 
 assumed that the end of the 26th year is adopted as the 
 redemption date. Four separate sinking funds have been kept 
 and each fund stands at the proper amount shown by the pro 
 forma account. This means that the accumulation of each 
 fund by way of income from investments has been equal to the 
 calculated amount, or that any deficiency has been made good 
 year by year out of revenue or rate. 
 
 In case there is a deficiency or a surplus in the fund at the 
 time of making the adjustment it may be accurately adjusted 
 if necessary by the methods fully described in previous chap- 
 ters, but as a general rule unless the discrepancy is of large 
 amount it is merged in the general adjustment about to be 
 made. In ordinary practice of course the present position of 
 the fund is ascertained from the actual records or books of 
 account, but in the present example the amount in the fund 
 must be found by actual calculation. 
 
 The first step therefore is to ascertain the amounts which 
 should stand to the credit of each of the individual sinking 
 funds relating to each of the four year's borrowings at the end 
 of the fourth year, this being the date when the maximum 
 instalment has been set aside in respect of the full amount of 
 the loan which has then been borrowed. 
 
 This may be done by the following arithmetical calculation 
 which is somewhat shorter than by the tables and logarithms 
 and which has the further advantage that it shows, although 
 in decimal form, the actual entries in the ledger. 
 
362 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE XXIX. D. 
 Loan of £26,495, borrowed over four years, repayable in one 
 
 sum on a specified date fixed after tlie fund lias been in 
 
 operation for a number of years, and an adjustment is 
 
 required. 
 Separate sinking funds. The amount in each fund at the 
 
 end of the fourth year, will be as follows : — 
 
 Amount set aside 
 
 at the end of the 
 
 following 
 
 financial years. 
 
 Sinking Funds in respect of loan borrowed at 
 of the following financial years. 
 
 First. Second. Third. 
 
 the beginning 
 Fourth. 
 
 First 
 
 Instalment 
 
 97T76 
 
 
 
 
 Second 
 
 Interest 
 
 :J-4U1 
 
 
 
 
 
 Instalment 
 
 97176 
 
 145-764 
 
 
 
 
 197-753 
 
 145-764 
 
 
 Third 
 
 Interest 
 
 6-921 
 
 5-102 
 
 
 
 
 Instalment 
 
 97-176 
 
 145-764 
 
 194-353 
 
 
 
 301-850 
 
 296-630 
 
 194-353 
 
 
 Fourth 
 
 Interest 
 
 10-565 
 
 10-382 
 
 6-802 
 
 
 
 Instalment 
 
 97176 
 
 145-764 
 
 194-353 
 
 242-941 
 
 
 409-591 
 
 452-776 
 
 395-508 
 
 242-941 
 
 
 
 
 
 
 395-508 
 452-776 
 409-591 
 
 
 
 
 
 
 
 Amount star 
 
 
 
 
 
 iding to the 
 
 credit of the four 
 
 
 
 
 sinking 
 
 funds at the 
 
 end of the 
 
 fourth 
 
 
 
 year of 
 
 borrowing 
 
 
 
 £1500-816 
 
 The accuracy of the above calculation may be proved by 
 assuming that only one sinking fund had been kept. Although 
 this is not recommended in the case of a loan repayable over a 
 series of years, there is an advantage in keeping only one fund 
 where the loan is repayable in one sum provided a pro forma 
 account of the operation of the fund is prepared when the full 
 amount of loan has been borrowed. The following table shows 
 the amount in the sinking fund at the end of the fourth year : 
 
 TABLE XXIX. E. 
 
 Loan of £26,495, as above. 
 
 One sinking fund only. The amount in the iuiid at the end 
 of the fourth year, will be as follows : — - 
 
THE DATEvS OF BORROWING AND REPAYMENT 363 
 
 Vniount added to the 
 fund at the end of 
 the following years. 
 
 First 
 
 Amount to credit 
 
 of fund at 
 beginning of year. 
 
 Annual accretions. 
 Interest :U % Instalment. 
 
 97176 
 
 Amount to credit 
 of fund at 
 end of year. 
 
 97176 
 
 Second ... 
 
 97176 
 
 a-401 
 
 242-940 
 
 343-517 
 
 Tliird ... 
 
 U-S-bll 
 
 12023 
 
 437-293 
 
 792-833 
 
 Fourth . . . 
 
 792-833 
 
 27-749 
 
 680-234 
 
 1500-816 
 
 
 
 43173 
 
 1457-643 
 
 1500-816 
 
 At the time of luakiiig the adjustment four sinking funds 
 are in operation, all relating to the repayment of a loan of 
 £26,495 borrowed in unequal amounts over a period of four 
 years, and each year's borrowings are repayable in 25 years 
 from the date of the original borrowing, the last portion of the 
 loan being repayable at the end of the 28th year. The actual 
 repayment of the total loan was originally spread over a period 
 of 4 years, but under the new conditions it is required to amend 
 the annual instalment of £680234 to be set aside for 21 years 
 followed by decreasing instalments for 3 years as shown in 
 Table XXVIII. B, In place of these varying instalments 
 required to repay the loan at the end of four successive years 
 it is necessary to ascertain the annual instalment which will 
 repay the whole of the loan of £26,495 at the end of 22 years 
 from the present time, bearing in mind that there is in the 
 fund an amount of £1500816 which can be applied in reduc- 
 tion of the future annual instalment. 
 
 The amended annual sinking fund instalment may be 
 found in the following manner which is similar in principle to 
 the annual increment (balance of loan) method described in 
 Chapter XXII: — 
 Amount of loan, repayable in 22 years from the 
 
 present time £26495-00 
 
 Deduct therefroin the amount of loan which will be 
 
 provided by the accumulation at 3^ per cent. 
 
 for 22 years of the £1500-816 now in the fund. 
 
 By standard calculation form No. 1 £3199-07 
 
 leavino: a balance of loan of £23295-93 
 
 to be provided by the accumulation at 3^ per cent, of the future 
 amended annual sinking fund instalment to be set aside for 
 22 years. 
 
 This amended annual instalment as may be found by 
 standard calculation form No. 3 x, is £720-59. 
 
364 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Proof of the above Adjustment. In ordinary practice, 
 of course, tlie best method of proving the above calculation is 
 to prepare the usual pro forma account so often recommended, 
 showing the amount which should be in the fund at the end of 
 each year, and which is required in order to control the 
 subsequent accumulation of the fund. This method, however, 
 is unsuited to a work of this nature, and it is preferable to 
 adopt a method of proof based upon actuarial principles. 
 
 To recapitulate the data. A loan of £26,495 is repayable 
 at the end of 22 years, and there is in the sinking fund the sum 
 of <£1500"816 which will accumulate at 3^ per cent. The 
 problem is to ascertain the sinking fund instalment to be set 
 aside and accumulated for the remaining 22 years. 
 
 The calculation is made in two stages as follows : — First 
 ascertain the annual instalment to be set aside and accumulated 
 as a sinking fund at 3^ per cent, to provide £26,495 at the end 
 of 22 years. This annual instalment is c£819'54. 
 
 The next step is to ascertain the annual sum or annuity by 
 which this instalment will be reduced by the amount of 
 £1500'816 now in the fund, or in other words the annuity for 
 22 years at 3^ per cent, which may be purchased by the above 
 sum of £1500' 816. This annual amount, using the author's 
 standard calculation form No. 5, will be found to be £9895. 
 
 The adjusted annual instalment therefore is: — 
 
 Annual instalment to repay the loan of £26,495 in 
 
 22 years £819-54 
 
 less the reduction therein due to the amount of 
 
 £1500816 now in the fund £98-95 
 
 Amended annual instalment as previously ascertained £720-59 
 
 In the foregoing example it has been assumed that the 
 amount of the loan remains unchanged and that the rate of 
 accumulation of the fund and the income from investments 
 will continue to be 3^ per cent, as in the original example in 
 Chapter XV. The only variation is in the period of repay- 
 ment. Tlie methods described in Cluipter XXIY, variation in 
 the period of repayment, cannot be adopted because tlie amended 
 annual instalment here required is to replace four instalments 
 to l)e set aside for varying ])eriods instead of one instalment 
 for one period. The method will apply equally to loans not 
 raised by tlie issue of stock if the whole of the loans are 
 repayable in one sum on a specified date. 
 
THE DATE OF BORROWING 365 
 
 CHAPTER XXX. 
 
 SINKING FUND PROBLEMS, RELATING TO THE 
 DATE OF BORROWING AND THE REDEMPTION 
 PERIOD, 
 
 Without any complication as eegaeds the life or duration 
 OF continuing utility of the asset created out of the 
 LOAN [continued). 
 
 Loan borrowed in one or more years in varying amounts at 
 various dates in each year, and it is required that 
 the revenue or rate account of each year shall be 
 charged with a proportionate part of the annual 
 sinking fund instalment. 
 
 The actual borrowings in any one year (whether in respect 
 of a loan borrowed entirely in one year, or borrowed over a 
 series of years depending upon the period of construction) are 
 often made at various dates during the year because the money 
 is not required or is not readily obtainable. To carry out the 
 strict letter of the obligation to repay the loan at the end of a 
 prescribed number of years from the date of borrowing would 
 be practically impossible if each individual borrowing had to 
 be treated separately. The general practice is to treat all the 
 sums received in any one year as if they had been borrowed 
 at the end of the financial year and not to set aside any sinking 
 fund instalment in respect of the broken period of the year of 
 borrowing, the provision of the first full annual instalment 
 being deferred until the end of the succeeding financial year, 
 which simplifies the working of the fund very considerably. 
 In the case of a small loan borrowed piecemeal in this fashion 
 in one year there is not any great objection to outweigh the 
 manifest advantages; and the same applies to loans borrowed 
 over a period of vears during construction in which the annual 
 amount borrowed is not large. The principle of deferring the 
 first annual contribution has been extended by Parliament in 
 certain cases, where the operation of the sinking fund has been 
 suspended for a specified niimber of years. 
 
366 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 But it may happen in the case of a local authority or a 
 commercial or financial vindertaking that the loan is of large 
 amount and may be borrowed during one year. It may be 
 necessary and equitable in such a case to charge the revenue 
 or rate accoimt of that year with the proper calculated propor- 
 tion of one year's annual instalment in respect of each separate 
 borrowing, based upon the part of one year for which the under- 
 taking has had the use of the money raised during the year, 
 and not defer the first annual contribution until the end of the 
 succeeding year. 
 
 Such an instance might arise in connection with the pur- 
 chase of an existing undertaking by a local authority where 
 the purchase money is payable by instalments spread over a 
 year and is borrowed as and when required, but the local 
 authority enters into possession immediately and takes the 
 whole of the profits. If no contribution to the sinking fund 
 were made during the first year the revenue or rate account of 
 that year would show a fictitious profit as compared with 
 subsequent years. In such a case it would appear not only 
 equitable but good accounting jDractice to charge the revenue 
 or rate account of that year with a portion of the annual 
 instalment commensurate to the amount of loan it has had the 
 use of during part of the year. 
 
 In the case of a commercial undertaking the revenue 
 account for the year would of course l>e charged with interest 
 upon the loan for the exact number of days the money had 
 been borrowed, and the same would apply to the revenue 
 account of a local authority where the accounts are kept upon 
 the " income and expenditure " as distinguished from the 
 " receipts and payments " system. 
 
 If the principle applies to interest upon the loan, it should 
 certainly apply to the annual contribution to the sinking fund, 
 especially in the case of a local authority where both amounts 
 are specific charges against revenue or rate. In the case of a 
 commercial undertaking the conditions as to a sinking fund are 
 much more elastic than is the case with the loans of local 
 authorities, and mucli would depend upon the actual conditions 
 laid down in tlie deed governing the loan, which would be taken 
 into account by the auditors before certifying the accounts. 
 
 In the case of local authorities it is impossible to lay down 
 any hard and fast rule. The conditions imposed upon such 
 authorities have of late years been of a uniform nature depend- 
 ing upon the probable life of the asset, but where powers are 
 granted by special Act of Parliament wider latitude has often 
 
THE DATE OF BORROWING 367 
 
 been allowed, and tlie special nature of the powers requires 
 careful scrutiny in eacli case. Attention may, however, 
 properly be directed to the magnitude of the loan; in some 
 instances the amount involved may be considerable, and may 
 point to the necessity of making some such adjustment, but to 
 insist upon it in all cases, irrespective of the amount of the loan, 
 might, and possibly would, involve considerable labour without 
 any corresponding advantage. 
 
 With regard to the actual adjustment, there are several 
 interesting points, and the problem is not so simple as it appears 
 at first sight. To find the actual proportion of the annual 
 instalment to be charged to the revenue account of the year of 
 borrowing it is first necessary to ascertain the annual instalment 
 to repay the total loan borrowed, having regard to the redemp- 
 tion period imposed. Seeing that an adjustment of this nature 
 is rarely made in the case of small loans, but is confined to 
 loans of considerable magnitude, it is very important that the 
 calculation should be made with extreme accuracy. Such large 
 loans are generally raised by the issue of stock redeemable on a 
 fixed date, and it often happens that the total amount of the 
 loan is borrowed over a period of years, rendering it necessary 
 to make a similar calculation of the proportionate part of one 
 year's annual instalment at the end of each year of borrowing. 
 In this manner varying amounts are added to the fund each 
 year, which departs from the normal growth of a sinking fund 
 by equal annual instalments. This will render it necessary 
 to set aside each year what may be termed temporary instal- 
 ments, and to adjust the fund when the whole of the loan has 
 been raised, by ascertaining the exact equal annual instalment 
 required to be set aside during the remaining years of the 
 redemption period to repay the loan on the prescribed date, 
 having regard to the amount in the fund at the time of making 
 the adjustment. 
 
 The problem will be illustrated by a sinking fund to repay 
 a loan of £11,355 in one sum, on a specified date (namely, at 
 the end of 25 years) with a rate of accumulation of 3| per cent., 
 the loan being borrowed in three equal annual sums of £3,785. 
 These amounts are borrowed at various dates during the several 
 financial years, and it is required that the revenue or rate 
 account of each year shall be charged with a proportionate part 
 of the sinking fund instalment in respect of the money borrowed 
 during the year. 
 
 A similar loan has already been used to illustrate the 
 example in Chapter XXIX, in which case the money was 
 
368 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 supposed to be borrowed on the first day of each financial year, 
 and the conditions shown in Table XXIX. A. will be adopted 
 in the present instance in order to show the effect as compared 
 with that example, although the amounts are small. But the 
 principle is the same, and the effect upon a larger loan will be 
 readily appreciated when it is remembered that, given the same 
 number of years and rate of accumulation, the annual instal- 
 ment is always proportionate to the loan. 
 
 A further imaginary factor must be assumed, namely the 
 precise date or dates in each year on which the loan was raised. 
 
 It is more than probable that the loan borrowed in any one 
 year will be raised in more than one sum. In such cases it is 
 sufficiently if not quite correct to proceed by the arithmetical 
 method and multiply the several amounts borrowed by the 
 number of days between the respective dates of borrowing and 
 the end of the financial year. The sum of these products 
 divided by the total amount borrowed during the year and the 
 result again divided by 365 is the proportion of the year 
 required. 
 
 The following example will make the matter clear : — 
 
 TABLE XXX A. 
 
 To ascertain the proportion of the annual instalment in respect 
 of the amounts borrowed during one year. 
 
 The Arithmetical Method. 
 
 
 Number of days 
 
 Product 
 
 Amount 
 
 to end of 
 
 of amount 
 
 borrowed. 
 
 financial year. 
 
 X days. 
 
 Date of borrowing. 
 
 January 31 £100 334 33,400 
 
 March 31 £200 245 49,000 
 
 June 30 £300 184 55,200 
 
 September 30 £400 92 36,800 
 
 Total £1000 174,400 
 
 The eqiiivalent proportion of one year for which the under- 
 taking lias liad the benefit of the £1000 is arrived at as 
 follows : — 
 
 174,400 174-4 
 
 ^-365 = 
 
 1000 365 
 
 and this proportion of the annual sinking fund instalment is 
 chargeable against the revenue or rate account of the year of 
 borrowing. 
 
 In order, however, to vsimplify the following calculation it 
 
THE DATE OF BORROWING 369 
 
 will be assumed that the loan was raised in each year in one 
 sum, and that the local authority had the use of the money for 
 the following portions of each year : — 
 
 First year one half of the year 
 
 Second year one third of the year. 
 
 Third year one quarter of the year. 
 
 The exact dates of borrowing during each year have a very 
 important effect upon the variation in the annual instalment 
 during the period of borrowing and the subsequent period of 
 repayment. If the amounts are borrowed during the early 
 part of the year, the proportionate part of one year's instalment 
 will be greater than if the money were borrowed during the 
 later part of the year. 
 
 The complete conditions are shown in the following table : — 
 
 TABLE XXX B. 
 
 Loan of £11,-355 borrowed over 3 years, repayable in one sum 
 on a specified date, by means of an annual sinking fund 
 instalment to accumulate at 3| per cent. The revenue or 
 rate account of each year to be charged with a propor- 
 tionate part of the annual instalment in respect of the 
 amount of loan borrowed during such year. 
 
 Annual amounts borrowed and yearly and proportionate instal- 
 ments . 
 
 Annual instalment. 
 Poi'tion of 
 year for Period Proportionate 
 
 Amount wliicli money in wliicli part of first 
 
 Year. borrowed, borrowed. repayable. Yearly. year's instalment. 
 
 First 3785 i 25 years 97-176 48-588 
 
 Second 3785 I 24 years 103-227 34409 
 
 Third 3785 i 23 years 109-836 27-459 
 
 £11,355 £310-239 
 
 ]SfoTE. — This table should be compared with Table XXIX. A. 
 
 The above annual instalments are calculated for even periods 
 of 25, 24 and 23 years respectively, and in the following 
 example it will be assumed that they are set aside during the 
 three years, at the end of Avhich period the necessary adjustment 
 will be made. This is the most practical way of dealing with 
 the matter, although it may properly be contended that the 
 above yearly instalments should be slightly reduced in conse- 
 quence of the proportionate parts set aside in respect of the year 
 of borrowing. The main object of the adjustment is to ensure 
 
370 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 that the revenue or rate accoimt of the year of borrowing 
 shall be charged with a proper proportion of the sinking fund 
 instalment rather than that subsequent years shall be charged 
 to a fraction with the exact mathematical amount. 
 
 The following table shows the complete and proportional 
 instalments which will be added to the fund during the three 
 years, and the amount which should be in the fund at the end 
 of the third year of borrowing. The broken year during which 
 the first amount was borrowed is not included in the period of 
 repayment, which is in effect extended by part of a year. 
 
 STATEMENT XXX C. 
 
 Loan of £11,355, borrowed over three years, repayable in one 
 sum on a specified date. 
 
 A proportion of each annual instalment to be set aside in respect 
 of the amounts borrowed during each year. 
 
 The amount in the sinking fund at the end of each year of 
 borrowing, will be as follows : — 
 
 Borrowing begins 
 at beginning of fi.rst year of fund : — 
 
 I of £97-176, instalment, first year... 48588 
 
 Eepayment period begins 
 
 at end of first year of fund : — 
 
 Interest on £48-588 1700 
 
 Instalment, first year 97-176 
 
 i of £10-3-227, instalment, second vear 34-409 
 
 133-285 
 
 181-873 
 
 at end of second year of fund : — 
 
 Interest on £181-873 6306 
 
 Instalment, first year 97-176 
 
 Instalment, second year 103-227 
 
 i of £109-836, instalment, third year 27-459 
 
 234-228 
 416101 
 
THE DATE OF BORROWING 371 
 
 Borrowing ceases 
 
 at end of third year of fund: — 
 
 Interest on £416101 14'564 
 
 Instalment, first year 9T"176 
 
 Instalment, second year 103"227 
 
 Instalment, third year 109-836 
 
 324-803 
 
 Amount in the fnnd at the end of the third year ... £740-904 
 
 The above amounts credited to the sinking fund are 
 contributed as follows : — 
 
 Charged to Interest 
 revenue from 
 
 account. investments. Total. 
 
 First year of borrowing 48-588 — 48-588 
 
 First year of repayment period 131-585 1-700 133-285 
 
 Second year of repayment period 227-862 6-366 234-228 
 
 Third year of repayment period 310-239 14564 324803 
 
 718-274 22-630 740904 
 
 as compared with the previous 
 example in Statement XXIX B. : 
 
 First year of repayment period 97-176 — 97176 
 
 Second year of repaVment period 200403 3-401 203-804 
 
 Third year of repayment period 310239 10534 320-773 
 
 607-818 13-935 621-753 
 
 or a surplus of 110-456 8695 119151 
 
 There is in the fund at the end of the third year 
 
 the sum of " ... £740-904 
 
 as compared with the previous example, 
 
 Statement XXIX. B. £621-753 
 
 a surplus of £119-151 
 
 being the accumulation of the proportionate parts of the instal- 
 ments set aside in respect of the years of borrowing, as may be 
 verified bv a similar calculation. 
 
372 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Seeing that the kjau is the same in amount and the 
 unexpired period is 22 years in each case this surplus will tend 
 to reduce the annual instalment of £310'239. 
 
 The reduced annual instalment may be found in the follow- 
 ing manner which is similar in principle to the annual incre- 
 ment (balance of loan) method described in Chapter XXII : — 
 
 Amount of loan repayable in 22 years from the 
 
 present time £llo5500 
 
 Deduct therefrom the amount of loan which will 
 
 be provided by the accumulation at 3^ per cent. 
 
 for 22 years of the £740904 now in the fund. 
 
 By standard calculation form, No. 1 £15T9'24 
 
 leaving a balance of loan of £9T75'T6 
 
 to be provided by the accumulation, at 31 per cent., of the 
 future amended annual sinking fund instalment to be set aside 
 for 22 years. 
 
 This amended annual instalment, as may be found 
 
 by standard calculation form, No. 3x, is £302'38 
 
 Peoof of the above Adjustment. The accuracy of the 
 above adjustment may be proved in a similar manner to that 
 adopted in Chapter XXIX. A loan of £11,355 is repayable 
 at the end of 22 years, towards which there is in the fund an 
 amount of £740-904, which will accumulate at 3^ per cent. 
 
 The annual instalment to repay the loan of £11,355 
 in 22 years at 3^ per cent., as may be found by 
 standard calculation form, No. 3x, is £351"~.3 
 
 but the amount of £740-904 now in the fund is 
 equivalent to an annual instalment for the 
 same period, as may be found by standard 
 calculation form, No. 5, of £48"85 
 
 leaving a reduced annual instalment, as previously 
 
 ascertained, of £30^"3(S 
 
 Two methods have now been described of repaying a loan 
 of £11,355 (borrowed over a period of 3 years) nt tlie end of 
 25 years und(>r two sets of condiiions, namely: — 
 
THE DATE OF BORROWING 
 
 373 
 
 A, where the annual instalment is set aside at the end of 
 
 the financial year following the year of borrowing, and 
 the revenue or rate account of the year of borrowing 
 is relieved of any charge in respect of the sinking fund 
 instalment. Chapter XXIX, Table XXIX. B. 
 
 B, where the revenue or rate account of each year of borrow- 
 
 ing is charged with a proportionate part of the annual 
 instalment. Chapter XXX, Table XXX. C. 
 
 The annual charges to revenue or rate account in each case 
 may be usefully compared by means of the following table : — 
 
 TABLE XXX. D. 
 
 Loan of £11,355 borrowed over three years, repayable in one 
 sum on a specified date. 
 
 A. Annual instalments only. 
 
 Table XXIX. B. 
 
 B. Annual and proportional instalments 
 
 Table XXX. C. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of the sinking fund instalment. 
 
 Amount charged to the 
 revenue or rate account. 
 
 First year of borrowing ... 
 First year of repayment 
 
 period 
 
 Second year of repayment 
 
 period 
 
 Third year of repayment 
 
 period 
 
 Total 
 
 each of the subsequent 22 
 years of the repayment 
 period 
 
 A. 
 
 Where the year of 
 
 borrowing is 
 
 relieved of any 
 
 charge in respect 
 
 of the 
 
 sinking fund 
 
 instalment. 
 
 Table XXIX. B. 
 
 Xil 
 
 97176 
 
 607-818 
 
 310-239 
 
 Where the year of 
 
 borrowing is 
 
 charged with a 
 
 proportionate part 
 
 of the 
 
 sinking fund 
 
 instalment. 
 
 Table XXX. C. 
 
 
 Excess 
 
 of B. 
 
 over A. 
 
 48-588 
 
 131-585 
 
 718-274 
 
 302-380 
 
 48-588 
 
 34-409 
 
 200-403 227-862 27-459 
 310-239 310-239 Nil 
 
 110-456 
 
 7-859 
 
374 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The effect of charging the revenue or rate account of the 
 year of borrowing with a proportionate part of the sinking fund 
 instalment instead of deferring any charge to the end of the 
 following financial year may be summarised as follows : — 
 
 The charge to revenue or rate account is antedated by one 
 year to the extent of the proportionate instalment in respect of 
 the first year's borrowings, and the same applies to each year 
 during which the borrowing takes place. The difference 
 between the two methods affects only the revenue or rate 
 accounts of the years of extended borrowing, but as the first 
 broken year of borrowing is not included in the repayment 
 period, the annual instalment charged to revenue or rate in 
 the third year of the fund is the same in both methods. 
 
 Having thus charged the earlier years with a greater part 
 of the repayment burden, it is obvious that the later years will 
 be correspondingly relieved, and the above table shows this 
 to be the case. But the increased burden to revenue or rate 
 account during the years of borrowing is spread over a smaller 
 number of years than the relief is obtained during the remainder 
 of the repayment period, in consequence of which the effect is 
 to charge the revenue or rate accounts of the years of borrowing 
 with a far greater annual amount than that by which subse- 
 quent years are relieved. 
 
 In the above example the three years of borrowing are 
 charged with an additional amount of £110-456, or an average 
 of £;j6'818 per annum, whereas the subsequent years are 
 relieved to the extent of <£7'859 per annum only. 
 
 The above amounts of additional burden during the earlier 
 years and the corresponding amounts of relief during the later 
 years must not be accepted as an exact ratio Avhich will apply 
 to all examples of this nature, because the dominant varying 
 factor in the foregoing adjustment is the actual date or dates 
 in each year upon which the loan was borrowed. If the loan 
 had been borrowed on the first day of the financial year the two 
 methods would yield exactly similar results, but if the loan had 
 been borrowed during the early part of the previous year the 
 results would have shown much more variation than the average 
 example used to illustrate the subject. 
 
 Tlie necessity to make an adjustment of this nature therefore 
 depends, primarily, upon the magnitude of the loan, and, 
 secondly, upon the portion of the year during which the money 
 borroAved has been utilised. 
 
Section VI. 
 
 The Life or Duration of Continuing Utility 
 
 of the Asset Created out of the Loan, 
 
 and its Relation to the Redemption Period 
 
 and the Incidence of Taxation. 
 
377 
 
 CHAPTER XXXI. 
 
 THE LIFE OE DURATION OF CONTINUING UTILITY 
 OF THE ASSET CREATED OUT OF THE LOAN, 
 AND ITS DELATION TO THE REDEMPTION 
 PERIOD AND THE INCIDENCE OF TAXATION. 
 
 lu the case of tlie loans of municipal or other local 
 authorities, there is a further factor which requires serious 
 consideration, namely, the periods allowed by Parliament (or 
 the Government Department concerned) for the repayment of 
 loans authorised for different classes of outlay having longer 
 or shorter lives or periods of duration or utility, and this 
 variation in the life of the asset may in its turn react upon 
 the period over which the loaii is borrowed or is repayable. 
 This factor gives rise to the necessity to equate the period 
 during which loans shall be repayable depending upon, 
 
 1. The life of the asset and the consequent period of 
 
 repayment. 
 
 2. The date or dates of borrowing, whether in one year or 
 
 spread over a period of years. 
 
 3. A combination of both periods, namely, of borrowing 
 
 or repayment. 
 
 This is the most difficult problem in municipal finance, upon 
 which there is much divergence of opinion, as is only natural 
 considering the extended and complicated nature of municipal 
 activity, which, as all who have paid attention to such matters 
 know, is ever widening. 
 
 Communities have not any capital beyond the liability of 
 each citizen of both the present and future generations to 
 contribute his rateable proportion of the cost of the benefits 
 which he receives from the joint efforts of the community. 
 Such benefits are received by each citizen in each generation 
 year by year, and should be paid for as and when received. 
 In a primeval community individual benefit is paid for by 
 individual labour, but such an ideal method of contribution 
 can only exist in a small community, and the difficulty of 
 
37S REPAYMENT OF LOCAL AND OTHER LOANS 
 
 apportioning the annual burden in a rapidly growing one is 
 intensified in a far greater ratio than the actual numerical 
 increase of population. 
 
 Year by year, as the community grows, the problem becomes 
 more complicated. Works of public utility, which in a small 
 community might be ignored or neglected, become of vital 
 importance, and must be carried out, and in doing so regard 
 must be had not only to the present requirements, but also 
 to the future growth. It is obviously useless to undertake 
 IHiblic works which it is well known will be utterly inadequate 
 to provide for the needs of future generations, and provision 
 must be made in advance. 
 
 This increases the cost of all works of public utility, and 
 involves the immediate spending of large sums of money which 
 cannot be found by, and cannot properly be charged against, 
 the present generation of ratepayers, either at once or spread 
 over comparatively iew years. Such outlay can only be met by 
 pledging the credit of the community for the purpose of raising 
 a loan. Consequently the repayment of the loan must be 
 spread over an extended period depending upon — 
 
 (1) the probable life of the asset upon which the money is 
 expended ; 
 
 (2) the liability of future generations to provide further 
 works of public utility which may then be required; and 
 
 (3) the judgment of those immediately responsible for the 
 adequacy of the present outlay, including in such term not only 
 the actual permanence of the work undertaken, but also the 
 probability that future advancements in knowledge may render 
 such works either inadequate in design or too costly in operation. 
 This throws the responsibility of the actual outlay upon those 
 who incur it, and it is now a generally accepted principle that 
 the cost of all outlay upon works of public utility should be 
 written oif, and the loans raised therefor actually repaid, out of 
 current revenue or rate during a period well within the life 
 of the particular works to provide which the loan is borrowed. 
 It is obvious, therefore, that the provision of public utilities 
 adequate to the needs of future generations in any individual 
 community is far too great a burden to be imposed upon the 
 present gem^ration of ratepayers, and that this involves pledging 
 the future credit of the community. By a parity of reasoning 
 the increase in size and number of communities, and the ever 
 widening sphere of local activities, renders it imperative that 
 the extent to which the present generation shall be allowed to 
 
THE LIFE OF THE ASSET 379 
 
 pledge the credit oi the iuture should be treated not as a local 
 but rather as a national question. At the present time, there- 
 fore, all loans raised by local authorities for purposes of public 
 utilities are subject to the final approval of Parliament, but 
 owing to the enormous increase in this direction Parliament has 
 been compelled to delegate its powers as to detail to Committees 
 and to certain Government departments. This has been a very 
 gradual process extending over many jyears, during which 
 time many Acts have been placed upon tlie Statute Book, with 
 the result that powers have been obtained under both General 
 and Special Acts, and this has led to considerable difference in 
 practice. The great disadvantage of this variation consists 
 in the fact that the larger municipalities, instead of seeking 
 powers under General Acts, may, in many cases, avoid the 
 careful scrutiny of the permanent Government departments 
 (which now proceed upon regularly defined principles) by 
 applying to Parliament for a Special Act. All such Special 
 Acts are referred to Committees composed of members of both 
 Houses of Parliament, but there is not any continuity in the 
 membership of such Committees, and as the permanent 
 Government departments are not represented thereon, there is 
 not any uniformity of practice, and the result is seen in the 
 extreme variation in the powers as to borrowing and repayment 
 now existing. The present general policy of Parliament and 
 of the Government departments charged with the duty of fixing 
 the respective periods of repayment operates in the direction of 
 equalising the period of repayment and the life of the asset, 
 although the conditions now in force vary considerably in 
 individual cases for the reasons already stated. 
 
 This principle is of modern growth. In the early days of 
 municipal government, i.e., prior to 1847, the Acts authorising 
 expenditures upon public utilities did not impose any obligation 
 of any kind to repay the loan out of annual rates to be levied 
 upon the community, and there are to-day many loans out- 
 standing in respect of which no such obligation exists, and the 
 debt and the interest payable thereon may for all practical 
 purposes be considered as a perpetual charge upon the annual 
 rates to be levied by the municipalities unless and until they 
 voluntarily provide for its redemjition by making annual 
 charges against revenue or rate. In some cases this provision 
 has been made on the initiation of those responsible for the 
 financial administration of the municipality, and in other cases 
 such delayed provision has been imposed by Parliament as a 
 condition precedent to the granting of further borrowing powers. 
 
38o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 There is now, however, a considerable body of municipal opinion 
 that where the money borrowed is expended in the purchase 
 of land in or near the centre of a city, and the erection thereon 
 of buildinirs of a substantial nature and of assured future 
 utility, the asset may be considered as of permanent and in 
 many cases even of improving value, and that there is not 
 therefore any necessity to burden the ratepayers of the present 
 or any future generation with any charge in respect of the 
 redemption of the debt beyond the annual interest payable upon 
 the loan which interest may, it is contended, properly be 
 considered as the equivalent of an annual rent. 
 
 In support of an argument of this nature it is contended 
 that local authorities may, and very often do, occupy lands and 
 premises as ordinary tenants, paying therefor the usual rents 
 demanded by the owners of the property, and such tenancies 
 may be of an annual nature or be by way of lease for a term of 
 years. Such leases for years may be of short duration, but, on 
 the other hand, they may, in certain districts, be for very long 
 terms, possibly longer than would be granted by Parliament 
 for the repayment of a loan authorised for the purchase of the 
 property. In such cases it is obviously to the advantage of the 
 local authority to acquire the property by way of lease rather 
 than by purchase, seeing that there will not be any burden in 
 respect of the sinking fund instalment for the redemption of 
 the loan. Especially does this apply to the acquisition of land 
 or buildings which do not immediately require any large outlay 
 or where the outlay is of such a character that it may be spread 
 over a number of years and be met by charging it direct to 
 current annual revenue or rate, or where the annual outlay may 
 be so arranged that it is less than the sinking fund instalment 
 to be set aside to repay the loan necessary to be raised to 
 purchase the property. Such conditions may not always exist, 
 especially in the case of outlay in respect of land required for 
 purposes of public parks or open spaces, or large 2)ublic build- 
 ings, such as town halls, requiring a large expenditure upon 
 buildings, but the principle is important and may be applied 
 to the occupation of land and buildings without imposing any 
 burden upon the present and future generation of ratepayers 
 for the acquisition of properties which may at any future time 
 be replaced by others, which may be not only as cheaply 
 acquired but may be more suitable for the purpose. As against 
 this it is argued that land in the centre of a city required 
 for the erection of a town liall, or land for public parks, increases 
 rapidly in value, and at the end of a long lease the fine or 
 
THE LIFE OF THE ASSET 381 
 
 premium payable on renewal of the lease would be very large, 
 and tbe probability of such a burden being laid by the present 
 upon the slioulders of a future generation would certainly not 
 be sanctioned by Parliament. 
 
 Markets. The argument appears to be equally strong when 
 applied to markets which generally occupy land near the centre 
 of the city and in respect of which the cost of the land is the 
 predominant factor, since the buildings are not usually of an 
 expensive character. In addition to the improving value of 
 the site, markets are a source of revenue consisting of tolls 
 upon produce and rents of floor space and buildings, which 
 revenue, after providing for all charges, yields a surplus which 
 is applied in aid of the rates levied upon the general body of 
 ratepayers. In most cases markets yield a surplus revenue 
 over and above all upkeep charges, and it seems only proper 
 that the present generation of ratepayers should out of such 
 surplus revenue provide an annual instalment towards the 
 redemption of the debt before applying any profits in aid of 
 their annual rateable contributions towards the upkeep of the 
 city. 
 
 Water. The provision of a permanent supply of pure 
 water for sanitary and other purposes is the prime necessity of 
 all communities for many weighty reasons, and demands special 
 consideration. The paramount factor in this case is the 
 imperative obligation to provide for the needs of the community 
 for a number of generations far in excess of that requisite in 
 the case of any other public utility; indeed, it may properly 
 be contended that it is the duty of the present generation to 
 ensure that a permanent supply of pure water sufficient for the 
 needs of the community shall continue for ever. Methods of 
 lighting, transportation, sewage disposal and other communal 
 necessities are being constantly improved, and any future 
 improvements in such comparatively minor utilities may be 
 carried out upon land already allocated to them and acquired 
 by the municipality. But with water supply the conditions are 
 the exact opposite. Owing to the rapid growth of cities involv- 
 ing increasing demands for water for sanitary and maniifactur- 
 ing purposes, the natural areas suitable for the supply of water 
 are being year by vear continually encroached upon and 
 reduced, and future improvements in methods of transportation 
 will enable manufarturinc; processes to be profitably carried on 
 far beyond the present city limits. Such conditions are favour- 
 able to the creation of vested interests in all land which is a 
 
382 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 natural water area, and such vested interests will be scattered 
 in such a mt^nner as to render their acquisition at some future 
 time practically impossible even at any jDrice. It is therefore 
 the duty of all municipalities to protect and preserve all natural 
 water areas for the public use and to expend money upon the 
 purchase far in advance of present requirements. There is here 
 an obligation to pledge the credit of the community for the 
 purchase of land and the construction of works to provide a 
 sufficient supply of water to meet the maximum needs of the 
 community, and yet the present policy of Parliament is to allow 
 a shorter p'eriod than formerly. Owing to the reasons already 
 mentioned such land is continually increasing in value, and 
 many existing water undertakings are now worth very much 
 more than their original cost. It is therefore argued that the 
 repayment of money borrowed to provide the cost of land for 
 water areas should be spread over a very long period of years, 
 even if it be not treated as a debt in perpetuity. 
 
 The argument as to the large amount expended in the 
 purchase of land is supported by the substantial and permanent 
 character of the works erected thereon, and it seems at first 
 sight sound policy to relieve the present generation of rate- 
 payers from what appears to be an undue burden by spreading 
 the redemption of the loan over a longer period than is at 
 present allowed by Parliament. As against this it is pointed 
 out that water works have failed, water areas have yielded a 
 decreased and insufficient supply, and works which were once 
 thought adequate have, owing to the large increase of towns, 
 become insufficient and have had to be supplemented by further 
 outlay. It is also contended that if the repayment of the debt 
 be spread over a very extended period the interest paid equals, 
 and soon exceeds, the amount of principal. This is not in 
 itself a very good reason against extended periods of repayment 
 seeing that its effect is to spread the burden over a greater 
 number of generations Avho derive benefit from the outlay, 
 provided always that the works continue to meet the needs of 
 the community and subsequent generations do actually derive 
 a benefit therefrom. 
 
 But it is common knowledge that very few works of public 
 utility last for more than a certain number of years. In some 
 cases the rateable value of a district falls, but in nearly all 
 cases the future demands of the community increase so rapidly 
 that it is imperative to put what may by some be termed an 
 undue burden upon the present generation in order to avoid 
 placing an iutolorablo l>ur(l('ii iqioii the future. The personal 
 
THE LIFE OF THE ASSET 383 
 
 element also enters largely into the matter, and it has been 
 found that the surest, if not the only, way to check undue 
 expenditure, if not extravagance, upon the part of local 
 authorities is to convert each £1,000 of capital outlay into a 
 definite proportion of the annual amount payable by the rate- 
 payer by way of rate, and, further, to educate the ratepayer 
 to appreciate this. There is also another interest to be con- 
 sidered, namely, the loanholder who finds the money and who, 
 in a great majority of cases, has not any local interests. 
 He looks solely to his security both for the annual payment of 
 interest and the ultimate repayment of his capital. His 
 security consists partly of the assets created out of his money 
 and partly of the annual revenues derived therefrom, but 
 in practice mainly of the future annual rates to be 
 levied upon the community. Seeing that the value of the 
 communal assets depends entirely upon the perpetual prosperous 
 existence of the community, such assets have really no value 
 unless the community is able to pay the future annual rates. 
 A bankrupt or insolvent community, if not an absolute im- 
 possibility, would not be able to pay any serious percentage 
 of its liabilities; and seeing that the security for its loan 
 indebtedness is a mixed fund of capital and revenue, of which 
 the latter is the chief, it seems not only reasonable, but just, 
 that revenue or rate should bear the greater proportion of the 
 burden. It follows, therefore, that the cost of the outlay 
 should be repaid within the productive life of the asset and be 
 charged against the annual rates levied by the local authority. 
 In the case of revenue earning undertakings it may, not very 
 unreasonably, be contended that any surplus profits should 
 partially, if not wholly, be applied in redemption of debt 
 instead of in aid of rate. If the whole of such profits were 
 applied in redemption of debt, it would avoid the present 
 anomaly of towns with equal annual rates but with widely 
 varying expenditures, due solely to the fact that the excess 
 expenditure in one case is concealed by the profit derived from 
 trading departments. In the case of tramways this profit is 
 fairly earned since there is a generally accepted level of fares 
 all over the country, but in the case of gas and electric lighting 
 undertakings there is such a wide divergence of charges as 
 between different municipalities, that a very high charge, levied 
 at will by the local authority, is called a profit, and is taken 
 out of the pockets of one class of ratepayers, namely, the gas 
 or electricity consumers, and applied in relief of the rates paid 
 bv the whole community. 
 
384 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 The foregoing remarks deal very fully with waterworks as 
 representing a class of outlay which lends itself most readily to 
 the argument in favour of a total abandonment of the annual 
 charge against revenue or rate in respect of the redemption of 
 debt, or at least in favour of a reduction in the annual charge 
 to the present generation of ratepayers to the possible and 
 probable detriment of future generations. They will have 
 their own burdens to bear both as to their then present, and 
 future obligations. Any relaxation of the present, as some think, 
 stringent regulations and practice will most probably give them 
 in addition a past burden to bear, which, owing to the foresight 
 of our local authority forefathers we have escaped. Conse- 
 quently the modern Parliamentary practice is right, namely, to 
 require the redemption of the loan to be spread over a period 
 well within the life of the asset created out of the loan and to 
 differentiate between various classes of outlay in fixing the 
 period to be allowed in respect of each. 
 
 Outlay on Manufacturing Plants. So far the enquiry 
 
 has been confined to capital outlay upon public works in which 
 
 the greater part of the cost is for land, which rarely depreciates, 
 
 and very often appreciates in value, or for buildings for which 
 
 a very long life may reasonably be expected seeing that 
 
 judicious outlay npon repairs will prolong the life considerably. 
 
 There is, however, a further class of outlay of a much more 
 
 complex nature where the proportion of the original cost 
 
 attributable to land is comparatively small, and the greater 
 
 part of the outlay is in respect of buildings, motive power, 
 
 plant and machinery, including in the latter term everything 
 
 in the nature of an engiiie, gas making plant, tramway plant, 
 
 electrical generating machines and all the subsidiary works 
 
 required. The necessity to exercise careful control over such 
 
 outlay arises from the fact that as the element of a probable 
 
 appreciation in value decreases, it is requisite to provide 
 
 for the very opposite conditions, namely, a probable fall in 
 
 value due to two causes, first, a gradual wasting of the asset 
 
 due to Avear and tear (which cannot be met by current repairs 
 
 and renewals charged to revenue account) and the further 
 
 probability that future advances in scientific and mechanical 
 
 knowledge may result in the discovery of new and improved 
 
 methods long before the oi'iginal ])lant, etc., is worn out and 
 
 the loan repaid. Local authorities as well as commercial 
 
 concerns are here confronted with a difficult problem and have 
 
 cai(>fullv to consider whether it is advisable to discard the 
 
THE LIFE OF THE ASSET 385 
 
 present obsolete plant which is costly in operation and deficient 
 in productive power, and replace it with more modern plant, 
 relying npon the saving in working charges and the increase in 
 production to recoup the annual burden imposed by installing 
 the modern outfit. In such an event there is a wide difference 
 between the conditions existing in the case of a commercial 
 undertaking and a local authority. A commercial undertaking 
 may set aside any part of its profits and so accumulate a 
 reserve fund of unlimited amount for such a contingency; 
 whereas, as a general rule, a municipality is restricted as to the 
 amount which may be so set aside as a reserve fund, as dis- 
 tinguished from a renewals fund. If a commercial concern 
 requires to undertake outlay of this nature there are not any 
 statutory or other difficulties in the way provided the credit 
 of the undertaking is good ; and it is not always under any 
 obligation to set aside part of the profits towards the redemption 
 of debt. On the other hand, a local authority is bound by 
 Statute to charge its annual revenue or rate account with a 
 fixed sum to be applied in redeeming its loan indebtedness, and 
 such obligation may not be released without the consent of 
 Parliament. Any further borrowing powers required to replace 
 obsolete assets or outlay before the original loan is repaid, have 
 to be granted by Parliament, and very severe scrutiny is made 
 into all the circumstances, because the grant of further powers 
 will lay a double burden upon the community until the original 
 loan is repaid. All this tends to support the present practice 
 of Parliament, namely, to fix the period of repayment at a 
 number of years well within the life of the asset; in other 
 words, to make the annual charge for the redemption of debt a 
 little more than equivalent to the normal rate of depreciation 
 which would be charged to revenue, or profit and loss, account 
 by a prudent trader. This practice supports the view that in 
 the case of a local authority there is not any necessity, in 
 respect of original outlay, to charge the revenue account with 
 depreciation or wear and tear in addition to the sinking fund 
 instalment. This question of depreciation (or wear and tear) 
 should be kept entirely distinct from the provision of a general 
 reserve fund to make good any capital losses due to obsolescence, 
 or to the provision of a renewals fund to meet repairs which 
 cannot be made year by year, such as the periodical relaying 
 of a tramway track. The statement that the sinking fund 
 instalment takes the place of an annual charge for depreciation 
 requires important modification in one respect. It has been 
 stated that an annual charge for depreciation may be omitted 
 
 A A 
 
386 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 ill the case of original outlay only. The ease is different when 
 once the original loan has been repaid and the asset becomes 
 the property of the local authority free from any debt and 
 without the necessity to set aside any annual instalment, or to 
 pay any interest upon the loan. It should here be remembered 
 that under the sinking fund method of repayment both these 
 annual charges are a burden upon the revenue or rate account 
 of each year of the repayment period, and that this annual 
 burden is equal during the whole of the period, seeing that 
 although part of the loan may have been repaid out of the 
 sinking fund, yet the interest upon the amount of loan so repaid 
 must be charged to the revenue or rate account and added to 
 the fund. On the final repayment of any loan the revenue or 
 rate account is immediately relieved of a heavy annual charge, 
 consisting of the instalment and interest upon the loan, and 
 the local authority is in possession of an undertaking which has 
 been provided out of the revenue or rate of previous years, and 
 in addition maintained in a state of efficiency by annual repairs 
 and renewals, and very possibly kept up-to-date by improve- 
 ments defrayed by means of additional charges to revenue or 
 rate. 
 
 It is therefore equitable to assume that it is obligatory upon 
 future generations of ratepayers to ensure that this asset shall 
 be maintained by them in an efficient state, as far as possible, 
 but since any expenditure upon repairs and renewals cannot 
 prevent a further loss in value, such wastage should be made 
 good bv charging future years with an annual sum in respect 
 of depreciation. This is a matter which is frequently over- 
 looked, but it is worthy of serious consideration. When, in 
 spite of all repairs and renewals, the asset becomes valueless, 
 or so nearly so that it cannot be worked at a profit or economi- 
 cally, it must be replaced and the depreciation fund in hand 
 may then be applied in relief of the cost of the new works, 
 leaving only the balance to be raised by further borrowing. 
 
Section VII. 
 
 The Equation of the Period ol 
 Repayment. 
 
389 
 
 CHAPTEE XXXII. 
 
 The equation or the period of repayment of loans repayable 
 
 AT various dates WHICH ARE REQUIRED TO BE REDEEMED 
 
 on one uniform date : 
 
 1. Where the loans are authorised in respect of 
 
 outlays of varying character, each having a 
 
 different life or PERIOD OF CONTINUING UTILITY 
 
 and consequent repayment. 
 
 2. Where the necessity to find the equated period 
 
 OF repayment arises on the consolidation of 
 
 EXISTING loans. 
 
 The arithmetical method of finding the equated period 
 
 KNOWN as the equation OF PAYMENTS, THE TRUE OR MATHE- 
 MATICAL method; and the error in the generally 
 
 ADOPTED arithmetical METHOD. 
 
 The Necessity for the Equation of the Period of Repay- 
 ment. In the early days of municipal loans they were relatively 
 small in amount as compared with what they are at the present 
 day, and as a general rule each loan was sanctioned for a 
 specific purpose and related to one class of outlay only. When 
 a sanction or authorisation included several classes of outlay 
 a definite amount of loan was authorised, and a definite period 
 was prescribed, for each class, carrying out the provision m 
 Section 234(1) of the Public Health Act of 1875, namely: — 
 
 " Money shall not be borrowed except for permanent works 
 (including under this expression any works of which the 
 cost ought, in the opinion of the Local Government Board, to 
 be spread over a term of years)." 
 
 Under this Act (Sec. 234 [4]) the period of repayment may 
 be fixed by the local authority with the sanction of the Local 
 Government Board. 
 
 Section 243 of the same Act dealing with loans to local 
 authorities by the Public Works Loan Commissioners, provides : 
 
390 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 " That in determiuing the time when a loan under this 
 section shall be repa^-able the Local Government Board shall 
 have regard to the probable duration and continuing utility 
 of the works in respect of which the same is required." 
 
 With the widening of the sphere of municipal activity to 
 include gas works, tramways, electric supply, hydraulic power 
 supply and other manufacturing (and in many cases profit 
 earning) utilities, the problem became more complicated, seeing 
 that the total loan authorised for any one undertaking neces- 
 sarily included outlays of very diverse character having widely 
 varying periods of utility and consequently varying periods of 
 repayment. Further difficulties were introduced, when, under 
 the Public Health Acts Amendment Act, 1890, local authorities 
 generally were empowered, subject to certain conditions laid 
 down in the Stock Hegulations of 1891, etc., to raise money 
 by the issue of stock redeemable on a specified date or at the 
 end of a given number of years. 
 
 During the earlier years, when each loan was authorised 
 for one class of outlay only with a definite repayment period, 
 all that was necessary was to keep a separate sinking fund for 
 each loan, when the whole amount was borrowed in one year, 
 and to keep a separate fund for each year's borrowings, when 
 the loan was borrowed over a series of years. The same applied 
 to loans authorised for one undertaking including various 
 classes of outlay, each having a difl:erent period of repayment, 
 so long as the sinking funds could be kept distinct for each class 
 of outlay or each year's borrowings, and the funds could mature 
 at the end of the respective periods and the loans then be 
 redeemed. But when it became possible to raise loans by the 
 issue of stock redeemable on a fixed date it at the same time 
 became necessary to so arrange the sinking fund instalments 
 that the total loan should be redeemed on the prescribed date 
 irrespective of tlie repayment periods imposed for the several 
 component parts of the outlay. 
 
 The difficulty is overcome by ascertaining the equivalent 
 average date of repayment of the Avhole of the loan, and 
 calculating the annual instalment required to be set aside and 
 accumulated in one instead of in several sinking funds. The 
 actual practice varies. In some cases the sanction states the 
 specific amounts to be borrowed for each class of outlay with 
 the period of repayment allowed for each class, and the duty 
 of fixing the average date falls upon the local authority. In 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 391 
 
 other cases the local authority submits a scheme to the Local 
 Government Board showing the various sums proposed to be 
 borrowed for each class of outlay, the respective periods of 
 repayment suggested and the proposed average date of repay- 
 ment of the whole loan. This is subject to revision by the 
 Government department concerned, especially as to the period 
 desired by the local authority, and this being fixed the average 
 or equated period is found by calculation in a manner which 
 will be discussed in detail. 
 
 These are the general considerations which are involved in 
 the equation of the period of repayment, but they may be 
 further complicated by reason that the amounts are borrowed 
 over a series of years, or that the loans in respect of the 
 component parts of the outlay are borrowed together at 
 irregular times, and without any definite allocation as between 
 the various classes of outlay. In many cases it is necessary 
 to set aside temporary instalments during construction, leaving 
 the final instalment to be ascertained by adjustment when the 
 total loan has been borrowed and the whole of the works carried 
 out and an apportionment made of the outlay. In the case of 
 very large undertakings this cannot be done until the engineer 
 has given his final certificate. 
 
 Another ditficulty arises in cases where the operation of the 
 sinking fund is suspended for a number of years, and it is often 
 almost impossible, owing to a combination of the above factors, 
 to decide upon the amount of the first instalment to be set aside. 
 The only permanent factor is the repayment of the whole of the 
 loan on a fixed date. The same considerations apply on the 
 consolidation of several existing loans repayable at various 
 dates, when it becomes necessary to fix a uniform date of 
 repayment and adjust the instalment, having regard to the 
 amounts now in the several sinking funds. 
 
 Space will not permit of the detailed treatment of any such 
 examples owing to the difficulty of stating a set of conditions 
 which would be generally applicable. Each case must be 
 dealt with on the individual facts, but any question likely to 
 arise may be treated by one or more of the methods described 
 in this book. As a general rule, where the conditions are at all 
 complicated, it is better to set aside, during the construction 
 period, temporary instalments of a general natiire and defer 
 any final adjustment until the whole of the loan has been 
 borrowed and the actual outlay under each head has been 
 certified by the engineer. 
 
392 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 The Methods of Finding the Equated Peeiod of Repay- 
 ment. Tlie foregoing remarks will uow be illustrated by tbe 
 following example whicli is of a simple character witliout any 
 of the complications previously referred to, and relates to 
 a loan of £56,000, raised by the issue of stock redeemable at 
 par in one sum on a date to be ascertained. The loan is 
 required for an undertaking comprising outlay of a variable 
 nature, each class of Avliich has a different life or period of 
 utility, and separate periods are prescribed for each. In order 
 to ascertain the date of redemption of the stock it is required 
 to find the equated period corresponding to the several prescribed 
 periods and amounts. This method will apply to ordinary 
 loans if it be required to repay the total debt on one date. 
 
 The classes of outlay, the amounts of loan authorised in 
 respect of each class and the prescribed periods of repayment 
 are as follows : the rate of accumulation is 8 per cent, per 
 annum. The rate of interest payable upon the stock does not 
 enter into the calculation. 
 
 TABLE XXXII. A. 
 
 Particulars of the Loan of £56,000. 
 
 Nature of outlay. 
 
 Amount of loan 
 authorised. 
 
 Prescribed period. 
 
 Class A 
 
 £10,000 
 
 repayable in 45 years 
 
 55 13 
 
 20,000 
 
 on 
 
 )5 55 ~''^ 5 5 
 
 ,5 C 
 
 24,000 
 
 55 55 1^ 55 
 
 55 ^ 
 
 2,000 
 
 55 55 '^ 55 
 
 £56,000 
 
 The Arithmetical Method. Under ordinary circum- 
 stances the above amounts of loan woiild be repayable at the 
 end of the respective periods by means of the usual sinking 
 fund instalments, as described in previous chapters, but under 
 the present conditions the whole of the loan is repayable on 
 one date, which has to be so fixed that the lender will receive 
 his money at a time equivalent to that at which he would have 
 received it if the original varying periods and amounts had 
 been adhered to. In arithmetic this is known as " the equation 
 of payments," and the rule is stated as follows : — 
 
 A/ulfiply each debt by tJie number of years which will 
 
 elapse before it becomes payable; add the results together; 
 
 divide this sum. by the sum of the debts; the quotient will 
 
 be the number of years in the equated time. 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 393 
 
 But it is stated in tlie books on arithmetic that this is only 
 approximately correct, and can only be taken as equitable when 
 the various times of repayment are not widely apart. The 
 error, it is pointed out, is in favour of the payer as it extends 
 the period of repayment. This arithmetical method will first 
 be applied to ascertain the equated period of repayment of the 
 above loan of £56,000, after which an investigation will be 
 made in order to ascertain the true equated period suggested in 
 the arithmetic book. This is the more necessary because in 
 the case of the loans of local authorities the various times of 
 repayment are very widely apart. The result of the investiga- 
 tion into the true equated period will show that it is shorter 
 than under the arithmetical method. In the case of a local 
 authority, however, the arithmetically equated period may be 
 preferred because it is slightly in favour of the payer (in this 
 case the revenue or rate account of the equated period) as it 
 extends the repayment beyond the time required by the true 
 equated method. The effect of equating several sinking fund 
 periods is to reduce the total period over which the repayment 
 is spread and thereby relieve part of the original period of any 
 charge whatever. The burden of this relief is thrown upon the 
 equated period taken as a whole, and any extension of this 
 period tends to redress the inequality caused by the equation. 
 With regard to the interest upon the loan, which will be 
 considered fully in Chapter XX XY, it should be remembered 
 that under the original conditions the annual interest charge 
 to revenue or rate will gradually be reduced as the loans with 
 shorter prescribed periods are repaid; whereas under the 
 generally adopted method of distributing the annual burden 
 after equation, interest upon the full amount of the loan is 
 payable equally during and charged equally against the revenue 
 or rate account of each year of the equated period. This will 
 be discussed in a later chapter, but the present subject of 
 enquiry relates solely to the method of finding the true e([uated 
 period. 
 
 The calculation of the equated period relating to the above 
 loan of £56,000 will now be made, by the arithmetical rule, 
 and, although the figures adopted give a period of an exact 
 number of years, yet in practice this will rarely be obtained. 
 It is in fact somewhat difficult to state original conditions 
 which will work out to an even number of years in the 
 equated period. 
 
394 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE XXXII. B. 
 
 The Arithmetical Method of Finding the Equated Period 
 or Repayment. 
 
 Required the equated period, at the end of wliicb the total 
 loan should be repayable, corresponding to the repayment 
 of the component parts of the loan at the end of the 
 respective periods prescribed for each. 
 
 iture of outlay. 
 
 Amount of 
 loan authorised. 
 
 Prescribed 
 periods. 
 
 Product of amount 
 
 of loan multiplied by 
 
 number of years. 
 
 Class A 
 
 £10,000 
 
 45 years 
 
 £450,000 
 
 „ B 
 
 20,000 
 
 ^9 ,, 
 
 580,000 
 
 „ c 
 
 24,000 
 
 15 „ 
 
 360,000 
 
 „ B 
 
 2,000 
 
 5 „ 
 
 10,000 
 
 
 £56,000 
 
 
 £1,400,000 
 
 Equated period : — 
 1,400,000 _ 
 
 25 years. 
 
 56,000 
 
 The Trie or Mathematical Method. The correctness of 
 the above arithmetically equated period will now be investi- 
 gated, as well as the eifect of the alteration upon the repayment 
 of the loans. 
 
 To do this it is first necessary to ascertain the exact 
 equivalent of the original conditions. This will be stated as 
 if those conditions had been carried out hj setting aside an 
 equal annual instalment in respect of each of the amounts of 
 loan and accumulating them in four separate sinking funds to 
 repay the several portions of the loan at the end of 5, 15, 29, 
 and 45 years respectively. But as the individual sinking funds 
 mature at different dates each annual instalment must be 
 reduced to its present value. The sum of such present values 
 represents the amount of money noAv required to purchase an 
 equivalent annuity or annual instalment for the equated period 
 of 25 years. For purposes of the comparison to be made in 
 Chapters XXXIV and XXXV it is necessary to know the 
 individual instalments to be set aside during the whole of the 
 above periods, and tlie annual instalment as well as its present 
 value will therefore be shown in each case. The calculations 
 are all similar to others which have been previously worked out 
 so that it is not necessary to show the actual working as in 
 earlier chapters. 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 395 
 
 TABLE XXXII. C. 
 
 The equation of the I'EKIOD of RErAYMENT OF LOANS, 
 
 repayable at various dates, wliicli are required to be re- 
 deemed on one uniform date. 
 
 Loan of £56,000, authorised for outlays of varying cliaracter, 
 each having a different life or period of continuing utility, 
 and a consequent period of repayment. 
 
 Eate of accumulation, 3 per cent. 
 
 Annual instalments required under the original conditions. 
 
 Equated period for a loan for public works consisting of outlay 
 having varying lives or periods of continuing utility. 
 
 Period 
 
 Nature of allowed for 
 
 outlay. repayment. Details 
 
 Class A 45 years — 
 
 „ B 29 „ - 
 
 ,, C 15 „ — 
 
 .. D 5 ,. — 
 
 Amount of loan Present value of 
 
 authorised. Annual Loan Future 
 
 instalment repayable at annual 
 
 Total. to repay loan, end of period. instalments. 
 
 10,000 107-85 2644-39 264439 
 
 20,000 442-29 8486-93 8486-93 
 
 24,000 1290-40 1540469 15404-69 
 
 2,000 376-71 1725-22 1725-22 
 
 Total 45 years — 56,000 2217-25 28261-23 28261-23 
 
 The present values in the above table are the present values 
 both of the amounts of loan repayable at the end of the 
 respective periods and also of the corresponding sinking fund 
 instalments, since the instalments, if accumulated, will, at the 
 end of the respective periods, amount to the respective loans. 
 
 The whole of the loans, although repayable at the end of 
 successive periods, have now been reduced to a common 
 measure, namely, a " present value " of £28261-23, which 
 represents the amount for which the various sinking fund 
 obligations might be redeemed at the present time, and upon 
 which the calculation of the true equated period will be based. 
 
 The folloAving argument is summarised in Table XXXII. D., 
 which may be referred to with advantage : — 
 
 If the arithmetical calculation of the equated period of 25 
 years be correct this sum of £28261-23 should in 25 years, at 
 3 per cent., amount to £56,000, and the annuity which it will 
 purchase (or the sinking fund instalment) should also amount 
 to £56,000 in that period. This, however, is not the case. 
 
396 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 It may be found by calculation that i>28261"23 will in 25 years, 
 at 3 per cent., amount to £591T2"75, or an excess of £3172'75. 
 And it may also be ascertained that the annuity or annual 
 instalment, which will amount to £591 72' 75 in 25 years, at 
 3 per cent., and of which £28261'23 is the present value, is 
 £1622"98 per annum. Since £28261'23 is also the present value 
 of the four annual instalments required to provide the com- 
 ponent parts of the loan of £56,000 at the end of the respective 
 periods of 5, 15, 29 and 45 years, it is obvious that the error 
 lies in the number of years in the equated period, as found by 
 the arithmetical method. 
 
 The next step is to calculate the actual annual instalment 
 (and also the present value of the instalment) required to repay 
 £56,000 in 25 years, the equated period as found by the 
 arithmetical method, but which there is reason to suspect is in 
 excess of the true period. It will be found, on making the 
 calculation, that the annual instalment to repay £56,000 in 
 25 years is £153595, and its present value £2674590 (which is 
 also the present value of £56,000 due at the end of 25 years). 
 This annual instalment of £1535'95 cannot, of course, be 
 compared with the four sinking fund instalments, amounting 
 together to £221725, to be accumulated for the original periods 
 because they are all for different numbers of years, but it has 
 been ascertained that they are equivalent to an annual instal- 
 ment of £162298 to be set aside for 25 years, and accumulated 
 at 3 per cent. It is therefore possible to compare the two 
 annual instalments of £153595 and £1622"98, and the result is 
 to prove that the arithmetically equated period gives an annual 
 instalment which is less by £87 03 than the exact equivalent of 
 the original instalments. In other words, the arithmetically 
 equated period is too long. The following table (XXXII. D.) 
 shows the above conclusions : — 
 
 TABLE XXXII. D. 
 
 The True or Mathematical Method of Finding the Equated 
 Period of Repayment, 
 
 Showing the annual instalments and their present values under 
 
 (1) the original conditions; 
 
 (2) the arithmetically equated period ; 
 
 (3) the true or mathematically equated period. 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 397 
 
 Sinking fund instalment ami 
 present value thereof for 
 
 Original periods of 
 repaymejit 
 
 Amount 
 of loan. 
 
 10,000 
 
 20,000 
 
 24,000 
 
 2,000 
 
 Sinking fund 
 instalment 
 Number per 
 
 ofj'ears. annum. 
 
 Present 
 
 value of 
 
 loan or 
 
 sinking fund 
 
 instalment. 
 
 45 107-85 2644-39 
 
 29 442-29 8486-93 
 
 15 1290-40 15404-69 
 
 5 376-71 1725-22 
 
 
 56,000 
 
 
 2217-25 
 
 28261-23 
 
 25 years, the equated 
 period as found by 
 the arithmetical 
 
 
 
 
 
 method : — 
 Amount of loan 
 
 5600000 
 
 25 
 
 1535-95 
 
 26745-90 
 
 Amovmt which will be 
 
 
 
 
 
 provided at end of 
 25th year 
 
 59172-75 
 
 25 
 
 1622-98 
 
 28261-23 
 
 Surplus 
 
 3172-75 
 
 25 
 
 87-03 
 
 1515-33 
 
 24 years : — 
 
 Amount of loan ... 
 
 56,000 
 
 24 
 
 1626-65 
 
 27548-28 
 
 23 years : — 
 
 Amount of loan 
 
 56,000 
 
 23 
 
 1725-58 
 
 28374-73 
 
 Amount which will be 
 
 
 
 
 
 provided at end of 
 23rd year 
 
 55,776 
 
 23 
 
 1718-68 
 
 28261-23 
 
 Deficiency 
 
 224 
 
 23 
 
 6-90 
 
 113-50 
 
 Having ascertained the exact error in the annual instalment 
 under the arithmetical method of equation the error in the 
 equated period itself may now be found. It has already been 
 ascertained that at 3 per cent. £26745-90 will amount to £56,000 
 in 25 years, and the problem is to ascertain in how many years 
 £28261-23 will amount to £56,000. As the present value of 
 the four original instalments, namely, £28261*23, is greater 
 than £26745-90, which is the present value of £56,000, it will 
 amount to £56,000 in a smaller number of years than 25. The 
 exact number of years may be ascertained by using the formula 
 relating to Table I, in standard calculation form, No. 1, to find 
 the amount of £1 in any number of years, but this will give 
 a result consisting of a number and a fractiou. In cases such 
 
398 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 as tlie present the exact fractiou of the year is required only 
 for the purpose of fixing- the nearest number of whole years, 
 so that the problem will work out in practice. The method of 
 finding the number of years by using the formula relating to 
 Table I is as follows, and may be compared with the standard 
 calculation form for the purpose given in Chapter X. 
 
 STATEMENT XXXII. E. 
 
 Required the number of years in which £28261-23 will amount 
 to £56,000 at 3 per cent. 
 
 By formula and logs. 
 
 A = PEN 56,000 = 28261-23 x 103N. 
 
 Log. amount at end of period ... 56,000 4-7481880 
 
 deduct Log. present value ... 2826r23 4-4511911 
 
 = Loo- RN ... 0-2969969 
 
 divide bv Log. E= 1-03 ... 00128372 
 
 To divide one Log. by another find 
 
 the(Logs. of the above Logs, as if 
 
 they were actual numbers, viz. : 
 
 Log. 2969969 = 6-4727516 
 
 Los-. 128372 = 5-1083703 
 
 1-3642813 
 
 which is the Log. of the number of years, viz. ... 23-136 
 
 In order to avoid the necessity of dividing one log by 
 another, the exact number of years may be ascertained by 
 means of Thoman's Table giving the logs of R^, at 3 per cent., 
 for various years, as follows: — 
 
 Proceed as in the above Statement by deducting 
 the log of the present value of the annual 
 instalments under the original conditions, 
 from the log of the amount of loan repay- 
 able at the end of the period. The remain- 
 der is the log of RN, from wliich the value 
 of N, mav be obtained. In the above case, 
 the loc. of RN is 0-2969969 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 399 
 
 On referring to Thoman's Table, tlie nearest logs of E^, above 
 and below this, are found to be as follows : — 
 
 at ;} per cent., log RN 24 years 0-30809;J4 
 
 23 years 0-2952562 
 
 a difference of 0-0128372 
 
 The next step is to find the difference between the log. of H^, 
 as found in the above calculation in Statement XXXII. E., and 
 from which it is required to find the value of N ; and the lower 
 of the above logs in Thoman's Table as follows: — 
 
 log of EN^ ill calculation, as above 0-2969969 
 
 log of EN 23 years, by Thonian 02952562 
 
 a difference of 0-0017407 
 
 and the fraction of a year above 23 years is : — 
 
 — or 0- 135598 as may be found by logs. 
 
 128372 
 
 The number of years therefore is 23-136, and agrees with the 
 calculation in Table XXXII. E., made by means of the formula. 
 Another method of making the calculation, after having 
 found the above difference of 0-0128372 in the logs, is to refer 
 to the tables of differences given in the margin of the ordinary 
 log tables, and under 128 the following amounts will be found : 
 
 •10 =0-12800 
 •07 =0-00900 
 •004 = 000051 
 
 174 = 013751 of 1 year, 
 
 which differs from the previous result by less than 1 day. 
 
 Summary of the True or Mathematical Method. In 
 order to ascertain the number of years in the true equated 
 period it is advisable to find, first, the approximate number 
 of years by the arithmetical method, in this case 25 years, and 
 then to find by calculation in the manner already described, 
 and shown in Table XXXII. C, the present value of the several 
 annual instalments under the original conditions before 
 equation ; in the present instance, £28261-23. The next step is 
 
400 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 to find the amount of loan, £59172"T5, wliicli will be provided 
 by the accumulation, at Ibe estimated rate, of the above present 
 value, for a number of years (25) equal to the equated period, 
 as ascertained by the arithmetical method. The amount of 
 loan which will be thereby provided should then be compared 
 with the actual amount of the loan. As a general rule the 
 amount of loan Avhich will be provided by the accumulation of 
 the present value of the annual instalments under tlie original 
 conditions before equation at the end of the equated period, 
 will be greater than the amount of the loan, and will denote 
 that the equated period as found by the arithmetical method 
 is in excess of the true equated period. 
 
 The enquiry is therefore confined to the present value of the 
 actual loan, at the estimated rate of accumulation, for periods 
 of years less than the number of years (25) as found by the 
 arithmetical method. 
 
 Eeference is nest made to the tables of compound interest in 
 order to ascertain the present value of the loan at the estimated 
 rate of accumulation for periods less than the arithmetically 
 equated period. 
 
 Reverting to the present example, a period of 24 years will 
 first be taken, and it will be found by calculation on standard 
 calculation form, No. 2, that the present value of £56,000 due 
 at the end of 24 years is £27548'28, requiring an aunual instal- 
 ment (as may be found by standard calculation form, No. 8x) of 
 £1626-65. 
 
 The above present value, £27548"28, as compared with 
 £2826r23, the actual present value of the original annual 
 instalments, before equation, is still insufficient, and a period 
 of 23 years is adopted. Similar calculations will show that 
 the present value of £56,000 due at the end of 23 years, is 
 £28374*73, which is very nearly correct. And therefore 23 years 
 is adopted as the nearest to the true equated period. The 
 future annual instalment to be spread equally over the equated 
 period may now be ascertained by calculation on standard form 
 3x, and will be found to be £1725-58. 
 
 The only conchislon which may properly be drawn from 
 the above facts, is, that an annual instalment of £1725"58 to be 
 accumulated for 23 years at 3 per cent., is, within a small limit 
 of error, the true mathematical equivalent of the four annual 
 instalments under the original conditions, amouutiug together 
 to £2217-25, as shoAvn in Table XXXII. C, to be accumulated 
 for the respective periods shown in that table. It has nothing 
 whatever to do with tlie incidence of Ibe bnrden upon the 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 401 
 
 revenue or rate accounts of the equated period, wliicli will be 
 fully considered in a later chapter. 
 
 The correct figures as to the equated period of repayment 
 of the above loan are therefore as follows : — 
 
 Amount of loan repayable at the end of 23 years £56,000 
 Present value thereof £283T4"73 
 
 Annual instalment of which £28374'73 is the 
 present value, and which will amount to 
 £56,000 in 23 years at 3 per cent 
 
 £1725-58 
 
 Owing to the fact that the equated period is fixed at the nearest 
 whole number of 23 years, instead of 23" 136 years, as shown in 
 Statement XXXII. E., the annual instalment of £1725-58 is 
 larger than the instalment £1718-68, which is the equivalent 
 of the present value £28261-23, of the original instalments 
 shown in Table XXXII. C. The following table shows the 
 error involved bv taking the nearest whole number of years : — 
 
 23 years 3 per cent, 
 based upon : — 
 
 Actual amount of loan 
 
 Actual j^resent value 
 
 of the original in- 
 
 Capital 
 
 Present 
 value. 
 
 Annual 
 instalment. 
 
 stalments 
 
 £56,000 £28374-73 £1725-58 
 
 £55,776 £28261-23 £1718-68 
 
 Difference 
 
 £224 
 
 £113-50 
 
 £6-90 
 
 The above table shows that £28261-23 will not amount to 
 £56,000 in 23 years but only to £55,776, requiring an annual 
 instalment of £1718-68, consequently it is not possible to arrive 
 at anything nearer than an approximation of the period. The 
 annual instalment to be set aside for 23 years corresponding to 
 the present value of £28261*23, namely £1718*68, is less by 
 £6-90 than the instalment required to repay £56,000, and would 
 fall short of repaying the loan by £224 at the end of 23 years. 
 The method is an approximation only and in actual practice 
 the arithmetical method would give a number of years contain- 
 ing a fraction, but the result is sufficientlv correct if the nearest 
 number of even years be taken. 
 
 The effect of adopting an equated period of 25 years, as 
 shown by the arithmetical method, instead of 23 years as shown 
 
 AB 
 
402 REPAYMICNT OF LOCAL AND OTHER LOANS 
 
 by the true equated method, may be seen from an inspection 
 of Table XXXII. D. It may be taken as a general rule that 
 the arithmetical method gives the longer repayment period, 
 and relieves the revenue or rate accounts of the equated period 
 as compared with the true method of equation which should 
 always be used when it is desired to accelerate the repayment 
 of the loan, or when extreme accuracy is required. 
 
 Further Proof. The previous example of the equation of 
 the period of repayment of a loan of £56,000 is not a case 
 occurring in actual practice. The amounts of outlay composing 
 the loan as well as the periods of repayment are all assumed, 
 and the problem has been treated purely from the theoretical 
 standpoint in order to show the difference between the 
 arithmetical and true methods of finding the e(|uated period. 
 The basis of the method there adopted is to ascertain, first, 
 the annual instalments required in respect of each part 
 of the loan and then to find the present value of such instal- 
 ments. The same present values may be obtained in one 
 operation by finding the present values of the several amounts 
 of loan repayable at the ends of the respective periods, where it 
 is not necessary to know the actual instalments, as was the case 
 in the foregoing example. 
 
 A further example will now be given, using figures occurring 
 in actual practice, but with a shorter period of ultimate repay- 
 ment than 45 years, and the results will be compared with the 
 previous example. The calculation will be made by the shortest 
 possible method. The actual example may be found in the 
 report by a Select Committee of the House of Commons upon 
 the Repayment of Loans by Local Authorities (1902), page 261. 
 This example of an equated period Avas put in evidence by the 
 Assistant Secretary of the Local Government Board to illustrate 
 the method adopted by the lioard in order to arrive at the 
 average period to be granted for the repayment of a loan to be 
 expended upon a gas undertaking where the component parts of 
 the outlay have a variable probable duration and continuing 
 utility. 
 
 The following table shows, in the first four columns, the 
 nature of the outlay, the period allowed for repayment in 
 respect of each, and the amount of loan to be expended in each 
 case. The fourth column shows the component parts of the 
 loan in respect of which the same period is allowed. The 
 details arc taken from the appendix to the above report. In 
 ordcT- to avoid repetition tlie table also contains the present 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 403 
 
 values of the component parts of tlie loan as found by 
 calculation. The annual instalments are not shown, because in 
 this case they do not enter into the calculation. This table may 
 be compared with Table XXXII. C. 
 
 TABLE XXXII. F. 
 
 The Equation of the Period of Repayment of Loans 
 repayable at various dates, which are required to be 
 redeemed on one uniform date. 
 
 Loan of £9105, authorised for outlays of varying character, 
 each having a different life, or period of continuing utility, 
 and a consequent period of repayment. 
 
 Rate of accumulation, 3 per cent. 
 
 Equated period for a loan for Gas-works purposes. 
 
 
 
 
 
 
 
 Present value of 
 
 Nature of outlay. 
 
 Period 
 allowed 
 
 for 
 repayment. 
 
 Amount of loan 
 authorised. 
 
 Details. Total. 
 
 Annual 
 
 instalment 
 
 to repay 
 
 loan. 
 
 Loan 
 repayable at 
 end of period. 
 
 Future 
 annual 
 
 instalments. 
 
 Buildings 
 
 30 
 
 years 
 
 250000 
 
 
 .siii 
 
 
 
 Mains 
 
 
 jj 
 
 124500 
 
 
 
 
 
 Gasometer 
 
 
 35 
 
 150000 
 
 
 lof 1 
 1 is 
 )f th 
 nt V 
 
 
 
 Condensers 
 
 
 jj 
 
 530-00 
 
 
 2.2^1 . 
 ^ ^ 3 *-< -*^ 
 
 
 
 
 30 
 
 years 
 
 
 57T500 
 
 
 2379-22 
 
 2379-22 
 
 
 
 
 Purifiers 
 
 20 
 
 years 
 
 
 100000 
 
 ^ 3 r/; ^ 
 
 553-67 
 
 553-67 
 
 Benches 
 
 15 
 
 5J 
 
 
 120000 
 
 ° " -^-3 
 
 770-23 
 
 770-23 
 
 Meters 
 
 10 
 
 5> 
 
 
 530-00 
 
 this 
 true 
 1 the 
 ad c 
 ,nnu 
 
 394-37 
 
 394-37 
 
 Retorts 
 
 2 
 
 )J 
 
 
 600-00 
 
 In 
 
 the 
 upon 
 inste 
 the a 
 
 565-55 
 
 565-55 
 
 
 30 
 
 years 
 
 
 9105-00 
 
 
 4663-04 
 
 4663-04 
 
 The report shows also the arithmetical method adopted to 
 arrive at the equated period of repayment of the total loan 
 authorised, and this method corresponds exactly with the 
 method laid down in the books on arithmetic and illustrated 
 by Table XXXII. B. The actual working is given in the 
 report and may be summarised as folloAvs : — 
 
404 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE XXXII. G. 
 
 The Arithmetical Method of Finding the Eqlated Peeiod 
 OF Eepayment of the Lo.\n. 
 
 Nature of outlay. 
 
 Amount 
 
 of loan 
 
 authorised. 
 
 Prescribed 
 periods. 
 
 Product of amount 
 of loan multiijlied 
 
 by 
 number of years. 
 
 Buildings, etc. 
 
 5,775 
 
 30 years 
 
 173,250 
 
 Purifiers 
 
 1,000 
 
 20 „ 
 
 20,000 
 
 Benches 
 
 1,200 
 
 15 ,, 
 
 18,000 
 
 Meters 
 
 530 
 
 10 „ 
 
 5,300 
 
 Retorts 
 
 600 
 
 
 1,200 
 
 Equated period : — 
 
 217750 
 9105 
 
 9,105 
 
 = 23-915, or 24 years. 
 
 217,750 
 
 The True or Mathematical Method of Finding the 
 Equated Period of Repayment. As already stated, the 
 method about to be described differs slightly from that 
 adopted in the previous example, and is shorter. The first step 
 is to find by calculation on standard calculation form, Xo. 2, 
 the present values of the component parts of the loan for the 
 respective periods allowed. These present values are shown in 
 the sixth column in Table XXXII. F., and amount together to 
 £4663'04. The next step is to find the number of years in 
 which £4463-04 will amount to the loan of £9,105, at 3 per 
 cent., in order to replace a gradual repayment of the component 
 parts of the loan by a simultaneous repayment of the whole. 
 In the previous example, three methods are described of finding 
 the number of years, one being by direct calculation by means 
 of the formula relating to the amount of £1 per annum in 
 standard calculation form, Xo. 3, which is illustrated by 
 Statement XXXII. E. The second method of finding the 
 number of years is by means of Thoman's table-, and is fully 
 described in the previous example. The third method is by 
 trial and error, based upon the approximate- ininiber of years 
 in the equated period found by the arithmetical method, 
 and this method will be applied to the present instance. T'sing 
 standard calculation form Xo. 1, it may be found that £4663'04, 
 accumulated at 3 per cent., will amount to: — 
 
THE EQUATION OF THE PERIOD OF REPAYMENT 405 
 
 in 24 years £9479-00 
 
 in 23 years £9202-90 
 
 in 22 years £8934-87 
 
 as compared with the equated period as found by 
 the arithmetical method : — 
 
 in 24 years £9105-00 
 
 It is obvious that the true period is nearer to 23 years than 
 to 22 years. If it be calculated exactly by either of the 
 methods used in the previous example it will be found to be 
 22-638 years; and therefore 23 years should be adopted in 
 practice instead of 24 years as found by the arithmetical 
 method above described. The actual difference between the 
 periods found by the two methods is 1-277 years. In the 
 previous example the actual difference was 1-864 years, but in 
 that case the longer repayment period was assumed to be 45 
 years, whereas in the present instance it is 30 years only. There 
 is not any common ratio existing between the original and the 
 equated periods, or between the two equated periods as found 
 by the arithmetical and true methods. The number of years 
 in the equated period depends upon the interaction of the 
 component parts of the loan and the respective periods pre- 
 scribed for their repayment. 
 
 Having found the true equated period in the above manner 
 the enquiry strictly comes to an end, but if the annual instal- 
 ment is required to be spread equally over the period it may 
 be found in the usual manner on standard calculation form 
 No. 3x. So far as the lender is concerned this is quite equit- 
 able, but having regard to the varying life of the assets created 
 out of the loan the question of the annual charges to revenue 
 or rate during the period becomes important and will be fully 
 considered in the following chapters. 
 
Section VIII. 
 
 The Equation of the Incidence of 
 Taxation. 
 
409 
 
 GHAPTEE XXXIII. 
 
 THE EQUATIOX OF THE IXCIDEXCE OF TAXATION. 
 
 Comparison of the total annual loan charges to revenue or 
 rate, before and after the equation of the period of 
 repayment, showing the unequal incidence of taxation 
 if the annual instalment and interest upon the total 
 loan be spread equally over the equated period. 
 
 The subject of enquiry in the previous chapter is the correct 
 method of finding tlie equated date of repayment of several 
 loans repayable at varying- dates, and the result of such enquiry 
 is to show that the generally adopted arithmetical method is 
 wrong in principle seeing that it tends to prolong the period. 
 Having found the equated date the next step is to ascertain 
 the annual instalments to be charged to revenue or rate account 
 during the equated period in substitution for the annual instal- 
 ments required under the original conditions before equation. 
 The present practice is to regard the matter purely from the 
 point of view of the loan holder and to set aside an equal 
 annual instalment to be spread over the Avhole of the equated 
 period without any regard to the incidence of taxation or the 
 life of the asset created out of the loan. Seeing, however, that 
 the annual instalments are accumulated in the sinking fund 
 and are not repaid to the lender until the end of the period it 
 is immaterial to him how the annual instalments are distributed 
 over the revenue or rate accounts of the equated period. On 
 the contrary it is a matter of concern to the ratepayer that the 
 annual contributions out of revenue or rate are borne equitaldy 
 by successive years, and this question will now be considered. 
 The permanent character of the security for local loans is shown 
 by the preferential nature of the redemption of part of the 
 loan out of the sinking fund before maturity. In the case of 
 financial and commercial undertakings any such redemptions 
 are made pro rata or by some method in which each loanholder 
 has an equal chance. . 
 
 The effect of the generally adopted method of equation of 
 the period of repayment is to reduce the annual instalment 
 
4IO REPAYMENT OF LOCAL AND OTHER LOANS 
 
 during the earlier years of the equated repayment period and 
 thus relieve the revenue or rate account of those years. This 
 may at hrst sight appear strange, when it is borne in mind that 
 under an equation of the period the total loan is repaid at an 
 earlier date although the mathematical result may be exactly 
 equal. 
 
 This will be found to be the case on referring to Table 
 XXXII. D. in Chapter XXXII, giving the annual instalments 
 required to repay a loan of £56,000 in an equated period of 
 2o years in substitution for periods of 5, 15, 29 and 45 years. 
 
 The following table (XXXIII. A.) shows the annual instal- 
 ments to be set aside, dividing the original periods into five, at 
 the end of four of which, part of the loan would have been 
 repaid, whereas the total loan is repayable at the end of the 
 23rd year under the equated method : — 
 
 TABLE XXXIII. A. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of the annual instalments under (1) the original conditions, 
 and (2) after equation where such annual instalments are 
 spread equally over the equated period. 
 
 
 (1) Original 
 periods. 
 
 (•2) Equated 
 period. 
 
 Equated period 
 as coni])ared with 
 original periods. 
 
 Periods of 
 equal incidence. 
 
 No. of 
 years. 
 
 Sinking fund 
 instalments. 
 
 No. of 
 years. 
 
 Sinking fund 
 instalments. 
 
 Increase. 
 
 Decrease. 
 
 5 years 
 
 5 
 
 2217-25 
 
 5 
 
 1725-58 
 
 — 
 
 491-67 
 
 10 years 
 
 10 
 
 1840-54 
 
 10 
 
 1725-58 
 
 — 
 
 114-96 
 
 8 years 
 
 8 
 
 550-14 
 
 8 
 
 1725-58 
 
 1175-44 
 
 — 
 
 6 years 
 
 6 
 
 550-14 
 
 — 
 
 — 
 
 — 
 
 55014 
 
 16 years 
 
 16 
 
 107-85 
 
 — 
 
 — 
 
 — 
 
 107-85 
 
 45 
 
 23 
 
 The above table shows that during the first five years of the 
 equated period the annual instalment is reduced by £491-67, 
 and that during the second period of 10 years there is a similar 
 
THE EQUATION OF THE INCIDENCE OF TAXATION 411 
 
 aunuai reductiou oi <£li4 y6. itie heaviest cliarge falls upon 
 the third and huai period of eight years, which is part of an 
 original period of i4 years. Uuring tiiis period tlie annual 
 instalment is greater by i;il75"44 than the corresponding 
 annual instalment under the original conditions before equa- 
 tion. This large increase in the annual instalment is due to 
 the fact that under the original conditions, before equation, 
 £26,000 of loan would have been repaid by the end of the 15th 
 year, being the end of the second portion both of the equated 
 and original periods. This amount of loan, having a short 
 period of repayment, naturally required a larger annual instal- 
 ment than the remaining loans having longer periods of repay- 
 ment. After the final repayment of the loan, at the end of the 
 equated period of 23 years, by means of the equal annual 
 equated instalment of £1725'58, the revenue or rate account is 
 relieved of all contributions both in respect of the annual 
 instalment and interest upon the loan. 
 
 As already pointed out, the actual figures in individual cases 
 will vary in accordance with the amounts of the respective 
 loans and the length of the various periods allowed for repay- 
 ment, but the generality of equations will follow the main 
 features here outlined. The results of an equation of the period 
 of repayment may be summarised as follows : — As regards the 
 annual sinking fund instalment to be charged to revenue or 
 rate account, the earlier and later years of the original repay- 
 ment period will be relieved and the resulting burden thrown 
 upon the middle portion of the original repayment period, 
 which is the final part of the amended equated period. This 
 relief will of course apply in full to that part of the original 
 repayment period beyond the equated period, seeing that the 
 whole of the loan will then have been repaid. 
 
 The following table, XXXIII. B., shows the result of the 
 equation of the period of repayment, as regards the interest 
 upon the loan, chargeable against the revenue or rate account 
 of each year of the equated period as compared with the corres- 
 ponding annual interest charges under the original conditions, 
 before equation. Under the original conditions the loan would 
 have been gradually repaid, thereby reducing the annual 
 interest charges against the revenue or rate accounts of subse- 
 quent years, but after the equation of the period of repayment, 
 the revenue or rate account of each year of the equated period 
 is charged with interest upon the total amount of the loan. 
 The equation of the annual charge for interest upon the loan 
 will be fully considered in Chapter XXXV. 
 
412 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 TABLE XXXIII. B. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of interest upon the loan under (1) the original conditions, 
 and (2) after equation. 
 
 Periods of equal 
 incidence. 
 
 (1) Original 
 periods. 
 
 Loan. Interest. 
 
 (2) Equated 
 period. 
 
 Loan. Interest. 
 
 Equated period 
 as compared with 
 original periods. 
 
 Increase. Decrease 
 
 5 years 
 
 56000 
 
 1960 
 
 56000 
 
 1960 
 
 
 
 10 years 
 
 54000 
 
 1890 
 
 56000 
 
 1960 
 
 70 — 
 
 8 years 
 
 ; 50000 
 
 1050 
 
 56000 
 
 1960 
 
 910 — 
 
 6 years 
 
 ;{0000 
 
 1050 
 
 — 
 
 — 
 
 — 1050 
 
 16 years 
 
 10000 
 
 350 
 
 — 
 
 — 
 
 — 350 
 
 45 years 
 
 
 
 
 
 
 The foregoing tables show that under the generally adopted 
 equated method the relief during the first period of 5 years is 
 solely .in respect of the annual sinking fund instalment and 
 that the annual interest charges are unaltered owing to the 
 fact that no part of the loan is repayable, under the original 
 conditions, until the end of the fifth year. During the second 
 period of 10 years there is a decrease in the annual instalment, 
 but there is an increase in the amount of the annual interest 
 charges, because the repayment of £2000 of loan which, imder 
 the original conditions, would have been made at the end of 
 the fifth year, has by the equation of the period of repayment 
 been deferred until the end of the 23rd year. The third period 
 of (S years, being the final portion of the equated period, has to 
 bear an increased annual charge of £2085' 44, being an increase 
 in the annual instalment of £1175'44 in addition to an 
 increased annual interest charge of £910. 
 
 The explanation of the large increase in the total annual 
 burden imposed upon this period is, that it has to bear, not only 
 the relief to the earlier periods of 5 and 10 years, but also tlie 
 total relief to that part of the original repayment period which 
 is beyond the equated period. The whole of the foregoing 
 conclusions are shown in the following table : — 
 
THE EQUATION OF THE INCIDENCE OF TAXATION 413 
 
 TABLE XXXIII. C. 
 
 Loan of £56,000 (authorised for outlays of varying nature, 
 liaving- prescribed periods of repayment) the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Showing the variation in the total annual charges to revenue 
 or rate in respect of the sinking fund instalment and 
 interest upon the loan, under (1) the original conditions 
 and (2) after equation, where such annual instalments are 
 spread equally over such equated periods. 
 
 A Summary of Tables A and B above. 
 
 Periods of equal 
 incidence. 
 
 5 years 
 10 years 
 
 8 years 
 
 6 years 
 16 years 
 
 45 years 
 
 Sinking fund 
 instalment. 
 
 Annual interest 
 on loan. 
 
 Total charge to 
 revenue or rate. 
 
 Increase. Decrease. Increase. Decrease. Increase. Decrease. 
 
 — 491-67 _ _ — 491-67 
 
 — 114-96 7000 — — 44-96 
 1175-44 ^ 91000 — 2085-44 — 
 
 — 55014 — 1050-00 — 160014 
 
 — 107-85 — 350-00 — 457-85 
 
 These results are so remarkable that some enquiry may 
 profitably be made into the matter, not only as to the necessity 
 to make the equation, but also as to the effect of the equation 
 upon the incidence of taxation. The necessity to make an 
 equation of the period may arise in several ways. The most 
 important is in order to provide for the repayment of a loan 
 raised by the issue of stock redeemable on a fixed date where 
 the loan is authorised for works having varying lives or periods 
 of utility. In the case of a loan to provide for outlay of one 
 character only, or for different classes of outlay having the same 
 life or duration of utility, there is not any necessity to make 
 an equation, the calculation being a simple one. The governing 
 factor is the unequal life or duration of utility of the works 
 authorised, upon which are based the periods allowed for tlic 
 repayment of the component parts of the loan. 
 
 A further need for the equation of the period of repaymeut 
 arises on the consolidation of several loans repayable at various 
 dates. This may be part of a large financial scheme undertaken 
 
414 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 with the object of generally simplifying the finances of a local 
 authority or on the issue of stock to replace a number of small 
 loans borrowed for short periods. The issue of such a stock 
 avoids the necessity of reborrowing a large number of small 
 sums continually falling due, and gives a permanency to the 
 outstanding debt. It also considerably simplifies the sinking 
 fund book-keeping and renders much easier not only the 
 investment of the sinking fund but also the redemption of part 
 of the loan during the operation of the fund. Further, investors 
 prefer a stock of large amount which is quoted in the Stock 
 Exchange list and is readily saleable. Looking at the matter 
 from the investor's point of view, it is a coincidence perhaps 
 that the equated period generally found necessary is about 
 20 — 30 3^ears, which is as long as investors generally approve. 
 Both the shorter and longer repayment periods allowed for the 
 repayment of local debt are not suitable for permanent invest- 
 ment as a stock, and local authorities are thus obliged to rely 
 upon the small investor who causes much more administrative 
 work than the holders of a stock. The renewal or reborrowing 
 of small loans falls upon the officials of the local authority, 
 whereas the burden of any change in the ownership of a 
 registered stock is borne by the holder except the registration 
 of the transfer and the preparation of the new certificate. 
 
 The investor may therefore be eliminated from the enquiry 
 because if he is willing to accept payment on the equated date 
 the arithmetical or mathematical methods of ascertaining that 
 date both give a sufficiently near approximation. The 
 investor, except in a very academic way, is not concerned with 
 the annual charges for redemption of the loan. The effect of 
 the equation of the period upon the incidence of taxation 
 therefore becomes the principal subject of enquiry, and the 
 above table (XXXIII. C.) shows that there is a very wide differ- 
 ence as between the original and the equated periods in regard 
 to the burden imposed upon successive years or periods of years. 
 It is here advisable to recapitulate the principles governing the 
 method of fixing the original periods of repayment. The 
 predominant factor in fixing the proper repayment periods to 
 be alloM'ed in respect of each individual class of outlay is found 
 in the principle laid down in the Statutes and adopted in the 
 practice of Parliament and the Government departments, 
 namely, that all loans shall be repaid during the period of 
 utility or duration of the works in respect of which ihe loan was 
 borrowed. But a local authority has not any capital and can 
 only repay the loan by annual contributions out of rate or out 
 
THE EQUATION OF THE INCIDENCE OF TAXATION 415 
 
 of the profits of its revenue earning undertakings. It may be 
 contended that revenue earning undertakings should be treated 
 in a different manner to purely spending departments, such as a 
 sanitary, highway or education authority, where the annual 
 expenditure, both for current expenses and debt redemption 
 charges, is taken direct from the pockets of the community by 
 way of a rate. Here it is important to adjust the incidence of 
 taxation very accurately — and this is the object of the careful 
 scrutiny by Parliament and the Government departments of 
 the periods of repayment allowed. The effect of this scrutiny 
 is seen in the original repayment periods allowed which are 
 generally fixed at a number of years well within the life of the 
 works for which the loan is authorised. If these periods are 
 properly allowed and the sinking fund instalments are based 
 upon them there is an equitable incidence of the annual burden, 
 and the annual instalments may then properly be considered as 
 the equivalent of an annual charge for depreciation — thereby 
 carrying out the principle laid down in an earlier chapter of 
 making each ratepayer contribute annually his due proportion 
 of the cost of the benefits he receives each year, whether that 
 cost be paid for during the year or be spread over a series of 
 years. 
 
 But the equation of the period of repayment is purely a 
 financial operation, and relates solely to the date of repayment 
 of the loan without any regard to the effect of such equation 
 upon the annual charges to the community by way of rate. 
 The case is different with a commercial or financial undertaking 
 where the repayments of debt are made out of the general assets 
 of the concern and are not charged against the profit and loss 
 account except and in so far as the operations of each individual 
 year cause a loss of capital due to wear and tear of the asset. 
 The repayment of debt and the annual charge to revenue are 
 in the case of such undertakings kept severely separate and 
 distinct. In the case of a local authority the conditions are 
 the exact opposite. In the first place, there is a careful and 
 searching enquiry by Parliament and the Government depart- 
 ments, with the object of fixing the annual amounts to be 
 charged to the revenue or rate accounts of successive years in 
 respect of the repayment of the debt and the consequent charges 
 for interest. These total charges are in many cases regularly 
 met out of revenue or rate during a part of the period so 
 allowed, and it may then become necessary or advisable to 
 make an alteration in the date at which the loan shall be 
 repaid, and an equation of the period of repayment is made 
 
4i6 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 resulting in such a drastic rearrangement of tlie total annual 
 charges that the original careful calculations as to the life of 
 the asset are ignored and rendered valueless. Reverting to the 
 present example, it will be seen from Table XXXIII. C. that, 
 although, in consequence of the equation of the period, the final 
 repayment of the loan is expedited, there is actually a decrease 
 in the annual burden for the next 15 years and an absolute 
 relief from any burden whatever during the later years of the 
 original period which were, under the conditions existing at the 
 time the loan was authorised, charged with their due proportion. 
 And the whole of the added burden is imposed upon the final 
 years of the newly ascertained or equated period at a time 
 when probably the undertaking may have to incur outlay on 
 renewals. 
 
 If it at any time becomes necessary or advisable to expedite 
 
 the repayment of the loans the calculation of the equated 
 
 period, and the resulting amended annual instalment should be 
 
 made in such a manner as to impose a proportionate part of the 
 
 additional burden upon each year of the equated period of 
 
 repayment, instead of, as is the present practice, relieving both 
 
 the earlier and the later years of the original repayment period 
 
 at the expense of the middle portion of that period which is also 
 
 the final portion of the equated period. A method of doing this 
 
 as regards the annual instalment will be described in Chapter 
 
 XXXIY, and as regards the interest upon the loan in Chapter 
 
 XXXY. Fp to this point the general question of the repayment 
 
 of debt has been treated from an actuarial or mathematical 
 
 standpoint only, but the disturbing element now introduced by 
 
 the necessity to accelerate the final repayment of the loan and 
 
 to vary the dates of repayment, by substituting therefor a 
 
 common 'date for the repayment of the whole of the loan, 
 
 depends upon circumstances which are generally of a variable 
 
 and somewhat accidental nature. It is therefore necessary to 
 
 find an equitable practical method of restoring the original 
 
 status, namely, to charge the revenue or rate account of each 
 
 year with its due proportion of the annual burden of redemption 
 
 and interest charges. At this stage it becomes advisable to 
 
 differentiate between the annual charges in respect of the 
 
 redemption of debt and the interest payable upon the loan, 
 
 because where loans are repaid by means of a sinking fund. 
 
 interest is payable upon the total amount of the loan during the 
 
 whole of the redemption period of whatever duration, and 
 
 ceases entirely at the end of that ])eiiod. Any reduction 
 
 therefore in the period of redemption will corrosjinndingly 
 
THE EQUATION OF THE INCIDENCE OF TAXATION 417 
 
 reduce the period during which there is any charge whatever 
 in respect of interest upon the loan, and it will, in the first 
 instance, be assumed that it is perfectly equitable to ignore the 
 relief of later years in respect of interest upon the loan due to 
 an increased sinking fund burden imposed upon the earlier 
 years in consequence of the accelerated final extinction of the 
 debt. Confining the enquiry therefore solely to the sinking fund 
 instalment, annually charged to the revenue or rate account, it 
 is necessary to revert to the primary factor in the redemption 
 of debt, namely, the life or duration of continuing utility of the 
 asset created out of the loan. A broad line is here required to 
 be drawn between the two objects of the annual instalment, 
 namely, the repayment of the debt and the charge to revenue or 
 rate of the cost of the asset during its life or period of utility. 
 
 Under the original conditions before equation the total debt 
 would have been gradually repaid at the end of a series of 
 periods up to the 45th year, whereas under the amended 
 conditions the total loan will be repaid in one sum at the end 
 of the 23rd year or approximately in about one half the 
 number of years. The magnitude of the annual instalment to 
 be set aside and charged to revenue or rate in order to redeem a 
 given loan depends primarily upon the period allowed for its 
 redemption, which is based upon the life of the asset; and 
 therefore if the total period be reduced, the periods allowed for 
 the repayment of the component parts of the loan should be 
 correspondingly reduced. 
 
 A c 
 
4i8 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 CHAPTER XXXIV. 
 
 THE EQUATION OF THE INCIDENCE OF TAXATION 
 
 {Continued.) 
 
 THE ANNUAL INSTALMENT. 
 
 The various methods of adjusting the annual charges to 
 revenue or rate during the equated period in propor- 
 tion to the life or duration of continuing utility of 
 
 the asset created out of the loan, viz : by charging 
 
 the revenue or rate account of each year of the 
 equated period with the annual instalment chargeable 
 against each year, before equation. and in addition 
 thereto a supplementary annual instalment : 
 
 [a] to be spread equally over the equated period, or 
 
 (6) to be proportionate, year by year, to the annual 
 instalments before equation. 
 
 The previous argument will now be applied to tlie example 
 under review, namely, tlie repayment of a loan of £56,000, 
 authorised for large public works, the component parts of wliicli 
 have varying periods of continuing utility and consequent 
 prescribed periods of repayment. Under tlie original conditions 
 the repayment of the loan Avas spread over a period of 45 years, 
 but under the altered conditions it is required that the whole 
 of the loan shall be repaid at the end of an equated period of 
 23 years. In this chapter the correct method of spreading the 
 actual burden equitably over the equated period is the subject 
 of enquiry and not the method of finding the true equated 
 period which has been fully discussed in Chapter XXXII. 
 
 The object is to distribute the annual sinking fund burden 
 equitably over the rcHluced period of repnyment instend of 
 imposing an undue burden upon the final yeors of the equated 
 period which is the effect of the mefliod generally adopted, as 
 shown in Table XXXIII. C. 
 
 For this purpose it Avill be an advantage to tal)u]ate the 
 original conditions in the example considered in Chapter 
 
THE INCIDENCE OF TAXATION 419 
 
 XXXII, and to show also tlie proportionate amended periods of 
 repayment under the amended conditions due to the equation 
 of, or alteration in, the final period of repayment, as follows : — 
 
 TABLE XXXIV. A. 
 
 Loan of £56,000 (authorised for outlays of varying nature, 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of the annual instalments under (1) the original conditions, 
 and (2) after equation if such instalments are increased 
 in proportion to the reduction in the repayment period. 
 
 Rate of accumulation, 3 per cent. 
 
 Number of years allowed Annual sinking fund 
 
 for repayment of loan. instalments. 3 per cent. 
 
 Amount Equated 23 years 
 
 Class of of loan Original period of 45 year equated 
 
 outlay. authorised, conditions. 23 years. period. period. 
 
 A 10,000 45 23 107-85 308-14 
 
 B 20,000 29 14 442-29 1170-53 
 
 C 24,000 15 8 1290-40 2698-95 
 
 D 2,000 5 3 376-71 64706 
 
 56,000 2217-25 4824-68 
 
 The above annual instalments under the 23 year equated 
 period will repay the component parts of the loan at the end 
 of the respective reduced periods of repayment, and are not in 
 any way equated instalments. They are placed here only to 
 show the effect of reducing the period from 45 to 23 years. 
 They illustrate forcibly the wide difference between the financial 
 obligation to repay the loan and the anticipated life of the 
 various classes of outlay. Both these factors are distinct, but 
 the generally adopted method of fixing the future annual 
 instalment after ascertaining the equated period does not make 
 any such distinction but treats them as being equivalent. 
 
 It may be at once stated that it is not possible to adjust the 
 unequal burden of the annual instalment as ascertained by the 
 generally adopted method, by reducing each of the periods 
 allowed for the component parts of the outlay in proportion to 
 the reduction in the final period of repayment as shown in the 
 above table. There are so many disturbing factors and the 
 conditions are so widely altered by reducing the periods by 
 
420 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 one half that the results obtained show wider differences than 
 exist under the method now in use. The author has worked out 
 the problem in detail, but the results are too long to give in 
 full and would not be of any practical A^alue. 
 
 It is therefore necessary to adopt another line of enquiry 
 in order to find a method of determining the annual instalment 
 or instalments to be charged to revenue or rate to repay the 
 total loan in one sum at the end of the e(|uated period. So far 
 as the investor is concerned the equated period of repayment 
 already found is comparatively correct, but a more important 
 matter is the equated annual incidence of the burden upon each 
 year's revenue or rate account. So long as the period of repay- 
 ment and the life of the asset are the same the two methods 
 yield identical results, but any alteration in the term of repay- 
 ment makes an important difference in the annual charges to 
 revenue or rate in successive years. By the method now 
 generally adopted and previously described the annual instal- 
 ment is spread over the whole of the equated period of repay- 
 ment, and this is considered to be an equitable substitute for a 
 gradually decreasing charge which has been ascertained after 
 careful enquiry as to the life of the asset. 
 
 Very little consideration will show that this is very far from 
 being correct; and the result of the previous enquiry is to show 
 that although the effect of the equation is to reduce the period 
 of repayment and correspondingly increase the total charge for 
 redemption, yet there is actually a reduction in such annual 
 charge during the earlier years of the equated period and an 
 absolute relief from any charge whatever during that part of 
 the original period beyond the equated period. These reduc- 
 tions in the charsres against the earlier and later vears of the 
 original period involve a severe additional annual burden upon 
 the later years of the equated period. Although this inequality 
 is not fully appreciated yet its effect has been mentioned in 
 several places in the reports of the parliamentary committees 
 which have enquired into the finances of local authorities where 
 it is pointed out that under an equated method, loans for out- 
 lays for which short terms are generally allowed are not repaid 
 until the end of the longer equated period, and consequently 
 further borrowing powers ought not to be granted when the 
 asset is exhausted. 
 
 The result of the previo\is discussion of the subject is to 
 em])hasise the fact that as regards the annual charge to revenue 
 or rate the most important factor is the life of the asset, and it 
 may naturally be concluded that if any change is made in the 
 
THE INCIDENCE OF TAXATION 421 
 
 final period of repayment the amended annual instalment to be 
 charged to revenue or rate account should continue to bear as 
 near as possible an approximate ratio to the original charge, 
 instead of, as is the present practice, spreading the burden 
 equally over the equated period without any regard to the life 
 of the asset. 
 
 A method of doing this will now be fully described, taking 
 as an example the loan of £56,000 already used to illustrate the 
 previous remarks in Chapter XXXII upon the method of 
 finding the true equated period. In comparing the two 
 methods it is important to bear in mind that where the periods 
 are not equated the several sinking funds will mature at 
 successive dates, at each of which portions of the original loan 
 will be repaid out of such funds, whereas under the equated 
 method the whole loan is repayable on one date. Further, the 
 equated period covers the date of final repayment of one or more 
 of the component parts of the original loan. In the present 
 chapter the sinking fund instalment only will be considered, 
 leaving out of account for the moment the interest payable 
 upon the loan which, as already pointed out, ceases entirely at 
 an earlier date under the equated method, although the later 
 years of the equated period bear a larger interest charge than 
 they do under the original conditions. This is shown by 
 Table XXXIII. B. 
 
 The following method of adjusting the annual charge to 
 revenue or rate after an equation of the period is based upon 
 the Relative periods allowed for the repayment of the component 
 parts of the loan, as expressed by the sinking fund instalments 
 originally found requisite to repay the several portions of the 
 loan at the end of the respective periods prescribed. In order 
 to make the adjustment it is first requisite to ascertain the 
 amount which would have been in the fund if the original 
 instalments had been allowed to accumulate until the end of 
 the equated period of 23 years instead of repaying £2,000 at 
 the end of the 5th year and a further £24,000 at the end of the 
 15th year. The original annual instalments are as follows : — 
 
 for the first five years £2217-25 per annum. 
 
 for the next ten years 1840-54 
 
 for the final eight years 550-14 ,, 
 
 and the following table shows the amount which would be m 
 the fund at the end of the 2ord year under the above conditions. 
 
422 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 STATEMENT XXXIV. B. 
 
 Loan of £56,000 (as above). 
 
 Sliowing tlie amount which, will be in the sinking fund at the 
 end of the equated period of 23 years if the original annual 
 instalments as shown in Table XXXIII. A., are set aside 
 for the periods originally prescribed and no part of the 
 fund is applied in repaying the loan. 
 
 Interest at 3 per cent. 
 
 1. Amount of iJ2217'25 per annum for 5 
 
 years 
 
 Amount thereof at the end of a further 
 18 years 
 
 2. Amount of £1840"54 per annum for 10 
 
 years 
 
 Amount thereof at the end of a further 
 8 years 
 
 3. Amount of £550'14 j)er annum for 8 years 
 
 11772 
 
 21100 
 
 20040 
 
 26728 
 4892 
 
 Total amount in the fund at end of 23rd year 51660 
 
 being : — 
 
 Loan repayable at end of 5th year 
 
 accretions for 18 years 
 
 Loan repayable at end of 15th year ... 
 accretions for 8 years 
 
 Annual instalment of £550" 14 accumu- 
 lated for 23 years 
 
 Amount as above 
 
 2000 
 1404 
 
 24000 
 6403 
 
 3404 
 
 30403 
 17853 
 51660 
 
 The above amount of £51,660 which would have been iu the 
 fund if the original annual instalments had been added thereto 
 and allowed to accumulate for 23 years, represents that portion 
 of the loan of £56,000 which would have been repaid by means 
 of the annual charges to the revenue or rate accounts of the 
 23 years. These annual instalments are based upon the 
 respective repayment periods proper to be allowed for the 
 component parts of the outlay and may be accepted as fair and 
 ])r()per charges against each year's revenue or rate, irrespective 
 of the equated period allowed for tlie repayment of the loan. 
 
THE INCIDENCE OF TAXATION 423 
 
 The balance of loau thus unprovided for is arrived at as 
 follows : — 
 
 Total amount of loan £56000 
 
 Amount which will be provided by the accumula- 
 tion of the original annual instalments for 
 23 years £51660 
 
 Amount which will be unprovided for £4340 
 
 This amount represents the deficiency stated in terms of 
 the loan, by which the accumulated amount of the original 
 annual instalments, based upon the life of the asset, will be 
 insufficient to repay the loan at the end of the amended or 
 equated period, and this amount has to be provided by a 
 supplementary instalment or instalments to be set aside in 
 some way during the equated period of 23 years. To be per- 
 fectly accurate, the annual instalments should bear the same 
 ratio one to another as the original annual instalments, and 
 this should be done where the amounts of loan in question are 
 large, seeing that the greater portion of the original burden 
 should be borne by the earlier years. This will be referred to 
 again in a later part of this chapter where the proper mathe- 
 matical method of making the adjustment will be fully 
 described. For the present it will be assumed that it is 
 equitable to distribute the supplementary annual instalment 
 equally over the equated period of 23 years, and the problem 
 therefore is to find the annual sinking fund instalment to repay 
 a loan of £4340 in 23 years at 3 per cent. This supplementary 
 instalment, as may be found by standard calculation form 3x, 
 is £133'73 per annum, and the ultimate annual instalments to 
 be added to the fund during the successive portions of the 23 
 years are obtained by increasing each of the original annual 
 instalments, prior to equation, by this amount, as shown in the 
 following table : — 
 
424 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE XXXIV. C. 
 
 Loan of £56,000 (authorised for outlays of varying nature, 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of the annual instalments under (1) the equated method 
 generally adopted, and (2) in which the annual instalments 
 as originally ascertained are supplemented by an equal 
 additional instalment spread over the equated period. 
 
 'eriods of equal 
 incidence. 
 
 Original 
 annual 
 
 instalments. 
 
 Additional 
 
 annual 
 instalment. 
 
 Total 
 
 annual 
 
 instalments. 
 
 Annual 
 instalment 
 as equated. 
 
 5 years 
 
 2217-25 
 
 133-73 
 
 2350-98 
 
 1725-58 
 
 10 years 
 
 1840-54 
 
 133-73 
 
 1974-27 
 
 1725-58 
 
 8 years 
 
 550-14 
 
 133-73 
 
 683-87 
 
 1725-58 
 
 23 years 
 
 
 
 
 
 By the aid of the above figures a comparison will now be 
 made between the annual instalments obtained by the three 
 methods, namely : — 
 
 (1) Instalments payable during the original prescribed 
 
 periods, providing for the gradual repayment of the loan 
 at the end of 5, 15, 29, and 45 years, and which are based 
 upon the life of the asset. 
 
 (2) Instalments payable during the equated period only, 
 
 based upon an equal annual charge to the revenue or rate 
 account of each year, according to the method generally 
 adoj)ted. In this case the life of the asset is not taken 
 into account in fixing the annual burden. It is true 
 that the periods originally prescribed enter into the 
 arithmetical calculation of the equated period, but the 
 effect of this is lost by spreading the instalment equally 
 over the whole of the equated period so ascertained. 
 
 (3) Instalments payable during the equated period only, but 
 
 which are not equal throughout the period but are based 
 upon the life of the asset and are approximately pro- 
 portionate to the instalments before equation. 
 
 In the following table the instalments (2) and (3) are 
 compared with (1) those found requisite under the original 
 conditions, and the increase or decrease in the annual charge 
 is shown in respect of each period of equal incidence. It will 
 be noticed that the final 8 years of the equated period alone 
 bear any increased charge. 
 
THE INCIDENCE OF TAXATION 
 
 425 
 
 TABLE XXXIV. D. 
 
 Loan of £56,000 (authorised for outlays of varying nature, 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the annual charges to revenue or rate in respect 
 of the annual instalments : — 
 
 1. Based upon the original repayment periods. 
 
 2. The equated method generally adopted. 
 
 3. The annual instalments, as in Table XXXIV. C. 
 The following increased or decreased annual charges are com- 
 pared with the amounts in Column 1, 
 
 Original 
 periods. 
 
 Periods of equal Annual 
 Incidence. instalment. 
 
 Equated nietliod usually 
 adopted. 
 
 Equated metliod previously 
 described. 
 
 5 years 
 10 years 
 
 8 years 
 
 6 years 
 16 years 
 
 2217-25 
 
 1840-54 
 550-14 
 550-14 
 107-85 
 
 Annual 
 instalment. 
 
 1725-58 
 1725-58 
 1725-58 
 
 Increase+ 
 Decrease— 
 
 -491-67 
 -114-96 
 + 1175-44 
 -55014 
 -107-85 
 
 Annual 
 instalment. 
 
 2350-98 
 1974-27 
 
 683-87 
 
 Increase+ 
 Decrease — 
 
 + 133-73 
 + 133-73 
 + 133-73 
 -550-14 
 -107-85 
 
 45 years 
 
 The above table shows that the method just described, 
 although it concerns the annual instalment only, removes the 
 gross inequality which exists in the method generally adopted, 
 in which each year of the third period of 8 years is charged 
 with £1 175-44 per annum more than is the case under the 
 original conditions before equation, based upon the life of the 
 asset. In the method above described, not only is there a 
 decreasing annual charge due to the fact that the classes of 
 outlay with shorter periods of utility are written off in the 
 earlier years but the total relief to the final 22 years of the 
 original period (or the post equated period) is charged against 
 the whole of the equated period. This is almost as near an 
 equalisation of the original annual burden as can be made, and 
 removes the objection that under the generally adopted method 
 of equation the redemption of loans authorised for outlay in 
 respect of which only short periods are granted is unduly 
 delayed. With regard to the annual charges in respect of 
 interest upon the loan, there is not any variation from the 
 conditions previously shown under the generally adopted 
 
426 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 method of equation, and Table XXXIII. B., showing the com- 
 parison will still apply. 
 
 At this point it is interesting to compare the total annual 
 charges for sinking fund instalment and interest upon the loan 
 by means of the following table which may usefully be com- 
 pared with Table XXXIil. C. : — 
 
 TABLE XXXIV. E. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Showing the variations in the total annual charges to revenue 
 or rate under the method described in Table XXXIV. C, 
 as compared with the original annual instalments before 
 equation. 
 
 This table should be compared with Table XXXIIL C. 
 
 Periods of equal 
 incidence. 
 
 Sinking fund 
 instalment. 
 
 Increase. Decrease. 
 
 Interest on 
 loan. 
 
 Increase. Decrease. 
 
 Total chai 
 revenue oi 
 Net 
 Increase. 
 
 ge to 
 rate. 
 
 Net 
 Decrease. 
 
 5 years 
 
 133-73 
 
 — 
 
 — 
 
 133-73 
 
 — 
 
 10 years 
 
 133-73 
 
 — 
 
 70 — 
 
 203-73 
 
 
 
 8 years 
 
 133-73 
 
 — 
 
 910 — 
 
 1043-73 
 
 
 
 6 years 
 
 — 
 
 55014 
 
 — 1050 
 
 — 
 
 160014 
 
 16 years 
 
 
 107-85 
 
 — 350 
 
 
 457-85 
 
 45 years 
 
 
 The principal points to be noticed in the above table as 
 compared with Table XXXIIL C, which shows the difference 
 between the total annual loan charges under the original con- 
 ditions and under the generally adopted equated method, are 
 (1) the reduction in the excess additional charge during the 
 third period of 8 years from £2085-44 per annum to £1043-73 
 per annum due solely to the reduction in the sinking fund 
 instalment, and (2) the additional burden imposed upon the 
 first 15 years. As already stated it has been assumed that it 
 is perfectly equitable to consider interest upon the total loan 
 as a proper charge against revenue or rate during the later 
 years, although the increase in the annual interest is due to the 
 delay in the repayment of the loan for purely financial reasons. 
 The following statement shows the final repayment of the loan 
 by means of the instalments in Table XXXIV. C, ascertained 
 in the above manner. 
 
THE INCIDENCE OF TAXATION 427 
 
 STATEMENT XXXIV. E. 
 
 Loan of £56,000 (as above). 
 
 Showing the final repayment of tlie loan by the operation of 
 the sinking fund at the end of the equated period of 
 2o years, by setting aside the original annual instalments 
 based upon the life of the asset, and a further additional 
 instalment spread equally over the equated period. Such 
 instalments are shown in Table XXXI\ . C. 
 
 1. Amount of ii2o5098 per annum for 5 years 12482 
 Amount thereof at tiie end of a further 
 
 18 years — 21^49 
 
 2. Amount of £19T4'27 per annum for 10 
 
 years 22633 
 
 Amount thereof at the end of a further 
 
 8 years 28670 
 
 3. Amount of c£68o'87 per annum for 8 
 
 years 6081 
 
 Amount of loan 56000 
 
 The amended annual instalments to repay £4340, the 
 balance of loan unprovided for by the accumulation of the 
 original instalments before equation, should properly be dis- 
 tributed over the equated period in such a manner that they 
 will be proportionate to the original instalments, but they may 
 be spread equally over the equated period without any great 
 injustice being caused. Table XXXIV. C. shows that the 
 additional annual instalment under these conditions is £133" 73. 
 It is, however, necessary to point out the correct method, in 
 order that it may be applied to cases where the magnitude of the 
 loan renders it desirable to make an absolute equation of the 
 incidence of the sinking fund instalment, as well as of the 
 number of years over which the equated burden should be 
 ■spread. If the original periodically decreasing annual instal- 
 ments are set aside for the respective repayment periods, and 
 are allowed to accumulate until the end of the equated period 
 they will provide £51,660 of the original loan of £56,000, 
 leaving £4,340 to be provided by the accumulation of supple- 
 mentary annual instalments to be set aside for similar numbers 
 of years and allowed to accumulate for the same periods. The 
 
428 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 first step in tlie calculation is to divide the £4,340 in the Bauie 
 proportions as the £51,660 as follows : — 
 
 (1) 5 years £20,Q40 £168:j-6 
 
 (2) 10 years 26,728 2245-4 
 
 (3) 8 years 4,892 411-0 
 
 23 years £51,660 £4340 
 
 The component parts of the £4,340 represent the amounts 
 which will be in the fund at the end of the period due to (1) an 
 annual instalment to be set aside for 5 years and then accumu- 
 lated for a further 18 years, followed by (2) an annual instal- 
 ment to be set aside for the next 10 years and then accumulated 
 for a further 8 years, followed by (3) an annual instalment to 
 be set aside for the final 8 years at the end of which period the 
 fund will mature. This latter period of 8 years is the one 
 which bears the undue burden under the method of equation 
 generally adopted and which it is the object of the present 
 adjustment to remove. Having analysed the component parts 
 of the deficiency of £4,340, the respective instalments are 
 ascertained by working backwards. In the case of item (1) it 
 is required to ascertain the amount of an annuity for 5 years, 
 which amount if accumulated for a further 18 years will 
 provide £1683-6. The first step therefore is to find by standard 
 calculation form No. 2 the present value of £1683-6 due at the 
 end of 18 years at 3 per cent., and having done so to find by 
 standard calculation form No. 3x the annuity for 5 years which 
 will amount to this sum. The calculation may be made direct 
 bv Thoman's method as follows: — 
 
THE INCIDENCE OF TAXATION 429 
 
 CALCULATION XXXIV. G. 
 
 To fiud the annual instalment to be set aside and accumulated 
 for a given number of years, at the end of which period 
 the amount thereof will continue to accumulate for a 
 further specified period and will then amount to a given 
 sum. 
 
 Required, the annuity for 5 years which will amount to [the 
 present value of £1683-6 due at the end of a further 18 
 years] . 
 
 By Thoman's Tables and Logs. 
 
 First period 5 years. Second period 18 years. 
 
 Log of the given future sum 1683-60 3-2262330 
 
 deduct Log. RN^ 3 per cent. 18 years 0-2310T00 
 
 Loff of 988-92 2-9951630 
 
 -"to 
 
 add Log. ft", 3 per cent. 5 years ... 93391623 
 
 12-3343253 
 deduct Log. RN^ 3 per cent. 5 years + 10 10-0641861 
 
 Log of annuity required 2-2701392 
 
 Annual instalment required . . . £186-26 
 
 Note. This calculation may be compared with XYI. D. 1 
 and XXYII. C. 
 
 The second item may be ascertained in a similar manner, 
 but the third calculation consists merely of finding the annual 
 instalment, and it may be performed on standard form No. 3x. 
 It is not necessary to give the actual details of the calculations, 
 but merely to state the results in the following table which 
 shows the manner in which the above deficiency of £4,340 will 
 be provided at the end of the equated period. 
 
430 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 STATEMENT XXXIV. H. 
 
 Loan of £56,000 (as above). 
 
 Showing the supplementary annual instalments to be set aside 
 and added to the sinking fund during the equated period 
 of 23 years, to repay £4,340 of the original loan un- 
 provided by the original annual instalments added to the 
 fund. 
 
 The following supplementary annual instalments are 
 proportionate to the original annual instalments based 
 upon the life of the asset, and are not equal during the 
 whole of the repayment period, as was the case in Table 
 XXXIV. C, and Statement XXXIV. F. 
 
 1. Amount of £186-26 per annum for 5 
 
 years ..." 988'92 
 
 Amount thereof at the end of a furtber 
 
 18 years. Calculation XXXIV. G. 1683-60 
 
 2. Amount of £154-62 per annum for 10 
 
 years 177260 
 
 Amount thereof at the end of a further 
 
 8 years 2245-40 
 
 3. Amount of £4622 per annum for 8 years 41100 
 
 4340-00 
 
 The above annual instalments may be usefully compared 
 with those previously obtained where the supplementary annual 
 instalment of £133-73 is spread equally over the equated period, 
 as shown in Table XXXIV. C. The following table shows 
 the animal iiistnlmeiits under the present method: — 
 
THE INCIDENCE OF TAXATION 431 
 
 TABLE XXXIV. J. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Showing the annual charges to revenue or rate in respect of 
 the annual instalments under (1) the equated method 
 generally adopted, and (2) in which the annual instalments 
 as originally ascertained are supplemented by additional 
 annual instalments spread over the equated j^eriod in pro- 
 portion to the original periods allowed. 
 
 riods of equal 
 incidence. 
 
 Original 
 
 annual 
 
 instalment. 
 
 Additional 
 
 annual 
 instalment. 
 
 Total 
 
 annual 
 
 instalment. 
 
 Annual 
 instalment 
 as equated. 
 
 5 years 
 
 2217-25 
 
 186-26 
 
 2403-51 
 
 1725-58 
 
 10 years 
 
 1840-54 
 
 154-62 
 
 1995-16 
 
 1725-58 
 
 8 years 
 
 550-14 
 
 46-22 
 
 596-36 
 
 1725-58 
 
 23 years 
 
 It is not necessary to give details of the actual calculations, 
 but the following statement has been prepared in order to show 
 the final repayment of the loan at the end of the equated period 
 of 23 years, by means of the annual instalments in Table 
 XXXIV. J., thereby proving the accuracy of the above method. 
 
432 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 STATEMENT XXXIY. K. 
 
 Loan of £56,000 (as above). 
 
 Showing" the final repayment of the loan by the operation of 
 the sinking fnnd at the end of the equated period of 
 23 years by annual instalments spread over the equated 
 period, with due regard to the life of the asset instead of 
 being spread equally over such period. 
 
 Table XXXIY. J. 
 
 1. Amount of <£2403'51 per annum for 5 
 
 years 127606 
 
 Amount thereof at the end of a further 
 
 18 years — 21724 
 
 2. Amount of ±*1995'16 per annum for 10 
 
 years 22872'3 
 
 Amount thereof at the end of a further 
 
 8 years 28973 
 
 3. Amount of £59636 per annum for 8 
 years 
 
 5303 
 
 Amount of loan 56000 
 
 Four methods have now been shown by which the loan of 
 £56,000 may be repaid before and after the eqiuition of the 
 period, and ihe total annual loan charges for instalment and 
 interest, under each method, will uow be summarised. 
 
THE INCIDENCE OF TAXATION 
 
 433 
 
 TABLE XXXIV. L. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 having prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Comparison of the total annual charges to revenue or rate 
 during the several periods of equal incidence forming part 
 of the original extended period of repayment, and now 
 constituting the equated period, under the following 
 methods : — 
 
 1. The generally adopted method of equation in which 
 
 the instalment is spread equally over the equated 
 period. 
 
 2. The instalments originally calculated before equation 
 
 based upon the life of the asset. 
 
 3. The method in which the original annual instalments, 
 
 based upon the life of the asset, are set aside during 
 the equated period, and any deficiency is made good 
 by a supplementary annual instalment spread 
 equally over the equated period. 
 
 Table XXXIY. C. 
 
 4. The method in which the instalments during the 
 
 equated period are exactly proportional to the life 
 of the individual assets and to the original annual 
 instalments based thereon. Table XXXIY. J. 
 
 Annual instalments. 
 
 Period of equal 
 incidence. 
 
 5 years 
 
 10 years 
 
 (S vears 
 
 Tlie equated 
 method. 
 
 1725-58 
 
 1725-58 
 1725-58 
 
 Method 
 
 (2) 
 above 
 
 Table 
 XXXIII, A. 
 
 2217-25 
 
 1840-54 
 550-14 
 
 Method 
 
 (3) 
 above 
 
 Table 
 XXXIV. c. 
 
 2350-98 
 1974-27 
 
 683-87 
 
 Method 
 
 (4) 
 above 
 
 Table 
 XXXIV. J. 
 
 2403-51 
 
 199516 
 
 596-36 
 
 Interest upon the loan. 
 
 See Table XXXIII. B. 
 
 Periods of 
 equal incidence 
 
 The equated 
 method 
 
 Method 
 
 (2) 
 above. 
 
 Method 
 
 (3) 
 above. 
 
 Method 
 
 (4) 
 above. 
 
 5 years 
 
 1960 
 
 1960 
 
 1960 
 
 1960 
 
 10 years 
 
 1960 
 
 1890 
 
 1960 
 
 1960 
 
 8 years 
 
 1960 
 
 1050 
 
 1960 
 
 1960 
 
 AID 
 
434 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Total annual charges for instalment and interest on loan. 
 
 Periods of 
 
 equal 
 incidence. 
 
 5 years 
 
 Instalment 
 Interest 
 
 Total 
 
 Instalment 
 Interest 
 
 Total 
 
 Instalment 
 Interest 
 
 Total 
 
 Instalment 
 Interest 
 
 Total 
 
 Instalment 
 Interest 
 
 Total 
 
 1. 
 
 The equated 
 method. 
 
 1725-58 
 1960-00 
 
 Method 
 
 (2) 
 above. 
 
 2217-25 
 196000 
 
 Method 
 
 (3) 
 above. 
 
 2350-98 
 196000 
 
 Method 
 
 (4) 
 above. 
 
 2403-51 
 196000 
 
 
 3685-58 
 
 4177-25 
 
 4310-98 
 
 4363-51 
 
 years 
 
 1725-58 
 1960-00 
 
 1840-54 
 189000 
 
 1974-27 
 1960-00 
 
 1995-16 
 1960-00 
 
 
 3685-58 
 
 3730-54 
 
 3934-27 
 
 3955-16 
 
 8 years 
 
 1725-58 
 196000 
 
 550-14 
 1050-00 
 
 683-87 
 1960-00 
 
 596-36 
 1960-00 
 
 
 3685-58 
 
 1600-14 
 
 2643-87 
 
 2556-36 
 
 6 years 
 
 — 
 
 550 14 
 105000 
 
 — 
 
 — 
 
 
 — 
 
 1600-14 
 
 — 
 
 — 
 
 .6 years 
 
 — 
 
 107-85 
 350-00 
 
 — 
 
 — 
 
 
 — 
 
 457-85 
 
 — 
 
 — 
 
 The conclusions to be drawn from the above results are, that 
 the generally adopted method of equating the burden upon 
 successive generations of ratepayers is unjust to the later years 
 of the equated period seeing that the acceleration of the final 
 repayment of the loan ought to impose a larger burden upon 
 each year of the equated period. It is also obvious that on the 
 contrary the generally adopted method of equation relieves the 
 earlier years instead of increasing the annual charge during 
 such years. The dates of repayment, as originally fixed before 
 equation, were based upon the life of the asset, and this should 
 not be lost sight of in amending the annual instalment after 
 the equation of the period, as it is in fact ignored, in the 
 generally adopted method in which the whole of the outlay is 
 treated as having an equal repayment period. The proper and 
 consistent method of apportioning the burden between the 
 several years of the <M]uated period is to adhere as closely as 
 
 I'll 
 
THE INCIDENCE OF TAXATION 435 
 
 possible to the original annual instalments before equation 
 because by this means alone can two important results be attained, 
 namely, the annual redemption charge to the revenue or rate 
 account of each year will be proportionate to the wastage of the 
 asset and, which is equally important, the loan in respect of 
 outlay of short duration, if not actually repaid, will be in the 
 sinking fund slightly earlier than the date at which it would 
 have been repaid under the original conditions. This will 
 remove an objection at present existing as to re-borrowing for 
 outlay of short duration where the repayment of the loan has 
 been delayed by the equation of the period. A final conclusion 
 seems to be that in making the adjustment in the annual instal- 
 ment the true mathematical method last described should be 
 followed although it may involve rather more intricate calcula- 
 tions. The equation of the incidence of the annual interest 
 charges will be considered in the following chapter. 
 
436 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 CHAPTER XXXV. 
 
 THE EQUATION OF THE INCIDENCE OF TAXATION 
 
 [continued) . 
 
 INTEREST UPON THE LOAN. 
 
 The method of adjusting the annual charges to revenue 
 OR rate during the equated period in proportion to 
 
 THE LIFE OR DURATION OF CONTINUING UTILITY OF THE ASSET 
 CREATED OUT OF THE LOAN. 
 
 By CHARGING THE REVENUE OR RATE ACCOUNT OF EACH 
 YEAR OF THE EQUATED PERIOD WITH THE ANNUAL AMOUNT OF 
 INTEREST PAYABLE BEFORE EQUATION, AND IN ADDITION 
 THERETO A SUPPLEMENTARY ANNUAL AMOUNT PROPORTIONATE 
 YEAR BY YEAR TO THE ANTSTUAL INTEREST CHARGES BEFORE 
 EQUATION. 
 
 A GENERAL SUMMARY OF THE RESULTS OBTAINED IN CHAPTERS 
 
 XXXIII, XXXIY AND XXXY. 
 
 The previous enquiry into tlie method of adjusting the 
 annual incidence of the h)an burden, after equation, is confined 
 solely to the annual instalment to be charged to revenue or rate 
 and added to the sinking fund. The result of the enquiry is to 
 prove that the final 8 years of the equated period are made to 
 bear not only the whole of the relief to the 22 years of the 
 original period beyond the equated period, but a certain amount 
 in relief of the earlier years of the equated period. The annual 
 instalments during the final 8 years of the equated period are 
 as follows : — • 
 
 Before equation, under the original conditions 
 
 Table XXXIII. A. £55014 
 
 After equation : 
 
 (1) by the method generally adopted 
 
 Table XXXIII. A. £172558 
 
 (2) by tlie true method just described 
 
 Table XXXIV. J. £596-36 
 
THE INCIDENCE OF TAXATION 437 
 
 The annual charge ior interest upon the loan during the 
 final 8 years of the equated period has already been referred to. 
 Under the original conditions, before equation, this annual 
 charge was £1,050 only, owing to the fact that £26,000 of 
 loan had been repaid by the end of the 15th year. Under the 
 equated method the whole of the loan is not repayable until the 
 end of the equated period, and consequently the interest upon 
 the total loan becomes an annual charge against revenue or rate 
 during the whole of the equated period. The effect is to 
 impose upon the final 8 years of the equated period an interest 
 burden of £910 per annum in addition to the annual charge of 
 £1,050 under the original conditions. The first period of 
 5 years is charged annually with the same amount of interest 
 upon the loan under both methods; and the second period of 
 10 years is charged with an additional £70 per annum only, 
 being interest upon £2,000 of loan which would otherwise have 
 been repaid at the end of the first period of 5 years. It is 
 therefore apparent that the final 8 years of the equated period 
 bears the greater portion of the interest of which the final 
 22 years of the original period is relieved, in addition to the 
 whole of the sinking fund instalment. 
 
 The interest charges against the revenue or rate accounts 
 of the final 8 years of the equated period are as follows : — 
 
 Before equation, under the original conditions 
 
 Table XXXIII. B. £1050 
 
 After equation, under the method generally 
 adopted, and also under the true method, 
 above described, if limited to the annual 
 instalment only Table XXXIY. L. £1960 
 
 The total annual loan charges during the final 8 years of 
 the equated period are therefore as follows (Table XXXIV. L.) : 
 
 Before equation, under the original conditions : 
 
 Instalment £550-14 
 
 Interest ^lO^OOO 
 
 £160014 
 
43S REPAYMENT OF LOCAL AND OTHER LOANS 
 
 After equation : — 
 
 (1) by the method generally adopted : 
 
 Instalment £172558 
 
 Interest £196000 
 
 
 method just 
 the annual 
 
 described, 
 instalment 
 
 . £596-36 
 . £196000 
 
 
 (2) by the true 
 relating to 
 only: 
 Instalment 
 Interest 
 
 £?5iifi-3fi 
 
 
 
 
 The method just described removes the injustice to the final 
 3'ears of the equated period so far as the annual instalment is 
 concerned, but leaves untouched the question of the interest 
 upon the loan. 
 
 The advisability of making a similar adjustment with regard 
 to the interest charges is a matter upon Avhich opinion may be 
 divided, and in order to elucidate the question it is necessary to 
 state a few general propositions. The practice now adopted in 
 the case of all original loans is to spread the repayment 
 of the princijjal over a series of years commensurate to the life 
 or period of continuing utility of the asset created out of the 
 loan, with certain limitations as to assets of long continuing or 
 permanent utility. Of the three alternative methods allowed 
 by statute, only one, the instalment method, involves an un- 
 equal annual charge in respect of interest upon the loan. In 
 the case of the annuity and sinking fund methods the total 
 annual charges for principal and interest are equal throughout 
 the period of repayment or redemption. But in both these 
 cases interest is payable during such redemption period only, 
 and of course ceases on the final repayment of the loan. In 
 other Avords, the total annual loan charges, both for instalment 
 and interest are spread over a period depending upon the life 
 of tlie asset, and the annual charge to revenue or rate is the 
 same in each year of the period. This principle is carried out, 
 and is considered equitable, in the case of individual loans 
 relating to outlay of one character only, repayable on fixed 
 dates, and in respect of which separate sinking funds are or 
 may be kept. 
 
 But the conditions are different in the case of the equation 
 of the period of repayment whether the equation is made on the 
 
THE INCIDENCE OF TAXATION 439 
 
 consolidation of existing loans having varying unexpired 
 periods of repayment, or whether it is made in respect of one 
 loan, relating to outlay of a varied character, each class having 
 different periods of continuing utility and consequent periods 
 of repayment. The effect of the equation of the period as 
 regards the loan holder has already been fully discussed in 
 Chapter XXXII, where it has been ascertained that the 
 arithmetical method of finding the equated period generally 
 adopted, although wrong in principle, may perhaps be con- 
 sidered sufficiently correct. 
 
 Prior to an equation due to either of the events already 
 described, the amount to be charged to the rate or revenue 
 account of each year has already been, or may be, ascertained, 
 and the result of any true equation should be that the future 
 substituted total annual burdens are in proportion to the 
 original obligations; any variations therefrom being due only 
 to the substituted period imposed, and since the original annual 
 burdens were not equal during the original periods they 
 certainly should not be equal during the substituted period. 
 
 The equation of the incidence of the sinking fund instalment 
 has been fully discussed in Chapter XXXIV, and a method 
 described in detail of finding the future instalments only. If 
 it be required to adjust the annual incidence of the interest 
 upon the loan with a similar object in view, the adjustment 
 cannot be made in quite the same manner, seeing that in this 
 case an equal annual amount of interest has to be paid to 
 the loan holder, but the burden of providing the interest has to 
 be spread unequally over the period. The method is in fact a 
 combination of the sinking fund and annuity methods of 
 repayment of debt, with varying instead of equal annual 
 charges to revenue or rate. 
 
 In order to simplify the method of adjustment and to bring 
 it as near as possible to the method of equating the annual 
 incidence of the sinking fund instalment, already described in 
 Chapter XXXIV, it is necessary to reduce the interest upon the 
 loan under both sets of conditions to its corresponding capital 
 value at the end of the equated period of 23 years. The capital 
 value might also be expressed in terms of the present value, but 
 it would not be so convenient because the loan is repayable at a 
 future date. The annual interest of £1,960, payable upon the 
 total loan during the 23 years of the equated period, will there- 
 fore be treated as if it were allowed to accumulate at compound 
 interest at 3 per cent, until the end of that period. The sum 
 to which £1,960 per annum will then amount may be ascer- 
 
440 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 tained in the usual way by standard calculation form, No. 3, 
 and will be found to be £6360770. 
 
 For the present purpose this amount may be treated in 
 exactly the same manner as the total loan of £56,U0U when 
 adjusting the annual incidence of the sinking fund instalment 
 only. In the same way that it was there required to find three 
 annual instalments of unequal amount, to be set aside and 
 accumulated for successive periods of 5, 10, and 8 years 
 resjjectively, it is now required to find three annual amounts of 
 interest to be charged to the revenue or rate account, such 
 annual interest charges to be of vmequal amount during each 
 of the above periods, and to bear a relation to the life of the 
 asset as expressed in the repayment periods originally pre- 
 scribed. 
 
 In the case of the adjustment of the annual instalment the 
 basis of the calculation was the original annual instalments 
 before equation and in the present example the basis is the 
 original charges for interest upon the loan. In this connection 
 it is important to point out, as will be seen on reference to the 
 second column in the third part of Table XXXIV. L., that 
 there is not, before equation, any definite ratio between the 
 annual instalment and the annual interest charges during any 
 period of equal incidence, consequently the two adjustments are 
 quite distinct. 
 
 The next step is to ascertain the accumulated amount at the 
 end of 23 years of the original annual interest charges before 
 equation shown in Table XXXIII. B. in the same way that the 
 amount in the sinking fund was ascertained at the same date 
 by the accumulation of the original annual iuvstalments shown 
 in Statement XXXIV. B. as follows: — 
 
 TABLE XXXV. A. 
 
 Loan of £56,000 (authorised for outlays of varying nature 
 liaving prescribed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Showing the accumulated iimount at the end of the e(in;\1ed 
 period of 23 years, of the original annual interest charges, 
 as shown in Table XXXIII. B. (Interest at 3 per cent.) 
 
 This statement should be compared with XXXIV. B. 
 
THE INCIDENCE OF TAXATION 441 
 
 (Ij Amount of i>l,9GU per annum for 5 
 
 years '... 10405-90 
 
 Amount thereof at tlie end of a further 
 
 18 years 17715-40 
 
 (2) Amount of £1,890 per annum for 10 
 
 years 2166670 
 
 Amount thereof at the end of a further , 
 
 8 years 27446-80 
 
 (3) Amount of £1,050 per annum for 8 
 
 years 9336-90 
 
 Total, being' the accumulated amount of the original 
 annual interest charges, before equation, at the 
 end of the equated period 54499-10 
 
 Up to this point it has been ascertained that the 
 accumulated amount, at the end of the equated 
 period of 23 years, of the equal interest 
 charges of £1,960 per annum after equation is £6360770 
 
 and the corresponding amount of the original 
 varying annual interest charges as shown by 
 the foregoing table £5449910 
 
 A deficiency of ... 
 
 £9108-60 
 
 which is exactly comparable with the deficiency of £4,340 m 
 the case of the annual instalment (after Statement XXXIV. B.). 
 The adjustment of the present deficiency may be made by 
 the method described in Chapter XXXIV leading up to 
 Calculation XXXIV. G., and Statement XXXIV. H., but as 
 the conditions as to period and rate per cent, are similar in 
 both cases, and differ only in amount it is possible to adopt a 
 shorter method by utilising the information there obtained 
 and increase the supplementary annual charges found in 
 Statement XXXIV. H., in the ratio that 4340 bears to 9108-60 
 as follows : — 
 
442 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 TABLE XXXV. B. 
 
 Showing tlie method of finding the .supplementary annual 
 charges to revenue or rate to be added to the original 
 annual interest charges before equation. 
 
 Periods of equal incidence. 
 
 Amount of deficiency... 
 
 Deficiency at end of 23 years. 
 Annual instalment. Interest on loan. 
 Statement XXXIV. H. As above. 
 
 Supplementary annual charges to 
 revenue or rate. 
 
 5 year period 
 
 10 year period 
 
 8 year period 
 
 23 years 
 
 Total 
 
 434000 
 
 9108-60 
 
 Annual Annual 
 
 instalments interest charges 
 
 Table XXXIV. J. Table XXXV. C. 
 
 186-26 390-93 
 
 154-62 324-51 
 
 46-22 9700 
 
 387-10 
 
 812-44 
 
 The above annual interest charges are ascertained from 
 the amounts in the first column by the ordinary rules of pro- 
 portion or by logs., of which it is not necessary to show the 
 actual working. The total annual interest charges to revenue 
 or rate during the equated period may now be stated in the 
 following table : — 
 
THE INCIDENCE OF TAXATION 
 
 443 
 
 TABLE XXXV. 0. 
 
 Loan of i;56,000 (authorised for outlays of varying nature 
 having prescri])ed periods of repayment), the whole to be 
 redeemed in one sum at the end of an equated period. 
 
 Showing the annual charges to revenue or rate in respect of 
 interest upon the loan under (1) the equated method 
 generally adopted, and (2) in which the annual interest 
 charges originally payable are supplemented by additional 
 annual amounts spread over the equated period in propor- 
 tion to the original interest obligations. 
 
 This table should be compared with Table XXXIV. J. 
 
 Equated annual interest charges. 
 
 Periods 
 of 
 ecjual incidence. 
 
 Original 
 
 annual 
 
 interest charges. 
 
 Additional 
 
 annual 
 
 interest charges. 
 
 Total 
 
 annual 
 
 interest charges. 
 
 Annual 
 
 interest charges 
 
 under the 
 equated method. 
 
 5 years 
 
 1960-00 
 
 390-93 
 
 2350-93 
 
 196000 
 
 10 years 
 
 189000 
 
 324-51 
 
 2214-51 
 
 196000 
 
 8 years 
 
 105000 
 
 97-00 
 
 1147-00 
 
 196000 
 
 23 years 
 
 
 Statement XXXIV. K. shows the final repayment of the 
 loan by means of the amended annual instalments to be spread 
 over the equated period with due regard to the life of the asset, 
 instead of being spread equally over such period, and thereby 
 proves the accuracy of the method adopted with regard to the 
 annual instalment. In a similar manner, although expressed 
 in different terms, the following Statement XXXY. D. proves 
 the accuracy of the method adopted in order to equate the 
 incidence of the annual interest charges. 
 
444 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 STATEMENT XXXV. D. 
 
 Loan of £56,000 (as above). 
 Sliowiiig- that the accumulated amount oi the amended annual 
 interest ckarges ascertained as in Table XXXV. C. will be 
 equal to the accumulated amount of the equal annual 
 interest charges after equation, both at the end of 23 years, 
 at 3 per cent, per annum. 
 
 This statement should be compared with Statement XXXIV. K. 
 
 (1) Amount of £2c550'9o jjer annum for 
 
 5 years 
 
 Amount thereof at the end of a further 
 
 18 years 21248-80 
 
 (2) Amount of £2214"51 per annum for 
 
 10 years 
 
 Amount thereof at the end of a further 
 
 8 years 32159-40 
 
 (3) Amount of £1147 per annum for 8 
 
 years 1019950 
 
 which is the accumulated amount of an annuity of 
 £1960 for 23 years at 3 per cent, as previously 
 ascertained £63607-70 
 
 The above calculations (1) and (2) have been made direct 
 by the " method by step " shown in Statement XXVII. C. 
 
 It will be gathered from the above Statement XXXV. D., 
 that tlie annual interest charges to revenue or rate during the 
 first two periods of 5 years and 10 years are greater than the 
 annual amounts of interest payable to the loanholders during 
 those periods after equation, as follows : — - 
 
 5 years (2350-93—1960) an increase of 390-93 
 10 years (2214-51—1960) an increase of 254-51 
 
 and that the annual amounts of interest payable to the loan- 
 holders during the final 8 years of the equated period are 
 greater than the amended annual amounts charged to revenue 
 or rate during that period, in Statement XXXV. D., as follows : 
 
 8 years (1960—1147), a decrease of 81300. 
 
THE INCIDENCE OF TAXATION 445 
 
 The correctness of the foregoing calculations is proved by the 
 following statement giving the accumulated amounts of the 
 above annuities at the end of the 2ord year, without details of 
 the actual calculations which are similar to XXVII. C. : — 
 
 Amount of £390-93 per annum for 5 years, accu- 
 mulated for a further period of 18 years at 3 
 per cent, per annum £3533'40 
 
 Amount of £254-51 per annum for 10 years, accu- 
 mulated for a further period of 8 years, at 3 
 per cent, per annum 
 
 £369600 
 
 £7229-40 
 which is equal to the 
 
 Amount of £813*00 per annum for 8 years at 3 per 
 
 cent, per annum £(-^^9 41) 
 
 This proves that the amounts charged to the revenue or rate 
 account during the first 15 years, in excess of the amounts 
 annually payable to the loanholders during that period, will, 
 if accumulated, be suflacient to provide the future annual 
 deficiencies in the amounts charged to revenue or rate account 
 during the final 8 years of the equated period. It also points 
 out the methods to be adopted as regards the actual book- 
 keeping, and indicates the opening of an account which may 
 be termed an : — 
 
 " Equated Loan Interest, Reserve Account," 
 
 and which will closely resemble the repayment of a loan by an 
 equal annual instalment of principal and interest combined, 
 or the annuity method, but with a varying instead of an equal 
 annual charge to revenue or rate. To the extent that the 
 annual charges to revenue or rate are, during the earlier years, 
 greater than the annual amounts payable to the loanholders by 
 way of interest, the account will also partake of the nature of 
 a sinking fund, and will therefore require the same careful 
 future supervision as to the amount standing to the credit of 
 the account, the rate of accumulation, and also the immediate 
 preparation of a pro forma account showing the ultimate work- 
 ing out of the account. 
 
 As regards the actual book-keeping the above interest reserve 
 account may be treated in two distinct ways, namely, by 
 
446 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 crediting the account with the total annual amounts of interest 
 charged to revenue or rate, as shown in Table XXXY. C. and 
 debiting the account with the annual interest payable to the 
 loanholders, which is the more scientific method as yielding an 
 exact record of the actual transactions. The other method is 
 to treat it as a reserve account pure and simple and credit it 
 only with the above excess annual amounts of £39093 and 
 <£254"21 charged to revenue or rate account during the periods 
 of 5 and 10 years respectively. During the third period of 
 8 years the interest reserve account would of course be debited, 
 and the revenue or rate account credited, with the difference 
 of £813 per annum already referred to. Unlike a sinking fund 
 proper, the amount to the credit of the interest reserve account 
 need not be separately invested, but may be merged in the 
 general assets of the undertaking, provided always that the 
 proper annual amounts of interest at the calculated rate of 
 accumulation are credited to the account, and charged to the 
 current year's revenue or rate account. If the account be kept 
 in this manner and compared annually with the pro forma 
 account there should not arise at any time any necessity to 
 make an adjustment so long as the repayment period remains 
 unaltered. The following pro forma account will illustrate the 
 method of keeping the interest reserve account applicable to the 
 foregoing example : — 
 
THE INCIDENCE OF TAXATION 447 
 
 TABLE XXXV. E. 
 
 Equated Loan Interest, Reserve Account. 
 
 Interest, 3 per cent, per annum. 
 
 Amount to 
 
 
 Interest 
 
 
 Interest 
 
 
 
 credit at 
 
 
 charged to 
 
 
 paid to 
 
 Balance 
 
 
 beginning 
 
 Interest 
 
 revenue 
 
 Total 
 
 loan 
 
 carried 
 
 
 of year. 
 
 thereon. 
 
 or rate. 
 
 credits. 
 
 holders. 
 
 forward. 
 
 Year. 
 
 1 Nil Nil 2350-93 2350-93 196000 39093 1 
 
 2 390-93 11-73 2360-93 2753-59 196000 79359 2 
 
 3 793-59 23-81 235093 316833 196000 1208-33 3 
 
 4 1208-33 36-25 235093 3595-51 196000 163551 4 
 
 5 1635-51 49-06 235093 403550 196000 207550 5 
 
 6 2075-50 62-26 2214-51 435227" 1960-00 2392-27 6 
 
 7 2392-27 7177 221451 467855 196000 271855 7 
 
 8 2718-55 81-56 221451 5014-62 1960-00 3054-62 8 
 
 9 3054-62 91-64 2214-51 536077 196000 3400-77 9 
 
 10 3400-77 102-02 221451 571730 1960-00 375730 10 
 
 11 3757-30 112-72 221451 608453 196000 412453 11 
 
 12 4124-53 123-73 221451 646277 196000 450277 12 
 
 13 4502-77 135-08 221451 685236 196000 489236 13 
 
 14 4892-36 14677 2214-51 7253-64 1960-00 5293-64 14 
 
 15 5293-64 15885 2214-51 766700 1960-00 5707-00 15 
 
 16 570700 171-21 114700 702521 196000 506521 16 
 
 17 5065-21 151-96 114700 636417 1960-00 4404-17 17 
 
 18 4404-17 132-12 114700 568329 196000 372329 18 
 
 19 3723-29 11170 114700 498199 1960-00 3021-99 19 
 
 20 3021-99 90-66 114700 425965 1960-00 229965 20 
 
 21 2299-65 6899 114700 351564 1960-00 1555-64 21 
 
 22 1555-64 4667 114700 274931 196000 789-31 22 
 
 23 789-31 23-69 1147-00 196000 1960-00 Nil 23 
 
448 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 A comparison lias already been made, in Table XXXIV. L., 
 
 between the total annual loan charges under the original 
 
 conditions and under the generally adopted method after 
 
 equation; and a final comparison will now be made between 
 
 those methods and the one just described. The following 
 
 results are worthy of careful study and show the very wide 
 
 difference between the incidence of the total annual loan 
 
 burden under the generally adopted method after equation on 
 
 the one hand as compared with the annual charges under the 
 
 original conditions before equation and also under the amended 
 
 method just described. Under the method generally adopted 
 
 the annual burden, both as regards the annual instalment and 
 
 interest upon the loan, is spread equally over the whole of the 
 
 equated period with a total disregard to the life of the asset and 
 
 the consequent repayment periods originally based thereon. 
 
 The following table shows that the original incidence of the 
 
 burden is departed from under the generally adopted method 
 
 after equation in that it relieves the earlier years, and throws 
 
 a severe additional burden upon the later years, of the equated 
 
 period. The result of the author's method is that the revenue 
 
 or rate account of each year of the equated period is charged 
 
 with an amount in respect both of annual instalment and 
 
 interest upon the loan, which is exactly in proportion to the 
 
 amount with which it would have been charged under the 
 
 original conditions. These amended annual charges are greater 
 
 than under the original conditions and include, as is equitable, 
 
 the relief to the post equated period, and such relief is imposed 
 
 rateably upon each year of the equated period instead of being 
 
 charged against the later period of 8 years only. 
 
THE INCIDENCE OF TAXATION 
 
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450 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 In order to make tlie foregoing results perfectly clear, three 
 charts have been prepared as follows : — 
 
 The Equation of the Incidence of Taxation. Chart. I. 
 
 Showing the total annual loan charges in respect of the sinking 
 fund instalment and interest upon the loan, during each 
 year of the original and equated periods. 
 
 (A). Under the original conditions, before equation. 
 (B). Under the equated method, as generally adopted. 
 
 (C). Under the author's method of equation relating to 
 
 the annual instalment only, as described in 
 
 Chapter XXXIV. 
 (D). Under the author's method of equation relating to 
 
 the annual instalment and interest upon the loan, 
 
 as described in Chapter XXXV. 
 
 The Equation of the Incidence of Taxation. Chart. II. 
 
 Showing the total annual loan charges in respect of the sinking 
 fund instalment and interest upon the loan, during each 
 year of the original and equated periods. 
 
 (A). Under the original conditions, before equation. 
 (B). Under the equated method, as generally adopted. 
 (D). Under the author's method of equation relating to 
 
 the annual instalment and interest upon the loan, 
 
 as described in Chapter XXXV. 
 
 The Equation of the Incidence of Taxation. Chart III. 
 
 Showing the difference between the total annual loan charges 
 in respect of the sinking fund instalment and interest upon 
 the loan, during each year of the original and equated 
 periods : — 
 
 (B). Under the equated method as generally adopted, 
 in which the charge is spread equally over the 
 period. 
 
 (D). Under the method described in Chapters XXXIV. 
 and XXXV., in which the revenue or rate account 
 is charged with annual sums based upon the life 
 of the asset, and proportionate, year by year, to 
 the annual charges before equation. 
 
THE INCIDENCE OF TAXATION 451 
 
 These charts show in graphic form : — 
 
 In Chart I, the total annual loan charges under each 
 method, during each period of equal incidence. These annual 
 charges are divided as between the interest upon the loan which 
 is shown in the lower part of the diagram, and the annual 
 instalment which is shown above it. The height of each 
 column represents the total annual loan charges, and the width 
 of the columns represents the number of years in the periods 
 of equal incidence. This chart brings out clearly the compara- 
 tively small relief to the earlier years of the equated period and 
 the large increased annual burden during the final eight years 
 of such period. 
 
 In Chart II, the total annual charges under each method 
 during each period of equal incidence are further compared, 
 but without any subdivision as between the interest upon the 
 loan and the annual instalment. The broken line shows the 
 equal annual burden under the generally adopted method after 
 equation. The thin unbroken line shows the annual burdens 
 under the original conditions before equation, which were 
 based, both as regards instalment and interest, upon the life 
 of the asset. The thick unbroken line shows the corresponding 
 annual charges under the author's method of equation. It will 
 be noticed that the two unbroken lines agree very closely and 
 differ widely from the broken line of the generally adopted 
 method. 
 
 In Chart III, the total annual loan charges under the 
 author's method of equation are taken as the standard or zero, 
 and are compared, as to the equated period, with the charges 
 under the generally adopted method after equation, and as to 
 the post equated period with the charges under the original 
 conditions. The area below the zero line represents, in the case 
 of the equated period, the relief afforded by the generally 
 adopted method after equation as compared with the author's 
 method, and as regards the post equated period, the absolute 
 relief afforded by the equation of the period irrespective of the 
 method in which the burden is distributed over the equated 
 period. The area above the zero line, which occurs only in the 
 final 8 years of the equated period, represents the additional 
 annual burden imposed upon this period under the generally 
 adopted method as compared with the author's method. 
 The actual amounts of relief and overcharge are taken from 
 Table XXXV. F., and relate to the loan of £56,000 used to 
 illustrate the problem, consequently any comparison based 
 
452 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 solely upon that table must be made witb tliis actual loan in 
 mind. In order tberefore to show tbe results in a form wbicb 
 will be readily appreciated, tbe above differences bave eacb been 
 expressed, in the chart, in terms of an annual rate. The basis 
 upon which this has been done is a statement by a witness 
 before one of the Parliamentary Committees appointed to 
 enquire into such questions, who proved that in a particular 
 ease the immediate effect of an equation of the period was to 
 reduce the rates by 3d. in the £ upon the annual value. This 
 reduction was of course, only between the annual instalments, 
 before and after equation, because during the earlier years of 
 the equated period, as shown by Chart I, there is not any 
 change in the amount of interest payable, no part of the loan 
 having then been repaid by the maturing of the sinking fund 
 for the shorter period. If, however, the comparison be made 
 between the amount payable after equation and the proper 
 amount which should have been payable under the author's 
 method as a consequence of such equation, the saving would 
 be 6'52 pence in the pound instead of 3 pence, and in the chart 
 the relief to the first part of the equated period of 5 years has 
 been taken at that figure. Fpon this basis, the effect of an 
 equation of the period, in the present instance, under the 
 method generally adopted, is to relieve the annual rate accounts 
 as follows : — 
 
 during the equnted period : — • 
 
 for a period of 5 years of 6"52 pence in the £. 
 for a period of 10 years of 3'20 pence in the £. 
 
 during the post equated period : — 
 
 for a period of 6 years of 9' 76 pence in the £. 
 for a period of 16 years of 2' 79 pence in the £. 
 
 and to impose an additional annual burden upon the final 
 8 years of the equated period of 11 'So pence in the £. 
 
 The above method of adjusting the annual incidence of the 
 total loan burden may, and undoubtedly will, appear com- 
 plicated when compared with the rough and ready method 
 now adopted. It will certainly increase the labour involved 
 upon the equation of the period of repayment of new loans 
 authorised for outlays of varying natures and also upon the 
 consolidation of existing loans, but it is sound in principle and 
 carries out the fundamental law of locnl finance, that the 
 
THE INCIDENCE OF TAXATION 453 
 
 present generation shall bear at least its due burden and not 
 transfer it to future years. It is very tempting to local leaders 
 of finance to pose as tbe benefactors of the present ratepayers 
 by adopting a method, having a high-sounding title, which has 
 the immediate effect of reducing the present burden at the 
 expense of the future ; but it ought to be recognised, that where 
 the only means of paying for municipal works is by annual 
 contributions out of revenue or rate, to be spread over a 
 prescribed period of years fixed after very careful enquiry as 
 to the life of the asset, any reduction in the period of repay- 
 ment, due solely to causes of a purely financial nature, 
 cannot possibly equitably reduce the annual burden but 
 must inevitably increase it. Any departure from this principle 
 is a violation of the recognised canons of local government. 
 
A. 
 B. 
 C. 
 D. 
 
 under original conditions — before equation. 
 
 under equated method, as generally adopted. 
 
 Author's equated method (annual instalment only). 
 
 Author's equated method (annual instalment, and interest on the loan) 
 
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 under equated method, as generally adopted. 
 
 Author's equated method (annual instalment only). 
 
 Author's equated method (annual instalment, and interest on the loan). 
 
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Appendix. 
 Calculations Referred to in the Text. 
 
Note. Tlie number of the calculation refers to tlie chapter to 
 which it relates. 
 
 The detailed working of the method (A) by formula is not 
 included in any of the following calculations, but may be 
 found by referring to the examples given in the text, namely : 
 
 Form No. 1. Amount of one pound. No. (lY) 3. 
 
 2. Present value of one pound. (Y) 2. 
 
 3. Amount of one pound per annum. (YI) 2. 
 3x. Sinking fund instalment. (^^^) 1- 
 
 4. Present value of one pound per annum. (YII) 2. 
 
 5. Annuity which one pound will purchase. (YIII) 2, 
 
 Full instructions as to the use of the author's standard 
 calculation forms are given in Chapter X. 
 
462 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 5. 
 
 No. (XV) 3. 
 
 To find the additional sinking fund instalment to be set aside 
 during the unexpired portion of tlie repayment period, to 
 compensate for a deficiency in the fund. Table Y. 
 
 Required the annual instalment to be set aside and accumulated 
 for 1'^ years at 3| per cent., which is equivalent to a 
 deficiency of £469'744 in the amount now in the fund. 
 
 (C) By Thoman's Table, 3^ per cent. Rule 3, Chapter YIII. 
 
 Log. Present sum 
 
 add Log. rt", 13 years 
 
 469-744 2-6718612 
 8-9870474 
 
 11-6589086 
 
 deduct 10 
 
 1-6589086 
 
 Required annual instalment 
 
 £45-5941 
 
 Standard For in. To. 1. 
 
 jNo. (XT) 4. 
 
 To find the portion of original loan which will be provided by 
 the future accumulation of the present investments repre- 
 senting the fund. Table I. 
 
 Required the amount of £9,463, at the end of 13 years, 
 accumulated at 3^ per cent. 
 
 (C) By Thoman's Table, 3^ per cent. Rule 3, Chapter IT. 
 
 Log. Present sum 
 
 add Log. R^\ 13 years... 
 
 Required future amount 
 
 9463 3-9760288 
 . ... 01942245 
 
 41702533 
 
 £14799-71 
 
APPENDIX 
 
 Standard Fortn, No. 3. 
 
 463 
 No. (XV) 5. 
 
 To find the portion of original loan which will be provided by 
 the accumulation of the original annual instalments to be 
 set aside during the unexpired portion of the redemption 
 period. Table III. 
 
 llequired the amount of an annual instalment of c£680"234, to 
 be set aside and accumulated for 13 years at 3^ per cent. 
 
 (C) By Thoman's Table, 3i per cent. Rule 3, Chapter VI. 
 
 Log. Annuity 680-234 
 
 add Log. E^^ 13 years, +10 
 
 deduct Log. a" 
 
 Required future amount 
 
 2-8326581 
 10-1942245 
 
 13-0268826 
 8-9870474 
 
 4-0398352 
 
 £10960-62 
 
 Standard Form. No. 1. 
 
 No. (XV) 6. 
 
 To find the portion of original loan which will be unprovided 
 if the present deficiency in a sinking fund be allowed to 
 remain uncorrected during the remainder of the redemption 
 period. Table I. 
 
 Required the amount of £469-744, at the end of 13 years, 
 accumulated at 3| per cent. 
 
 (B) By Table I, 13 years, ^ per cent. Rule 2, Chapter IV. 
 
 Log. Amount of £1 
 
 add Log. present sum ... 
 
 1-56395 0-1942245 
 469-744 2-6718612 
 
 2-8660857 
 
 Required future amount 
 
 £734-659 
 
464 REPAYMENT OF LOCAL AND OTHER LOANS 
 Standard Form, .To. .3x. No. (XVI) 1. 
 
 To find the sinking fund instalment required to provide the 
 amount of loan represented by a deficiency in the fund. 
 
 Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3^ per cent., to provide £734'659 at the end of 13 years. 
 
 (C) By Thoman's Table, 3i per cent. Eule 3, Chapter XIII. 
 
 Log. Amount of loan ... 
 
 add Log. fl", 13 years. 
 
 deduct Log. RN+iq 
 
 Required annual instalment... 
 
 734-659 2-8660857 
 8-9870474 
 
 11-8531331 
 ,.. 10-1942245 
 
 1-6589086 
 
 £45-5941 
 
 Standard Form, No. 3. 
 
 Tso. (XVI) 2. 
 
 To find the amount of loan which will be provided by the 
 
 accumulation of the augmented annual instalment. 
 
 Table III. 
 
 Required the amount which should stand to the credit of a 
 sinking fund representing the accumulation of an annual 
 instalment of £725-828 for 13 years at 3| per cent. 
 
 (]V) By Table III, 13 years, 3i per cent. Rule 2, Chapter YI. 
 
 Log. Amount of £1 per annum ... 16-11303 1-2071771 
 ad^ Log. annuity 725-828 2-8608337 
 
 40680108 
 
 Required future amount 
 
 £11695-29 
 
APPENDIX 
 
 Standard Fonn, No. 2. 
 
 465 
 No. (XVI) 3. 
 
 To find the acciiiiiulated anioimt of au annual instalment to be 
 set aside for a limited period only ; at the end of that period 
 the amount so found to be accumulated for a further pre- 
 scribed period. Method by " step." Table II. 
 
 Eequired the present value of £7o4"659 due at the end of 8 years 
 at 3^ per cent. 
 
 (C) By Thoman's Table, 3i per cent. 
 
 Eule 3, Chapter V. 
 
 Log. Future sum 734-659 2-8660857 
 
 deduct Loo-. RN^ 8 years 0-1195228 
 
 Required present value 
 
 2-7465629 
 
 £557-908 
 
 Standard Form, No. 3x. 
 
 No. (XVI) 4. 
 
 To find the annual instalment which will amount to a given sum 
 if accumulated for a prescribed number of years. Table III . 
 
 Required the annual instalment to be set aside and accumulated 
 at 3i per cent, to provide £557 908 at the end of 5 years. 
 
 (C) By Thoman's Table, 3i per cent. Rule 3, Chapter XIII. 
 
 Log. amount of loan 
 
 add Log. a", 5 years 
 
 deduct Log. RN+10 ... 
 Required annual instalment... 
 
 557-908 2-7405629 
 9-3453372 
 
 12-0919001 
 100747017 
 
 2-0171984 
 
 £1040395 
 
466 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 1 
 
 No. (XVI) 5. 
 
 To find the amount of loan wliicli will be provided by the future 
 accumulation of the present investments representing the 
 fund. Table I. 
 
 Required the amount of £9,46-3 at the end of 5 years, accumu- 
 lated at 3^ per cent. 
 
 (C) By Thoman's Table, 3i per cent. Rule 3, Chapter lY. 
 
 Log. Present sum 
 
 add Log. R^, 5 years 
 
 9463 3-9760288 
 . ... 00747017 
 
 4-0507305 
 
 Required future amount 
 
 £11239-07 
 
 Standard Form, No. 3. 
 
 No. (XVI) 6. 
 
 To find the amount of loan which will bo provided by the future 
 accumulation of the augmented annual instalment. 
 
 Table III. 
 
 Required the amount of an annual instalment of £784-273 to be 
 set aside and accumulated for 5 years at 3^ per cent, 
 
 (C) By Thoman's Table, 3^ per cent. Rule 3, Chapter YI. 
 
 Log. Annuity 784-273 2-8944673 
 
 add Log. RN, 5 years, +10 100747017 
 
 deduct Loff. a" 
 
 12-9691690 
 9-3453372 
 
 3-6238318 
 
 Re(iuired future amount 
 
 £4205-637 
 
APPENDIX 
 
 467 
 
 Standard Form, No. 1. 
 
 No. (XVI) 7. 
 
 To find tlie amount of loan wliicli will be provided by tlie 
 accumulation of the amount in the fund. Table I. 
 
 Required the anu)unt of £15444"71 at the end of 8 years, 
 accumulated at o\ per cent. 
 
 (13) By Table I, 8 years, S^ per cent. Rule 2, Chapter IV. 
 
 Log. Amount of £1 
 
 add Log. present sum ., 
 
 .. 1-31681 0-1195228 
 ..15444-71 4-1887798 
 
 4-3083026 
 
 Required future amount 
 
 £20337-74 
 
 Standard Form, No. 3. 
 
 No. (XVI) 8. 
 
 To find the amount of loan which will be provided by the 
 accumulation of the original annual instalment. Table III. 
 
 Required the amount of an annual instalment of £680-234 to be 
 set aside and accumulated for 8 years at 3| per cent. 
 
 (B) By Table III, 8 years, ^ per cent. Rule 2, Chapter VI. 
 
 Log. Amount of £1 per annum ... 905168 0-9567296 
 add Log. annuity 680-234 2-8326581 
 
 3-7893877 
 
 Required future amount 
 
 £6157-26 
 
468 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3x. 
 
 No. (XVI) 9. 
 
 To find the future annual increment to be accumulated to 
 provide tlie balance of loan not provided by tlie present 
 investments representing the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3^ per cent, to provide £17,032 at the end of 13 years. 
 
 (C) By Thoman's Table, 3| per cent. 
 
 Log. Amount of loan 
 
 add Log. a", 13 years ... 
 
 deduct Log. EN + 10 
 
 Rule 3, Chapter XIII. 
 
 . 17032 4-2312656 
 8-9870474 
 
 Required annual instalment 
 
 13-2183130 
 10-1942245 
 
 3-0240885 
 
 £1057-033 
 
 Standard Form. No. 3,r 
 
 No. (XYT) 10. 
 
 To find the anniuil sinking fund instalment previoush' set aside 
 in error to provide the deficient amount now in the fund. 
 
 Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3^ per cent, to provide £9,463 at the end of 12 years. 
 
 (B) By Table III, 12 years, 3i percent. Rule 2, Chapter XIII 
 
 Log. Amount of loan 9463' 3-9760288 
 
 <:?eJwcf Log. amount of £1 per ann. 146019 11044112 
 
 2-8116176 
 
 Re(|uired aiinual instalment 
 
 £648-0636 
 
APPENDIX 
 
 Standard Form, No. 5. 
 
 469 
 No. (XVII) 1. 
 
 To find tlie amount by whicli the annual sinking fund instalment 
 may be reduced in consequence of a payment into the fund 
 of proceeds of sale of part of the security for the loan. 
 
 Table Y. 
 
 Required the annuity which may be purchased with £4,560 
 for 13 years at 3^ per cent. 
 
 (B) By Table V, 13 years, 3^ per cent. Eule 2, Chapter VIII. 
 
 Log. Annuity £1 will purchase ... 0-097061 2-9870474 
 a^^ Log. present sum 4560 3-6589648 
 
 Required annuity ... 
 
 2-6460122 
 
 £442-6008 
 
 Standard Form, JSo. 1. 
 
 No. (XVII) 2. 
 
 To find the portion of the original loan which will be provided 
 by the future accumulation of the present investments 
 representing the fund. Table I. 
 
 Required the amount of £9932*744 at the end of 13 years, 
 accumulated at 3^ per cent. 
 
 (C) By Thoman's Table, 3| per cent. Rule 3, Chapter IV. 
 
 Log. Present sum 9932-744 3-9970693 
 
 aiZfZ Log. RN, 13 years 01942245 
 
 41912938 
 
 Required future amount 
 
 £15534-375 
 
470 REPAYMENT OF LOCAL AND OTHER LOANS 
 Stcmdard Form,, No. 1. No. (XVII) 3. 
 
 To find the portion of the original loan which will be provided 
 by the future accumulation of the proceeds of sale of assets 
 paid into the fund. Table I. 
 
 Required the amount of £4,560 at the end of 13 years, accumu- 
 lated at 3^ per cent. 
 
 (B) By Table I, 13 years, at 3^ per cent. Rule 2, Chapter IV. 
 
 Log. Amount of £1 1-66395 01942245 
 
 a<Z^ Log. present sum 4560 3"6589648 
 
 3-8531893 
 
 Required future amount £7131*64 
 
 Standard Forvi, No. 3x. No. (XVII) 4. 
 
 To find the annual instalment which will provide the amount of 
 loan represented by the proceeds of sale of part of the 
 security paid into the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3| per cent, to provide £7131-64 at the end of 13 years. 
 
 (C) By Thomun's Table, 3i per cent. Rule 3, Chaper XIII. 
 
 Log. Amount of loan 7131-64 38531893 
 
 a^Z^/' Log. a«, 13 years 8-9870474 
 
 12-8402367 
 cZe^act Log. RN+10 101942245 
 
 2-6460122 
 
 Required annual instalment £4426008 
 
APPENDIX 471 
 
 Standard Form, No. 3. No. (XVII) 5. 
 
 To fiud the amount of loan which will be provided by the future 
 accumulation of the reduced annual instalment consequent 
 upon the payment into the fund of the proceeds of sale of 
 part of the security. Table III. 
 
 Required the amount of an annual instalment of <£237'633 to 
 be set aside and accumulated for 13 years at 3| per cent. 
 
 (C) By Thoman's Table, 3^ per cent. Rule 3, Chapter YI. 
 
 Log. Annuity 237-633 2-3759068 
 
 add Log. RN, 13 years, +10 101942245 
 
 12-5701313 
 deduct Loff. a" 8-9870474 
 
 3-5830839 
 
 Required future amount £3823*987 
 
 Standard Form, No. 3x. No. (XVII) 6. 
 
 To find the future annual increment to be accumulated to 
 provide the balance of loan not provided by the investments 
 representing the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3| per cent, to provide £1200226 at the end of 13 years, 
 
 (B) By Table III, 13 years, 3^ percent. Rule 2, Chapter XIII. 
 
 Log. Amount of loan 1200226 4-0792630 
 
 deduct Log. amount of £1 per aun. 16-11303 1-2071771 
 
 2-8720859 
 
 Required annual instalment £744-879 
 
472 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Standard Form, Xo. 3it. No. (XVIII) 1. 
 
 To fiud the aniouut by which the original annual instalment 
 may be reduced in consequence of the withdrawal of part 
 of the loan from the operation of the fund by reason of its 
 conversion into ordinary share capital or stock. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at '6\ per cent, to provide £5,000 at the end of 13 years. 
 
 (C) By Thoman's Table, o\ per cent. Rule 3, Chapter XIII. 
 
 Log. Amount of loan 5000 3-6989700 
 
 add Log. a'S 13 years 8-9870474 
 
 12-6860174 
 Je^m-t Log. RN+io 10-1942245 
 
 2-4917929 
 
 Required annual instalment £310-308 
 
 Standard Form, No. 3a',. >'«• (XVIII) 2. 
 
 To find the amended annual instalment which will provide the 
 balance of loan not provided by the future accumulation 
 of the present investments and the original annual instal- 
 ment, after withdrawal of part of the loan from the opera- 
 tion of the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at ^ i)er cent, to provide £6695-30 at the end of 13 years. 
 
 (13) By Table III, 13 years 3i per cent. Rule 2, Chapter XIII. 
 
 Log. Amount of loan 669530 3-8257700 
 
 deduct Log. amount of £1 per ann. 1611303 1*2071771 
 
 2-6185929 
 
 Required annual instalment £415-520 
 
APPENDIX 473 
 
 Standard Form, Xo. 3x. No. (XVIII) 3. 
 
 To find tlie future annual increment to be accumulated to 
 provide the balance of loan not provided by the present 
 investments representing' the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at o\ per cent, to provide i^l2,0o2 at the end of 13 years. 
 
 (C) By Thoman's Table, ^ per cent. Rule 3, Chapter XIII. 
 
 Lo^. Amount of loan 12032 4-0803378 
 
 aJ(Z Log. (/«, 13 years 89870474 
 
 13-0673852 
 ^(■J,/rf Log. RN + 10 101942245 
 
 2-8731607 
 
 Required annual instalment £746-725 
 
 Standard Form, No. 3. ^'o- (XVIII) 4. 
 
 To find the amount which should stand to the credit of the 
 sinking fund, lable 111. 
 
 Required the amount of an annual instalment of £7,500 to be 
 set aside and accumulated for 7 years at 3 per cent. 
 
 (B) By Table III, 7 years, 3 per cent. Rule 2, Chapter YI. 
 
 Log. Amount of £1 per annum ... 7-6625 0-8843684 
 a(^(Z Loff. annuity 7500 3-8750613 
 
 4-7594297 
 
 Required future amount £57468-48 
 
474 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form-, No. Sx. >'o. (XVIII) 5. 
 
 To find the annual instalment for an even number of years 
 which ajDproximates to the instalment of specified amount 
 not found by calculation. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £150,000 at the end of 16 years. 
 
 (B) By Table III, 16 years, 3 per cent. Rule 2, Chapter XIII. 
 
 Log. Amount of loan 150,000 51760913 
 
 deduct Log. amount of £1 per ann. 20' 1569 13044233 
 
 3-8716680 
 
 Required annual instalment £74416285 
 
 Standard Form, No. 3. No. (XVUI) 6. 
 
 To find the amount of loan which will be provided by the 
 instalment of stated amount at the end of the approximate 
 period of even years. Table III. 
 
 Required the amount of an annual instalment of £7,500 to be 
 set aside and accumulated for 16 years at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter VI. 
 
 Log. Annuity 7500 38750613 
 
 add Log. RN, 16 years, +10 10-2053956 
 
 14-0804569 
 deduct Log. r/« 8-9009723 
 
 5-1794846 
 
 Required future amount £15117659 
 
APPENDIX 
 
 475 
 
 Standard Form, No. 3. 
 
 No. (XVIII) 7. 
 
 To find the portion of the original loan, being the accumulation 
 of an intentional error in the sinking- fund instalment 
 assumed in the adjustment. Table III. 
 
 Required the amount of £58-3715 per annum for 16 years at 
 3 per cent. 
 
 (B) By Table III, 16 years, 3 per cent. Rule 2, Chapter VI. 
 
 Log. Amount of £1 per annum ... 201569 1-3044233 
 a^Z^ Log. annuity 58-3715 r7662008 
 
 30706241 
 
 Required future amount 
 
 £1176-58 
 
 Standard, Form, No. 3x. 
 
 No. (XVIII) 8. 
 
 To find the amount by Avhich the original annual instalment 
 may be reduced in consequence of the withdrawal of part 
 of the loan from the operation of the sinking fund by 
 reason of its conversion into ordinary share capital or 
 stock. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £45,000 at the end of 9 years. 
 
 (B) By Table III, 9 years, 3 per cent. Rule 2, Chapter XIII. 
 
 Log. Amount of loan 45000 4-6532125 
 
 ^eiZwci Log. amount of £1 per ami. 101591 1-0068555 
 
 3-6463570 
 
 Required annual instalment. 
 
 £4429-523 
 
476 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Staiulard Form, Xo. 3. 
 
 No. (XVIII) 9. 
 
 To find the aiiiuiiut wliieli sliuuld stand to the credit of the 
 fund. Table III. 
 
 Required the amount of an annual instalment of £7441'6285 to 
 be set aside and accumulated for 7 years at -3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. 
 
 Log. Annuity 
 
 add Log. E.N^ 7 years, + 10 
 
 deduct Log. a" 
 
 E-ule o, Chapter YI. 
 
 7441-6285 3-8716680 
 100898606 
 
 Required future amount 
 
 13-9615286 
 9-2054922 
 
 4-7560364 
 £57021-21 
 
 Standard Form, No. 5. 
 
 No. (XVIII) 10. 
 
 To find the amount by which the annual instalment may be 
 reduced in consequence of a surplus in the fund. Table Y. 
 
 Required the annuity which may be purchased with £44727 for 
 9 years at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter YIII. 
 
 Log. Present sum 
 
 add. Log. a", 9 years 
 
 447-27 2-6505698 
 9-1086795 
 
 11-7592493 
 
 deduct 10 ... 
 Required annuity ... 
 
 1-7592493 
 
 £57-4446 
 
APPENDIX 
 
 477 
 
 Standard Form, No. 1. 
 
 ISo. (XVIII) 11. 
 
 To find tlie amount of loan wliirli will be provided by tbe fiiture 
 accumulation of the present investments of the amount 
 in the fund. Table I. 
 
 Eequired the amount of £57468-48 at the end of 9 years, 
 accumulated at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Cliapter lY. 
 
 Log. Present sum 59468-48 4-7594297 
 
 rtfZfZ Log. RN 9 years 0-1155350 
 
 4-8749647 
 
 Required future amount 
 
 £74983-335 
 
 Standard Form, No. 3. 
 
 No. (XVIII) 12. 
 
 To find the amount of loan which will be provided by the future 
 accumulation of the reduced annual instalment in conse- 
 quence of the withdrawal of part of the loan from the 
 operation of the fund. Table III. 
 
 Required the amount of an annual instalment of £295466 to be 
 set aside and accumulated for 9 years at 3 per cent. 
 
 (B) By Table III, 9 years, 3 per cent. Rule 2, Chapter YI. 
 
 Log. Amount of £1 per annum ... 101591 1-0068555 
 «^^ Log. annuity 2954-66 3-4705075 
 
 4-4773630 
 
 Required future amount 
 
 £30016-70 
 
478 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3. 
 
 No. (XVIII) 13. 
 
 To find the amount of loan represented by the adjustment to be 
 made in the annual instalment in consequence of an inten- 
 tional error introduced for purpose of calculation. Tablelll. 
 
 Required the amount of an annual instalment of £40"215 to 
 be set aside and accumulated for 9 years at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter YI. 
 
 Log. Annuity 40-215 r6043881 
 
 «^cZ Log. RN, 9years, +10 10-1155350 
 
 deduct Log. a" 
 
 Required future amount 
 
 11-7199231 
 91086795 
 
 2-6112436 
 
 £408-549 
 
 Standard Form, No. 4. 
 
 No. (XVIII) 14. 
 
 To find the present amount to be paid into the fund to 
 compensate for the intentional error introduced for purposes 
 of calculation. Table IV. 
 
 Required the present value of an annuity of £40215 for 9 years 
 at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. 
 Log. Annuity 
 
 Rule 3, Chapter y II. 
 40-215 1-6043881 
 
 add 10 
 
 deduct Log. a'*, 9 years ., 
 
 Required present value 
 
 11-6043881 
 9-1086795 
 
 2-4957086 
 
 £313-118 
 
APPENDIX 
 
 479 
 
 Standard Form, No. 3x. 
 
 No. (XVIII) 15. 
 
 To find the future annual increment to be accumulated to 
 provide tlie balance of loan not provided for by the present 
 investments representing the fund. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £47531-52 at the end of 9 years. 
 
 (C) By Thoman's Table, 3 per cent. Eule 3, Chapter XIII. 
 
 Log. Amount of loan ... 
 add. Log. a", 9 years 
 
 deduct Log. EN +10 .. 
 
 Required annual instalment 
 
 47531-52 4-6769818 
 9-1086795 
 
 13-7856613 
 10-1155350 
 
 3-6701263 
 
 £4678-71 
 
 Standard Form, No. 3. 
 
 No. (XIX) 1. 
 
 To find the portion of original loan which will be provided by 
 the future accumulation of the annual income from the 
 present investments. Table 111. 
 
 Required the amount of an annual income of £347 648 to be 
 added to the sinking fund and accumulated for 13 years 
 at 3 per cent. 
 
 (B) By Table III, 13 years, 3 per cent. Rule 2, Chapter VI. 
 
 Log. Amount of £1 per annum ... 15-6178 1-1936196 
 a^cZ Log. annuity 347648 2-5411397 
 
 3-7347593 
 
 Required future amount 
 
 £5429-494 
 
48o REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3. 
 
 >o. (XIX) 2. 
 
 To find the portion of tlie originai loan wliicli will be provided 
 by the future accumulation of the original instalment. 
 
 Table III. 
 
 Required the amount of an annual instalment of £680'234 to be 
 set aside and accumulated for l-J years at o per cent. 
 
 (C) By Thoman'sTable, 3per cent. 
 
 Rule 3, Chapter VI. 
 
 Log. Annuity 680-234 2-8826581 
 
 arZ^Z Log. RN, 13 years, + 10 10-1668839 
 
 deduct Log. a'^ .. 
 
 Required future amount 
 
 12-9995420 
 8-9732643 
 
 4-0262777 
 
 £10623-75 
 
 Standard Form, No. 3x. 
 
 ^0. (XIX) 3. 
 
 To find the additional sinking fund instalment to compensate 
 for a reduction in the rate of accumulation. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £509- 02 at the end of 13 years. 
 
 (B) By Table III, 13 years, :{ per cent. Rule 2, (liapter XIII. 
 
 Loff. Amount of h)an ... 
 
 509-02 2-7067348 
 
 deduct Log. amount of £1 per ami. 156178 1-1936196 
 
 1-5131152 
 
 Required annual instalment 
 
 £32-5923 
 
APPENDIX 
 
 481 
 
 Standard Form, No. 3. 
 
 No. (XIX) 4. 
 
 To find tlie portion of the orio^inal loan which will be provided 
 by the aceumulation of the amended annual increment. 
 
 Table III. 
 
 Required the amount of an annual increment of £1060'474 to 
 be added to the sinking- fund and accumulated for 1^ years 
 at 3 per cent. 
 
 (B) By Table III, 13 years, 3 per cent. Rule 2, Chapter YI. 
 
 Log. Amount of £1 per annum ... 15"617T9 1"1936196 
 add Log. annuity 1060-474 3-0255000 
 
 4-2191196 
 
 Required future amount 
 
 £16562-26 
 
 Standard Form, No. 3x. 
 
 No. (XIX) 5. 
 
 To find the amended annual instalment to repay the balance of 
 loan at the end of the period of repayment. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £1656226 at the end of 13 years, 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter XIII. 
 
 Log. Amount of loan 16562-26 4-2191196 
 
 flfZfZ Log. ««, 13 years 8-9732643 
 
 deduct Log. RN + 10 . 
 
 Required annual instalment 
 
 13-1923839 
 10-1668839 
 
 30255000 
 
 £1060-474 
 
 A G 
 
482 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Forw, No. 3. ^o- (XX) 1. 
 
 To find the amount of loan wliicli will be provided by the futiire 
 accumulation of the income from the present investments 
 representing the fund. Table III. 
 
 Required the amount of an annual income of £297'984 to be 
 added to the sinkino- fund and accumulated for 13 years 
 at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter YI. 
 
 Loc.. Annuity 297-984 2-4741929 
 
 add Lo^. RN, 13 years, +10 
 
 10-1668839 
 
 12-6410768 
 deduct Loo-, ^n 89732643 
 
 Required future amount 
 
 3-6678125 
 £4653-85 
 
 Standard Form No. 3x. ^»- (XX) 2. 
 
 To find the additional annual instalment to provide the amount 
 of loan unprovided for owing to a reduction in the rate of 
 income from the present investments. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £77564 at the end of 13 j-ears. 
 
 (B) By Table III, 13 years, 3 per cent. Rule 2, Chapter XIII. 
 
 Log. Amount of Loan 775-64 2-8896602 
 
 deduct Log. amount of £1 per ann. 1561779 1-1936196 
 
 1-6960406 
 
 Required annual instalment £49-664 
 
APPENDIX 
 
 483 
 
 Standard Form, No. 3. 
 
 No. (XX) 3. 
 
 To find the amount of loan wliirli will bo provided by the 
 accumulation of the future annual increment. Table III. 
 
 Required the amount of an anniml increment of £1060'4T4 to 
 be added to the sinking fund and accumulated for 13 years 
 at 3 per cent. 
 
 (B) By Table III, 13 years, 3 per cent. Eule 2, Chapter YI. 
 
 Log. Amount of £1 per anniini ... 156178 1-1936196 
 ^«W Loo-, annuity 1060474 3-0255000 
 
 Required future amount 
 
 4-2191196 
 £16562-26 
 
 Standard Form, No. 3x. 
 
 No. (XX) 4. 
 
 To find the amended annual instalment consequent upon a 
 variation in the rate of income upon the present invest- 
 ments, but without any variation in the rate of accumula- 
 tion. Table III. 
 
 Required the annual increment to be added to the sinking 
 fund and accumulated at 3 per cent, to provide £16562-26 
 at the end of 13 years. 
 
 (C) V>y Thoman's Table, 3 per cent. Rule 3, Chapter XIII. 
 
 Log. Amount of loan 1656226 4-2191196 
 
 rtrffZ Lop-. r/« 13 years 8-9732643 
 
 deduct Lost. R^+IO 
 
 Required annual instalment. 
 
 13-1923839 
 ,. 10-1668839 
 
 3-0255000 
 
 .. £1060-474 
 
REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3. No. (XXI) 1. 
 
 To find tlie amount of loan wliicli will be provided by the 
 accumulation of the annual income from tbe present invest- 
 ments under tbe altered conditions. Table III. 
 
 Required the amount of an annual income of <£297'984 to be 
 added to tbe sinkinfr fund and accumulated for 13 years at 
 2^ per cent. 
 
 (C) By Thoman's Table, 2^ per cent. Rule 3, Cbapter YI. 
 
 Log. Annuity 297-984 2-4741929 
 
 add Log. RN^ i:> years, +10 101394103 
 
 12-6136032 
 deduct Log. «« 8-9592717 
 
 3-6543315 
 
 Required future amount £4511-6094 
 
 Standard Form, No. 3. No. (XXI) 2. 
 
 To find tbe amount of loan wbicb will be provided by tbe 
 accumulation of tbe original annual instalment under tbe 
 altered conditions. Table III. 
 
 Required tbe amount of an annual instalment of £680234 to be 
 set aside and accumulated for 13 years at 2^ per cent. 
 
 (B) By Table III, 13 years, 2^ per cent. Rule 2, Cbapter VI. 
 
 Log. Amount of £1 per annum ... 1514044 1*1801386 
 «J^I Log. annuity 680234 28326581 
 
 40127967 
 
 Required future amount £10299038 
 
APPENDIX 485 
 
 Standard Form, No. 3. No. (XXI) 3. 
 
 To find tlie amount of loan wliicli will be provided by the 
 accumulation of the additional annual instalment under 
 the altered conditions. Table III. 
 
 Required the amount of an annual instalment of £32"592 to be 
 set aside and accumulated for 13 years at 2\ per cent. 
 
 (C) By Thoman's Table, 2\ per cent. Rule 3, Chapter VI. 
 
 Log. Annuity 32592 1-5131152 
 
 add Log. RN, 13 years, +10 101394103 
 
 11-6525255 
 deduct Log. a'' 89592717 
 
 2-6932538 
 
 Required future amount £493-462 
 
 Standard Form, No. 3x. No. (XXI) 4. 
 
 To find the annual instalment required to provide the balance 
 of loan which will be unprovided for owing to a reduction 
 in the rate of accumulation, etc. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 2i per cent, to provide £1258-15 at the end of 13 years. 
 
 (B) By Table III, 13 years, 2^ per cent. Rule 2, Chapter XIII. 
 
 Log. Amount of loan 125815 30997325 
 
 deduct Log. amount of £1 per ann. 15-14044 1-1801386 
 
 1-9195939 
 
 Required annual instalment £83-0986 
 
486 
 
 REPAYMENT OF LOCAL AND OTHER LOANvS 
 
 Standard Fortn, No. 3. 
 
 IVo. (XXI) 5. 
 
 To prove tliat the amended annual increment as ascertained 
 will complete tlie linal repayment of tlie loan under the 
 altered conditions. Table III. 
 
 llequired the amount of an annual increment of £1093'909 to he 
 added to the sinking fund and accumulated for V-^ years at 
 2\ per cent. 
 
 (B) By Table III, I'i years, 2^ per cent. Eule 2, Chapter VI. 
 
 Log. Amount of £1 per annum ... 15-14044 1-1801386 
 «^'/ Log. annuity 1093-909 3-0389812 
 
 4-2191198 
 
 llequired future amount 
 
 £16562-26 
 
 Standard Form, No. 1. 
 
 ]No. (XXIV) 1. 
 
 To find the anunint of loan which will be provided l)y the future 
 accumulation of the present investments representing the 
 fund. Table I. 
 
 llequired the amount of £9932-744 at the end of 8 years, 
 accumulated at 3^ per cent. 
 
 [\\) By Table I, 8 years, 3| per cent. llule 2, Chapter I\'. 
 
 Log. Amount of £1 
 
 add Log. present sum .. 
 
 1-31681 0-1195228 
 9932-744 3-9970693 
 
 41165921 
 
 llequired future amount 
 
 £13079-53 
 
APPENDIX 
 
 487 
 
 Standard Form, No. 3. 
 
 >o. (XXIV) 2. 
 
 To find the amount of loan whicli will be provided by tbc future 
 accumulation of tbe original annual instalment. Table III. 
 
 Required tbe amount of an annual instalment of £680'234 to be 
 set aside and accumulated for 8 years at 3| per cent. 
 
 (C) By Tboman's Table, 3i per cent. Rule 3, Chapter YI. 
 
 Log. Annuity 680-234 2-8326581 
 
 «^rf Log. RN 8 years, + 10 101195228 
 
 deduct Log. ci^ ... ... 
 
 Required future amount 
 
 12-9521809 
 ... 9-1627932 
 
 3-7893877 
 
 ... £6157-2614 
 
 Standard Form, No. 3x. 
 
 No. (XXIV) 3. 
 
 To find tbe additional annual instalment to be set aside and 
 added to tbe fund to compensate for tbe reduction in tbe 
 redemption period. Table 111. 
 
 Required tbe annual instalment to be set aside and accumulated 
 at 3i per cent, to provide £725821 at tbe end of 8 years. 
 
 (C) By Tboman's Table, 3^ per cent. Rule 3, Chapter XIII. 
 
 Log. Amount of loan ... 
 add Log. fl", 8 years 
 
 7258-21 3-8608294 
 9-1627932 
 
 deduct Log. RN+10 
 
 Required annual instalment... 
 
 130236226 
 101195228 
 
 2-9040998 
 
 £801-8624 
 
488 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3. 
 
 No. (XXIY) 4. 
 
 To find the amount of loan whicli will be provided by tlie future 
 accumulation of the present annual income from invest- 
 ments for the unexpired portion of the repayment period. 
 
 Table III. 
 
 Required the amount of an annual income of ^347'648 to be 
 added to the sinking fund and accumulated for 13 years at 
 3| per cent. 
 
 (B) By Table III, 13 years, 3^ per cent. Rule 2, Chapter YI. 
 
 Log. Amount of <£1 per annum ... 1611303 1-2071771 
 flfZfZ Log. annuity 347'648 2-5411397 
 
 Required future amount 
 
 3-7483168 
 £5601-66 
 
 Standard- Form, No. 3. 
 
 No. (XXIV) 5. 
 
 To find the amount of loan which will be provided by the future 
 accumulation of the present annual income from invest- 
 ments at the end of the substituted period of repayment. 
 
 fable III. 
 
 Required the amount of an annual income of £347-648 to be 
 added to the sinking fund and accumulated for 8 years at 
 3^ per cent. 
 
 (Cj By Thoman's Table, 3^ per cent. Rule 3, Chapter A'l. 
 
 Log. Annuity 
 
 add Log. 1{N, 8years, +10 
 
 deduct L()<r. a^ ... 
 
 Recjuired future amount 
 
 347-648 2-5411397 
 10-1195228 
 
 12-6606625 
 9-1627932 
 
 3-4978693 
 
 £3146-81 
 
APPENDIX 
 
 489 
 
 Standard Form, No. 3. 
 
 No. (XXIY) 6. 
 
 To find tlie amount of loan which will be provided at the end 
 of the substituted period of repayment by the accumulation 
 of the amended annual increment. Table III. 
 
 Required the amount of an annual increment of £1829" 744 to 
 be added to the sinking fund and accumulated for 8 years 
 at 3| per cent. 
 
 (B) By Table III, 8 years, 3i per cent. Rule 2, Chapter VI. 
 
 Log. Amount of £1 per annum ... 9-05169 0-9567296 
 acZtZ Log. annuity 1829744 ;j-2623904 
 
 Required future amount 
 
 4-2191200 
 £16562-26 
 
 Standard Form, No. 3. 
 
 No. (XXVI) 1. 
 
 To find the amount of loan which will be provided by the future 
 accumulation of the present annual instalment at the end 
 of the substituted period of repayment. Table III. 
 
 Required the amount of an annual instalment of £712-826 to be 
 set aside and accumulated for 8 years at o^ per cent, 
 
 (C) By Thoman's Table, o^ per cent.. Rule 3, Chapter VI. 
 
 Log. Annuity 
 
 add Log. RN, 8 years, + 10 . 
 
 deduct Loo;, a" ... 
 
 Required future amount 
 
 712-826 2-8529836 
 10-1195228 
 
 12-9725064 
 9-1627932 
 
 3-8097132 
 
 £6452-28 
 
490 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, No. 3x. No. (XXVI) 2. 
 
 To find the additioual annual instalment to be set aside and 
 added to the fund in consequence of a reduction in the 
 period of repayment accompanied by an increase in the 
 rate of accumulation. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3| per cent, to provide £696-j-19 at the end of 8 years. 
 
 (B) By Table III, 8 years, ^ per cent. Kule 2, Chapter XIII. 
 
 Log. Amount of loan 6963-19 3-8428082 
 
 deduct Log. amount of £l per ann. 9-06169 0-9567296 
 
 2-8860786 
 
 llequired annual instalment £769-270 
 
 Standard Form, No. 3. No. (XXVI) 3. 
 
 To find the amount of loan which will be provided by the 
 accumulation of the present annual increment. Table III. 
 
 Kcfjuired the amount of an annual increment of £1060-474 to be 
 added to the sinking fund and accumulated for 8 years at 
 3 per cent. 
 
 (B) ]Jy Table III, 8 years, 3 per cent. Kulc 2, Chapter VI 
 
 Log. Amount of £1 per annum ... 889234 09490159 
 add Log. annuity 1060474 3-0255000 
 
 3-9745159 
 
 llequired future amount £943009 
 
APPENDIX 
 
 491 
 
 Standard Form, No. 3. 
 
 No. (XXVI) 4. 
 
 To find the amount of loan wliieli will l>e provided by tlie 
 accumulation of the amended annual instalment. Table III. 
 
 Required the amount of an annual instalment of £816'232 to be 
 set aside and accumulated for 8 years at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter VI. 
 
 Log. Annuity 
 
 add Log. RN, 8 years, +10 
 
 deduct Loa;. a'^ 
 
 Required future amount 
 
 816-232 2-9118125 
 101026978 
 
 13-0145103 
 91536819 
 
 3-8608284 
 
 £7258-19 
 
 Standard Form, No. 3x. 
 
 No. (XXVI) .5. 
 
 To find the amount by which the annual instalment may be 
 reduced in consequence of a surplus of loan which will be 
 provided by an excessive annual instalment. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £126-04 at the end of 8 years. 
 
 (Cj By Thoman's Table, 3 per cent. Rule 3, Chapter XIII. 
 
 Log. Amount of loan 126-04 
 
 flc/^ Log. a", 8 years 
 
 deduct Loff. RN+10 
 
 Required annual instalment... 
 
 21005084 
 9-1536819 
 
 11-2541903 
 101026978 
 
 11514925 
 £14-174 
 
492 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, iVo. 3. 
 
 No. (XXVII) 1. 
 
 To find the portion of original loan wliicL. will be provided by 
 the future accumulation of the varying annual income from 
 the present investments, being the first stage in the method 
 by step Table III. 
 
 Required the amount of an annual income of £-)4T'648 to be 
 added to the sinking fund and accumulated for 8 years at 
 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter YI. 
 
 Log. Annuity 
 
 add Log. RN, 8 years, +10 
 
 deduct Log. a^ 
 
 Required future amount 
 
 347-648 2-5411397 
 10-1026978 
 
 12-6438375 
 91536819 
 
 3-4901556 
 
 £3091-403 
 
 Standard Form, 3 o. 1. 
 
 No. (XXVII) 2. 
 
 To find the accumulated amount, at the end of the unexpired 
 period, of the amount found in the previous calculation, 
 being the second stage in the method by step. Table I. 
 
 Required the amount of £3091-403 at the end of 5 years 
 accumulated at 3 per cent. 
 
 (C) By Thoman's Tabh', .'*, i)er cent. Rule 3, Chapter lY. 
 
 Log. Pieseiit sum 
 
 add Log. R^, 5 years .. 
 
 Required future amount 
 
 3091-403 3-4901556 
 00641861 
 
 3-5543417 
 
 £3583-783 
 
APPENDIX 493 
 
 Standard Form, No. 3. IVo. (XXVIT) 3. 
 
 To find the portion of original loan which will he provided hy 
 the futnre accumulation of the reduced annual income from 
 the present investments during the second part of the 
 unexpired repayment period. Table III. 
 
 Eeqiiired the amount of an annual income of £297"984 to he 
 added to the sinking fund and accumulated for 5 years at 
 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter VI. 
 
 Log. Annuity 297-984 2-4741929 
 
 acU Log. RN, 5 years, +10 10-0641861 
 
 12-5383790 
 deduct Loff. a« 9-3391623 
 
 31992167 
 
 Required future amount £1582-037 
 
 Standard Form, No. Sx. No. (XXVII) 5. 
 
 To find the additional annual instalment required in consequence 
 of a reduction in the rate of income from investments 
 during the later years of the unexpired period of repay- 
 ment. Table III. 
 
 Required the annual instalment to be set aside and accumulated 
 at 3 per cent, to provide £26367 at the end of 13 years. 
 
 (B) By Table III, 13 years, 3 per cent. Rule 2, Chapter XIII. 
 
 Log. Amou2it of loan 26367 2-4210607 
 
 deduct Log. amount of £1 per ann. 156178 1-1936196 
 
 1-2274411 
 
 Required annual instalment £16-8827 
 
 \ 
 
494 
 
 REPAYMENT OF LOCAL AND OTHER LOANS 
 
 Standard Form, ]\^o. 3.t. 
 
 No. (XXVII) 6. 
 
 To fiiul tlie equated annual inoome to he received over the whole 
 of the unexpired period which is equivalent to the varying 
 amounts of income to he received during the first and 
 second parts of such period respectively. Tahle III. 
 
 Eequired the annual instalment to he set aside and accumulated 
 at 3 per cent, to provide £5165"82 in 13 years. 
 
 (C) By Thoman's Table, 3 per cent. Rule 3, Chapter XIII. 
 
 Loo". Amount of loan ... 
 add Log. (7", 13 years 
 
 51G5-82 3T131393 
 8-9732643 
 
 dedmct Loo^. E^+IO 
 
 Eequired annual instalment.. 
 
 12-6864036 
 10-1668839 
 
 2-5195197 
 
 £330-765 
 
 Standard Form, No. 3. 
 
 No. (XXTII) 7. 
 
 To find the amount which will he in the fund at the end of the 
 first part of the unexpired period of repayment, being the 
 accumulation of the amended annual increment during 
 that period. Table III. 
 
 Eequired the amount of an annual increment of £1077-357 to be 
 added to the sinking fund and accumulated for 8 years at 
 3 per cent. 
 
 (B) By Table III, 8 years, 3 per cent. Eule 2, Chapter YI. 
 
 Log. Amount of £1 per annum ... 889234 0-9490159 
 r/rW Locj. annuitv 1077-357 3-0323597 
 
 Eecjuired future amount 
 
 3-9813756 
 £9580-22 
 
APPENDIX 
 
 495 
 
 Standard Form, No. 1. 
 
 No. (XXVII) 8. 
 
 To find the amount of loan wliich will be provided at the end 
 of the repayment period, bein^ the accumulation durinf,^ 
 the second part of such period of the amount in the fund at 
 the end of the first part. Table I. 
 
 Required the amount of £9580'220 at the end of 5 years 
 accumulated at 3 per cent. 
 
 (B) By Table I, 5 years, 3 per cent. Eule 2, Chapter IV. 
 
 Log. Amount of £1 
 
 add Log. present sum 
 
 Required future amount 
 
 11593 
 9580-22 
 
 0-0641861 
 3-9813756 
 
 40455617 
 
 £1110610 
 
 Standard Form, No. 3. 
 
 No. (XXVII) 9. 
 
 To find the amount of loan which will be provided by the 
 accumulation of the amended annual increment during the 
 second part of the unexpired repayment period. Table III. 
 
 Required the amount of an annual increment of £1027-693 to 
 be added to the sinking fund and accumulated for 5 years 
 at 3 per cent. 
 
 (C) By Thoman's Table, 3 per cent. 
 
 Log. Annuity 
 
 q^fZ Log. RN^ 5 years, + 10 
 
 Rule 3, Chapter VI. 
 
 1027-693 30118636 
 10 0641861 
 
 deduct Loo;, a^' 
 
 Required future amount 
 
 130760497 
 9-3391623 
 
 3-7368874 
 
 £5456165 
 
Index 
 
INDEX. 
 
 Accounting methods. See, Book-keeping methods. 
 
 Accounts. /See Bank accounts. 
 
 investment accounts. 
 pro forma accounts, 
 sinking fund accounts, 
 suspense accounts. 
 
 Accounts of Local Authorities, report of Departmental Committee (1907), xii. 
 
 Accounts, standard forms of, British, gas works, xii. 
 electric lighting, xii. 
 tramways, xii. 
 
 Accumulating sinking fund, referred to in all cases as the sinking fund unless 
 term non-accumulating is used, 126. Sec Sinking fund. 
 
 Accumulation, rate of. See Rate of accumulation. 
 
 Additional burden imposed, on equation, on the final years of the equated 
 period, as regards, the annua! instalment, 410. 
 interest upon the loan, 412. 
 the total annual loan charges, 413. 
 charts or diagrams, 454-7. 
 undertaking may have to incur outlay on renewals, 416. 
 
 Adju.stment, causes of, 7, 146. 
 
 Adjustment, methods of, 8, 17. 
 
 annual increment (balance of loan) method, 8, 152, 260. 
 
 annual increment (ratio) method, 9, 151, 263, 279. 
 
 deductive method, 224. 
 
 direct method, 237. 
 
 statement shewing full details of each adjustment will be found at the end 
 
 of each chapter, 8. 
 summary of methods will be found at the head of each chapter, 8. 
 
 Advantage of method by formula, 159. 
 
 Advantage of use of logarithms, 3, 21. 
 
 Algebraical formulpe. See Formulae. 
 
 Algebraical theory of indices, and its relation to logarithms, 24. 
 
 Alternative nature of the methods of repayment in Sec'tion 234 of the Public 
 Health Act, 1875, 6, 109, 110. 
 
 Amended annual increment. See Annual increment. 
 
 Amended annual instalment. See Annual instalment. 
 
 American experts to the National Civic Federation of New York, 1906, xi. 
 
 American readers, to, ix. • 
 
 Amount, as used in published tables, must not be confounded with a sum ot 
 money; a better term would be "accumulated amount" or "accumu- 
 late," 31. 
 
500 INDEX 
 
 Amount due at the end of any number of years : to provide the same by an 
 annual sum or annuity to be accumulated at compound interest, 126. 
 
 Amount in the sinking fund, deficiency, 154, 171. 
 surplus, 186, 199. 
 investments, present, 147. 
 
 Amount of an annuity. Sue Amount of one pound per annum. 
 
 Amount of one pound, the, in any number of years, arithmetical method, 37, 38. 
 derivation of formulje : mathematical, 29, 38. 
 
 Thoman's, 77. 
 formula : mathematical, 36. 
 
 Thoman's method, 77. 
 Inwood's table, No. 1, 36. 
 logarithmic methods of calculation, 36. 
 rate per cent, per annum, to find, 89. 
 
 ditto, standard calculation form, 89. ^ 
 
 rules for calculations : by formula, rule 1, 36. 
 
 by published tables of compound interest, rule 2, 36. 
 
 by Thoman's method, rule 3, 37. 
 standard calculation form, author's, No. 1, 40, 41, 88. 
 years, number of, to find, 89. 
 
 ditto, standard calculation form, 89. 
 
 Amount of one pound per annum, in any number of years, arithmetical method, 
 
 54-56. 
 derivation of formulte : mathematical, 55. 
 
 Thoman's, 77. 
 formulae : mathematical, 50. 
 
 Thoman's method, 73, 77. 
 Inwood's table. No. 3, 50. 
 logarithmic methods of calculation, 50. 
 rate per cent, per annum, to find, 94. 
 
 ditto, standard calculation form, 94. 
 rules for calculations : by formula, rule 1, 51. 
 
 by published tables of compound interest, rule 2, 51. 
 
 by Thoman's method, rule 3, 51. 
 standard calculation form, author's, No. 3, GO, 61, 93. 
 yea.rs, number of, to find, 94. 
 
 ditto, standard calculation form, 94. 
 
 Amount of one pound, and of one pound per annnum at end of one year, 
 comparison, 33. 
 
 Annual charges to revenue or rate, under the annuity method, 122. 
 instalment method, 113. 
 sinking fund methods, the accumulating fund, 139. 
 
 the non-accumulating fund, 137. 
 comparison under all jnethods, 140. 
 
 Annual incidence of taxation. Si-e Equation of the period. See Incidence of 
 taxation. 
 
 Annual income from inve.stments. See Income from investments. 
 
INDEX 501 
 
 Annual increment, advantages of methods, 149. 
 amended annual increment, definition of, 260. 
 balance of loan method, 152, 260. 
 definition of terms, 151, 175, 260. 
 future annual increment, definition of, 260. 
 methods, 8, 262. 
 
 present annual increment, definition of, 260. 
 problems, 259. 
 ratio method, 9, 151, 263. 
 variation in rate of income included in adjustment, 150. 
 
 Annual increment (balance of loan) method, compared with application of part 
 of fund in redemption of debt, 153, 175. 
 
 definition of terms, 260. 
 
 description of, 152. 
 
 rates of income and accumulation must be uniform during whole of repay- 
 ment period, 175. 
 
 summary of method, 261. 
 
 under this method amount in fund may vary from proper calculated 
 amount, 205. 
 
 Annual increment (ratio) method, assumed that the fund stands at the proper 
 
 calculated amount as shewn by the pro forma account, 227. 
 definition of terms, 260, 281. 
 description of method, 151, 263. 
 principles of method, 259, 279, 311. 
 
 rate of income from investments, merged in annual increment, 2? 8. 
 will apply equally to an increase or reduction, 282, 299. 
 rate of accumulation only : summary of method, 277. 
 
 rule, 277. 
 
 derivation of rule and formula, 278. 
 
 formula, 281. 
 
 pro forma account, 235. 
 
 method described, 279. 
 period of repayment only : summary of method, 283. 
 
 rule, 295. 
 
 derivation of rule and formula, 296. 
 
 formula, 299. 
 
 pro forma account, 294. 
 
 method described, 296. 
 rate of accumulation and period of repayment in combination : summary of 
 
 method, 300. 
 
 rule, 301. 
 
 derivation of rule and formula, 304. 
 
 formula, 306, 307. 
 
 pro forma account, 321. 
 
 method described, 303. 
 
 proof of method, 307. 
 
 Annual instalment, equal, of principal and interest combined : «n of Thoman 
 gives log values of, 118. 
 annuity method of repayment, 114. 
 annuity which one pound will purcliase, 67. 
 
5o:J INDEX 
 
 Annual instalment, equal, of principal and interest combined : apportionment 
 
 between capital and income, 122, 133. 
 compared with sinking fund instalment, 118. 
 formulae, 67. 
 
 hire purchase sy.stem. 111. 
 method of calculating, 115. 
 number of years, to find, 104 
 
 ditto, standard calculation form, 105. 
 rate per cent., to find, 105. 
 
 ditto, standard calculation form, 105. 
 relation to sinking fund instalment, 118. 
 repayment of loan by the annuity method, 120, 122. 
 rules, 114, 115. 
 
 standard calculation form, No. 5. 103. 
 statement shewing final repayment of loan, 122. 
 Thoman's method, 67, 68. 
 Annual instalment of the sinking fund : amended annual instalment, definition 
 
 of, 261. 
 apportionment of, in respect of loan borrowed for part only of first year, 
 
 365. 
 before and after equation of the period, 15, 418. 
 compared with equal annual instalment of principal and interest combined, 
 
 118. 
 formulae, 126, 127. 
 
 functions of annual instalment, 417. 
 future annual instalment, definition of, 261. 
 method of calculating, 131. 
 number of years, to find, 97. 
 
 ditto, standard calculation form, 97. 
 present annual instalment, definition of, 261. 
 problems relating to the, 145. 
 
 pro forma account of the normal accimiulation of the fund, 168. 
 rate per cent., to find, 98. 
 
 ditto, standard calculation form, 98. 
 relation to equal annual instalment of principal and interest combined, 118. 
 rules, 127. 
 standard calculation form. No. 3x, 96. 
 
 Annual instalment of the sinking fund, after equation : spread equally over the 
 equated period without any regard to the life of the asset, 14, 409. 
 should be spread over the equated period in proportion to the life of the 
 
 a.sset, 14, 416. 
 when it may be considered the equivalent of depreciation, 415. 
 
 Annual instalment of the sinking fund, before equation : spread over the period 
 
 in proportion to the life of the asset, 14, 378. 
 Annual in.-^talment of the sinking fund, fir.st. See Fir.^t annual instalment. 
 Annual sum, formula relating to, 2, 50. 
 Annuities set aside at end of year, 33. 
 
 Annuities, tables relating to : amount of, 50. 
 present value of, 62. 
 purchased with one pound, 67. 
 
INDEX 503 
 
 Annuities or other periodic payments, 51. 
 
 cannot be considered a pure geometrical progression, 52. 
 
 Annuity, amount of. Ste Amount of one pound per annum. 
 
 Annuity method of repayment, 114. 
 
 advantages and disadvantages, 115. 
 
 annual incidence of taxation, 122. 
 
 annual charges to revenue or rate compared with instalment and sinking 
 fund methods, 140. 
 
 apportionment of annual instalment as between principal and interest, 
 122, 133. 
 
 commercial form, hire purchase system, 111. 
 
 instalment method compared, 115, 121. 
 
 loanholder may out of equal annual instalment of principal and interest 
 combined, provide a sinking fund for redemption of his capital, 115, 133. 
 
 relation between the equal annual instalment and the sinking fund instal- 
 ment, 118. 
 
 sinking fund methods compared, 133. 
 
 table shewing the operation of the method and the annual incidence of 
 taxation, 122. 
 Annuity of which one pound is the present value. See Annuity which one 
 pound will purchase. 
 
 may be found by Thoman's factor an, 118. 
 
 Annuity, present value of. Sec Present value of one pound per annum. 
 
 Annuity which one pound will purchase, the; or of which one pound is the 
 
 present value : annuity method of repayment, relation to, 69. 
 derivation of formulfe : mathematical, 69. 
 
 Thoman's, 76. 
 equal annual instalment of principal and interest combined, relation to, 69. 
 formulae : mathematical, 67, 68. 
 
 Thoman's method, 73, 76. 
 Inwood's table. No. v, 67. 
 logarithmic methods of calculation, 67, 68. 
 rate per cent, per annum to find, 105. 
 
 ditto, standard calculation form, 105. 
 rules for calculations : by formula, rule 1, 68. 
 
 by published tables of compound interest, rule 2, 68. 
 
 by Thoman's method, rule 3, 68. 
 standard calculation form, author's. No. 5, 71, 72, 103. 
 Thoman's log factor an, 76. 
 years, number of, to find, 104. 
 
 ditto, standard calculation form 104. 
 
 Antilogarithm, 26. 
 
 Appendix, calculations, after Chapter xv, 459. 
 
 Application of sinking fund, in repayment of loans or redemption of stock. 
 
 See Loans repaid out of the sinking fund. 
 Apportionment : annual instalment or rent, method of finding apportioned 
 part, 29. 
 
504 INDEX 
 
 Apportionment : as between capital and income of equal annual instalment of 
 
 principal and interest combined, 122, 133. 
 effect of apportioning sinking fund instalment during year of borrowing, 
 
 365. 
 equal annual in.stalment of principal and interest combined, 122, 133. 
 method of finding any number of days' proportion of an annual instalment 
 
 or of an annual rent, 29. 
 of bank interest between several sinking funds, 352. 
 
 ditto, method of avoiding when only one bank account is kept, 352. 
 of income from investments as between several sinking funds, 352. 
 
 method of avoiding, etc., 352. 
 of part of instalment to be charged again.st year of borrowing, may be 
 
 advisable in case of large loans, 10, 366. 
 
 ditto, is generally ignored in case of small loans, 10, 365. 
 sinking fund instalment, method of finding apportioned part, 29. 
 
 Appreciation of assets, 381. 
 
 Arithmetical method of finding : equated annual income, 334. 
 
 error in method, 335. 
 
 example, 334. 
 
 pro forma account, 341. 
 period of repayment, 360, 392, 404. 
 
 error in method, 397, 405. 
 
 example.?, 392, 404. 
 
 may be preferred on equation as it extends period, 393, 402, 439. 
 proportionate part of annual instalment, 368. 
 
 Arithmetical progression : definition of, 22. 
 relation to geometrical progression, 22. 
 
 Assessment or local rate in Great Britain : levied on annual and not capital 
 value, xiii. 
 charged with any deficiency of loan charges not provided by profits of 
 trading departments, xv. 
 
 Asset : after equation of period, redemption charges should be governed by life 
 
 of, 416, 439. 
 appreciation of, 381. 
 depreciation of, 346, 385, 415. 
 
 life of, and its relation to the redemption period, 11, 377. 
 life of, generally ignored in fixing total redemption charges after equation 
 
 of period, 416. 
 
 ditto, should govern total redemption charges after equation of period, 
 416. 
 obsolescence, 346, 385. 
 proceeds of sale of, 186, 189. 
 supersession, 346, 385. 
 wastage of. See Depreciation. 
 
 Author's standard calculation forms. Set Calculation forms. 
 
 Balance of loan method of adjustment, and the annual increment, summary of, 
 152, 261. 
 
 Balance sheet, standard forms of, xii. 
 
INDEX 505 
 
 Bank accounts : only one account required for the whole of the sinking funds 
 of each department of the local authority, 351. See. Book-keeping 
 methods. 
 
 Bank, gain by, on discount of bills, 34. 
 
 Bank interest, apportionment of, between several sinking funds where only one 
 bank account kept, 352. 
 apportionment of, to avoid, 352. 
 
 Bankrupt or insolvent community, 383. 
 
 Bills, discount on : difference between practical discount and true or mathe- 
 matical discount, 34. 
 published tables of present values will not apply, 35. 
 
 Book-keeping methods : bank accounts, number of, 351. 
 bank interest, apportionment of, 352. 
 
 ditto, to avoid, 352. 
 commercial undertakings, 160. 
 
 deficiency to be adjusted annually by revenue or rate account, 352 
 equated loan interest reserve account, 445. 
 income from investments, apportionment of, 352. 
 
 ditto, to avoid, 352. 
 interest on loan after equation, reserve account, 445. 
 
 pro forma account, 447. 
 interest (sinking fund) suspense account, 352. 
 investment accounts, number of, 351. 
 
 loans redeemed out of sinking fund, interest to be added to the fund, 129. 
 loans redeemed out of sinking fund, should be treated as investments, and 
 
 not debited to the sinking fund, 351. 
 non-accumulating sinking fund, 138. 
 number of sinking funds required : loans repayable at various dates, 349. 
 
 objections to keeping only one account, 350. 
 
 stock redeemable on one date, 362. 
 pro forma accounts, preparation of, 5. See pro forma accounts. 
 
 should be kept in a separate book and not in the current ledger, 5. 
 separate sinking funds required, 362. 
 
 ditto, not required, 353, 363. 
 simplified on consolidation of loans, 414. 
 sinking fund interest suspense account, 138. 
 sinking funds should be earmarked, 350. 
 verification of calculations, by alternative method of proof, 18. 
 
 Borrowing : construction period, extended borrowing during, 347, 356, 391. 
 each year's borrowings treated as separate loan with separate sinking fund, 
 
 348. 
 first year of, and proportion of annual instalment, 365. 
 general practice, 345. 
 
 in advance of requirements, results of, 347. 
 may be complicated by variation in life of asset, 346. 
 relation between date of borrowing and redemption period, 345. 
 loan relating to outlay of one character only, borrowed over several years, 
 in one sum in each year, each year's borrowing being repayable 
 in prescribed period from date of borrowing, 345. 
 
5o6 INDEX 
 
 Borrowing : example to illustrate, 348. 
 particulars of loan, 349. 
 
 alternative methods of keeping sinking funds : one fund with increas- 
 ing instalment, 349. 
 objection to method, 350. 
 
 separate fund for each year's borrowings, 350. 
 each fund should be earmarked, with department, date of sanction and 
 
 loan, and period allowed, 350. 
 bank account, only one required, 351. 
 investment account, only one required, 351. 
 book-keeping methods, 352. 
 loan relating to outlay of one character only, borrowed over several years, 
 in one sum in each year, repayable in one sum on a specified date. 
 Where date of repayment is known at the time money borrowed, 
 354. 
 summary of methods, 354. 
 how it may arise, 356. 
 example to illustrate, 357. 
 particulars of loan, 358. 
 one sinking fund only required, 358. 
 increasing annual instalments, 358. 
 final repayment of loan, 359. 
 loan relating to outlay of one character only, borrowed over several years, 
 in one sum in each year, repayable in one sum on a specified date, 
 fixed after fund has been in operation a number of years, 359. 
 summary of methods, 355. 
 consolidation of existing loans, 356. 
 temporary instalments during construction, 357. 
 equation of period of repayment, 360. 
 example to illustrate, 360. 
 particulars of loan, 361. 
 sinking funds required, 362, 363. 
 proof of method, 364. 
 loan relating to outlay of one character only, borrowed at various dates 
 in one or more years, and it is required that the revenue or rate 
 account of each year shall be charged with a proportionate part of 
 the annual instalment, 
 adjustment not required in case of small loans, 365, 367. 
 may be necessary in case of large loans, 366. 
 method of adjustment, 367. 
 example to illustrate, 367. 
 
 arithmetical method of finding proportionate part of year, 368. 
 proof of method, 372. 
 
 compari.son with instalments when adjustment not made, 373. 
 effect of adjustment, 374. 
 
 Buildings, outlay upon, 380. 
 
 Burden. S'u: additional burden. 
 
 See earlier years of the equated period. 
 See final years of the equated period. 
 Sec post equated period. 
 
INDEX 507 
 
 Calculations, are numbered to shew, in bold type, the chapter to which they 
 relate, 
 on author's standard forms are given at the end of each chapter, 
 after Chapter xv are given in appendix only, 459. 
 
 should in all cases be verified by an alternative method of proof, 5, 17. 
 for periods other than years, 44, 81. 
 calculations other than on standard forms, 83, 84. 
 method by step, 183, 338, 429. 
 
 Calculation forms, author's standard, 4, 73. 
 advantage of, 81. 
 thi'ee methods on each form, 82. 
 amount of one pound. No. 1, 40, 41, 88. 
 
 calculations made on form, nature of, 87. 
 
 rate per cent., to find, 89. 
 
 number of years, to find, 89. 
 amount of one pound per annum, or annuity. No. 3, 60, 61, 93. 
 
 calculations made on form, nature of, 92. 
 
 rate per cent., to find, 94. 
 
 number of years, to find, 94. 
 annuity method of repayment, No. 5, 103. 
 
 rate per cent., to find, 105. 
 
 number of years, to find, 104. 
 annuity which one pound will purchase. No. 5, 103. 
 
 calculations made on form, nature of, 102. 
 
 rate per cent., to find, 105. 
 
 number of years, to find, 104. 
 present value of one pound. No. 2, 91. 
 
 calculations made on form, nature of, 90. 
 
 rate per cent., to find, 89. 
 
 number of years, to find, 89. 
 present value of one pound per annum, or annuity. No. 4, 65, 66, 99. 
 
 calculations made on form, nature of, 98. 
 
 rate per cent., to find, 101. 
 
 number of years, to find, 100. 
 rules for calculations. Stu liules. 
 sinking fund method of repayment, No. 3x, 96. 
 
 calculations made on form, nature of, 95. 
 
 rate per cent., to find, 98. 
 
 number of years, to find, 97. 
 special calculation forms, list of, 83. 
 
 Calculation of a typical fund, 158. 
 
 Calculations, rules for. Ser Rules for calculations. 
 
 Capital of communities, consists of power to levy a rate, 377 . 
 
 Causes of adjustment, 7, 146. 
 
 Causes of an equation of the period, 13, 413. 
 
 Cessation of annual contributions, 221. 
 
 Characteristic, the integral part of a logarithm, 25, 26. 
 negative, 26. 
 
 ditto, to divide or multiply a log with a, 26. 
 
5o8 INDEX 
 
 Charges, annual, to revenue or rate, under instalment method, 113. 
 annuity method, 122. 
 sinking fund methods, accumulating sinking fund, 139. 
 
 non-accumulating sinking fund, 137. 
 comparison under all methods, 140. 
 
 Charts, or diagrams, 454-457. 
 
 Clauses, model, of the Local Government Board, 6, 110. 
 
 Commercial and financial, or private undertakings : annual in.stalment not 
 
 always a charge upon protits, 191. 
 auditors and position of fund, 161. 
 book-keeping, sinking fund, methods, 160. 
 
 conditions as to repayment much more elastic than local authorities, 285. 
 conversion of loan debt into ordinary share capital or stock, 199, 203. 
 deferred payment system, 111. 
 deficiency in the sinking fund, 160. 
 general practice as to sinking funds, 109. 
 hire purchase system, 111. 
 incorporation of, xiv. 
 
 investment of fund in outside securities, 147, 160. 
 limited liability, xiv. 
 loan debt of, xiv. 
 
 loan repayment and depreciation kept distinct, 415. 
 methods of fixing annual instalment to sinking fund : by calculation, 203. 
 
 by a fixed annual sum, 203. 
 methods of repayment, 109. 
 
 outside investment, if not required, transactions are book entries only, 160. 
 parliamentary companies, xiv. 
 period of repayment, variation in, 285. 
 proceeds of sale of assets paid into fund, compared with local authorities, 
 
 189. 
 redemption by drawings, 220. 
 
 repayment of loans distinct from depreciation, 415. 
 reserve funds unlimited in amount, 385, 415. 
 share capital or stock, methods of raising, xiv. 
 sinking funds compared with those of local authorities, 202. 
 sinking fund not always specifically invested, 160. 
 surplus in fund and how corrected, 199. 
 
 surplus in fund arising on conversion of loan into ordinary share capital or 
 stock : where original instalment found by calculation, 204. 
 
 where original instalment a stated sum, 210. 
 unlimited liability, xiv. 
 withdrawal of part of loan from operation of fund, 203. 
 
 Committee on Investigation, National Civic Federation of New York, ix. 
 Companies or corporations in Great Britain, xiii. 
 
 Comparison of : amounts and i)resent values of one pouml and of one pound per 
 annum at end of one year, 33. 
 annual charges to revenue or rate inider all methods of repayment, 140. 
 annual instalments under annuity and sinking fund methods, 118. 
 
INDEX 509 
 
 Comparison of : arithmetical and mathematical methods of finding the equated 
 date of repayment, 392, 397. 
 
 depreciation and the sinking fund instalment, 385, 415. 
 
 local authorities and private undertakings, conditions imposed as to repay- 
 ment of loan debt, 109. 
 
 incidence of taxation before and after the equation of the period of repay- 
 ment, 413. 
 
 instalment method and non-accumulating sinking fund method, 135. 
 
 Compound interest, applied to : 
 
 a sum of money : amount of one pound, 36. 
 
 present value of one pound, 42. 
 an annual or other periodic payment : amount of one pound per annum, 50. 
 
 present value of one pound per annum, 62. 
 annuity which one pound will purchase or of which one pound is the 
 
 present value, 67. 
 derivation of formula relating to, 29. 
 formula, relating to a sum of money, 2. 
 
 ditto, an annual sum, 2. 
 geometrical progression, a, 29. 
 
 ditto, algebraical formula for, 30. 
 mathematical formula derived from algebraical formula for geometrical 
 
 progression, 31. 
 .symbols, explanation of, 30, 84. 
 tables of, varieties, 3. 
 Thoman's tables, 3. 
 
 Thoman's method and formulae are stated at the head of the Chapter 
 relating to each Table and also in Chapter ix. 
 Compound interest, published tables of. See Tables of compound interest. 
 Con.'^olidation of existing loans, additional burden on final years of equated 
 period, 410, 412, 413. 
 advantages of, 413. 
 amount in the sinking funds, 13, 361. 
 cause of an equation of the period of repayment, 11. 
 claim that it resulted in an immediate saving in rates, 17, 452. 
 effect on different departments of local authority, 11. 
 equitable method of adju.sting the annual incidence of taxation after 
 
 consolidation, 409, 418, 436. 
 incidence of taxation, annual, effect upon, 434, 449. 
 
 ditto, previous to consolidation should not be varied, 435. 
 incidence of loan charges on different departments of the local authority 
 
 should not be varied on consolidation, 11. 
 increased burden upon final years of equated period, after consolidation, 
 
 449. 
 interest upon the loan before and after consolidation, 412. 
 loanholder, consent of, 12. 
 
 method of ascertaining the equated date, 389. 
 powers of Local Government Board, 285. 
 relief to early years of equated period, 413. 
 
 ditto, post-equated period, 413. 
 sinking fund instalment before and after consolidation, 410. 
 total annual loan charges before and after con.solidation, 413. 
 
5IO INDEX 
 
 Consols, variation in the rate of interest on, 225, 327. 
 
 Construction period, and its relation to the sinking fund instalment, 347, 356. 
 adjustment of fund required when works completed, 356, 391. 
 estimated amounts only can be set aside during construction, 391. 
 matter complicated when works include portions with varying periods of 
 
 continuing utility, with varying periods of repayment, 346. .See 
 
 Temporary instalments. 
 
 Continuation of annual instalments, 222. 
 
 Continuing utility. ,SV^ Asset. Life of asset. 
 
 Conversion of part of loan debt of a private undertaking into share capital or 
 stock, and its effect upon the sinking fund, 199. 
 
 Corporations in Great Britain, include local authorities and private under- 
 takings, xiii. 
 
 Correct method of di.'itributing annual burden after equation of period, 427, 
 436, 449. 
 
 Correction of a deficiency in the sinking fund of a local authority or private 
 undertaking : by additional annual instalment spread equally over 
 whole of unexpired portion of repayment period, 173. 
 by additional annual instalment spread over earlier part only of unexpired 
 
 portion of repayment period. 179. 
 by an immediate payment into the fund, 162. 
 
 Correction of a surplus in the sinking fund : of a local authority, 186. 
 of a private undertaking, 199. 
 
 Co.sts of obtaining powers : effect of equation of period to extend repayment to 
 full term, 15. 
 periods allowed for repayment generally short, 129. 
 
 Dates of borrowing : in one sum in each year, over several years, each yearly 
 
 amount being repayable over a similar period, 345. 
 ditto, the whole loan being repayable in one sum on a specified date, 354. 
 in several sums in one year, and it is required to charge the revenue or 
 
 rate account of the year of borrowing with an apportioned part of one 
 
 year's annual instalment, 365. 
 may be complicated by a variation in the life of the asset, 346. 
 apportionment of annual instalment in year of borrowing, 365. 
 relation between date of borrowing and the redemption period, 345. 
 sinking fund, loan generally treated as being borrowed at end of financial 
 
 year and full annual instalment set aside at end of following year, 365. 
 
 See also Construction period. 
 
 Date of repayment, 10, 285, 345, 354. 
 
 Deductive method of adju.stment, 224, 283. 
 
 Deferred payment system, a commercial form of the instalment method of 
 repayment, 111. 
 
 Deferred sinking fund, 391. 
 
INDEX 511 
 
 Deficiency in the fund : causes, 7, 154, 161. 
 
 compared with method of adjusting a surplus, 193, 219. 
 
 depreciation in value of investments, 161. 
 
 if small in amount should be corrected annually, 146. 
 
 income from investments, decrease in, 161. 
 
 methods of adjustment, summary of, 155. 
 
 methods will apply to a surplus, 226. 
 
 methods of correcting, 161. 
 
 by payment of deficiency into fund, 162. 
 
 by additional instalment spread over whole of unexpired period, 162. 
 summary of method, 171. 
 method described, 173. 
 pro forma account, 178. 
 by additional instalment spread over early years only of unexpired 
 period, 162. 
 summary of method, 172. 
 method described, 179. 
 pro forma account, 185. 
 Stock Eegulations, County, 1891, Art. 11 (2), 161. 
 
 Departments of local authority : one investment account and one bank account 
 only required for each, 351. 
 should be kept distinct on consolidation of loans, 11. 
 
 Departmental committee on the accounts of local authorities, report dated 
 
 1907, xii. 
 Depreciation of assets : and its relation to the sinking fund instalment, 346. 
 may be considered as provided by the annual instalment when the period 
 
 of repayment is fixed with regard to the life of the asset, 385, 415. 
 should be charged against revenue or rate account when total loan has 
 been repaid by means of the sinking fund, 386. 
 Derivation of formute and rules (annual increment (ratio) method) : rate of 
 accumulation, variation in, 277. 
 period of redemption, variation in, 295. 
 
 rate of accumulation and period of redemption in combination, variation 
 in, 301. 
 Derivation of formula (mathematical methods) : amount of one pound, 36. 
 amount of one pound per annum, 50. 
 annuity method of repayment, 114. 
 annuity which one pound will purchase or of which one pound is the 
 
 present value, 67. 
 equal annual instalment of principal and interest combined, 67. 
 present value of one pound, 42. 
 present value of one pound per annum, 62. 
 sinking fund method of repayment, 126. 
 Diagrams or charts, shewing incidence of taxation before and after the equation 
 
 of the period, 454-457. 
 Difference between the amounts of £1 and of £1 per annum at the end of one 
 
 year, 33. 
 Direct method of adjustment, 237. 
 
512 INDEX 
 
 Discount, mathematical or true, compared with practical discount, 34, 35. 
 Discount on bills, compared with mathematical or true discount, 34. 
 Discount practical, published tables of present values will not apply, 35. 
 Division, by logarithms, 23, 24. 
 
 Drawing.?, redemption of loans by, in case of commercial and financial under- 
 takings, 220. 
 Duration of continuing utility, or the life of the asset and its relation to the 
 
 redemption period and the incidence of taxation. 111, 377, 389. 
 main factor in fixing the original period of repayment of the loan, 378, 
 
 390, 413. 
 Local Government Board, powers of, as to determining the period, 111, 390. 
 ignored generally when fixing the equated period of repayment of loans in 
 
 respect of outlays of varying nature ; and also on consolidation of 
 
 existing loans, 416. 
 Earlier years of the equated period : relief, on equation, annual instalment 
 
 only, 410, 449. 
 interest on loan, unaltered on equation, 412, 449. 
 total annual loan charges, 413, 449. 
 corresponding burden on final years of equated period, 413. 
 
 Early provisions as to repayment of local loans, 109, 379. 
 
 Effect of the generally adopted method after equation as regards : the annual 
 
 instalment, 15, 410, 418. Chart I, 454-5. 
 the interest on the loan, 16, 412, 436. Chart I, 454-5. 
 the total annual loan charges, 16, 413, 449. Charts II and III, 456-7. 
 the incidence of taxation, 15, 16, 409. 
 saving in annual rates claimed on equation, 17, 452. 
 loans in respect of outlay having a short life, not repaid by the time the 
 
 asset is worn out, 420. 
 reborrowing in respect of assets having a short life, 15, 420. 
 
 Electric lighting undertakings : standard forms of account, xii. 
 
 wide divergence in charge by different municipalities, 383. 
 
 See also Revenue earning undertakings. 
 Enquiry into the municipal and private ownership and operation of public 
 
 utilities in Great Britain ; by the National Civic Federation of New York 
 
 (1906), ix. 
 Equal annual instalment of principal and interest combined : may be found by 
 Thoman's factor an, 89. 
 
 See Annuity which one pound will purchase, 67. 
 
 Equated annual income from investments. 330. 
 arithmetical method of finding, 334. 
 error in method, 335. 
 pro forma account, 341. 
 
 Equated date, method of ascertaining, adopted by Local Government Board, 
 402, 403. 
 generally fixed by method known as the equation of payments, 392, 404. 
 error in method, 397, 405. 
 true or mathematical method described, 394, 399, 404. 
 
INDEX 513 
 
 Equated loan interest, reserve account, after equation of period, 445. 
 pro forma account, 447. 
 
 Equated period : necessity to fix, how arising, 13, 389. 
 
 arithmetical method or the equation of payments, 14, 334, 392, 404. 
 arithmetical method gives longer period, 402. 
 true or mathematical method, 394, 404. 
 
 summary of method, 399. 
 on the consolidation of loans, 389. 
 for repayment of loans, raised by the issue of stock, in respect of outlay 
 
 upon works having varying periods of continuing utility, 390. 
 relief to early years on equation of period in respect of the : annual instal- 
 ment, 15, 410, 449. 
 interest on the loan, 412, 449. 
 total annual loan charges, 413, 449. 
 charts or diagrams, 454—457. 
 further borrowing powers after equation, 420. 
 
 additional burden imposed after equation of period upon the final years of 
 the equated period in respect of the : annual instalment, 15, 410, 
 449. 
 interest on the loan, 16, 412, 449. 
 total annual loan charges, 413, 449. 
 charts or diagrams, 454-7. 
 number of years in period depends upon amount of loan and respective 
 
 repayment periods, 405. 
 See Earlier years of equated period. 
 
 Equation, method of : as generally adopted, 14, 410. 
 effect of, 15, 413. 
 true or mathematical method, 394, 399, 404. 
 
 Equation of payments, the arithmetical method in general use for finding the 
 equated date, 14, 334, 392, 404. 
 
 Equation of the incidence of taxation : before equation the annual instalment 
 and interest on the loan are borne by future years in proportion to the 
 life of the asset, 438. 
 after equation, life of asset generally ignored, 434. 
 
 the unequal incidence of taxation if the annual instalment and interest on 
 the loan are spread equally over the equated period without any 
 regard to the life of the asset, 409. 
 remedy as to the annual instalment, 15, 431. 
 ditto, interest on the loan, 16, 443. 
 method of continuing the original annual instalments during the equated 
 period, and spreading the supplementary annual instalment, represent- 
 ing the relief to the post-equated period, equally over the equated 
 period, 15, 16, 424. 
 method of spreading the burden over the equated period strictly in propor- 
 tion to the life of the asset before equation, 16. 
 the annual instalment, 431. 
 the intere-st on the loan, 443. 
 the total annual loan charges, 449. 
 claim to reduce rates on consolidation, 17, 452. 
 
514 INDEX 
 
 Equation of the incidence of taxation : cannot be made by reducing each period 
 in proportion to the reduction in the final period, 419. 
 difference between financial obligation to repay loan, and the charge to 
 
 revenue or rate based upon life of asset, 419. 
 charts, 454—457. 
 
 Equation of the period of repayment : and the life of the asset, 11, 389. 
 
 and the incidence of taxation, 15, 409. 
 
 on the consolidation of existing loans, 391. 
 
 claim to reduce rates on consolidation, 17, 452. 
 
 in respect of a loan authorised for outlays having varying periods of con- 
 tinuing utility and consequent repayment, 309. 
 
 and effect in case of renewals, 416. 
 
 postponed repayment of loans in respect of outlay having a life shorter 
 than the repayment period, 15, 420. 
 
 causes giving rise to equation, 13, 413. 
 
 life of asset enters into calculation on equation but effect lost by equal 
 annual instalment, 424. 
 
 a financial operation only, 409, 415. 
 
 effect of equation, 409, 411, 434. 
 
 Evolution by logarithms, 24. 
 Expenses of sinking fund, 129. 
 
 Final years of the equated period. See Additional burden, on equation, and 
 Equation of the incidence of taxation. 
 
 Final repayment of the loan, 8. .S'ec Statements which will be found at the 
 end of each chapter dealing with an adjustment in a smking fund, ^'ee 
 also the several pro forma accovmts. 
 
 Finance Act, 1888, 225, 327. 
 
 Financial details of undertakings in Great Britain examined by the Committee 
 of Investigation of the National Civic Federation of New York in 1906, x. 
 
 Financial nature of the equation of the period, 415, 453. 
 
 Financial undertakings. See Commercial undertakings. 
 
 First annual instalment to sinking fund : method of ascertaining the propor- 
 tionate part of the annual instalment in respect of moneys borrowed 
 for part of a year to be charged against the rate or revenue account of 
 the year of borrowing, 365. 
 usually set aside at the end of the first complete year and no charge made 
 against the year in wliich the loan is borrowed, 365. 
 
 Forms of accounts. See Accounts. 
 
 Forms for calculations, author's standard, these forms are fully described and 
 explained in Chapter x. 
 
 FormuliE (annual increment ratio method) : rate of accumulation, variation in, 
 
 281. 
 period of redemption, variation in, 299. 
 rate of accumulation and period of redemption in combination, variation 
 
 in, 307. 
 
INDEX 515 
 
 Formula : advantages of method by, 3, 158. 
 annuities or other periodic payments, 50. 
 geometrical progression, 2, 30. 
 compound interest, 2, 31. 
 
 simple interest, 28. 
 differ from those in actuarial works, 3. 
 
 Formulae (mathematical methods) : amount of one pound, 36. 
 amount of one pound per annum, 50. 
 annuity method of repayment, 67, 114. 
 annuity which one pound will purchase or of which one pound is the 
 
 present value, 67, 68. 
 equal annual instalment, of principal and interest combined, 67, 68. 
 present value of one pound, 42. 
 present value of one pound per annum, 62. 
 sinking fund method of repayment, 127. 
 derivation of. See derivation of formulfe. 
 
 Future amount, present value of. See Present value of one pound. 
 
 Future annual income from investments, definition of, 260. -S'ee Income from 
 investments. 
 
 Future annual increment, definition of, 260. 
 
 Future annual instalment, definition of, 261. 
 
 Future generations, liability to provide further utilities, 378, 384. 
 safeguards to, 14, 378, 384, 410, 412, 413. 
 
 Future rate of accumulation, definition of, 260. 
 
 Gas works, profits from, applied in aid of rates, 383. 
 
 wide divergence of charges by different municipalities, 383. 
 
 standard forms of account, xii. 
 
 See also under Revenue earning undertakings. 
 
 Geometrical progression : algebraical formula for, 2, 30. 
 basis of formula relating to compound interest, 2, 30. 
 definition of, 22, 30. 
 
 derivation of formula relating to compound interest, 30, 31. 
 example of, 22. 
 relation to logarithms, 22. 
 
 Goschen (Finance Act, 1888), 225, 327. 
 
 Hire purchase system, a commercial form of the annuity method of repayment, 
 111. 
 
 Incidence of loan charges on different departments of a local authority, on 
 
 consolidation of loans, 11. 
 Incidence of taxation. See Ratepayer. 
 Incidence of taxation, equation of the. See Equation of the incidence of 
 
 taxation. 
 
5i6 INDEX 
 
 Incidence, annual, of taxation, under various methods of repayment : instal- 
 ment method, 113. 
 annuity method, 122. 
 sinking fund methods : the accumulating fund, 139. 
 
 the non-accumulating fund, 137. 
 
 comparison under all methods, 140. 
 
 Income from investments : consols, variation in the rate of interest on, 225, 327. 
 future annual income, definition of, 149, 260. 
 present annual income, definition of, 260. 
 non-accumulating sinking fund, 135. 
 
 rate of, variation in : uniform during whole of period, 236. 
 not, ditto, 322. 
 See also Variations. 
 
 Increment, annual, 148. See Annual increment. 
 
 Indices, algebraical theory of, 24. 
 
 Injustice to ratepayers, of final years of equated period. ^ee Additional 
 burden, ratepayer. 
 
 Instalment, equal annual, of principal and interest combined, may be found by 
 Thoman's factor an, 118. See Annual instalment, equal, of principal and 
 interest combined. 
 
 Instalment method of repayment, 109. 
 
 annual incidence of taxation, under this method, 113. 
 
 annual charges, to revenue or rate, compared with : annuity method, 115, 
 121. 
 
 sinking fund methods, 128. 
 commercial form, the deferred payment system. 111. 
 decreasing annual charge to revenue or rate, 112. 
 insurance companies, 111. 
 loanholder, frequent reinvestment by, 112. 
 Public Works Loan Commissioners, 111. 
 no accumulating sinking fund, 113. 
 
 relation to the non-accumulating sinking fund method, 135. 
 statement shewing the operation of the method and the annual incidence 
 
 of taxation, 113. 
 
 Instalment, annual, to sinking fund, problems, 145. 
 
 Interest : compound, a geometrical progression, 29. 
 simple, an arithmetical progression, 28. 
 
 Interest, bank. See Bank interest. 
 
 Interest from investments. See Income from investments. 
 
 Interest of one pound, for one year, 45, 52. 
 
 basis of finding tlie formula relating to the accumulation of an annual or 
 
 other periodic sum, 52. 
 Symbol r, 32, 85. 
 values and corresponding logs for 49 rates per cent., from ^ per cent, to 
 
 7 per cent.. Table V, A, Chapter V, 48, 49. 
 
 Interest suspense account, sinking fund, method of keeping, relating to one 
 bank account, and one investment account for several sinking funds, 352. 
 
INDEX 517 
 
 Interest upon the loan : after repayment of loan out of the accumulatmg sink- 
 
 ijig fund to be added to the fund, in whole or in part, 129. 
 annuity method, apportionment of equal animal instalment as between 
 
 principal and interest, 122. 
 annuity method, decreasmg annual amount of interest included in the 
 
 equal annual instalment repaid to the lender, 115. 
 annuity method, loanholder may provide a sinkmg fund to equalise his 
 
 annual interest, 133. 
 charge to revenue or rate account, under all methods of repayment, 140. 
 
 before and after consolidation, 412, 436. 
 
 before and after equation of period, 412, 436. 
 
 effect of the equation of the period, 416. 
 consent of loanholder, on consolidation, 12. 
 instalment method, decreasing annual amount of interest paid to the 
 
 lender, 112, 113. 
 need not be added to non-accumulating fund after repayment of loans, 137. 
 the incidence of taxation, 416, 436. 
 
 reserve account, to equalise the incidence of taxation after the equation of 
 the period of repayment, pro forma account, 447. 
 
 book-keepmg methods, 445. 
 sinking fund method, constant annual amount of interest paid to the 
 
 lender, 129. 
 variation in the mcidence of taxation after the equation of the period of 
 
 repayment, 436. 
 
 Introduction, 1. 
 
 Investment accounts : only one investment account required for the whole of 
 the sinking funds of each department of a local authority, 351. 
 should be raised in the ledger for all loans repaid or redeemed out of the 
 sinking fund, 351. iSee also Book-keeping methods. 
 
 Investment of the sinking fund in loans of the same local authority, 130. 
 
 Investments representing the sinking fund and the income arising therefrom. 
 
 See Income from investments. 
 Investments, value of, an important factor in any adjustment of a sinkuag 
 
 fund, 147. 
 Investments, rate of income from. -S'ee Kate of income from investments. 
 Involution, by logarithms, 24. 
 
 Inwood's tables, of compound interest, etc., new edition by Schooling, 3. 
 Irredeemable stock, 12. 
 Irregular contributions to the sinking fund in earlier years. See Construction 
 
 period. See Temporary instalments to the sinking fund. 
 
 Land, appreciation in value, 382. 
 acquisition by way of lease, 380. 
 period allowed for repayment of loan, 380. 
 
 Law relating to the repayment of the loan debt of local authorities, is not 
 included in this book except so far as it relates to the actual methods of 
 repayment, 1. 
 
5i8 INDEX 
 
 Leasehold projjerties, outlay upon, by local authorities, 380. 
 
 Life, or duration of continuing utility, of the asset created out of the loan, IL 
 and its relation to the redemption period, 377. 
 after the equation of the period of repayment, -116, 439. 
 See also Assets. 
 
 Limitation of the period of repayment, in the case of works of almost per- 
 manent utility, 14. 
 
 Limited Liability Acts (Great Britain), xiv. 
 
 Loan charges, annual, charged against profits of trading departments : any 
 deficiency made good out of annual rates or assessments, xv. 
 
 Loan debt of local authorities, how raised : mortgages for short terms, 414. 
 stock, 390. 
 security for, 383. 
 
 Loan debt of local authorities, how repayable : by instalment method, 109. 
 by annuity method, 114. 
 by sinking fund methods, 126. 
 at end of short periods, mortgages, 414. 
 on a specified date (stocks), 390. 
 out of the sulking fund, 129. 
 See reborrowing. 
 
 Loan debt of private undertakings, how repayable. See Loan debt of local 
 authorities, commercial undertakings. 
 
 Loan, final repayment of. See Statements shewing, etc., pro forma accounts. 
 
 Loanholder : and the equation of the period of rejaayment, 414. 
 
 annuity method, difficulty in ascertaining the amount of principal included 
 
 in each annual in.stalment, 115. 
 ditto, effect of annual repayments, 115. 
 may provide a sinkmg fund for the redemption of his capital out of the 
 
 equal annual mstalment of principal and interest combined, 115. 
 consent required on consolidation of existing loans, 12, 383. 
 fluctuating nature of investment under the instalment and annuity methods, 
 
 112, 115. 
 interests considered on equation of period, 409. 
 instalment method, effect of repayment by, 112. 
 commercial and financial undertakings, conversion of loan into ordinary 
 
 share capital or stock, 203, 210, 213. 
 security for local loans, 383. 
 
 preferential nature of repayment out of sinking fund, 409. 
 reinvestment, frequent, of caj)ital, under the instalment and annuity 
 
 methods, 112, 115. 
 sinking fund method, a permanent investment, 134. 
 option to convert into ordinary share capital or stock, 203. 
 
 Loans, consolidation of. See Consolidation of loans. 
 
 Loans, conversion of, of a private undertaking, into ordinary share capital or 
 stock, 199. 
 
 Loans fund, definition, 129. 
 
INDEX 
 
 519 
 
 Loans of local authorities : how raised, and how repayable. Sac l^oan debt of 
 local authorities. 
 
 Loans in respect of outlay having a short life, not repaid after equation by the 
 time the asset is worn out, 15, 420, 435. Sac also Reborrowing. 
 
 Loans repaid and stock redeemed out of the sinking fund : application of whole 
 
 or part of fund authorised by Public Health Act, 1875, Sec. 234 (5), 
 
 129. 
 ditto, non-accumulating sinking fund, 137. 
 interest to be added to the fund to make up the deficiency in the fund 
 
 which would be caused by such application of the fund, 130. 
 interest to be added to the fund to be equivalent to the interest which 
 
 would liave been produced by the amount so applied, 130. 
 need not be added in the case of the non-accumulating fund, 137. 
 should be treated as an investment of the fund, and not debited to the 
 
 sinking fund account, 28, 130. 
 Stock purchased at a premium ; par value only may be taken out of the 
 
 fund, the premium to be charged against the current year's revenue 
 
 or rate account or against a reserve created for the purpose, i30. 
 
 Local authorities, accounts of, report of the Departmental Committee, 1907, xii. 
 
 Local authorities, principles as to borrowing and repayment laid down in the 
 Public Health Act, 1875, 110. 
 
 Local authority, definition of term, xiii. 
 
 Local Government Board : auditors, and the pro forma account, 5. 
 County Stock Regulations, 1891, deficiency in fund, 161. 
 Local Government Act, 1888, 285. 
 
 method of equating the period of repayment, actual example, 402. 
 model clauses (1893), See. 
 
 not represented on parliamentary committees, 379. 
 powers as to period of repayment. 111. 
 powers as to permanency of works, 110. 
 powers, on consolidation of loans, 285. 
 practice as to fixing periods of repayment, 1. 
 pro forma accounts, copies should be sent to Board, in relation to all loans 
 
 coming under the supervision of the Board, 5. 
 Public Health Act, 1875, powers under, 110. 
 Public Health Act, Amendment Act, 1890, powers under, 285. 
 sale of assets, proceeds of, 190. 
 supervision by Board avoided by proceeding by Special Act, 379. 
 
 Local rate, or assessment in Great Britain, levied on the annual value, and 
 not on the capital value, xiii, 17. 
 
 Logarithms, Chapter II. : advantages of, 3, 21. 
 algebraical theory of indices, 24. 
 antilog, to find, 26. 
 
 arithmetical progression, definition of 22. 
 Briggean or decimal logs, 22. 
 characteristic, definition of, 25. 
 
 of numbers greater or less than unity, rules, 26. 
 common or decimal logarithms, 22. 
 
520 INDEX 
 
 Logarithms : definition of, 23. 
 
 divide one log by another, to, 27, 398. 
 division by, 23, 24. 
 earliest published tables of, 21. 
 evolution by, 24. 
 
 fractional part or mantissa, always positive, 24. 
 geometrical progression, algebraical formula, 2. 
 ditto, definition of, 22. 
 ditto, relation to compound interest, 2. 
 history of, 21. 
 
 interest of one pound for one year for 49 rates per cent, from ^ per cent. 
 7 per cent, logs of. Table V, A, 48, 49. 
 
 integral part or characteristic may be either positive or negative, 26 
 involution by logs, 24. 
 
 mantissa, definition of, 24. 
 always positive, 24. 
 
 multiplication by logs, 23, 24. 
 
 negative characteristic, rule to multiply or divide a log with, 26. 
 
 ratios for 49 rates per cent, from ^ per cent, to 7 per cent. Table, Va, 48, 49. 
 
 relation between arithmetical and geometrical progression, 22, 23. 
 
 tables of common logarithms, 21. 
 
 Thoman's logs of an increased by 10 to avoid negative characteristic, 75, 79. 
 
 to find, of even powers of 10, 23. 
 Logarithmic formulae and rules. iSce Formulae, rules. 
 
 Logarithmic tables, of compound interest and annuities, Thoman's, 73. 
 of numbers from 1 to 108,000 to seven places of decimals, 26. 
 
 Loss on sale of investments representing the sinking fund, 161. 
 Mantissa, the fractional part of a logarithm, 24. 
 Manufacturing plants, outlay upon, 384. 
 Markets, outlay upon, 381. 
 
 increasing value of site, 381. 
 Mathematical formulae and rules. Sec Formulte. 
 Mathematical method of equation of period, 394, 399, 404. .'Sec Equated date, 
 
 methods of obtaining, 
 ^lathematical or true discount, compared with practical discount, 34. 
 Mathematical method by formula, indispensable if rate per cent, not given in 
 
 any published table, 4, 5, 159. 
 Mathematical principles, 2, 19. 
 Mathematical tables, varieties of, 3. Schooling, 3. Inwood's, 3. Thoman's, 73. 
 
 Sprague's, 3. 
 Method by step, of finding the amount of an annuity for a term of years accu- 
 mulated for a further period, 183, 338, 429. 
 Methods of adjustment described : annual increment methods, 8. 
 
 annual increment (balance of loan) method, 8, 152, 260. 
 
 ditto, (ratio), 9, 151, 263, 279. 
 
 deductive method, 224. 
 
 direct method, 237. 
 
 statement shewing full details of each adjustment will be found at the end 
 of the Chapter, 8. 
 
 summary of methods will be found at the head of each Chapter. 
 
INDEX 521 
 
 Methods of ascertaining equated date. ISee Equated date. 
 
 Methods of book-keeping. 6'ee Book-keeping. 
 
 Methods of calculation, 82. 
 
 Methods of repayment of loan debt, 107. Sat Loan debt of local authorities. 
 
 Model clauses of the Local Government Board, 6, 110. 
 
 Mortgages, for short terms, 414. 
 
 Multiplication, by logarithms, 23. 
 
 Municipal and private ownership and operation of public utilities in Great 
 
 Britain ; enquiry by the National Civic Federation of New York, 1906 ; 
 
 report, 1907, ix. 
 Municipal trading, ix. 
 
 National Civic Federation of New York, ix. 
 
 enquiry into the municipal and private ownership and operation of public 
 
 utilities in Great Britain, 1906, ix. 
 financial details of undertakings examined, x. 
 committee of investigation, list of members, ix. 
 experts engaged on the enquiry, xi. 
 report, dated 1907, xi. 
 Non-accumulating sinking fund : accumulation of the fund under this method, 
 137. 
 advantages and disadvantages, 136, 137, 138. 
 annual incidence of loan charges, 136. 
 application of part of fund in repayment of debt, 137. 
 book-keeping methods, 138. 
 
 compared with instalment and annuity methods, 136. 
 description of the method, 135. 
 first introduced in the model clauses of the Local Government Board, 
 
 1893, 6, 110. 
 income from investments, how treated, 135. 
 objects of, 135. 
 
 relation to the instalment method, 135. 
 
 statement shewing the operation of the method and the annual incidence 
 of taxation, 137. 
 Number of bank accounts required. -S'ee Bank accounts. Book-keeping methods. 
 Number of investment accounts required. See Investment accounts. Book- 
 keeping methods. 
 Number of sinking funds required : in respect of loan borrowed over a series 
 of years and repayable at different dates over a similar period, objec- 
 tions to keeping one sinking fund only, 350. 
 in respect of a loan borrowed over a series of years, redeemable in one sum 
 on a certain specified date, 353. 
 Number of years to find. .See Calculation forms. 
 
 may be found exactly by formula in the following cases : amount of one 
 pound, 89. 
 present value of one pound, 89. 
 amount of one pound per annum, 94. 
 sinking fund instalment, 97. 
 
522 INDEX 
 
 Number of years to find : may be found approximately by reference to the 
 published tables in the following cases : present value of one 
 pound per annum, 100. 
 annuity one pound will purchase, 104. 
 method fully described, 397. 
 in equated period depends upon two factors, 405. 
 
 Objections to keeping one sinking fund for loans repayable at different dates, 
 350. 
 
 Obsolescence of assets, 346. See Assets. 
 
 Occupation of lands by local authorities by lease rather than purchase, 380. 
 
 One pound, amount of. See Amount of one pound. 
 
 One pound, interest of, for one year. See Interest of one pound, etc. 
 
 One pound, present value of. See Present value of one pound. 
 
 One pound per annum, amount of. See Amount of one pound per annum. 
 
 One pound per annum, present value of. See Present value of one pound per 
 
 annum. 
 Outlay having short period, and effect of equation, 15, 420, 425. 435. 
 
 Outlay having varying periods of repayment included in one sanction, 346, 390. 
 
 Outlay upon manufacturing plants, 384. 
 
 Outlay upon renewals during later years of equated period, 416. 
 
 Ownership and operation of public utilities in Great Britain, ix. 
 
 Parks and open spaces, outlay upon, 380. 
 
 Parliament, variation in conditions allowed, 379. 
 investment of the sinking fund, 130. 
 final approval required for all loans, 379. 
 committees of, 379. 
 policy as regards repayment periods, 382. 
 
 Parliamentary Companies in Great Britain, xiv. 
 
 Past rate of accumulation, definition of, 260. See Rate of accumulation. 
 
 Payments, equation of. See Equation of payments. 
 
 Periodic payments. See Annuities. 
 
 Periods shorter than one year, method of calculation, 44. 
 
 Period of construction and relation to sinking fund. See Construction 
 period. 
 
 Period of continuing utility. See Asset, life of asset. 
 
 Period of repayment, equation of. See Equation of the period. 
 
INDEX 
 
 Period of redemption or repayment : 
 
 commercial and linanciai undertakings, 285. 
 determined with relation to the life of the asset, 377. 
 equation of the period a financial operation, 415, 453. 
 may be varied on consolidation of loans, 285. 
 rarely altered in case of individual loans of local authority, 285. 
 relation to the date of borrowing, 345. 
 sub.stituted repayment period, definition, 261. 
 unexpired repayment period, definition, 261. 
 variable nature of conditions now existing, 379. 
 
 variation in the period without any variation in the rate of accumu- 
 lation : summary of method, 283. 
 
 rule, 295. 
 
 derivation of rule and formula, 296. 
 
 formula, 299. 
 
 pro forma account, 294. 
 
 method de.scribed, 286, 296. 
 variation in the period with a variation in the rate of accumulation : 
 
 summary of method, 300. 
 
 rule, 301. 
 
 derivation of rule and formula, 304. 
 
 formula, 306, 307. 
 
 pro forma account, 321. 
 
 method described, 303. 
 
 proof of method, 307. 
 
 Permanence of works. See Asset, life of. Local Government Board. 
 
 Permanent utility, works of almost, limit imposed as to period of repay- 
 ment, 14, 380, 382. 
 
 Perpetual debt of local authorities, 379. 
 
 Plants, manufacturing, outlay on, 384. 
 
 Post equated period, relief to, should be spread over the equated period 
 
 in proportion to the life of the asset, 416, 449. 
 under present method on equation is spread equally over the equated 
 
 period, 409, 449. 
 relieved of all charges in resiject of annual instalment and interest on 
 
 loan, 413, 449. 
 charts or diagrams, 454-7. 
 
 Practical discount compared with true or mathematical discount, 34, 35. 
 published tables of present values will not apply, 35. 
 
 Preface, v. 
 
 Preface, to American readers, ix. 
 
 Preferential nature of repayment by means of sinking fund, 409. 
 
 Preliminary stages in sanction of a loan, care bestowed to find the proper 
 
 period to be allowed for repayment, 14, 415. 
 Premium, redemption of stock, at a, 221. 
 
 Premium paid on purchase of stock out of sinking finid, 12. 
 may not be taken out of sinking fund, 130. 
 
 523 
 
524 INDEX 
 
 Present annual income from investments. Sec Income from investments. 
 
 definition of, 260. 
 Present annual increment, definition of. Sec Annual increment. 
 Present annual instalment, definition of. .S'er Annual instalment. 
 Present investments, value of, 147. 
 
 Present sum, to find the amount of. See Amount of one pound. 
 Present value of one pound, the, due at the end of any number of years, 42. 
 
 derivation of formulae, mathematical, 43. 
 Thoman's, 44. 
 
 formulae, mathematical, 42. 
 Thoman's method, 73, 77. 
 
 Inwood's table, No. 2, 42. 
 
 logarithmic method of calculation, 42. 
 
 rate per cent, per annum, to find. 89. 
 
 ditto, .standard calculation form, 89. 
 
 rules for calculations, by formula, rule 1, 43. 
 
 by published tables of compound interest, rule 2, 43. 
 by Thoman's method, rule 3, 43. 
 
 standard calculation form, author's. No. 2, 46, 47, 91. 
 
 years, number of, to find, 89. 
 
 ditto, standard calculation form, 89. 
 
 Present value of one pound per annum, the, for any number of years, 62. 
 derivation of formulae, mathematical, 63. 
 
 Thoman's, 78. 
 formulae, mathematical, 62. 
 
 Thoman's method, 73, 78. 
 Inwood's table. No. 4, 62. 
 logarithmic methods of calculation, 62. 
 rate per cent per annum, to find, 101. 
 ditto, standard calculation form, 101. 
 rules for calculations, by formula, rule 1, 63. 
 
 by published tables of compound interest, rule 2, 63. 
 
 by Thoman's method, rule 3, 63. 
 standard calculation form, author's, No. 4, 65, 66, 99. 
 years, number of, to find, 100. 
 ditto, standard calculation form, 100. 
 
 Principles governing the borrowing and repayment of the loan debt of 
 
 local authorities, xi, 110. 
 Private ownership and operation of public utilities, ix. 
 Private undertakings. See Commercial undertakings. 
 
 Problems, sinking fund, the annual instalment, Sec. III. 
 the amount in the fund, 145. 
 
 a deficiency, 154, 171. 
 
 a surplus, 186, 199. 
 the rate of income from investments, 236, 322. 
 the rate of accumulation, 223. 
 the rates of income and accumulation, 247. 
 
INDEX 
 
 525 
 
 Problems, sinking fund, the annual increment. Section IV. 
 
 the rate of accumulation, 259, 277. 
 
 the redemption period, 283, 295. 
 
 the rate of accumulation and the period in combination, 300. 
 Proceeds of sale of assets, forming part of the security for the loan, 7, 189. 
 
 applied in repayment of loan, 189. 
 
 may be paid into sinking fund and accelerate final repayment, 190. 
 
 may be applied in reduction of future annual instalment over whole 
 or part of unexpired period, 190. 
 Profits of trading departments applied in aid of rate, 381, 383. 
 Profit and loss account, standard forms of, xii. 
 Profit on sale of investments representing the sinking fund, 188. 
 Pro forma accounts, should be prepared in all cases to shew final repay- 
 ment, 5, 17, 145. 
 
 should be kept in separate book, 5. 
 
 amended account should be prepared after any future adju.stment of 
 the fund, 5. 
 
 copy should be sent to the Local Government Board, 5. 
 
 Local Government Board Auditors, 145. 
 Pro forma accounts : — 
 
 1. Normal accumulation of the fund, 168. 
 
 2. Correction of a deficiency by additional annual instalment spread 
 
 over whole of unexpired period, 178. 
 
 3. Correction of a deficiency by additional annual instalment spread 
 
 over early years only of unexpired period, 185. 
 
 4. Correction of a surplus by a reduced annual instalment spread 
 
 over whole of unexpired period, after payment into the fund of 
 proceeds of sale of assets, 198. 
 
 5. Correction of a surplus by a reduced annual instalment spread over 
 
 whole of unexpired period, after withdrawal of part of loan 
 from operation of fund, 209. 
 
 6. Correction of a surplus by a reduced annual instalment spread 
 
 over whole of unexpired period, after withdrawal of part of 
 loan from operation of fund. 219. 
 
 7. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of unexpired period, consequent upon a reduction in 
 the rate of accumulation, without any variation in the rate of 
 income from investments, 235. 
 
 8. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of unexpired period consequent upon a reduction in 
 the rate of income from investments without any variation in 
 the rate of accumulation, 246. 
 
 9. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of unexpired period consequent upon a variation in 
 the rates of income from investments and accumulation, 255. 
 
 10. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of substituted period consequent upon a variation 
 in the period of repayment, 294. 
 
 11. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of .substituted period consequent upon a variation in 
 the rate of accumulation and in the period of repayment, 321. 
 
526 INDEX 
 
 Pro forma accounts : — 
 
 12. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of unexpired period consequent upon a variation in 
 the rate of income from investments to occur at a known future 
 date, 339. 
 
 13. Correction of a deficiency by an increased annual instalment spread 
 
 over whole of unexpired period consequent upon a variation in 
 the rate of income from investments to occur at a known future 
 date, based upon the equated annual income, 3-tl. 
 ' Equated loan interest. Reserve account : shewing the method of 
 charging the equal annual amounts of interest payable to the 
 loanholders, after equation, to the revenue or rate accounts of 
 each year, in proportion to the life of the a.sset, 447. 
 
 Progressions, arithmetical, 28. 
 
 geometrical, 29, 30. 
 Proportionate part, of annual instalment, 10. 
 
 Public Health Act, 1875, alternative methods of repayment, 110. 
 expenses of sinking fund, 129. 
 interest on part of sinking fund applied in redemption of debt to be 
 
 paid into the fund, 129. 
 limit to total amount to be borrowed. 111. 
 limitation of period for purposes of the Act, 111. 
 Local Government Board to decide for what purposes money may be 
 
 borrowed, 389. 
 methods of repayment, 110. 
 
 permanency of works must be taken into account. 111, 390. 
 power to apply whole or part of sinking fund in redemption of debt, 
 
 129. 
 power to borrow limited, 110. 
 provisions as to borrowing and repayment of loans of local authorities, 
 
 6, 109. 
 regulations as to exercise of borrowing powers, 109. 
 
 Public Health Acts Amendment Act, 1890, 285, 390. 
 
 Public utilities, early Acts of Parliament and repayment obligations, 379. 
 manufacturing plants, 384. 
 
 many loans outstanding without any obligation as to repayment, 379. 
 markets, 382. 
 
 National Civic Federation of New York, enquiry, ix. 
 outlay by local authorities, a national question, 379. 
 outlay upon works of permanent utility .should not be charged against 
 
 present generation, arguments for and against, 380. 
 provided by pledging credit of community, 377. 
 regard for future requirements, 378, 381. 
 
 repayment of outlay spread over period equivalent to life of asset, 378. 
 safeguard to future generations, 14, 378, 384, 410. 412, 413. 
 sewage disposal, 381. 
 
 should be paid for in proportion to annual benefit received, 377. 
 tramways, 383. 
 
 want of uniformity of practice in loan conditions now existing, 379. 
 waterworks, 382. 
 
INDEX 527 
 
 Public Works Loan Commissioners, and life of asset, 390. 
 and Local Government Board, 390. 
 instalment method. 111. 
 
 Published tables of compound interest. Sec Tables of compound interest. 
 
 Rate of accumulation, general considerations, 225. 
 
 should be estimated on the low side, in the case of loans with long 
 
 repayment periods, 148, 326. 
 may be estimated slightly lower to allow for a reduction in the future 
 
 rate of income from investments, 326. 
 future rate of accumulation, definition, 260. 
 past rate of accumulation, definition, 260. 
 
 variation in, without any variation in the rate of income from invest- 
 ments, summary of methods of adjustment, 223. 
 
 method described, 230. 
 
 pro forma account, 235. 
 
 annual increment (ratio) method, 277. 
 variation in, and also in the rate of income from investments, sum- 
 mary of methods of adjustment, 247. 
 
 methods described, 248. 
 
 pro forma account, 255. 
 variation in, and also in period of repayment, summary of methods 
 of adjustment, 300. 
 
 methods described, 303. 
 
 pro forma account, 321. 
 
 annual increment (ratio) method, 300. 
 
 Eate of income from investments : 
 
 equated annual income, 330. 
 arithmetical method of finding, 334. 
 error in method, 335. 
 • pro forma account, 341. 
 variation in, without any variation in the rate of accumulation, 
 summary of methods of adjustment, 236. 
 methods described, 237, 239. 
 pro forma account, 246. 
 variation in, and also in the rate of accumulation, summary of methods 
 of adjustment, 247. 
 methods described, 248. 
 pro forma account, 255. 
 not uniform during unexpired repayment period, 322, 327. 
 variation known at time of making the adjustment, summary of 
 method, 323. 
 method described, 328. 
 pro forma account, 339. 
 variation anticipated, but uncertain as to time and amount, summary 
 of method, 325. 
 method described, 332. 
 future equated annual income, 330. 
 pro forma account, 341. 
 
528 INDEX 
 
 Ratepayer, accumulating sinking fund method, effect upon annual rates, 
 134. 
 additional burden on equation, 410, 412, 413. 
 annual contribution towards benefits received, 377, 409. 
 annuity method, effect upon annual rates, 115, 134. 
 depreciation should be charged against future generation?, 386. 
 immediate relief on equation of period, 409. 
 injustice to future generations after equation, 15, 410, 412, 413. 
 instalment method, effect upon annual rates, 112. 134. 
 interests on equation of period, 11, 12, 409. 
 
 non-accumulating sinking fund, effect upon annual rates, 136. 
 profits in aid of rate, 381, 383. 
 
 protection of future generations, 14, 378, 384, 410, 412, 413. 
 provision in advance for future generations, 378. 
 relief to post equated period, 409, 413, 449. 
 renewals during final years of equated period, 416. 
 repayment for assets of permanent utility, 380. 
 want of interest in municipal matters, 13, 383. 
 
 Rate per cent, general considerations as to, 225. 
 
 Rate per cent, to find. See Calculation forms. 
 
 may be found exactly by formula in the following cases : 
 
 amount of one pound, 89. 
 
 present value of one pound, 89. 
 may be found approximately by reference to the published tables in 
 the following cases : 
 
 amount of one pound per annum, 94. 
 
 sinking fund instalment, 98. 
 
 present value of one pound per annum, 101. 
 
 annuity one pound will purchase, 104. 
 
 Rate per cent., variations in, rate of accumulation only, 223. 
 rate of income from investments only, 236. 
 rate of accumulation and income in combination, 247. 
 statement showing original and amended conditions in all variations 
 considered, 265, 267. 
 
 Ratio, definition of, in algebraical formula relating to a geometrical 
 
 progression, 30, 45, 52. 
 in mathematical formula relating to compound interest, 31. 
 values and corresponding logs, for 49 rates per cent, from \ per cent. 
 
 to 7 per cent.. Table V, A, 48, 49. 
 
 Ratio method of adjustment. The annual increment. See Annual incre- 
 ment (ratio) method. 
 
 Ready reckoner tables of compound interest, etc., 3. 
 
 Reborrowing, in respect of assets having a short life which are worn out 
 before the equated date, 15, 420, 425, 435. 
 
 Redemption of .stock at a premium, 221. 
 Redemption of loan in part, 221. 
 
INDEX 
 
 529 
 
 Redemption by drawings, in case of commercial and financial under- 
 takings, 220. 
 
 Redemption fund, definition of, 129. 
 
 Redemption of stock by purchase out of the sinking fund. Sec Loans, 
 etc., repaid out of the sinking fund. 
 
 Redemption period. Sec Period of redemption or repayment. 
 
 Relation between amount of one pound and of one pound per annum, 53. 
 
 Relation between life of asset and period of repayment, 14, 377. 
 
 Relief to early years of equated period as regards the : 
 annual instalment, 410. 
 interest on the loan, 412. 
 total annual loan charges, 413. 
 interest on the loan is the same as before equation, 412. 
 relief is entirely in respect of the annual in!5taiment, and is at expense 
 of later years of equated period, 411, 449. 
 
 Relief to rates on consolidation of loans, instance given in evidence before 
 a Parliamentary Committee, 17, 452. 
 
 Renewals, effect upon later years of equated period, 416. 
 
 Repairs and renewals fund, 385. 
 
 Repayment, date of, 9, 10, 345. 
 
 Repayment of debt and its relation to depreciation. See Depreciation of 
 assets. 
 
 Repayment of loans out of moneys in the sinking fund. See Loans repaid 
 out of the sinking fund. 
 
 Repayment, final, of the loan. See Statements, which will be found at the 
 end of each chapter dealing with an adjustment in a sinking fund. 
 See also the several pro forma accounts. 
 
 Repayment period. Sec Period of redemption or repayment. 
 
 Repayment, period of, equation of the. See Equation of the period of 
 repayment. 
 
 Repayment of loans by local authorities (1902), report upon, 402. 
 
 Reports, Departmental Committee on the Accounts of Local Authorities, 
 1907, xii. 
 
 National Civic Federation of New York on the municipal and private 
 
 ownership and operation of public utilities in Great Britain, ix. 
 Repayment of Loans by Local Authorities (1902), 402. 
 
 Reserve account for interest on loans after equation, 447. 
 
 Revenue account, standard forms of, xii. 
 
 Revenue earning undertakings : appreciation in value, 384 
 compared with .spending departments, 415. 
 
 depreciation, covered by sinking fund when repayment period within 
 life of asset, 385, 
 
 A J 
 
530 INDEX 
 
 Revenue earning undertakings: depreciation should be charged when original 
 loan repaid out of sinking fund, 385. 
 land, outlay upon, 384. 
 obsolete plant, 385. 
 outlay on manufacturing plants, 384. 
 outlay upon repairs, 384. 
 profits applied in aid of rate, 383, 415. 
 renewals fund, 385. 
 standard forms of accounts, xii. 
 wastage of asset, 384, 385. 
 wide divergence of charges by different municipalities, 383. 
 
 Rules for calculations (annual increment (ratio) method) •. 
 rate of accumulation, variation in, 277. 
 period of redemption, variation in, 295. 
 
 rate of accumulation and period of redemption in combination, vana 
 tion in, 301. 
 
 Rules for calculations (mathematical) : 
 amount of one pound, 36. 
 amount of one pound per annum, 51. 
 annuity method of repayment, 68, 114. 
 annuity which one pound will purchase, or of which one pound is 
 
 the present value, 68. 
 equal annual ini=talment of principal and interest combined. 68, 114. 
 present value of one pound, 42. 
 present value of one pound per annum, 63. 
 sinking fund method of repayment, 126. 
 Thoman's methods, are given at the head of each chapter. 
 
 Sale of assets, ^ee Proceeds of sale of assets. 
 
 Sale of investments representing the sinking fund, loss on sale, 161. 
 profit on sale, 188. 
 
 Sanction of a loan, preliminary stages in the ; care bestowed to find the 
 proper period to be allowed for repayment, 14, 415. 
 
 Sanction for works comprising various classes of outlay may specify 
 amount and period in respect of each class, or only total amount with 
 an equated period, 346, 390. 
 
 Saving in annual rates on consolidation, 17, 452. 
 
 Schooling, William, F.R.A.S., author of the new edition of Inwood's 
 Tables (1899), 3. 
 
 Security for municipal loans, a mixed fund of capital and revenue, 383. 
 
 Separate sinking funds. See Number of sinking funds. 
 
 should be kept in all cases where loans are repayable at various 
 dates, 350, 362. 
 
 Share capital or stock of private undertakings, methods of raising, xiv. 
 preferences and priorities, xiv. 
 conversion of loan debt into, 199. 
 
 Short-term mortgages, 414. 
 
INDEX 531 
 
 Simple interest, arithmetical progression, 28. 
 formulae and rules, 28. 
 
 rule to find number of days' proportion of an instalment or other 
 annual sum, 29. 
 
 Sinking fund : 
 
 accounts, number of, 10, 350, 353. 
 accumulation, rate of,. See Rate of accumulation, 
 accumulating sinking fund, 128. 
 adjustment, methods of. See Adjustments, 
 amount in the fund, deficiency, 154, 171. 
 
 surplus, 186, 199. 
 annual charges to revenue or rate, 137, 139. 
 annual incidence of taxation, 139. 
 annual income from investments. Sec Income, 
 annual increment, 149, 175, 239, 262. 
 balance of loan method, 152, 260. 
 ratio method, 151, 263, 279. 
 annual instalment, method of calculating, 131. 
 before and after equation, 15, 418. 
 first, 365. 
 annuity method, comparison, 133. 
 application of, in repayment of debt, 129. 
 apportionment of one year's instalment, 365. 
 balance of loan method of adjustment, 152, 260. 
 bank accounts, 351. 
 bank interest, 352. 
 book-keeping methods, see. 
 borrowing, dates of, see. 
 calculation of a typical sinking fund, 158. 
 commercial undertakings, see. 
 
 comparison with annuity and instalment methods, 133. 
 consolidation of loans, sec. 
 construction, period of, 347, 391. 
 correction of a deficiency in the fund, 162, 173, 179. 
 correction of a surplus in the fund, 186, 199. 
 dates of borrowing, see. 
 deductive method of adjustment, 224, 283. 
 deficiency in the fund, 7, 154, 162, 173, 179. 
 depreciation, relation to, 346, 385, 415. 
 derivation of formulae, 131. 
 direct method of adjustment, 237. 
 equation of the incidence of taxation, 409. 
 equation of the period of repayment, 389. 
 expenses of, 129. 
 
 final repayment of the loan, 139, 168. 
 financial undertakings, see commercial, 
 first annual instalment, 365 
 formulae, 127. 
 
 incidence of taxation, 377, 409. 
 income from investments, see. 
 instalment, annual, sec annual instalment. 
 
53i INDEX 
 
 Sinking fund (continued) : 
 
 interest suspense account, 352. 
 
 investment accounts, 351. 
 
 investments in loans of same local authority, 130. 
 
 investments of the fund, 130, 147, 351. 
 
 investments, rate of income from, see income. 
 
 investments, sale of, 188. 
 
 investments, value of, 147. 
 
 irregular contributions to the fund, 346, 356, 367, 391. 
 
 loan, final repayment of, 139, 168. 
 
 Loans Fund, 129. 
 
 loans repaid out of the sinking fund, 129. 
 
 loss on sale of investment of the, 161. 
 
 method of repayment, 126. 
 
 methods of adjustment, see adjustments. 
 
 normal accumulation of the fund, pro forma account, 168. 
 
 non-accumulating sinking fund, 6, 135. 
 
 ditto, compared with instalment method, 135. 
 
 number of bank accounts, 351. 
 
 number of investment accounts, 351. 
 
 number of sinking funds required, 349. 
 
 number of years, to find, 97. 
 
 period of construction, 347, 391. 
 
 period of repayment, 283, 295, 300, 345. 
 
 period of repayment, equation of the, see. 
 
 position of, to ascertain, 160. 
 
 premium paid on redemption of stock, 130. 
 
 present investments, 147. 
 
 private undertakings, ^ee commercial. 
 
 problems relating to the sinking fund, 145. 
 
 proceeds of sale of assets, 7, 189. 
 
 profit on sale of investments, 188. 
 
 pro forma account, normal accumulation, 168. 
 
 Public Health Act, 1875, see. 
 
 rate of accumulation, see. 
 
 rate of income from investments, see rate of income. 
 
 rate per cent., to find, 98. 
 
 ratio method of adjustment, sec annual increment (ratio) method. 
 
 Redemption Fund, 129. 
 
 redemption of debt, out of sinking fund, 129. 
 
 redemption period, 283, 295, 300, 345. 
 
 repayment, final, of the loan, 139, 168. 
 
 rules for calculating annual instalment, 127. 
 
 sale of assets, 7, 189. 
 
 sale of investments, 188. 
 
 separate funds for each year's borrowings, 348, 350. 
 
 statements shewing the final repayment of the loan, 139, 168. 
 
 Stock redeemed out of the sinking fund, see loans repaid, etc. 
 
 summary of methods, see methods. 
 
 surplus in the .sinking fund, 186, 199. 
 
 suspense account for bank interest and income from investments, 352, 
 
inde:x 
 
 53^ 
 
 Sinking fund (continued) : 
 
 temporary instalments to the sinking fund, 347, 391. 
 typical sinking fund, 158. 
 variations in the sinking fund, see. 
 years, number of, to find, 97. 
 
 Sinking fund method of repayment, annual charges to revenue or rate 
 compared with the instalment method, 133. 
 the annuity method, 133. 
 table shewing the operation of the method and the annual incidence 
 of taxation, 139. 
 
 Sinking fund problems, the annual instalment, 145. 
 the annual increment, 259. 
 the date of borrowing and the redemption period, 345. 
 
 Spending departments compared with revenue earning undertakings, 415. 
 
 Sprague's tables of compound interest, 3. 
 
 Standard calculation forms, author's. See Calculation forms. 
 
 Standard forms of account, xii. 
 
 Statement shewing the final repayment of tlie loan will be found after 
 each adjustment. Sea pro forma account. 
 
 Statement shewing the methods of adjustment will be found at the end 
 of each chapter. 
 
 Step, method by, of finding the amount of annuity for a term of years, 
 accumulated for a further period, 183, 338, 429. 
 
 Stock or .share capital of private undertakings in Great Britain, xiv. 
 
 Stock redeemed out of the sinking fund. Sec Loans redeemed, etc. 
 
 Stock regulations, cessation of annual contributions, 222. 
 a deficiency in the fund, 161. 
 necessity to equate period of repayment, 390. 
 
 Stocks issued by local authorities, the principal cause of the equation of 
 the period of repayment, 13, 390. 
 
 Stocks of local authorities generally, .b'ec under the various headings of 
 "Loans," "Loan Debt," etc. 
 
 Subject matter of book, 1. 
 
 Substituted repayment period, definition, 261. 
 
 Summary of methods will be found at the head of the chapter dealing with 
 each adjustment, 
 annual increment (balance of loan) method, 8, 152, 260. 
 annual increment (ratio) method, 9, 151, 263, 279. 
 deductive method, 224 
 direct method, 237. 
 
 Supersession of assets, 346. 
 
534 
 
 INDEX 
 
 Surplus in the fund, causes of, 7, 188. 
 
 compared with methods of adjusting a deficiency, 193, 219 
 excessive past accimiulation of fund, 188, 189. 
 methods of adjustment, summary of, 186. 
 methods will apply to a deficiency, 226. 
 of a local authority and how applied, 189, 190. 
 payment into fund of proceeds of sale of assets, 189. 
 summary of method, 187. 
 method described, 189. 
 realised profit on sale of an investment, 188. 
 summary of method, 187. 
 method described, 189. 
 withdrawal of part of loan from operation of fund, 188, 199. 
 summary of methods of adjustment, 199. 
 original annual instalment found by calculation : 
 summary of method. 199. 
 method described, 204. 
 original annual instalment, a stated sum : 
 summary of method, 200. 
 method described, 210. 
 
 Suspense or reserve account for interest on loans after equation, 447. 
 
 Suspense account, sinking fund interest ; to avoid apportionment, where 
 only one bank account and one investment account are kept in respect 
 of several sinking funds, 352. 
 
 Symbols used in the various formulae, 3. 
 explanation of, 30, 84, 85. 
 derivation of, 30. 
 
 formulae relating to compound interest and annuities, 31. 
 geometrical progression, 30. 
 
 Tables of compound interest, etc., amount of one pound. Table I, 36. 
 amount of one pound per annum. Table III, 50. 
 annuity method of repayment, Table V, 67, 114. 
 annuity which one pound will purchase, or of which one pound is 
 
 the present value, Table V, 67. 
 equal annual instalment of principal and interest combined, 67, 114. 
 interest of one pound for one year (?) from ^ to 7 per cent, and 
 
 logarithms, 48. 
 Inwood's Tables, 3, 33. 
 present value of one pound. Table II, 42. 
 present value of one pound per annum. Table IV, 62. 
 ratios, from ;^ to 7 per cent, and logs., 48, 49. 
 Schooling, 3. 
 
 Sprague's Tables (American), 3. 
 Thoman's Logarithmic Tables, 3, 73. 
 varietie.- of, 3. 
 
INDEX 535 
 
 Taxation, annual incidence of, Tables shewing, under : 
 the instalment method, 113. 
 the annuity method, 122. 
 the accumulating sinking fund method, ^^9. 
 the above methods compared, 140. 
 the non-accumulating sinking fund method, 137. 
 after equation. See equated period, etc. 
 
 Taxation, equation of the incidence of. -S'ee Equation of the incidence of 
 taxation. 
 
 Temporary sinking fund instalments, 356, 367, 391. See Construction 
 period. 
 
 Thoman, Fedor. Logarithmic tables of compound interest and annuities, 
 3, 73. 
 
 Thoman 's formulae, methods and rules, advantages of, 5, 75. 
 amount of one pound, 36, 73, 77. 
 amount of one pound per annum, 50, 73, 77. 
 
 an log. of, stated by the addition of 10 to the characteristic, 75, 78, 79. 
 an represents the annuity which one pound will purchase, or the 
 
 equal annual instalment of principal and interest combined, 76. 
 annuity method, 67, 114. 
 
 annuity one pound will purchase, 67, 73, 78. 
 equal annual instalment of principal and interest, 67, 114. 
 formulae compared with formulae relating to compound interest, 73. 
 present value of one pound, 42, 73, 77. 
 present value of one pound per annum, 62, 73, 78. 
 rules are stated both generally and in logarithmic form at the head 
 
 of each chapter, 
 sinking fund method, 127. 
 symbols used by Thoman, 75. 
 tables are contained in Inwood's tables, 
 tables give log. values only, 79. 
 
 Town Halls, outlay upon, 380. 
 
 Trading undertakings. See Revenue earning undertakings. 
 
 Tramways, profits of undertaking, 383. 
 
 renewals fund, 385. 
 
 standard forms of account, xii. 
 
 See also Revenue earning undertakings. 
 True or mathematical method of tinding equated period of repayment, 394, 
 
 399, 404. 
 
 Typical sinking fund, 8, 158. 
 
 Unexpired repayment period, 261. 
 Utility of asset. See Asset, life of asset. 
 
 Value of investments, 147. 
 
53^ INDEX 
 
 Variations in, accumulation, rate of, without any variation in the rate of 
 income from investments, 223, 277. 
 with a variation in the rate of income from investments, 247. 
 with a variation in the period of repayment, 300. 
 consols, rate of interest upon, 225, 327. 
 income from investments, rate of : 
 
 without any variation in the rate of accumulation or the period 
 of repayment, 236. 
 where the future variation in the rate of income from invest- 
 ments is definite both as to date and amount, 322, 328. 
 where the future variation is anticipated but is uncertain 
 both as to date and amount, 322, 334. 
 with a variation in the rate of accumulation, 247. 
 period of repayment : 
 
 without a variation in the rate of accumulation, 283, 295. 
 with a variation in the rate of accumulation, 300. 
 separate effect of the variation in the : 
 rate of accumulation, 317. 
 period of repayment, 316. 
 
 Verification of calculations by alternative method of proof, 17 
 
 Wastage of asset. Sec depreciation. 
 
 Waterworks, 381. 
 
 argumenc against extended period of repayment, 382. 
 
 argument in support of extended period of repayment, 382. 
 
 importance of outlay beyond present requirements, 381. 
 
 large amount invested in land, 382. 
 
 large cost of water areas, 381. 
 
 permanent character of works, 382. 
 
 See also under Revenue earning undertakings. 
 
 Works, permanence of. See asset, life of asset. 
 Years, number of, to find. See number of years. 
 
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