INTEREST AND D VALUES | M, A. MACKENZIE THE UNIVERSITY OF ILLINOIS LIBRARY St TS O\W-8- M1924 19199; 1417 Return this book on or before the Latest Date stamped below. A charge is made on all overdue books. University of Illinois Library MAR 12 1966 L161—H41 HNe xe Sit AND VALUES BOND BY M. A. MACKENZIE, M.A., F.I.A., A.A.S. SECOND EDITION REVISED AND ENLARGED PRICE TWO DOLLARS UNIVERSITY OF TORONTO PRESS 1917 COPYRIGHT, CANADA, 1912, By M. A. MACKENZIE. OtePT ZS Net Hedrick PREFACE TO THE FIRST EDITION. This little book is not to be regarded as a treatise on the Mathematical Theory of Interest—that theory which has been so ably expounded by Mr. King in The Theory of Finance and by Mr. Todhunter in the Text Book of the Insti- tute of Actuaries, Part I. It is merely an explanation of the Interest Tables and Tables of Bond Values now in common use, and an attempt to instruct students concerning them. While an elementary knowledge of Algebra is a powerful aid to the intelligent appreciation and use of such tables, and Algebra has not been excluded from the following pages, yet it is believed that nearly everything contained therein may be _ followed by anyone who will take the trouble to learn the meanings of the standard interest symbols. Mainly written for the use of the author’s own elasses in the elementary mathematics of finance, it is hoped that the book may also be of value, not only to actuarial students but also to that increasing number of men who are finding it a business necessity to thoroughly understand the tables referred to THE UNIVERSITY, Toronto, January, 1912. PREFACE TO THE SECOND EDITION. The author desires to thank many friends for valuable suggestions. He hopes that all errors have been corrected and that in other respects also the book has been rendered more useful especially for classroom purposes. THE UNIVERSITY, Toronto, January, 1917. TABLE OF CONTENTS. CHAPTER I.—ON INTEREST AND DISCOUNT. Definitions. Notation. Interest Tables. Extension of Tables of (1+7)” and v”. Rule for time in which money will double itself. Frequency of compounding. Simple Interest. True Interest. Discount. ‘‘Banker’s Discount.”’ CHAPTER II.—ON PERIODICAL PAYMENTS. Definitions of SF and ar Value of SJ in terms of and . Examples. Value of az in terms of 1 and m. Examples. Extension of Tables of s;, and a5. Definitions of s— and ax" Examples. Increasing and Decreasing Payments. Equivalent Payments. CHAPTER III.—ON STRAIGHT TERM BONDS. Description. Factors determining investment rate. Bond Tables. Examples as investment. Capital written up or down. Sinking fund to replace premium. Algebraic investigation. Extension of Bond Tables. Bonds with yearly or quarterly coupons. Bonds repayable at a premium. Makeham’s formula. Serial Bonds. Bonds repayable by an accumulative sinking fund. Bonds bought between coupon dates. An unusual bond. Given the price, to find the yield. Bond Tables used as Interest Tables. CHAPTER IV.—ON ANNUITY BONDS. Typical schedule. Examples as investment. Payments made yearly or quarterly. Algebraic schedule. Redemption Price. Annuity Bond Issue with coupons. Values of Annuity Bonds from Straight Term Bond Table. Annuity Bonds bought between payment dates. CHAPTER V.—FROM THE ISSUER’S POINT OF VIEW. The Bond Rate. Premium or Discount. The Bond Term. Choice between different forms of issue. An example. CHAPTER VI.—SOME PROBLEMS. TABLES. EXERCISES. INTEREST AND BOND VALUES. GHAPTER si INTEREST AND DISCOUNT. 1. We are all aware of the fact that men and corporations of undoubted ability to pay may generally be found who are willing to pay more than a dollar at some future date in re- turn for a dollar today. The excess payment made when the borrowed dollar is returned is called interest. We are all equally aware of the corresponding fact that banks and similar institutions will give something less than a dollar today for a good promise to pay a dollar at some future date. The “something less”’ differs from the dollar by what is called discount. Interest is quoted at so much per cent. per annum, is cal- culated on the sum lent, and is payable at the end of the year or at the ends of such sub-divisions of the year as may be agreed upon. Discount is quoted at so much per cent. per annum, is cal- culated on the sum to be paid in the future, but is itself always payable in advance. These are facts of common knowledge. Our theory of interest is based on these facts and has nothing whatever to do with the speculations of the Economist who searches for the reasons for these facts. 2. Interest calculations must be as old as civilization. There were money lendersin Thebes and Babylon. Nowadays such calculations commonly occur all over the world. It is therefore not surprising that a world wide system of interest 6 _ Interest and Bond Values. symbols should have been developed. The elements of this notation are as follows:— 4 is the interest on 1 for one period (saya year). 1 at interest for one period will amount to 1+4. 1 at interest for two periods will amount to (1+7)?. 1 at interest for three periods will amount to (1-+1)'. &e. &c. &c. &c. 1 at interest for 2 periods will amount to (1-+7)”. Thus if the rate of interest be 5% perannum, 1=.05 or .05 is the interest on 1 for one year. 1 at interest for one year will amount to 1.05. 1 at interest for two years will amount to (1.05)? =1.10250. 1 at interest for three years will amount to (1.05)? = 1.15763. 1 at interest for » years will amount to (1.05)”. Again, v 1s the present value of 1 due at the end of one period. v? is the present value of 1 due at the end of two periods. v® is the present value of 1 due at the end of three periods. ‘&e. &c. &c. &c. v’ is the present value of 1 due at the end of periods. 3. Since 1 is the present value of 1+7 due at the end of one period, and v is the present value of 1 due at the end of one period, therefore 1: v ::1+7: 1, orv (147) =1. or v= aise and 1+1 = Sad L-t v Thus if the rate of interest be 5% per annum 1 ic gee naa tes ae Tt 1.05 v =.95238 is the present value of 1 due one year hence. v’ = .90703 is the present value of 1 due two years hence. v® = .86384 is the present value of 1 due three years hence. &c. &c. &c. &c. = .95238, so that ‘1 1 ~ (1.05)" is the present value of 1 due years hence. Interest and Discount. 7 4. d is the discount on 1 due one period hence, and obviously d must = 1—v. So we have Since v at interest for one period amounts to 1, the interest on v for one period must be 1-v or d: in short vi=d. In other words the present value of the interest on 1 is d. Therefore payments of d in advance each period are equivalent to payments of 7 in arrears each period. Again, 1 is the present value of 1+7 due one period hence. v is the present value of 1 due one period hence. .. l-v=d is the present value if 7 due one period hence. or 1—d is the present value of 1 due one period hence. 5. The three symbols 1, d, v, are related as shewn in the following schedule. The in terms of Value of 1 d | v | d ae : : 1—d v | i d —— a l—v i+72 1 v Si ae 1—d v 1+: 6. The ordinary interest tables, such as those issued by Colonel Oakes or by Mr. Archer, give the values of (1+7)” and v” for numerous values of 7 ranging from 34 of 1% up to 10%, and for values of x ranging from 1 period up to 200 periods. 8 Interest and Bond Values. Typical extracts might be:— Amount of 1 at interest for 7 periods. (1+1)” 5 | 1.05101 1.10408 | 1.15927 | 1.21665 | 5 10 | 1.10462 | 1.21899 | 1.34392 | 1.48024 |10 15 | 1.16097 | 1.34587 | 1.55797 | 1.80094 {15 20} 1.22019 | 1.48595 | 1.80611 |} 2.19112 |20 and Present value of 1 due 7 periods hence yt js 1% or 2% or 3% or 4% or i=.01 i=.02 1=.03 i=.04 51 .95147| .90573 | .86261 | __.g9193 | 5 Ao: .90529 82035 . 74409 .67556 10 at 86135 74301 .64186 PODozG 1 Po .81954 .67297 .55368 .45639 20 7. Although interest is always quoted at so much per cent. per annum, it is usually payable more frequently than once in each year. In interest calculations when interest is payable only once a year it is said to be compounded yearly, or com- pounded with yearly rests, or to be convertible yearly; but if the interest is payable twice or four times a year it is said to be compounded half yearly or quarterly. 8. It should be noted that many tables use the word ‘‘years’’ in place of the word “‘periods.’’ This is unfortunate since interest is usually compounded more frequently than once a year, and the word ‘‘years’’ must be understood to mean ‘‘half years’ or ‘“‘quarters’”’ as circumstances demand,- Interest and Discount. 9 ’ These tables are of course applicable toany currency. From the extracts given above we see that at 4% interest compound- ed yearly for 5 years, 1000 dollars will amount to 1,216.65 dollars, or 1000 pounds will amount to 1,216.65 pounds, or 1000 francs will amount to 1,216.65 francs. But if interest be compounded half yearly, then $1000 will amount to $1,218.99. While if interest be compounded quarterly, the $1000 will amount to $1,220.19. Similarly, a good promise to pay $1000 fifteen years hence is worth $555.26 if the rate of interest be 4% compounded yearly. 9. Such tables not only shew the amount to which $1. will accumulate in any time atany rate of interest, and the present value of a $1. due at any time in the future at any rate of in- terest, but will also, if entered inversely, answer questions similar to the following. i. In what time will $1. amount to $1.75 at 3% compounded half yearly? From the 1%% table we see the answer to be a little less than 38 periods, i.e. 19 years. ii. At what rate % compounded quarterly will $1. amount to $2.50 in 18 years? From the values opposite 72 periods we find the periodic rate required to be about lsz% i.e. 54% per annum. 10. To obtain a value of (1+7)” or of v” when 1 lies beyond the range of the table we have only to multiply together two or more values that are given. Suppose for example that we want the present value of $1000 due 40 years hence at 4% com- pounded quarterly. It will be $1000 v'!° at 1%. If our tables do not go beyond 100 periods, we have v!© = y® Xy®, or =v! X v®, or any pair of suitable factors. Now v!=.36971, and v®= .55045; therefore v!=.20351 and the value required is $203.51. 11. An inspection of the tables of (1+7)” will shew the truth of the common rule—To find the time (number of 10 Interest and Bond Values. periods) in which money will double itself at interest, divide 70 by the rate per cent. per period. The proof of this rule is quite simple by the aid of Napier’s logarithms. 2=(1+i)" .. log, 2=n log (1-7) orn = sue = ae + .35 ti es Se a &c s ) 3 e very nearly. The rule is almost exactly true for rates of between 1% and 2% per period, e.g. for 4% compounded half yearly or quarter- ly or for 38% compounded half yearly. For higher periodic rates the results of the rule are slightly too small, but even for 8% per period the error is only one quarter of a period, or 3 months if this rate be compounded yearly. 12. When interest is compounded more frequently than once a year the result is to produce an “‘effective”’ rate in excess of the ‘‘nominal’”’ or quoted rate. For example, if the nominal rate be 4% and interest be compounded half-yearly, $1 will amount to $1.02 at the end of six months and to $(1.02)?= $1.0404 at the end of the year; or a nominal rate of 4% com- pounded half yearly gives an effective rate of 4.04%. Simi- larly, if the nominal rate be 4% compounded quarterly $1. will amount to $(1.01)*=$1.0406 at the end of a year; or a nominal rate of 4% compounded quarterly gives an effective rate of 4.06%. In general, a nominal rate of 7 per annum compounded m times a year gives an effective rate of 1 where ¢ ++) =1+i. m When m becomes infinitely large the nominal rate is called the ‘‘force of interest’’ and is written 6. In this case Interest and Discount. 11 a+a=(+z) Spree q + EO (d eee aed ele Vite. (5, eS: =e) or a m is infinitely re or 6= log (1+72) Ba Be ee hy ee — it CLC, 7 age =1+j+—~,™ The following table illustrates the effect of frequency of compounding. Nominal Effective rate when compounded. Se re ea eT Interest|| half-yearly; quarterly | monthly daily 24% || 2.5156% | 2.5235% | 2.5288% | 2.5315% | | | SE ET ——— | | | | | SSS 34% || 3.53806% | 3.5462% | 3.5567% | 3.5620% ee | | f | SS LT 4% || 4.0400% | 4.0604% | 4.0742% | 4.0811% i SSE 4% || 4.5508% | 4.5765% | 4.5940% | 4.6028% a || | [| 5% || 5.0625% | 5.0945% | 5.1162% | 5.1271% Thus, if the rate be 38% per annum, and the sum at interest be $10,000.00, Compounding yearly the interest will be $300.00 a year, Compounding half yearly the interest will be $302.25:a year, Compounding quarterly the interest will be $303.39 a year, Compounding monthly the interest will be $304.16 a year, Compounding daily the interest will be $304.54 a year. Or, the maximum advantage which can be gained by fre- quent compounding is only $4.54 a year on a sum of $10,000.00 he - Interest and Bond Values. at 3% and practically half this advantage is gained by com- pounding twice a year, while only another quarter of it is secured by compounding four times a year. These propor- tions apply, as may be seen, to all ordinary rates of interest. 13. There is nodifference in essence between capital and in- terest. Each is money. Just as capital may be the interest accumulations of the past, so interest paid to-day and in- vested to-morrow will become nominally capital. Invested capital grows by the operation of interest and the rate of growth is called the rate of interest. A rate of growth is usually quoted at so much per period, as for example, so much per annum; but growth isacontinuous process—not proceed- ing by isolated jumps. To illustrate—when we speak of a population increasing at a uniform rate of 3.65% per annum, we do not mean that for each 1,000,000 at the beginning of the year 36,500 people were suddenly added at the end of the year; nor do we mean that 100 persons were added per day, for that would imply that 1,000,000 people were increased by 100 during the first day of the year, and that 1,036,400 people were only increased by 100 during the last day of the year, which would not beauniform rate of increase. What we do mean is that the population was increasing uniformly, that the increase was something less than 100 per day during the first days of the year and something more than 100 per day during the last days of the year, but that 36,500 people were added during the whole year. The rough assumption of 100 per million per day is nearly true, but obiously it is not exactly true. Such uniform daily increase—100 per million per day—would result in an increase of about 37,170 per million per annum or nearly 3.72%. SIMPLE INTEREST. 14. If the rate of interest be 7 per unit per annum, 1 will amount to 1+172 at the end of a year. Therefore 1 should amount to +i at the end of = of a year, or to (ih aye st the end of — of a year. Therefore the interest on 1 for > Interest and Discount. 13 m of a year is (1+7)"—1, or upon $1. for 71 days at 5% per annum the true interest would be ${ (1.05)2*=—1).} In order to avoid difficulties of calculation it is assumed in practice that sss of a year’s interest will accrue each day, or that if the rate be 3.65% per annum and the sum be $1,000,000 the interest will be $100 per day. The result of this assump- tion is called ‘‘simple interest.”’ It is of course equivalent to compounding at the date of the calculation and introduces the gain due to such compounding. To show the amount of the error introduced by the assumption of simple interest, let us say that the rate is 3% per annum compounded half-yearly, and that the sum at interest is $1,000,000, for 73 days. me Des CTESt 715 1c, «3. vis v oee eho kine: ee ark '. $6,000.00 , 2 The true interest is $1,000.000{ (1.015) >—1}.... 5,973.21 PEIMELEO EIT CX CESS Oli ae ego) haa nuk cer oe $26.79 Simple interest always produces an error in excess, that is to say, simple interest always exceeds true interest. This error is a maximum when the number of days for which simple interest is calculated is about half the period of compounding, and the error isleast when the number of days for which it is calculated is either very small or very nearly equal to the period of compounding. Algebraically, if 2 be the periodic rate, the true interest on 1 ; 1 ; 78 Ye ee ob a period is (1+i)" -l=— — (ae et eta ade NS: (n—1) (2n—1) (i \8_ (n—1)(2n—1)(8n—-1) (i \4, aes 3: =) ea Cone ae This is a rapidly converging series for all ordinary values of 1, and the assumption of simple interest that its value may be represented by its first term does not introduce a serious error unless the amount at interest be very large. 14 Interest and Bond Values. Most of the prevalent confusion regarding simple interest arises from the Arithmetics, wherein it is usually assumed that simple interest is based on one theory and compound interest on another. Problems are even set in which the assumption is made that money paid as interest can earn no interest itself. This absurdity comes from the attempt to carry the idea of simple interest past a period of compounding, quite ignoring the fact that simple interest is only a ready approximation to true interest for a period less than a period of compounding. As an approximation for a broken period it is excellent, readily calculated and very nearly accurate. To carry the method beyond a period of compounding is to misunderstand it. DISCOUNT. 15. It is usually said that discount is interest paid in ad- vance: that is quite true, but it is not the whole truth. When a banker quotes a rate of discount he is quoting a rate at which he will sell money, that is to say, at which he will sell the im- mediate right to draw cheques. His rate of discount is so much per cent. per annum and is calculated on the sum guar- anteed by his customer to be paid in the future, but the dis- count itself is always payable in advance. Also, the rate of discount charged by the banker is not the rate of interest paid by his customer. The Banker who discounts a three months bill for $1000.00 at 6% will credit his customer with $985.00. That is to say the customer borrows $985.00 and guarantees the bank $1000.00 in payment at the end of three months. This represents an interest rate of over 6.09%. It should be noted that the banker did not quote an interest rate of 6% but a discount rate of 6%. If now, the same customer wished to deposit money in the same bank, the bank would quote him a rate of interest, probably 3%. Most banks quote both rates, i.e. a discount rate (d) at which they will turn aright to future money into a right to a smaller amount of present cash, and also an interest rate (4) at which they will accept cash and give a right to a larger amount of money in the future. The Interest and Discount. 1109) Bank of England quotes a d—a discount rate—but gives no interest on deposits i.e. does not quote anzatall. Owing toa failure to appreciate the fact that the Bank was quoting a dis- count rate (d) and not an interest rate (1), our school arithme- tics contain a curious distinction between ‘‘ Banker’s discount”’ and ‘‘true discount.’’ On the false supposition that the Bank was quoting an 2, the schoolmaster truly calculated that the discount rate should be and accused the bank of charging 1 more than it was entitled to. Thus, say the bank discount rate was 4% i.e. d=.04, the schoolmaster assumed wrongly that the 4% was an interest rate, i.e, 1=.04, and on this false assumption correctly deduced that d should be equal to rer) 39469) STi will berememibered that the Bank Use 1.04 of England was originally a Whig institution and as such was opposed by the Church—then, even more than now, strongly Tory. The Schoolmasters of the eighteenth century were usually clergymen and were the authors of the prototypes of our modern school text books. In sympathy with their party these clerical authors made disparaging comments on the Bank, and, jumping to the conclusion that the bank rate was an in- terest rate, they invented the distinction between true dis- count and banker’s discount to shew how the Bank was over- charging its customers. This prejudice has long since passed away, and modern arithmetics are written or edited by men who are often neither Tories nor members of the Church of England, yet the old misconception has been copied and re- copied—a wonderful example of the persistence of formal error. Just as Simple Interest gave a ready approximation to true interest for a broken period, so with discount calculations the same approximation is used. If the rate of discount be d per period, the discount on 1 for wigs a period is practically taken n | d as equal to —. The true discount of course would be n 16 Interest and Bond Values. 1-a-ae= 44 +tsty4 (n—1).Qn—1) ioe 1) (ay ee 1)(2n—1)(8n—1) ee SRA ()+.--. This is a rapidly converging series and may, without serious loss of accuracy, be represented by its first term unless the amount at discount be large. The error is always an error in defect, i.e. simple discount is always less than true discount. The error is, like that of simple interest, a maximum when the number of days for which discount is calculated is half the period for which the rate is quoted. To illustrate the amount of the error let us say that the discount rate is 5% per annum and that the sum subject to this rate is $1,000,000, due 73 days hence. True discount is $1,000,000 {1 — (.95) > } TAPER ahr $10,206.22 Simple: discounts ss. ene ee oe ee eee 10,000.00 An-error in defect.Gis. > 41s 2 see eee $206.22 Bankers may be, doubtless are, guilty of many sins from their customers’ point of view; but so far as the rates of in- terest and discount are concerned, the banker gives too much interest and charges too little discount on every transaction covering a period less than that for which his rates are quoted. INTEREST AND BOND VALUES. GCHAPFER II. PERIODICAL PAYMENTS. 1. In the previous chapter reference was made to well known interest tables in which are published the values of such interest functions as (1+7)” and v” for various values of z and n. The tables also usually contain two other interest func- tions applicable to periodical payments. These are Sq, = the accumulated value of a series of 7 past payments of 1 each, made on the dates at whch interest was com- pounded, the last payment having just been made. a; = the present value of a series of m future payments of 1 each, to be made on the dates at which interest will be compounded, the first payment to be made one interest period hence. These are commonly referred to as Sq) = the amount of 1 per period. ay, = the present value of 1 per period. But the assumptions of the above definitions must be under- stood in detail. 2. To express sjj in terms of 7 and 2:— The payment just made is of course worth 1. The payment made a period ago is now worth (1-41). The payment made two periods ago is now worth (1-7). &c. &c. &c. &c. The first payment made (z—1) periods ago is now worth (1-+-1)*~! or sq=1+(1+i)+(1+i)?+...... +(1+1)""? Weal Gaeta em 1 Or (1-+-2)"—1=1 sy. This is a statement, the truth of which may be easily seen without algebra. : , the sum of the geometric series. 18 3 Interest and Bond Values. If A lent 1 to Bn periods ago at rate7z per period and B has made no periodical payments of interest, then B owes A (1-++12)” today. Had B however paid 7 at the end of each period mak- ing his last payment today, he would owe A only the original 1, the capital of the loan. The difference (1+7)” —1 must be the accumulated value of the interest payments of 2 each period, i.e. 7 sz. : Or (1+1)*—1=1 sq. ‘Reference to the tables will show that the first value of sq, i.e. sj, for any rate of interest is 1. The second value will be 1+(1+2) or at 2%, 1+1.02 =2.02. at 5%,.1+1.05 =2.00. Obviously the table of sjj may be constructed from the table of (1+72)” by continuous addition. (1+72)"—1 4 Also since S= (1+ 4)" — 14 af i | (1+-1) Selina na-+1 er and the table may be constructed by continuous multiplica- tion. It must be noted that the table is based on the assumption that the periodicity of the payments coincides with that of the compounding of the interest. For example, if the interest rate be 4% compounded half yearly, then payments of $100 at the end of each quarter must be regarded as payments of $201 at the end of each half year; while payments of $100 at the end of each year must be regarded as payments of $49.505 at the end of each half year. Since 49.505+49.505 x 1.02 = 100. A payment of 1 at the end of each year is equivalent, at a rate of 7 per year, to a payment of x at the end of each half year where x-++x (1+3)=1. A few examples will illustrate the uses of the table of sy. Periodical Payments. 19 (i) The value of 10 payments of $100 each, made annually during the past 10 years, the last payment having just been made, is, at 4% compounded yearly. $100 sz at 4% = $1200.61. (ii) The value of 10 annual payments of $100 each, the last payment hav ng been made 5 years ago is, at 5% compounded yearly, $100 (sjgi—sH) at 5% =$1605.29. Or $100 sz X (1-+72)5 at 5% = $1605.29. (iii) The value of 20 semi-annual payments of $100 each, the last payment having just been made, is, at 3% compounded half yearly, $100 sag at 114% =$2312.37. (iv) The value of 20 quarterly payments of $100 each, the last payment having just been made, is, at 4% compounded yearly, assuming simple interest for the broken periods, $(103+102+101+100) sz at 4%. = $406 sq at 4% =$2199.03. The error introduced by the assumption of simple interest for the broken periods is only 27 cents, the true value being $2198.76. t t aaa The true value is $100 {+ (1+7) +(1+7) +..+(14+2) | (1+1)5-1 (t= 1 (1+7)? may be obtained either by logarithms or by expansion by the binomial theorem. = $100 = $2198.76. (v) To find the value of 20 yearly payments of $100 each, the last payment having just been made, at 344% compounded half yearly. The equivalent half yearly payment at this rate of interest is found to be $49.566, so that the value required is $49.566 sg at 134% =$2,836.87. Or, without using equiva- lent half yearly payments, the value required is 20 Interest and Bond Values. = $100{1+-(1+7)"+ (1+7)4+ (1+4)*..+(1+7)*} at 134% (1+7)°— 1 SH] A - = $100 —. at 13 = $2836.87. (1+7)?—1 S93] *4 70 The table of s;j may also be used inversely to answer ques- tions similar to the following. = $100 (i) At what rate must annual deposits of $1 each have been accumulated to amount to $20 in 15 years, the interest having been compounded yearly and the last deposit having just been made? A reference to the values of sj5| will shew that the. rate required is a trifle under 4%. (ii) In how many years at 3% compounded half yearly will deposits of $1 each six months, amount to $100? A reference to the 114% values of sz will shew that sg = 98.66 and sg =101.14. This means that immediately after the 61st de- posit the amount will be $98.66 and six months’ interest on this will bring it up to $100.14. The required time is there- fore 31 years, entailing 61 deposits in all. 3. To express a3 in terms of 7 and n. The present value of the first payment is v. The present value of the second payment is v?. &e. &c. &e. &c. The present value of the last payment is v”. Or a eas BECO TOD aN Ea =U 1e5 = , the sum of the geometric ser‘cs. 1 Or 1=12 ay+v”. This is a statement the truth of which may be easily seen without algebra. If A lends B 1 today at rate z per period for m periods, B will make periodical payments of iand thena single payment of 1 being the return of the capital. The present value of the periodical payments of 7 is 1a7. The present value of the repayment of the 1 is v”. The sum of these must equal the 1 that A lent B, or 1=ta;y)-+0”". Pertodical Payments. 21 If 2 be infinite,that is to say if the loan is to last forever, there will be no repayment of the capital and q 1 1=1 de Or dx = —, 1 so that the value of a perpetuity of 1 is at SCE We Ue ped ors vg +y", ayj=v, az;=v-+v*, and so on, the a; table may obviously be constructed from the v” table by continuous additon. 1—v" Also since az] = = Ci ee esi ss 1am and the table may be constructed by continuous multipli- cation. In using the az table it should be noted that, just as with the sj table, the periodicity of the payments must coincide with that of the compounding of theinterest. If the payments for which we need the table in any case do not so coincide we must use equivalent payments that do coincide. See the table on page 32 A few examples will illustrate the use of the table of aj. (i) The value of 10 annual payments of $100 each to be made during the next ten years, the first payment to be made one year from now, is, at 4% interest compounded yearly, $100 ajg at 4% =$811.09. (ii) The value of 10 annual payments of $100 each, the first payment to be made5 years hence, is, at5% interest compound- ed yearly, $100 (ajq—az]) at 5% = $635.27. Or $100 ajgX vt at 5% =$635.27. 72 Interest and Bond Values. (iii) The value of 20 semi-annual payments of $100 each, the first payment to be made six months hence, is, at 3% — compounded half yearly, $100 az at 114% =$1716.86. : (iv) The value of 20 quarterly payments of $100 each, the first payment to be made three months hence, is, at 4% com- pounded yearly, assuming simple interest for the broken periods, $(103-++102+101+100) ax at 4% = $406 az at 4% =$1807.44. The error introduced by the assumption of simple interest for the broken periods is only 22 cents, the true value being $1807.22. The true value is $100 ‘en se ait on 4) 1—v =$100 —————-at 4% = $1807.22. ae (1-+i)* may be obtained either by logarithms or by expan- sion by the binomial theorem. (v) To find the value of 20 yearly payments of $100 each, the first payment to be made one year hence, at 3%% com- pounded half yearly. We must first find what half yearly payment at this rate of interest is equivalent to $100 a year. It is found to be $49.566, so that the value required is $49.566 agg at 134% =$1417.31. Or, without using the equivalent half yearly payments, the value required is $100 (v?+ v!+v°+....+v%) at 134% a 1—y* 0 = $100 100 ete ee (1 +72)?-1 Sz 7470 The tables of aj may also be used inversely to answer such questions as the following :— (i) At what rate of interest compounded half yearly will $20 be the value of 25 future half yearly payments of $1 each, Periodical Payments. | 23 the first payment to be made six months hence? A reference to the values of aj opposite 25 periods shews that the rate is nearly double 116% or nearly 33% per annum—more ac- curately 332% per annum. (ii) For how many years will a cash payment of $20 produce an annuity of $1 a year at 4%, the annuity to be payable at the end of each year and interest to be compounded yearly? A reference to the 4% values of aj will shew that the answer is 41 years. 4. We can always find values for sq and aj, when 1 lies outside the range of the tables we are using, by means of the formulae. Sam] = Sait (1+4)” sim Antm| =A tv” Oa: For example, if our tables run to only 50 periods S95] = So, + (1 +1) sagj Ag7| = 250, +™ a37}- The truth of such statements should be quite obvious. 5. The reciprocals of the functions sz and az are also fre- quently tabulated and may be defined as follows :— eee 1 = Sai = The sinking fund payment to be made at the end n| of each interest period in the future for ” periods in order to accumulate to 1 m periods hence; the first sinking fund payment to be made one interest period from now and the last one 7 periods hence. 1 2S . —=a5, = The future periodical payment to run for ” periods n| that can be purchased by the payment of 1 now, the first of the periodical payments to come in at the end of one interest period from now. These are commonly referred to as a = Sinking fund required to produce 1. ax = Periodical payment which 1 will purchase 24 Interest and Bond Values. But the assumptions of the above definitions must be under- stood in detail. Alocornigiliveeeen eters n| wie (1 BE ar and eeu z Say So Chaea eae Laer Tee te Pas, Seah ne eed eee -1 or at — sa! =7. n| n| This is a statement the truth of which may be seen without algebra. If A lends 1 to B now, B can repay principal and interest by nu periodical payments of oz each. Therefore a; must consist of, first, ithe interest 7 on the 1 lent and, second, the sinking fund sa needed to Soa the 1 at the end of the 2 periods, i.e. aq {aca Or Gay ‘sz == Therefore it is not necessary to tabulate both as and ss Many tables contain only a= from which any value ‘of s can be found by ee pases E.g. at 344% Ox, = .07036108, 4 == 03D: s= = .03536108. A few ee will illustrate the uses of these tables. same? (i) In order to produce a fund of $10,000 at the end of 10 years by semi-annual sinking fund payments which will ac- cumulate at 3% compounded half-yearly, there should be deposited at the end of each half year into this sinking fund $10,000 sz! at 134% = $432.46. If deposits are to be made at the beginning of each half year $432.46 +1.015 = $426.07 will suffice. Or, directly from the tables, we have $10,000 10,000 (ssj—1)7? = $ (sai) = 58 4706 = $426.07. Periodical Payments. 25 (ii) An investment of $10,000 now will purchase semi- annual payments beginning six months hence and running for thirty years, on an interest PEG of 4% compounded half yearly, amounting to $10,000 a at 2% =$287.68 each. If the annuity is to be payable half yearly in advance the semi-annual payments will be only $287.68 + 1.02 = $282.04 each. Or, directly from the tables, we have $10,000 35.4561 The same remarks regarding periodicity of payment and compounding apply of course to the tables of se and a= as $10,000(1 + am) ~* = = $282.04. apply to those of sjj and aj). 6. To find the accumulated value of 1 per annum paid p times a year for the past 7 years, the last payment having just been made, at a nominal rate of 7 per annum compounded g times a year. This is equivalent to Hes the accumulated eile of np periodical payments of ; each, the last payment having just been made, at a periodic rate of ; where Grae) h Therefore the accumulated value required is 7 sir at rate on p h np Lye fe eet Toolecuen) ee Co Ba ie eee ? ous P é +t )r—1 p q Similarly to find the present value of 1 per annum to be paid p times a year for the next 7 years, the first payment to be made 1 p of a year from now, at a nominal rate of 7 per annum com- 26 Interest and Bond Values. pounded g times a year is equivalent to finding the present value of np periodical payments of p each, the first payment cae h to be made one period from now, at a periodic rate of p 4) qg where é +4) = ( o! h Therefore the present value required is — 5 Om at rate D pn qn ik Os (rey 1 p h p (1 (+5) ae ; q 7. If the periodical payments are increasing or decreasing by a fixed amount each period, that is to say, if the amounts of these payments form an arithmetical series, we can find either the accumulated value of such payments in the past or the discounted value of such payments to be made in the future. Suppose that we are dealing with past payments and that x was deposited n—1 interest periods ago, x-+y was deposited n-—2 interest periods ago, x-+2y was deposited n—3 interest periods ago, &e. &e. &e. &c x-+(n-2)y was deposited 1 interest period ago, and x+(n-1)y was deposited today. The accumulated value of these deposits now is Vex (1+7)"-'+ (x+y) (1+ i)" 7+... wo. H(xtn — 2.y) (14+7)+(x+n — 1.4). (1+i)V=x Cee a Care apal) ar oo + (x+n—2.y) (1+72)?+(«+n — 1-y) (1+7). 4V= x (1+i)" Foy ati" 411)" 7+... eqn 1} ce es 4 Bh =x {(+i)"-1} +9 (sq—n). or Veesqt— (Satine): 1 Now suppose that we are dealing with future payments and that x will be received 1 interest period hence, Periodical Payments. od x+y will be received 2 interest periods hence, x+2y will be received 3 interest periods hence, &c. &e. &c. &c. x+n — 2 y will be received n—1 interest periods hence, x+n — 1ywill be received n interest periods hence. The present value of these future payments is V=xut+(x+y)v?+ (x+2y)8+..... .(x+n —2.y)0" T+ (x- ae .y)v". (1+2)V= paneer Vee (x + Qy)v?+.. ese. yon ta 1 yt ey Voy yr +y3+... +0" '+y") —(x-+ny)v”. Sh Vids, Ve. or V=xaqj + oe (Gy, 1 8. If the periodical payments form a geometrical series we can find expressions for either the accumulated value of such a series of payments made in the past, or the discounted value of such to be made in the future. Suppose that we are dealing with past payments and that x was deposited n—1 interest periods ago, xy was deposited n --2 interest periods ago, xy? was deposited n—3 interest periods ago, &c. &e. odes xy"? was deposited 1 interest period ago, and xy”~' was deposited today. The accumulated value of these deposits now is V=x(1 +i)” t+ay(t +i)" F+xy? (1 +i)" 8+... Abxy" 2(1 +2) +xy" =x Laer jes Tepes (1+72) —y = \n-1 AG am): Or V = (1-42) feta) +* Ga +... n-2 we 95 a ++) tee Ca) = (141)""". x Sa 28 Interest and Bond Values. where sa jis to be taken at rate j such that 1+-j= ae 1 OrV aU te eee) ~*~ where %,-4| is to be taken at rate j such that These alternatives enable one to avoid a negative rate of interest. . If we are dealing with future payments the discounted value of such a series is ete ace vt... xy 2 ot t4xyt! bey os ee yo Oni =i ~ {xoy-+eo? y+ oe tapes cle eaten cade xv" ty" 1 4 yytyt » x fie,” ee a y where a is to be taken at rate j such that 1+ 7 = ets > Or V=v{x+xyotaeyv+...... ep? "FL gent yet) =a OE where 53 mj is to be taken at rate 7 such that [tae eats Re Ts For example, the present value of a series of 50 annual payments beginning at $100 one year hence and increasing by 2% per annum is, at 5% compounded yearly, 1—(1.02)59(1.05) >” 7652835 either $100 = $100 ————_ = $2,550.95 1.05 — 1.02 .03 or $—— ag at 2.94% = $2,550.95 Periodical Payments. 29 or $—— Si at — 2.86% = $2,550.95. 9. To trace the growth of savings deposits intended for investment is interesting as well as important. A fund is to be built up as follows—deposits of 1 made at the end of each period are to be accumulated at 7 per period for nm periods. The accumulated amount is then to be in- vested in securities bearing j per period, (j >7). During the next 7 periods the interest from the securities is to be deposited with the 1 each period for another 7 periods, and the accumu- lated amount again invested as before. This process is to be continued. After ” periods the fund will amount to sjj at rate 7, which we will write as plain sz. After 2” periods the deposits will amount to (1+ 7 sq) sqj and the whole fund to syj+(1+7 sz) sa- After 3n periods the deposits will amount to {14+ 9 sq til +i sm) sa} sm= (U4 sa)? sap Midetieswhole itnd to. 5,)4- (4-9 Sui) Sait (LE Sa? Sue &c. &c. Sec; This schedule may iliustrate the matter. 2 ge . Accumulation Funds invested at Periodical d t Os eae : of the 2 rate 7 by the end periodical deposits} of the periods. First 1 1 Sn] Snj Periods. Second | 1+7 snj (1+ jsx]) suf Sn] (1 +1-+7sn)) Periods. Third n | (1+) sy)? (147 sn)2smy sal{ 1+(1+jsqj)+ Periods. (1+j sai)? The rthn | (1-+7 suj)’—* (L+j sa)’—! sep | Gtgsap'-1 Periods. J 30 Interest and Bond Values. After rn periods the whole fund will amount to Sahel ed sap Cl Foe ei Pi ction cael Omer nC ieee 4 Gems j = Reale) ic where (1+h)"=1+7 sq, a h “ = —— Sz, where Sz is taken at rate h. So that the fund might have been built up by uniform de 7 : nh : : j : posits of —— to increase at rate # without any reinvesting. 4 10. Reference has already been made to the frequent necessity for finding the half yearly payment equivalent to a given yearly or quarterly payment, or vice versa. A Table of equivalent payments at different periods is given below for ready reference. The method of obtaining the values may be illustrated, at 5% for example, as follows: Quarterly payments of $250 each are equ‘valent to half yearly payments of $250+$253.125 = $503.125. Therefore half yearly payments of $500 are equivalent to 500. quarterly payments of ae opis of $250. = $248.447. Or, half yearly payments of $500 are equivalent to quarterly payments of $500 $F; at 14% %=$248.447. Quarterly payments of $250 each are equivalent to yearly payments of $250.+$253.125+$256.25+$259.375 = $1018.75. Therefore yearly payments of $1000 are equivalent to quarterly payments of Pee of $250 =$245.399, allowing no interest on interest throughout the year. Periodical Payments. 31 Or, yearly payments of $1000 are equivalent to quarterly payments of $1000 SF at 14% = $245.361 allowing for interest on interest throughout the year. Half yearly payments of $500 each are equivalent to yearly payments of $500.+$512.50 = $1012.50. Therefore yearly payments of $1000 are equivalent to half- 1000. 1012.50 Or, yearly payments of $1000 are equivalent to half- yearly payments of $1000 SH at 214% =$493.827. yearly payments of of $500 = $493.827. EQUIVALENT PERIODICAL PAYMENTS. Each payment made at the end of the period. (Simple Interest). Rate of Interest % 25 .30 .40 .50 .60 70 15 80 90 .90 .10 20 .25 .30 40 50 60 10 15 .80 OW) .00 7k0 .20 25 30 .40 .50 . 60 rcO 15 .80 .90 .00 .10 20 25 30 .40 .50 .60 70 75 .80 .90 .00 $500 a half-year is $250 a quarter is $1000 a year is equal : equal to equal to Rate ot : =F Sa eee el iterest Half-yearly| Quarterly |} Yearly | Quarterly| Yearly |Half-yearly % paym nts of paym'nts of|paym'nts of| paym'nts of|paym'nts of|paym'nts of $497 20) $247.91/$1005.62) $249.30/31008.44) $501.41; 2.25 497.14) 247.86] 1005.75) 249.28] 1008.62). 501.44) 2.30 497.02} 247.77} 1006.00; 249.25} 1009.00} 501.50} 2.40 496.89} 247.68} 1006.25) 249.22) 1009.37] 501.56) 2.50 496.77| 247.59} 1006.50} 249.19} 1009.75} 501.62} 2.60 496.65} 247.49) 1006.75) 249.16) 1010.12) 501.69) 2.70 496.59} 247.45) 1006.87; 249.14; 1010.31} 501.72} 2.75 496.52} 247.40) 1007.00; 249.13) 1010.50} 501.75} 2.80 496.40) 247.31} 1007.25; 249.10) 1010.87) 501.81] 2.90 496.28} 247.22) 1007.50} 249.07} 1011.25} 501.87) 3.00 496.15} 247.13} 1007.75) 249.04] 1011.62} 501.94) 3.10 496.03} 247.04} 1008.00; 249.00} 1012.00; 502.00) 3.20 495.97} 247.00) 1008.12) 248.99) 1012.19} 502.03} 3.25 495.91} 246.94) 1008.25; 248.97) 1012.37! 502.06} 3.30 495.79) 246.85] 1008.50! 248.94) 1012.75) 502.12} 3.40 495.66} 246.76} 1008.75) 248.91} 1013.12} 502.19) 3.50 495.54} 246.67} 1009.00) 248.88; 1018.50; 502.25} 3.60 495.42} 246.58] 1009.25) 248.85} 1018.87; 502.31} 3.70 495.36} 246.53} 1009.37} 248.83} 1014.06) 502.34) 3.75 495.29 246.49) 1009.50} 248.82] 1014.25) 502.37} 3.80 495.17) 246.40) 1009.75) 248.79| 1014.62} 502.44} 3.90 495.05} 246.31} 1010.00) 248.76} 1015.00} 502.50) 4.00 494.93} 246.21) 1010.25) 248.73} 1015.37} 502.56] 4.10 494.80} 246.12} 1010.50} 248.69} 1015.75) 502.62) 4.20 494.74! 246.08} 1010.62) 248.68} 1015.94) 502.66) 4.25 494.68} 246.03) 1010.75) 248.66} 1016.12} 502.69} 4.30 494.56} 245.94) 1011.00; 248.63) 1016.50} 502.75) 4.40 494.44} 245.85) 1011.25} 248.60} 1016.87) 502.81} 4.50 494.32} 245.76) 1011.50) 248.57) 1017.25) 502.87] 4.60 494.19} 245.67) 1011.75) 248.54} 1017.62} 502.94; 4.70 494.13} 245.62} 1011.87) 248.52) 1017.81} 502.97) 4.75 494.07} 245.58) 1012.00) 248.51} 1018.00} 503.00] 4.80 493.95) 245.49] 1012.25] 248.48] 1018.37} 503.06] 4.90 493.83} 245.40) 1012.50) 248.45) 1018.75) 503.12} 5.00 493.71} 245.31} 1012.75) 248.42; 1019.12} 503.19) 5.10 493.58} 245.22) 1013.00; 248.39) 1019.50; 503.25) 5.20 493.52} 245.17) 1013.12} 248.37] 1019.69} 503.28) 5.25 493.46} 245.13} 1013.25) 248.35) 1019.87; 503.31} 5.30 493.34] 245.04} 1013.50} 248.32) 1020.25} 503.37, 5.40 493.22) 244.95) 1018.75} 248.29) 1020.62; 503.44} 5.50 493.10) 244.86) 1014.00} 248.26} 1021.00} 503.50) 5.60 492.98} 244.77) 1014.25) 248.23) 1021.37} 503.56) 5.70 492.91} 244.72) 1014.37; 248.22) 1021.56} 503.59) 5.75 492.85) 244.68! 1014.50; 248.20) 1021.75) 503.62) 5.80 492.73) 244.59) 1014.75} 248.17) 1022.12} 503.69} 5.90 492.61} 244.50) 1015.00} 248.14) 1022.50) 503.75] 6.00 FNTEREST AND, BOND VALUES. CHAPTER III. THE STRAIGHT TERM BOonp. 1. The straight term or ordinary coupon bond is a promise to pay a definite sum on a definite future date, and has attach- ed to it a number of separate promises called coupons to pay ’ interest in the meantime on the above sum at a specified rate. These coupons are meant to be cut off and presented for pay- ment on the successive interest dates printed upon them. The first coupon to be cut off is usually dated six months after the issue date of the bond itself: the last coupon to be cut off being the one which bears the due date of the bond. These bonds are not subscribed for at a fixed price like stock, nor is the holder under any obligations such as are imposed upon the stock subscriber. They are bought as an investment to yield the purchaser a rate of interest appropriate to the circumstances of the bond-issuing corporation and the details of the issue. This znvestment rate has no connexion whatever with the bond rate at which the coupons are payable. 2. In arriving at a decision as to what price he is willing to bid for a bond, the purchaser should have in mind a number of considerations amongst which the following may be mentioned. — (i) The extent and nature of the security mortgaged for the redemption of the issue. (ii) The ability of the issuing corporation to earn the annual coupon payments. (iii) The extent to which personal supervision of his invest- ment may become necessary, and his capacity to exercise such supervision. oan Interest and Bond Values. (iv) The breadth of the market in which his investment will be saleable in the event of need. (v) The current prices of apparently similar securities. The examination of these matters results in the purchaser reaching a decision as to the rate of interest the investment ~ should yield him under all the circumstances considered. Once that decision has been reached a mere reference to a book of Bond Values will indicate the price he can bid. 3. A sample extract from a Bond Table will shew the usual arrangement of values. 4% BOND. with half yearly coupons. Investment Rate % Time to Maturity. compounded half-yearly | 28 years |2814 years} 29 years |29%4 years 3.50 108.878 | 108.971 | 109.063 | 109.153 3.55 107.944 | 108.026 | 108.107 | 108.187 3.60 107.020 | 107.092 | 107.163 | 107.233 3.65 | 106.106 | 106.169 | 106.230 | 106.290 3.70 105.203 | 105. 256) 105.308 | 105.359 The sequence of values may be thus illustrated :— Purchase price of a 4%bond with 29% years to run houghttosyieldigi2 gor amet anne $109.153 Add halfa year's:interest at 334%. . us -.22. 542s eee 111.063 Deduct valué’ of cotipon.-4) 6. 62 oh ee ee 2.000 Value of bond with 29 years torun............... 109.063 Add ‘halt a:vear's anterest-at.3 14% ss. oe oe ee 1.908 FOrward ooicient oe nee te Le ee eee 110.971 The Straight Term Bond. 35 CET OE WAL Ge noe 6 tit vie oo oe oe gS $110.971 BemmeVALIC ON COUDOI oy rac tie ke ce he lence cee ws 2.000 Value of bond with 2814 years torun............. 108.971 Add-half a year’s interest at 314%... 2... cece. 1.907 110.878 MeV AIGL OL COUDOIN. .8 ic. o50 sar ci sivie ocls de pee oda 2.000 Value of bond with 28 years torun............... 108.878 4, The value of a bond for a unit due z coupon periods hence and bearing 1 coupons at rate j is, at an investment yield of 7 per period, V,,=jaz,)-+v" where aj and v” are taken at rate 7. One coupon period after the purchase, we have Va (1 +i) —j= Vn-1 Since the investment will have earned interest at rate 7, will have produced a coupon worth 7, and will then stand at Money; or (Jaq+e") (141) -j=jagzyto" so V,-; (1+7)-j=V,-2 &c. = &c. until we have (Vi=jaqt) (+i) —-j=1=,. The values printed in a bond table are of course not calculated independently, also of course the results must be checked before publication. The Bond rates usually tabulated range from 24% by steps of 4% to 6% or 7%. In some cases also values of 234%, 3%4%, etc., bonds are tabulated. The Investment or yield rates usually cover about the same range as the bond rates, but the steps or intervals are very much smaller—a common interval being .05%. The value of a bond for a unit due 2 coupon periods hence and bearing m coupons at rate j per period is, at an investment rate of 2 per period. j a4+v” when az and v” are taken at rate 1 36 Interest and Bond Values. -Fas+%=L4 G-4) De 4 4 4 or j dq tv"=j ag+1-i ag 1 ij) ah eee (B) Either of these formule are suitable for use on a multiplying machine. To check the values we have the following :— The value at yield rate 7 of a bond for ” periods with coupons at rate 7 is 7 djj+v", but if the coupons had been at rate j+6 the value would have been (j+6) ay-+v”. The difference is 6 dy. Therefore if the bond rates form an arithmetical series, as they almost always do, the prices for the same period and yield will also form an arithmetical series. In other words the price of a 5% bond is exactly half-way between the prices of a4% bond and a 6% bond for the same period and at the same yield; and the difference between the prices of a 4% bond and a 44%4% bond will, if added to the price of a 6% bond, give the price of a 6%% bond, all being for the same period and at the same yield. Many tables of Bond Values have been published and some of them are probably quite accurate, but the writer has in his possession the, thirteenth edition of a widely used table which contains over one hundred errors many of which are by no means trivial. The values given in any table should be check- ed before using, either by an independent table, or, better, by the use of the interest tables we have been discussing. 5. Consider a $10,000 bond, due exactly 20 years hence and bearing half yearly coupons at 5%. The holder of such a bond will receive (i) $250. at the end of each half year for 20 years. (ii) $10,000 at the end of 20 years. The Straight Term Bond o7 To find the value of such a bond on an interest basis of 414% compounded half yearly, we have only to discount the benefits to be received. POU real O41 6 rea eles cee veins $6,548.38 Mien O00 1 at 214%) sos. Bho 4,106.46 $10,654.84 in all. Or we may argue as follows:—Had the bond borne coupons at 414%, i.e. of the value of $225 each, its value would have been $10,000. But since the coupons are for $250 each, the value of the bond exceeds $10,000 by $25 ag at 214% =$654.84, or the value is $10,654.84. And this is the price which an investor should pay to yield him 414% compounded half yearly. Immediately after the purchase the investment will be standing in the purchaser’s ledger at its cost, $10,654.84. Six months later a coupon will be due and will produce $250. But the investment was made to yield 414%. Therefore it should only be debited for interest with $239.73 which is half a year’s interest at the investment rate on the purchase _ price. The balance of the coupon, namely $10.27 is a return of capital. So that after the cashing of the coupon the in- vestment will stand at $10,644.57 = 250 a3g| + 10,000 wehbe next half year’s interest will be $239.50 and the second coupon will contain $10.50 of capital repaid. By continuing this process the investment will be written down little by little, but more and more, each half year until at the end of 1914 years and just after cashing a coupon it will stand at $10,024.45 =250 ayj+10,000 v. The last coupon will con- sist of $225.55 interest, and $24.45 capital, which with the $10,000 then repaid will just balance the investment. It may be worth noting that if the investor could find an- other investment for $10.27 each half year to yield him 414% compounded half yearly, he could leave the bonds in his ledger at their purchase price and, regarding $239.73 as a uniform interest income from the bonds, place the $10.27 in 38 Interest and Bond Values. the other investment each half year. This second invest- ment, if allowed to accumulate, would amount on the due date of the bond to $10.27 sq at 214% =$655.08 which together with the $10,000 then payable would practically balance the purchase price. Sometimes in connexion with trust funds it is not desirable to write the investment down, and it is usually impossible to find another investment for the small periodical repayments of capital at the same investment rate. In the purchase we have been considering it may be imperative that the capital should remain intact, and yet it may be impossible to invest a sinking fund at better than 8% compounded half yearly. Under such conditions the sinking fund will demand $654.84 Sia at 1%% = $12.07, in place of $10.27 out of each coupon, thus reducing the investment yield to $237.93 each half year on a capital of $10,654.84 which will remain intact This is equivalent to an investment rate of 4.466% instead of 414%. 6. On the other hand suppose that the investor had bought $10,000 of 4% bonds due 20 years hence and bearing half yearly coupons to yield him 44% % compounded half yearly. A bond table will shew the price to be $9,345.16. This should be checked by discounting the benefits as follows $200 ag at 214% =$5,238.70 $10,000 v® at 244% = 4,106.46 $9,345.16 in all. Or, we may argue that had the coupons been for $225 each the value of the bond would have been at par. The coupons being only for $200 each, the value will be at a discount of p20 ay at 234% =$654.84. The value is $9,345.16. The first half year’s interest should be 214% on the price, that is $210.27 but the coupon would only produce $200. Debiting the account with the interest due and crediting it with the coupon would write up the investment by $10.27 to $9,355.43. The next half year’s interest would be $210.50. So The Stratght Term Bond. 39 that the investment would have to be still further written up by $10.50. Continuing this process we should find that six months before the due date of the bond, and immedi- ately after the cashing of the penultimate coupon, the invest- ment would be standing at $9,975.55, on which the last half year’s interest would be $224.45. The last coupon would produce $200 and the balance of the interest due would bring the investment up to the $10,000. then payable. Again, it is worth noting that if the investor could find someone to lend him $10.27 each half year, on the security of his bonds, and allow such a series of little loans to accumulate at 444% compounded half yearly, he would, on the due date of his bonds owe this lender $10.27 sq at 2144%=$655.08. So that the bonds might have remained on the purchaser’s books at their purchase price, $9,345.16, producing $210.27 each half year; and when they were paid off, the difference between the $10,000 and the purchase price would practically wipe out the accumulation of small loans effected to make up the interest. Here also it may be not only undesirable to write up this investment but it may be important to maintain an annual income as large as possible consistent with the preservation of the capital; and yet small periodical borrowings such as we have indicated might be only possible at, say, 6%. Now the premium of $654.84 at which the bonds will be repaid in excess of the purchase price will at this rate only permit of semi- annual borrowings for interest to the extent of $654.84 SH at 3% =$8.68 each half year. By this means the investment will yield $208.68 each half year ona uniformcapital of $9,345.16 or at the rate of 4.466% instead of 4%%. 7. It will be noticed that if the bond rate exceeds the invest- ment rate of interest, the bond will be selling at a premium, but if the bond rate is less than the investment rate, the bond will be selling at a discount. The algebraic investigation of this premium or discount is instructive. Consider a bond for 1 due 7 periods hence bearing coupons at j and selling at 1+, i.e. at a premium of #, to yield 1. 40 Interest and Bond Values. Then 1+ =j a;)]+v”" at rate 7. But 1=1 a;j+v" at rate 1. Therefore =(j—1) az at rate 1. Or the premium is the present value of the excess interest in the coupons. Similarly if 7 be less than 7 and the bond is selling at a dis- count of d we have 1—d =j a;j+v" at rate 7. 1 =1 a;)+v" at rate 7. .. d=(t—J) az at rate 7. Or the discount is the present value of the shortage of interest in the coupons. Moreover the Bond Tables might have been constructed from the interest table of ajj without reference to that of (1-+72)”. Again, a perpetual bond for 1 bearing coupons at j would be worth J at an investment rate of 1. 4 Therefore a bond due 7 periods hence, and bearing coupons at j(>1),isworth less than J by the cash value of the differ- 1 ence between the a and the 1 which the bond will ultimately 1 produce; or the price should be Z_(L_1)y= 1+ G—iaq 1 1 So a bond due m periods hence, and bearing coupons at j(,%, it should sell at a discount equivalent to the shortage of interest in the coupons. This discount will be $(302.50 —300) a3q at 234%. = $2.50 aaj = $50.62 The price therefore is $11,000—$50.62 = $10,949.38. The Straight Term Bond. 45 Consider the case of a $10,000 bond bearing half yearly coupons at 5% for only the last 15 years of its term, and re- payable 20 years hence at a premium of 25%. To find the value of such a bond to yield the investor 434% compounded half yearly. The bond tables are not directly applicable. Discounting the benefits by the use of interest tables we have $250 (aq —ajy) at 238% = $6,707.68 $12,500 v at 238% = 4,888.25 ———— $11,595.98. 12. The general theorem regarding the value of a loan has been established by the late Mr. Makeham of London as follows: Let C be the capital repayable including any premium on repayment; j be the nominal rate of interest payable on C or on any un- paid portion of C; 1 be the lender’s investment rate; K be the present value of C at rate 17; while A is the purchase price of the loan. Now A isthe present value of Catratez, that is K, plusthe pre- sent value at rate zof theinterest payments to bemadeatratej. Therefore the present value at rate 7 of the interest pay- ments to be made at ratej7 is A—K. Now had theloan borne a nominal rate 71 an investor would have paid par for it, i.e. A would have been equal to C. There- fore the present value at rate 2 of interest payments at rate 1 is C—K. Therefore the present value at rate 2 of interest payments at rate j is (C—K). 4 So that A =K+ (C—K), which is Makeham’s formula. 4 We may illustrate this formula by applying it to an ordin- ary bond for 1 due 7 periods hence and bearing coupons at rate 7. 46 Interest and BARE Values. In this case C=1, and K =v" Therefore the price is v”+ ea —v") 4 | =1- (1- ~~) (-»%. A convenient formula for the calculation of bond values. Since 1 — (1— +) 1-0") =1—-G—J) oar 4 =1+(j—1) az), we have again deduced that a bond will sell at a premium or at a discount equal the present value of the difference be- tween the coupons actually payable and those that would be payable under the yield rate. Arithmetically Makeham’s formula is of great value in cases where the capital is repayable by instalments. A series of bonds issued to repay capital by instalments is called a Serial Issue. Consider a loan for $10,000 at 6% payable half yearly, €500 of the capital to be repayable with each payment of in- terest, and suppose we want to know the price to yield the investor 534% compounded half yearly. In this case C=$10,000. K=$500 aay at 278% =$7,525.44 C—K =$2,474.56 J (C—K) =# of $2,474.56 = $2,582.16 4 Sees ee US, »,A=K+ 1 (C—K)=$10,107.60. 4 ————$ —___. Again, consider a loan of $10,000 at 6% payable half yearly to run for 10 years and then to be redeemed by yearly instal- ments of $1000 at a premium of 10%, the last instalment of capital to be repaid 20 years after issue; and suppose we want to know the price of such a loan to yield the investor 514% com- pounded half yearly. The Straight Term Bond. 47 In this case C= $11,000 and j =55 %. K =the present value of the instalments of $1100 a year equivalent at the yield rate to $542.54 each half year = $542.54 (aqj—aag) at 234% = $ 4,801.94 . C=K = $6,198.06 J (CK) = 12° of $6,198.06 = $6,146.84 4 Ged — Ke (CK) = $10,948.78. 4 Ee eee 13. There remain to be considered those bonds which are repayable by the operation of an accumulative sinking fund. The following is a case in point. A foreign government issues a loan of $1,000,000. at 5% with yearly coupons and agrees to set aside 614% of the par value of the bonds each year to pay the coupons and for the re- demption of the issue by annual drawings—the numbers on the bonds to be redeemed each year being chosen by lot. Here we have an annuity of $65,000 a year devoted to the service of the loan until it is all redeemed. At the end of the first year $50,000 will be needed to pay coupons, and $15,000 of the bonds will be drawn and cancelled. At the end of the second year only $49,250 will be needed for coupons and $15,750 will be available for redemption purposes, and so on. Since the $1,000,000. must be the cash equivalent of the annuity of $65,000, discounted at 5%, the bonds will be all paid off in 2 years where 1,000,000. = 65,000 az] on a 5% basis or = .065 in the 5% table. A reference to the table shews that 1 is a little over 30 years, which is of course quite independent of the issue price of the loan. To find the amount of the issue that will be outstanding after any, say the 18th, annual drawing, we have only to find the cash value at the bond rate of the remaining payments of $65,000 a year, that is $65,000 aj at 5% =$576,111. 48 Interest and Bond Values. To find the issue price of such a loan to net the purchaser of the whole tssue 514% yearly, we have only to find the cash equivalent of $65,000 a year for 30 years on a 54% basis, that is $65,000 agq at 514% =$944,693. The 30 years is slightly less than the true time, but a purchaser at 94% would be practically buying on a 514% basis. The gamble introduced by the annual drawings makes it impossible to say what investment rate would be realised by the holder of any one bond of the series. Suppose that a purchaser of one bond who bought at 96. found that his bond was drawn for cancellation at the end of the first year. He would get $105 after one year for an investment of $96, that is 9%. On the other hand if his bond was not drawn till the end of the 30 years he would only obtain an investment yield of about 544%. 14. Loans of this character are sometimes complicated by the introduction of a temporary sinking fund to provide for redemptions less frequently than once a year. Consider a loan of $1,000,000 at 5% with half yearly cou- pons. For the service of this issue $32,500 is to be set aside each half year, out of which the coupons are to be paid and the balance deposited in a sinking fund which accumulates at 4% compounded half yearly. At the end of every third year » during the currency of the loan the sinking fund is to be ap- plied to the redemption of bonds at a premium of 10%, the numbers on the bonds to be cancelled to be chosen by lot. To find in what time the issue will be redeemed by the operation of such a sinking fund. In this case we may regard the issue as one of $1,100,000 of bonds bearing 43°7%. The half yearly coupons will call for $25,000, leaving $7,500 for the sinking fund. The bonds redeemed each three years are purchased at par for the sinking fund, and the coupons on the bonds so purchased will increase the semi-annual con- tributions to that fund, which will grow as follows :— End of 3rd year:—$7,500 sqj at 2%. (See page 29.) End of 6th year :—$7,500 {set +] 56) ss} where j= 277% The Straight Term Bond. 49 End of 9th year:—$7,500 sqj {1+(1+j sq) +(14+j sep?} and so on. End of 3nth year:—$7,500 sq {1 +(1+j sop+(1+j sept... +(1+] Sapiro (L+J $67) =} _ 97 599 At 7) = 1-Fy Sei J If the whole $1,100,000. be then in the sinking fund we have = $7,500 53] _ 25,000 _ 10 G RBOO 3% ones log 4.3333 i log 4.33333 log (1+277% ofsgj) — log 1.14337 _ 6368221 0581867 Therefore the issue will be redeemed in 32.83 years: which means that the sinking fund will have more than enough money epson G3) (1+71 11 t $s) =C3—Ce; so (C3— Ces) (1 +44 ts) =Cy—Co. &c. &c. That is to say the triennial redemptions of capital are in- creasing in geometrical ratio with a constant factor (1+7% is). The first term in the series of redemptions is Co— C3, which is ti(p—12Co)s. Therefore there must be » drawings before the whole loan is redeemed where The Stratght Term Bond. 51 (1+ti1s)"—1 Ti (p—1 Co)s x ey Sartrg or (1-+19i s)"=1+ Berio ech me sng ea pa ek | ay (p—t Co)s p—1Ce in figures, (1++5 of 214% of 6.30812)”=1+ enh or (1.14337)”" = 4.33333 _ log 4.383333 — . 63682 _ log 1.14337 .05819 or the time required is 8n years=382.8 years. The progress of this loan might be scheduled as follows: 2 The semi-annual pay- ona et Bonds | Total Amount of ‘| ment of $32,500. ee ae redeemed eae Bonds out- & at end of ( demption dj S1 for foe [She Period riod, | 28d OE | a ot o e perio . 3 S tl d. “3 Coupons | Sink. fund 8) oe i (4) X it pier the period. (1) (2) (3) “(4) (5) (6) (7) 1 | $25,000. $7,500. | $47,311. | $43,010. | $43,010. | $956,990. 2| 23,925. 8,575. 54,094. 49,176. 92,186. | 907,814. 3| 22,695. 9,805. 61,849. 56,226. | 148,412. | 851,588. 4) 21,290. 11,210. 70,716. 64,287. | 212,699. | 787,301. 5| 19,683. 12,817. 80,854. 73,504. | 286,203. | 713,797. &c &c. &c. &c. &c. &c. &c. It should be noticed that each value in the 5th column may be obtained from the preceding value by increasing the latter by 14.337%. It may reasonably be objected that bonds are almost al- ways in fixed denominations and that fractional amounts could not be redeemed as is assumed in the above calculations. In order to test the error introduced by this assumption let us say that the bonds are in denominations of $1000 each. a 52 Interest and Bond Values. Then the actual working of the sinking fund would be as follows: First period—$7,500 s¢j = $47,310.91, enough to redeem $43,000 of the issue leaving $957,000 outstanding. Second period — $10.91 (1.02)§+$8,575 sg = $54,104.43, enough to redeem $49,000 of the issue leaving $908,000 outstanding. Third period—$204.43 (1.02)® + $9,800 s¢j = $62,049.81, enough to redeem $56,000 of the issue leaving $852,000 outstanding. Fourth period—$449.81 (1.02)° + $11,200 sgj= $71,157.52, enough to redeem $64,000 of the issue leaving $788,000 out- standing. Fifth period—$757.52 (1.02)® + $12,800 56] = $81,597.04, enough to redeem $74,000 of the issue leaving $714,000 out- standing against which there is a balance of $197. in the sink- ing fund. That is to say a net debt of $713,803. Our un- practical assumption resulted in shewing a net debt of $713,797 at the same date:an error of only $6. Practically, the sole cause of the error is the difference between the 5% bond rate and the 4% sinking fund rate on the small sums left between drawings to accumulate in the sinking fund. 15. So far we have been considering bonds bought either at issue or at a regular coupon date. But bonds are of necessity bought between coupon dates and when this happens the seller is obviously entitled to some portion of the value of the current coupon. There is no difficulty when the bond is sold at a flat price which is supposed to include the adjustment in question; nor should there be any difficulty where the bond is sold at a price ‘‘and interest’? which means accrued bond interest and is easily ascertained. It is when the bond is sold on a yield basis that difficulties have arisen. There is a want of uni- formity among bond dealers in this matter, although of recent years most of them, when dealing on a yield basis, are using the second of the methods to be now described. 16. If an ordinary coupon bond is sold between two coupon an ~ The Stratght Term Bond. 53 dates to yield the purchaser a definite investment rate of in- terest, it is sold on a yield basis. To find the price. True Price=Value as at last. coupon date, plus true interest for the fraction of a period elapsed since that date. First method of approximation: Price = Vo+J—D. where Vo = Value as at last coupon date. I=Simple interest on the face value of the bond at the bond nominal rate for the time since the last coupon date. D=Banker’s discount on J at the investment rate for the time to the next coupon date. Second method of approximation: Price= V)»>+J—D. where Vp = Value as at last coupon date. I =Simple interest on Vo at the investment rate for the time since the last coupon date. D =Banker’s discount on J at the investment rate for the time to the next coupon date. Third method of approximation: Price =(WitC) +(1+JD) Where V,=Value as at next coupon date. C=Face value of current coupon. I=Simple interest on 1 at the investment rate for the time to the next coupon date. Fourth method of approximation: eee (VCH YV) n Where Vo, Vi and C have the same values as before 1 3 : “and — = the fraction of a coupon period that has n elapsed since the last coupon date. All four methods are, or were recently, in actual use. 17. Discussing the problem algebraically we will consider a bond for 1 bearing coupons for j per period, sold of a period after a coupon date to yield the purchaser 7 per period. 54 Interest and Bond Values. If V,be the value at the last coupon date and V, be the value at the next coupon date we have Vo(1 +1) Rh Vi. The true sale price is Vo (1 Lie —1 1) (2 ie = Vi(1¢ 2 Boing COMED pte.) 2n” Gin The first method gives Vo+ ae engar: 1 ) n n n—l.. _ ale n n? The second method gives Vo {1 35 +(1 asfiSeen 1 )} nN nN & Vo{1+—— ee it}. n? = ot The third method gives (V; + 7) ( Udo i ) n Ve 12) (-* 4 4 Oa a (x—1)3 t+ —&c. ) n—1 i wri 1)? n* fe ee ~ +&c. }. The fourth method gives Vo + a (Vitj— Vo) n Shy7ie = {Yo(1+4)— Vo} 4 = m(1+—). 18. Arithmetically we may take the following example $1,000,000 of 4% bonds due 1 July 1940, with half yearly coupons 1 July and 1 January, bought on 1 September 1910 to yield 5% compounded half yearly. True Price = (Tabular value as at 1 July 1910) X(1.025)# = $845,456.72 x 1.00826484 = $852,444.28. The Straight Term Bond. 55 True Price X (1.025) 4 = $852,444.28 x 1.01659798 = $866,593.14 = $846,593.14-+ $20,000.00 = Tabular value as at 1 January 1911+-value of coupon then due. First Method: maauervalue as at | July 1910. 2.202.024... ©. $845,456.72 Two months’ interest at 4% on oS EOE Senay) Bree al tet Sot? $6,666 . 67 Less 4 months’ ‘‘discount”’ at 5% BEMEPIH DUG U ini e ts.) oe ower eke. LIV es OD don00 Erp OrOxIMIAtes TICE sg, 22 han aos in eee $852,012.28 PELPOUrTIEGOIECT 5.8 35th Se eee ee 432.00 Meiemirigey tas ht eee) ka Le ae $852,444 . 28 Cost price as at 1 September, 1910............. $852,012.28 4 months’ interest thereon at 5%............ 14,200.20 $866,212.48 Value of coupon due 1 January, 1911........... 20,000.00 Purchaser’s value as at 1 January, 1911........ $846,212.48 Epa aaL ACL OS ULC uate, ald skc Weds yo SC ee 380. 66 aviiar vale as at |: January, 191). 3 2. 220s $846,593.14 Second Method: Sitiarevalueds atl July 1910... 2...) 0.8 ..% $845,456.72 Two months’ interest at 5% on CN cf a a a ee $7,045.47 Less 4 months’ ‘‘discount’”’ at 5% a TYG ly a et 117.438 6,928.04 3 “sitet ea beh ag Cerca ay Re $852,384.76 - Pere TMGLE CUA tdi ice Gee ie? eee a 59.52 RE ACP op ONIN, “Sit roa we . at cate Rigen ss $852,444.28 ‘2 Se 56 Interest and Bond Values. . Cost price as at 1 September, 1910 $852,384.76 4 months’ interest thereon at 5%............ 14,206.41 $866,591.17 Value of coupon due 1 January, 1911 20,000.00 ~ Purchaser’s value as at 1 January, 1911....... $846,591.17 Erronan detectwnis eh aie ah Re eee 1697 Tabular value as at 1 January, 1911........... $846,593.14 Third Method: Tabular value as at 1 January 1911 with coupon. $866,593 . 14 Four months’ interest on $1. at 5% is .016...... Approximate price = $866,593.14+1.016........ $852,386.69 E’rfonin- celect) one o out ee eee 57.59 rues Priced isis ee i ah ees cre er ie $852,444 .28 - Cost price as at 1 September, 1910............. $852,386 . 69 Four months’ interest thereon at 5%.......... 14,206.45 $866,593.14 Value of coupon due 1 January, 1911........... 20,000.00 Tabular value as at 1 January 1911 (no error)... $846,593.14 Fourth Method: Vabular-value.asiat) ki July 1910: Fos eae $845,456.72 Two months’ interest thereon at 5%........... 7,045.47 ANDroximate price enc.s teacay es one eee $852,502.19 ErrOt 10 GXCOSS 9 8 Here et ait pee 57.91 ‘Prites Pace seo oxi Pt et is Ce ee $852,444.28 a a ite er rN The Straight Term Bond. 57 Cost price as at 1 September, 1910............. $852,502.19 Four months’ interest thereon at5%.......... 14,208 . 37 | $866,710.56 Value of coupon due 1 January, 1911........... 20,000.00 Purchaser’s value as at 1 January, 1911....... $846,710.56 ME TIICNOCGS teat in at ele ra ir a rien ati 117.42 Tabular value as at 1 January, 1911........... $846,593.14 ——— 19. In the first method of approximation the underlying argument seems to be that the only change in value from one - coupon date to the next is due to the accrued bond interest. This is of coursea serious error unless the bond rate is close to the investment rate. The method produces an error in defect when the bond rate is less than the investment rate and vice versa: this error increases with the time since the last coupon date. The method probably owes its origin to a confusion be- tween selling ona yield basisand selling at a price and interest. In the second method of approximation the underlying argument is sound, but the attempt to correct the error of simple interest can hardly be called successful. The method always produces an error in defect. The error is roughly equal to half the ‘“‘discount”’ item, and it is greatest about half way between two coupon dates. In the third method of approximation the underlying argu- ment contains the error of ‘“‘simple interest’’. The method always produces an error in defect which is uniformly slightly less than the error of the second method. In the fourth method of approximation the underlying argument contains the error of “simple interest’. The method always produces an error in excess nearly equal in amount to the opposite error of the second method. The advantage of the error in defect is that it will be more or less compensated by the error that the purchaser will make at the next coupon date when he adds simple interest for the 58 Interest and Bond Values. time he has held the bond. The third approximate method is such that this ‘‘compensation”’ is complete. The second approximation is, as already mentioned, the one in common use in Canada today; yet it is frequently used with a bond table to only two places of decimals. The result of using so “‘short”’ a table is'to introduce initially an error which may be as great as $50.00 on $1,000,000.00 of bonds or 5.000n 100,000.00 of bonds or? -7.50:on 10,000.00 of bonds. Not a very serious error, but one whichrenders the subsequent items of ‘‘interest’”’ and ‘‘discount”’ carried out to the nearest cent an absurdity. This lack of ‘‘ the sense of the fitness of things’’ may be seen on almost any statement of any bond- dealer who sells on a yield basis. If the bond value used be true to the nearest $1.00, then the interest and discount items should also be carried out only to the nearest $1.00. 20. It is usual for the coupon dates to correspond with the due date of the bond. For example if the bond be due on a 5th May, the coupons are usually due on 5th May and 5th November in each year. But such a correspondence is not always observed. Consider the following case which is not an imaginary one. Bonds for $100,000 at 4% due on 1 January 1930, had half yearly coupons dated 1 May and 1 November in each year, and were sold on 1 March 1910 to yield the purchaser 5% com- pounded half yearly. To find the price, we will use the second method of approximation referred to. With the capital there must be payable coupons for two months’ bond interest, that is, $666.67. The value as at 1 January 1910 of the capital and these coupons is $100,666.67 v#° at 2%% =...... $37,491. Ada twovwmonths interest a wa ae $312.4 less:dour. months: discount <2. oo eee 5.2 307. x Value of Capital and last coupons as at 1 March 1910 $37,798. es ee The Straight Term Bond. 59 The value of all the other coupons as at 1 November BeOS 2,000 Asya 22% eg ena ees $50,206. Bron amign tis NtCreSts.: ose 60. s 5 eke oa $836.8 fees ormnonlens Ciscount . 270. 3. Ae (fal) 830. Value of the other coupons as at 1 March, 1910.... $51,036. Motalvalue-or- bond as atl’ March, 1910... .7....2%.., $88,834. 21. The inverse problem—given the price to find the yield— is practically of very great importance and while the interest and bond tables available gave investment rates only at considerable intervals the problem was by no means a simple one. The best results were given by the formulae of Finite Differences. Now that bond and interest tables are published for investment rates proceeding by one twentieth of 1% the problem has practically lost its difficulty. Consider a 4% 40-year bond which sells at 9314. The Bond tables shew that such a bond would yield 4.85% if sold at 93.39; or yield 4.30% if sold at 94.30. That is an increase of .91 in the price produces a drop of .05 in the yield. Therefore a price of 93.50 corresponds to a yield of 4.35 — .006 =4.344%. 22. If we know the price at which any loan is issued we can use Makeham’s formula to find a close approximation to the yield. Changing the formula algebraically into the form we fo Ge A 1=J9+ C—K To use this formula, we make the best guess we can ati and 4 will give the best results. interpret the right hand side of the equation in accordance with that guess. The result should be very near the truth. Suppose that a 5% debenture with half yearly coupons and redeemable at 105 in 20 years is sold at 113.67. What will be the yield to the purchaser ? 60 Interest and Bond Values. 42, will be a fair guess. At this rate K =105 v® at 2% =47.55 C=105 : C-A=—8.67 : C—K=57.45 ae EL 105 1 & = 04762 — soe xX .64 57.45 = .04762—.00604 = .04158 or 4.158%. Our guess was toolow. Letustry4%4%. At this rate K=105 v® at 244% = 45.28. -C—A =—8.67 and j=.04762 as before. C—K=59.72. 8.67 . 4=.04762 — —— X .0425 59.72 = .04762 —.00617 = .04145 or 4.145%. A trial rate of 4% produced 4.158%. A trial rate of 444% produced 4.145%. .. the true rate is (4-+x) % where *:4%::.158—x: .013 or x=.150. The required rate is 4.150%. 23. A readier method of finding the yield would be as follows: Since the bond, bought at 113.67, will be redeemed at 105, there will be a loss of 8.67 on redemption. To make good this loss there should be set aside out of each coupon a sum equal to 8.67 a at the yield rate. Taking 4% as our first guess at this yield, we have 8.67 Sam at 29 =ai4oos But each coupon is worth 2.5. This leaves 2.38565 as a uniform half yearly return on a capital of 113.67 which will remain intact; that is, a yield of 4.146%. Trying 44% we have 8.67 s= at 2% %=.1397. This leaves 2.3603 as the 40] half yearly return on the unimpaired 113.67 ; that is, a yield o1 4.153%; A trial rate of 4% produced 4.146%. A trial rate of 44% produced 4.153%. Therefore the true rate is (4+x)%, The Straight Term Bond. 61 where x: .25 ::x—.146 : .007, or x=.15, and the required rate is 4.15%. 24. When neither a bond table nor an interest: table is available a fairly close approximation to the rate may be obtained by an algebraic formula. Consider a bond for 1 bearing coupons at j and due 7 periods hence, bought at1+ . To find 2, the yield rate. Now p=(j—1) az) at rate 2. pee ee in (1-24, approxi- heat & v 2 mately. oe 3 isd 58) = ( ~ aa 1 ) =] pee $ a approximately, 2(jn— p) n(2+p)+p Example: A 4% bond with half yearly coupons, and having 25 years torun is bought at 118%. To find the yield. Here 7=.02 ; n=50 ; p=.13 whence1 = 50 X2.13+.13 106.6 ora yield of 3.26% per annum. The true yield is 3.24 % per annum. If the bond be sold at a discount, the formula becomes _ 2(jntd) 2 ded: Example:—A 4% bond with half yearly coupons, and hav- ing 25 years to run is bought at 921%4%. To find the yield. Here 7=.02 ; 2=50 ; d=.075 mueteo dt 01D) eee et i 50 X 1.925—.075 96.175 or a yield of 4.47% per annum. The true yield is 4.50%. 62 Interest and Bond Values. 25. The values of the interest functions v” and aj may readily be deduced for all the investment rates of interest and all the unexpired terms of the bonds scheduled in the bond ~ table. For example the bond table contains the following: Investment Bond Unexpired Price Rate Rate term 3.60% 4% 40 years 108. 444650 which means that at 1.8% 100 v®+-2 agg = 108.444650 but 100v%-+-1.8 agj=100. . .2 ag = 8.444650 OF ag = 42.22325 ", 2 agy= 84.4465 -. 100 v®=23.99815 or v8 = 2399815 or using the bond table to obtain both equations we have Investment Bond Unexpired Price Rate Rate term 3.60% 4% 40 years 108. 444650 3.60% 6% 40 years 150.667900 which means that at 1.8% 100 v®-+-2 agj=108.444650 100 v®°+-3 ag =150.667900 - .. Ay] = 42.223250 as above. INTEREST AND BOND VALUES. — CHAPTER IV. ANNUITY BONDs. 1. As we saw ina previous chapter, a loan of $10,000 at 5% compounded half yearly, may be repaid by ten semi-annual payments of $10,000 ar at 214% =$1,142.59 each. At the end of the first half year the borrower will pay $1,142.59, consisting of $250 interest and $892.59 which must be areturn of capital, leaving $9,107.41 capital still outstanding. At the end of twelve months the interest on this will be only $227.69 so that the second payment of the annuity will con- tain $914.90 capital. This process may be scheduled as follows: Schedule illustrating the repayment of a loan of $10,000. at 5% compounded half yearly by a five year semi-annual annuity of $1,142.59 each half year :— Capital Capital repaid still out- | % Interest Capital to date standing |Yr. Semi-ann. payment of $250.00 ; $892.59 | $892.59 | $9,107.41 1 227.69 $14.90 | 1,807.49 | 8,192.51 | 2 204.81 937.78 | 2,745.27 | 7,254.73 ed 181.37 961.22 | 3,706.49 | 6,293.51 3 Al 157 .34 985.25 | 4,691.74 | 5,308.26 ee 132.71 | 1,009.88 | 5,701.62 | 4,298.38 ay oe 107.46 | 1,035.13 | 6,786.75 | 3,263.25 ahead tin Leek tbe la eer ey Mas | AN mc —— ——— — — — —_————eeEeSSFMN | eee 10 27.88 | 1,114.71 |10,000.C0O 0.00 | 10 64 Interest and Bond Values. 2. Bonds under which loans are repaid in this manner are known as Annuity Bonds. They are issued for a nominal amount and at a nominal rate of interest, but are of course bought as an investment to yield an appropriate investment rate of interest. 3. Annuity bond tables have been published and doubtless are used, but they are not satisfactory. Annuity bonds fre- quently provide for yearly payments, but the conventional yield basis is a rate compounded half yearly. It will be found more satisfactory to deal with such bonds by using the ordinary interest tables. Consider a twenty year annuity bond for $10,000 issued at a nominal 5% with half yearly payments and bought by an in- vestor to yield him 54% % compounded half yearly. The bond will be an obligation to pay $10,000 oF at 244% = $398.36 each half year for 20 years. The purchaser will pay $398.36 aq at 234% = $9,592 to obtain the required yield. The first payment of $398.36 will consist of $263.78 being half a year’s interest on the investment, and $134.58 being a repayment of capital. The capital then outstanding will be $9,457. The next payment of $398.36 will consist of é $260.07 being half a year’sinterest on capital outstanding, and $138.29 being another repayment of capital. The capital is thus written down from half year to half year until at the end of the 20 years it will be all written off. Theoretically the investment might have been treated as one producing $263.78 each half year for interest and $134.59 asa sinking fund payment towards the reproduction of the capital at the end of the investment period. Such a sinking fund would at 5%% compounded half yearly, produce $134.58 sq at 234% at the end of the twenty years. And $134.58 sp at 234% =$9,592. which is the amount of the capital invested. But sinking funds can seldom be accumulated at the high rates yielded by such investments, so that problems such as the following must be solved. Annutty Bonds. 65 4. A municipality issues a twenty year semi-annual annuity bond for $25,000, at anominal 5%. What should a purchaser offer in order that he may secure 544% on his investment for the whole time and reproduce his capital intact by a sinking fund which he can accumulate at only 4%? All ratesaretobe compounded half yearly. A reference to the Lien table will shew, that the munici- pality must pay $25,000 ar at 214% = $995.91 each half year for twenty years. But for every $10,000 invested by such a purclece he snore get each half year $262.50 for interest and also $10,000 i at 2% =$165.56 towards his sinking fund; or for each $428.06 in the semi-annual payment to be received, he can afford to pay $10,000. Therefore he will DENG, Ban a of $10,000 = $23 266. To test this result: The half yearly annuity payment is............. $995.91 Half a year’s interest at 544% on $23,266 is...... 610.73 Therefore amount available for sinking fund is ... $385.18 and $385.18 sq at 2% =$23,266, as it should. 5. To value an annuity bond with annual payments by the conventional half yearly compounded investment rate, we must find the equivalent half yearly payment at this rate. Consider a 25-year annuity bond for $50,000 at 5% with yearly payments to be valued on a 44%% basis compounded half yearly. The bond will produce $50,000 ax at 5% = $3,547.62 at the end of each year for 25 years, and this is equivalent at the investment rate to $1,754.08 at the end of each half year, as may be seen from the table of equivalent payments. Therefore the value required is $1,754.08 az at 214% = $52,331.92. Had the bond been issued with quarterly payments on the same nominal Petey these quarterly payments would have been $50,000 an vat 1 144% =$878.71 each, and the equivalent 66 Interest and Bond Values. half yearly payment at the investment rate would be $1,767.31, so that the value required would have been $1,767.31 ag at 214% = $52,726.63. 6. The repayment of a loan by an annuity which includes capital was well as interest may be shown algebraically as follows: Schedule illustrating the repayment of a loan of anil ie S ui by an annuity of 1 for 1 periods :— a The 7th payment 34 2h Period) of 1 consists of Total Capital repaid after the rth ‘a.9 3 ‘ Interest. |Capital. ee Aah C 3 5 1 1 ater ei vy” Uv" =07]/— Ana i an] 2 ope gt v+y" =azy Oni Ang] Be Pee ye yn? ott Tg" “= Ay] — An] A723] RG Rare &e. &c &c r togterti gn-rtl | "ame a 1b A0 Qn-r| ae a &c ‘e &e. &e &c n l—-v vy jvty+tr+... fo" l +0" =aq 0 It will be noticed that the repayments of capital form an increasing geometrical series with a common ratio 1+17; or the capital contained in any annuity payment exceeds the capital contained in the previous payment by the interest on the latter capital, since the interest in any payment is less than that in the previous payment by the interest on the capital repaid by that previous payment. | 7. When a loan of this character has been in progress for some years it may happen that the question of redemption will arise, and, in the absence of any pre-arranged and definite > Annutty Bonds. 67 agreement, disclose a wide divergence of views. Consider the following case. Five years ago a loan of $10,000 was made on a 5% basis to be repaid by a 15-year semi-annual annuity. The semi- annual payments which included capital as well as interest were $10,000 az at 244% =$477.78 each. The tenth payment having just been made, the question of redemption has arisen. The borrower would certainly regard $477.78 agj at 244% = $7,448. as the capital outstanding, and this is the amount that would be shewn by a schedule such as that on page 63. The _ lender however might refuse to discount the future payments at more than, say, 4%, claiming that he could not reinvest the money with equivalent security to yield him more than that; and $477.78 agg at 2%=$7,812. which is considerably in excess of the borrower’s idea of the capital outstanding. Now suppose that the lender sells the security to a third party who buys it at a price to yield him 444% compounded half yearly and to allow for the redemption of his capital by a sinking fund which he can accumulate at only 4%. For every $1000 in the purchase price, the semi-annual payment should BePTAULLOP Interests, 6c hie. ck Ore $21.25 and for the sinking fund $1000 sy at 2% =$41.16. BINGtELDLEHOP CACH ok wi ls ecethe oe ee $62.41 in the semi- annual payment such a purchaser would give $1000. 477. 18°. 6 ¢1000 =$7,656. for the 62.41. security and each payment of $477.78 will give him $162.69 for interest and $315.09 towards his sinking fund. Suppose further that after another two years and im- mediately after the fourteenth semi-annual payment, the ques- tion of redemption arises between the borrower and the present holder of his debt. The borrower will on the same basis as formerly estimate the capital outstanding to be $477.78 aig at 244% =$6,237. The owner of the security will however properly regard it as being worth his purchase price less the accumulated amount of his sinking fund, that is $7,656. — $315.09 sq at 2% He will therefore pay 68 Interest and Bond Values. = $7 656 — $1,299 = $6,357. Indeed the holder may even refuse to discount the future payments due him at more than 4% at which rate he knows he can reinvest. On this basis he would claim $477.78 ajg at 2% = $6,487. In the absence of any prior agreement, it would appear that if the borrower is anxious to repay, he can do so only on terms agreeable to the holder of his security. On the other hand, should the desire for redemption come from the holder of the security, he must accept the borrower’s terms unless he can sell to better advantage somewhere else. 8. The annuity bond possesses obvious advantages from the issuer's point of view when the redemption of the capital must be effected by charges against income; but it is not a suitable investment, generally speaking, for private funds owing to the fact that the capital comes back in small sums unsuitable for reinvestment, even if the holder has the skill to separate each payment into its component parts. In the hope of making the issue more attractive, some muni- cipal corporations have adopted an ingenious plan by which the issuing corporation obtains all the advantages of an annuity bond, and yet each holder of any part of the security has a straight term bond with the usual.coupons. Consider an issue of $50,000 of 5% bonds to run for 30 years. The equivalent half-yearly payment is $50,000 a5, at 214% =$1,617.67. But the first half year’s interest is only $1,250.00 Therefore the issue is divided as follows :— One Bond for $367.67 due six months after issue and carry- ing one coupon for $9.19. One Bond for $376.86 due 1 year after issue and ‘carrying two coupons for $9.42 each. One Bond for $386.28 due 1% years after issue and carry- ing three coupons for $9.66 each. One Bond for $395.94 due 2 years after issue and carrying four coupons for $9.90 each. &c. &c. &e. &c. Annuity Bonds. 69 One Bond for $1,578.21 due 30 years after issue and carry- ing 60 coupons for $39.46 each. Under such an issue there will fall to be paid one bond and a diminishing number of coupons each year but the one bond and the coupons payable will always aggregate $1,617.67 each half year. 9. One other unusual form of annuity bond may be illus- trated. An issue of $20,000 at 5% with the usual half yearly coupons for the first five years; after 5 years the loan to be repaid by a semi-annual annuity running for twenty years longer. Such a bond will produce $500 each half year for 5 years, to be followed by semi-annual payments of $20,000 ax, at 24% = $796.72 each for the next twenty years. The value of such an issue on a 544% basis compounded half yearly is obviously Us gO, se A ea = $ 4,347.90 +$796.72 (agj—aip) at256%......... = 15,114.71 $19,462.61 10. Annuity bond tables giving prices on a percentage basis are, as has been said, not at all satisfactory to the man who fully understands the assumptions underlying the given prices; and to one who does not clearly appreciate these assumptions such bond tables are dangerous tools to use. It should be noted however that the ordinary straight term bond tables enable one to value any annuity bonds, since from such tables the interest functions v” and azj may be readily deduced as indicated on page 62. 11. When an annuity bond is bought between two payment dates on a yield basis the interest adjustment may be made in one of three ways, all regardless of the nominal bond rate. Consider an annuity bond, under which 20 annual payments of $1000. each remain to be made, bought 3 months after a payment to yield the purchaser 4% compounded half yearly. 70 Interest and Bond Values. We should first ascertain the equivalent half yearly payment at 4%—this will be $495.05. First Method: The value as at last payment date : = $495.05 azprat 299 25 pa icies ee ee Hide = $13,542.33 add 3 months’ interest at 4% on $13,542.33 = $135.42 less 3 months’ discount at 4% on $185.42= 1.35 184.07 Price Bo ee ee ee ee $13,676.40 s Second Method: The value as at last payment date = $495.05: aagat 29905 o,f. a ie oe ene $ 13,542.33 Add 3 months’ interest at 4% on $13,542.33..... z 135.42 Brice oy ae ees or ene aie ar ae ee $13,677.75 Third Method: | The value as at next payment date = $495.05 (1+-a3g) at 2%.. ......... . +. =$18 S138 One dollar at interest to next payment date WHA IMOUNE £0002 rt ee oe $1.01 Therefore price is $13,813.18+1.01.......... . = $13,676.41 The true price is $13,542.33 x (1.02) = $13,542.33 X 1.009955 = $13,677.14 The error of the first method is 74 cents in defect. The error of the second method is 61 cents in excess. The error of the third method is 73 cents in defect. The first method is the one generally used and, as with straight term bonds, it will always giveanerror in defect which will always be about half the ‘‘discount”’ item. INTEREST AND BOND VALUES. Sri P ER ov. BONDS FROM THE ISSUER’S POINT OF VIEW. 1. We have been looking at bond issues from the investor’s point of view. It may not be out of place to consider such issues from the point of view of the issuing corporation. 2. As to the nominal bond rate, it is within limits immaterial theoretically what bond rate is used. If the corporation can borrow money at 4% compounded half yearly, then a 4% bond can be issued at par. A 314% bond will sell below par and a 444% bond will sell above par. The following prices will be found in any table of bond values. A3%% 20-year bond to yield 4% sells at 93.16. A 4 % 20-year bond to yield 4% sells at 100.00. A4%% 20-year bond to yield 4% sells at 106.84. The difference 6.84 in either case is merely the cash value of the difference in the coupons. The 3%% bond will produce less cash and entail a correspondingly smaller annual charge for coupons. The 4%4% bond will produce more cash and en- tail a correspondingly larger annual charge for coupons. The corporation cannot borrow more cheaply by printing a smaller bond rate in the securities itis about toissue. It will generally be advisable, however, to insert a bond rate as near as may be to the investment rate at which the securities will sell, since in many cases investors find it inconvenient to buy at either a considerable premium or a considerable discount. Bonds that are selling to the public at a discount ‘‘look cheap” and may prove attractive on that score to some buyers; but, as most bond-salesmen know, many private investors do not like bonds quoted at a discount, fearing that such a quotation implies a weakness in the security; while others compare the coupons with the price regardless of the fact that the coupons last for only a limited time and that the bond will 72 Interest and Bond Values. ultimately be paid off at par. Toa man of the latter type the 414% bond selling at 106.84 seems to give a better return than the 3%% bond selling at 93.16. 3. It is worth noticing that if the issuing corporation must meet the bonds at maturity by means of a sinking fund accumulating at a rate lower than that at which they can sell the issue, then to issue the bond at a premium will be the cheaper proceeding. Consider a bond for 1 due periods hence and bearing coupons for h per period. The bond will sell to yield the purchaser 2 per period, but the sinking fund can be accumu- lated at only 7 per period where 7 <1. The selling price is h az,+v”" at rate 1. The periodic charge is h+s— at rate 7. ~ should be a minimum. at Say ae BST aay RET ve n n| (8) std h dai @tr"@) Sea pen Fy Now a= () depends on 7 which is beyond the issuer's con- trol. aNe since 7 <1, SF G> Sa G) and therefore the fraction Sa ss (3) ot h+s S| (i) Suppose that in the issue we are considering it is possible to accumulate the sinking fund at only 3% compounded half yearly. Then $1,000 Si at 144% =$18.43 will be required for the sinking fund toretire each $1,000. of the issue no matter what the bond rate may be. The 314% bond will entail a semi-annual charge of $35.93 per $1000. for coupons and sinking fund and will produce $931.60 in cash on a 4% basis. The 4% bond will entail a semi-annual charge of $38.43 in return for $1000.00 cash on a 4% basis. The 41%4% bond will entail a semi-annual charge of $40.93 in return for $1068.40 on a 4% basis. will be a minimum when h is a maximum. Bonds from the Issuer's Point of View. 73 But eee? = 08857 931.60 _ 38.43 _ ogg49 1000.00 hcp Vee CEST 1,068.40 This shews that to issue the 444% bond will be really the cheapest way to borrow in this case. 4. As to the term for which the bonds should run it is obviously generally better to issue long term bonds when in- vestment rates are low and vice versa. The standard illus- tration usually referred to is the action of the British Govern- ment in selling 8% annuities during the struggle against Napoleon at apparently reckless discounts, instead of putting out short term bonds to be redeemed as the public credit rose by refunding issues at better prices. The exchequer bonds issued during the Boer War and the numerous recent short term notes are cases in point. Similarly a small town in a rapidly developing district of the North West may in a few years be in a much stronger position financially and might be well advised to issue bonds for only a moderate term in spite of a general low level of investment rate that may be ruling at the time. For many years it was commonly believed that the general trend of the interest rate from high class securities was to be continuously and indefinitely downwards; but that trend had been reversed for several years before the war and the serious rise in the price level of commodities or fall in the purchasing power of money had induced a belief that lenders would be forced to protect themselves by demanding higher investment rates of interest. Such speculations regarding the future of the rate of interest are however quite beyond our scope. 5. The choice between annuity bonds or serial bonds and the popular straight term issue, must frequently present itself for decision by the issuing corporation. The straight term 74 Interest and Bond Values. bond undoubtedly commands a broader market and is often supposed to indicate a certain financial prestige, so that municipal issues for example from the larger centres are usually in this form, while the annuity bond is generally adopted by the smaller municipalities. Of course if the security to be pledged should be of a de- preciating nature, such as railway rolling stock, it may well be that the issuing corporation has no choice—that only a serial issue or annuity bonds would be acceptable to the purchaser. But such questions lie outside our range. Assuming that the ~ test of figures is the only test, we can always decide the point. Consider the following case. It is intended to issue $10,000 of 20-year 5% bonds and it has been ascertained that if the issue be made in the form of straight term bonds they can be sold on a yield basis of 478%; but if the issue is to be in the form of annuity bonds they can only be sold on a yield basis of 54%. Assuming that the straight term issue will demand a sinking fund that must be accumulated at 4%, which will be the cheaper form of the issue ? We will assume that all interest rates are as usual to be com- pounded half yearly. The annuity bonds will entail a semi-annual charge of $10,000 az at 214% = $398.36. The straight term issue will demand each half year For:the Coupons? 0c en aes cel eae $250.00 For the sinking fund, $10,000 sx at 20 sca eee 165.56 $415.56 The annuity bonds will sell for $398.36 aq at 258% = $9,793. The straight term bonds will sell for the price quoted in a bond table, $10,159. But SRB = .04068 9,793. and BL DeD0 = .04091. 10,159. So that the annuity bond will be really the cheaper method of borrowing in this case. Le Bonds from the Issuer's Point of View. 75 But if the sinking fund can be invested in the bonds it is meant to redeem or in other securities of the issuing corpor- ation there should be no inducement to put out an annuity _ bond. 6. The treatment of the bond issue on the books of the issuing corporation is hardly within our scope; but it may be well to point out that any premium or discount at which the bonds may be sold should never find its way into a revenue account. If an issue of $50,000 is sold at 97, the initial out- standing debt in respect of this issue is $48,500 not $50,000; and each coupon as it is met will not pay the full interest on the debt. The difference between the full interest payable and the coupon will increase the debt at each coupon date, until at the due date of the bond the amount will be increased _ to the $50,000 then payable. Cn the other hand if the bonds be sold at 104, the initial debt is $52,000 and each coupon will contain some repayment of that debt in addition to the full interest payable upon it, until at the due date of the bond the debt has been diminished by $2,000 leaving only the $50,060 then payable. INTEREST AND BOND VALUES. CHAPTER VI. SOME PROBLEMS. 1. Twenty thousand dollars was deposited witha Trust Com- pany 30 years ago. Each half year, during the past 25 years, $500 has been drawn out, the last payment of $500 having just been made. How much should now be to the credit of the account, assuming 4% compounded half yearly ? Had nothing been drawn out, there would have been’ $20,000 (1-1-2) at: 2% 7 sy ae $65,620.62 But the half yearly drawings now amount to POO Ssof AtA My 1. oh has hs oe ee ee $42,289.70 leaving’a; balance of % As: ea ee eee $23,330.92 Or, five years after the original deposit it would have grown by interest to $20,000 (1-+1)!° at 2%=$24,379.89. Half a year’s interest on this at 4% is =$487.60. By drawing out $500, the interest was overdrawn by $12.40. These excess payments would in 25 years amount to $12.40 sip at 2%=$1,048.78. Deducting this from the ~ $24,379.89 would leave $23,331.11. This result is 19 cents in excess of the more accurate result above. The error is due to the fact that amounts are taken to the nearest cent, and so may in any item contain an error not greater than a half cent. The interest item of $487.60 should strictly be $487.5978. 2. A is lending $10,000 to B for 5 years at5% and B wishes to have the right of redemption at the end of any year during the currency of the loan at a pre-arranged price. Draw upa schedule shewing the redemption price at the end of each year so as to secure to A the full 5% on the whole loan for the whole ~ Some Problems. Pete time though he can reinvest at only 3%. Interest is payable half yearly. At the end of the first year after the interest has been paid the redemption price should be Pru PLO ag, at lie. ny. ee $10,748.59 At the end of the second year $10,000 + $100 agj= 10,569.72 At the end of the third year $10,000+$100 aq. . = 10,385.44 At the end of the fourth year $10,000+ $100 ag). = 10,195.59 To check the fourth year redemption price. A has had his interest up to date. He now receives and de- RRs sete tr ou LNESUNN OF 6 oe eels vce kin ets $10,195.59 Parsi months interest ato% 22.5). 02 oh 152.93 ; $10,348.52 Deduct A’s half yearly interest agreed upon.. 250.00 $10,098.52 Add six months’ interest at 3%............ 151.48 $10,250.00 Deduct A’s half yearly interest agreed upon 250.00 $10,000.00 ee 3. Five years ago two loans of $10,000 each were granted at 4%, the one repayable by an annuity for twenty years to include principal and interest and the other by equal instalments of $500 a year with interest on the principal outstanding. A third party has agreed to take over both loans ona 3%% basis. What should he pay the originallender? Interest isto be com- pounded yearly. Under the first loan there are 15 yearly payments of $10,000 ax at 4% = $735.82 each yet to be made. 20) The value of these on a 34%% basis is $735.82 aj at 314% =$8,474.74. Under the second loan there are 15 yearly payments of 78 Interest and Bond Values. principal, $500 each, with interest at 4% on the outstanding principal from time to time. Makeham’s formula is here most suitable. C= $7,500.00 - Kee G50 a ato oie. tector aide ee = $5,758.71 C—K =$1,741.29 \ Few tio BRR = sat RE de yr = 1,990.05 i 3% 7 i ( ) ‘hetvallie beings vita cy ein stein nk eae $7,748.76 4. A 6% bond with half yearly couponsisdueon 1 July 1925 and will be redeemed at 105. What will the bond yield if bought at 112% on 1 May 1911 by a purchaser who will set up a sinking fund at 3% compounded half yearly to replace the premium at which he bought? Taking 1 point to roughly represent interest on the invest- ment for the 2 months before the next coupon date. The capital invested as at 1 July 1911 is 118% —3=110.5. To replace the 5% points by which this exceeds the redemp- tion price the sinking fund will demand 5.5 ss, at 114% =.1595 out of every coupon, leaving 2.8405 for half a year’s interest on an investment of 110.5 or 2.57% i.e. 5.14% per annum. Our initial rough assumption of 1 as 2 months’ interest on 112% was at the rate of 54%. To test the accuracy of our result—5.14% ‘Lhe price.as at TsMay 191! was .22Gao 5 nae See 112.560 Add/2 months;interest at'5,14% ) au were ee . 964 113.464 Dedtict coupon due Dijuly ce 57a oe is ge Sat 3. Capital invested as at 1 July 1911............ 110.464 Sinking fund requirement =5.464 Ss at 14% = .1585 Some Problems. 79 Therefore the investment produces 2.8415 each half year on a capital of 110.464 which will remain intact. But this is at the rate of oy wane 5.14%. TY. 10464 '5. The debentures of acompany which bear interest at 5% payable half yearly are redeemable in 10 yearstime at 110, and are quoted at a price which yields 5% compounded half yearly. It is proposed to change these debentures into a per- petual 4%% debenture stock with interest payable half yearly. How much of the new debenture stock should be given for each $100 of the old issue ? The price of the present debentures must be 110 v°+2.5 agg at 244% = 67.130+38.973 =106.108. _ If the new debenture stock will sell on the same yield basis its price will be 90. Therefore each $100 of the old debentures should be worth 10,610.3 (See eee 90. be effected at the rate of 118 of the new for 100 of the old. = $117.89 of the new stock, and the change might 6. A bond redeemable in 12 years at par and bearing interest at 5% payable half yearly is bought at 105. Find the yield to the purchaser who sets up a sinking fund at 3% compounded half yearly to replace the decrease in capital. The sinking fund will demand 5 Sa at 144% =.17462 out of each coupon, leaving 2.32538 for a half year’s interest on a capital of 105 which will remain intact. That is 4.429%. 7. A 5% debenture due 8 years hence and carrying half yearly coupons for interest is quotedin the marketat109. Itis proposed to convert these debentures into 44% % debentures of the same amount with the same security. When should the new issue be made payable so as not to disturb the market price? 80 Interest and Bond Values. Reference to a bond table shews that the quoted price corres- ponds to a yield of 3.69%. Atthis yielda 414% debenture should have 14% years torun. 8. A Building Society grants loans repayable by 20 equal half yearly instalments including principal and interest, upon the basis of charging 7% interest on the total sum advanced and allowing 3% interest on sums repaid. The society ad- vances two-thirds of the value of his property to a customer, but borrows half the value by pledging its mortgage at 5%. What rate of interest does the society make on such trans- actions? Consider a property worth $3,000. The society will lend $2,000 and borrow $1,500. The society’s customer must pay each half year $70.00 interest and also 86.49 = $2,000 a at 1%%, towards redemption. Or $156.49 in all each half year. The society must pay $37.50 each half year for interest. At the end of the 10 years the customer’s balance in the re- demption account will wipe out his loan. _ Regarding this as a single transaction and assuming that the 3% on sums repaid by the customer can be obtained from the Bank, the society gets $32.50 each half year for an in- vestment of $500. which is 138%. Regarding this as one of a number of similar transactions and assuming that the society can reinvest the sums repaid by its customers in other loans of the same character, the rate of interest made is that at which (156.49 — 37.50) ag = 500 + 1500v, which is about 22=%, or 4434% per annum. 9. The value of an annuity for 20 years of which the pay- ments are successively 20, 19, 18, etc. is 150. What rate of interest does this return ? Here 150 = 20v-+19v? +182? + . . +3v18-+ 2y!9+-y%0 . (1 +2) 150 = 20+190+18v?+ . . +30!74- 2018+ v9 “. 150% =20—v—v?—v — .. . — 9 — 0 = 20 — ax. Some Problems. 6 ST The required rate is that at which zs ee 720! == 150, 1 Sega el rac Te AS) A NC 1 .05 At 54%, 7 Ne te 0028 72 149 64 ee ii = ee %o about 572%. 10. A certain stock of the nominal value of $100 bears a semi-annual dividend of 30 cents and a quinquennial bonus of $3.00. Thirty months after the last payment of the bonus and immediately after a semi-annual dividend has been paid the stock is quoted at 23. Find the investment yield at this price. At a yield of 2 per half year, the quinquennial aie of $3 each may be turned into half yearly bonuses of — ee, $3 270} each, the first one to come in six months hence. i 5 Therefore 23 is the value of a perpetuity of .30+ eee ORE SS Ax) 5 Ono3 ps sb ai] 5 Byes aa ed $ -.—o To get a rough approximation to 1 we may regard the quin- quennial bonuses as being worth 30 cents each half year, giving 231=.60 or 1 =2.6% roughly. Try 28%. At this rate the price would be .30 2.63544 .02625 . 22826 Try 2%%. At this rate the price would be 30, it 2.65155 .025 . 2188 = 11.429 + 11.546 = 22.975. = 12.000 + 12.119 = 24.119. 82 . Interest and Bond Values At 2.625% the price would be 22.975. At 2.500% the price would be 24.119. Therefore the true yield is (2.625—x)% where x : .125::.025 : 1.144 2003125 a pane 1.144 and the yield required is 2.6223 each half year or 5.245% per annum. 3 11. A government loan of $1,000,000 bearing interest at 514% payable half yearly is to be redeemed at 110 per cent. by a half yearly annuity of fixed amount including principal and interest extending over 10 years. What is the amount of the fixed half yearly payment? Regarding the loan as one of $1,100,000 at 5% to be re- deemed at par by a half yearly annuity of fixed amount in 10 years, the half yearly payment should be $1,100,000 a5 at 234% =$70,561.84. 12. An issue of $1,000,000 of 414% bonds with half yearly coupons is redeemable at 105 by annual drawings spread equally over 5 years; the first drawing to take place 3 years after issue. Find the issue price to yield 5% compounded half yearly. Using Makeham’s formula, we have C= $1,050,000 K =$210,000 (v§+v8-+-v%+v"+ 0") at 214% — 100 5 6 7 13 14 POE of $210,000 (v' ++’. . +v8-+y14) = 1035703270 {ajn— diye. ee $822,260.50 Now j=4% on 105 or 47% . 2 (CS Ri) = “ of $227,739.50..... 195,205.29 4 Ula del salsa dba etc $1 017,465.79 13. A loan of $100,000 is to be paid offin 30 years by quin- quennial instalments, the first of which is to be made at the Some Problems. 83 end of 5 years. The loan bears interest at 4% payable half yearly and the semi-annual sum set aside for the service of the loan is $2,912.25. Find the rate of interest at which the sinking fund should accumulate by half yearly compounding during each quinquennium. The semi-annual coupons call for $2,000. leaving $912.25 for the sinking fund. Each fifth year the sinking fund will be invested in the loan itself, so that the coupons on the bonds purchased for the sinking fund will increase the half yearly contribution to that fund. For each unit in $912.25 the sink- ing fund will grow as follows:— End of 5th year:—sjq at rate 1, the unknown rate. End of 10th year:—sjq+(1+-j sig) sig) Where j =.02, the bond rate. End of 15th year:—sij{ 1+(1+ 7 sig) +47 sig)? } and so on. End of 30th year:—sjq {1+(1+ 7 si) +(145 sig)?+ .... +(14+ 7 sig)® } Be Cte Sig) be aL ioe tes Cr areas ener { (1+ siq)*—-1 } . 912.25 x { (1+7 sqq)*—1 } =7 X 100,000 = 2,000. Peres 2,91 2020 “. (+7 sig)®= “O10 BE or, 6log (1+ 7 sig) = log 2,912.25 — log 912.25 = .5041148 log (1+ sj) = .0840191 “. 14.02 Xsjgj= 1.21344. The required rate is therefore that at which sig = 10.672. Reference to the tables shows this to be 1;5,% per half year or 278% per annum compounded half yearly. 14. A financial house has agreed to underwrite a foreign government loan of $1,000,000at97. Theloan will bearinterest at the rate of 4% perannum payable yearly. It is under dis- cussion whether repayment should be made by means of an 84. Interest and Bond Values. accumulative sinking fund of 2% or by uniform annual draw- ings of $20,000. What difference would there be in the rates of interest paid over the whole transaction by these methods ? | If the accumulative sinking fund is adopted the loan will be redeemed in 2 years where 100=6 ay] at 4% i.e. in 28 years, and the rate of interest paid is that rate at which 97 = 6 az] OF og| = 16.167 At 444%, d23| =16.193. At 438%, az] = 15.966. . a close approximation to the desired rate will be 4M + SS of + = 4.264%. If redemption is to be ets by uniform annual drawings we can approximate to the rate by using Makeham’s formula i aa eee dO t=jt+ RG —A. a (see page 59 spre Saal We may try 4 Yq, as our rough guess. At this rate K =2 a5| =42.06 C—A=100—97=3 : and C—K =57.94 a= .04 + Pea xX .04125 57. 94 = .04+ .00214 = .04214 or 4.214%. Our guess was too low: try 44%. _ At this rate K=2 ajoj = 41.19. C—K=58.81 §=.04-+ — 2 x ,0425 58.81 = .04+ .00217 = .04217 or 4.217% A trial rate of 4% % gave 4.214%. A trial rate of 414% gave 4.217%. .. The true rate is (4% +x)%, where x : .125 ::x—.089 : .003 or .003 «=.125x*—.011 ND = .090 Seae Some Problems. 85 And the rate required is 4.125+.090 =4.215%. By the former method the government would be paying 4.264% for its money and by the latter 4.215%, a difference of 049%. . 15. An issue of bonds of the nominal value of $100,000 is made on 1 April 1911. The bonds bear interest at 4% per annum, payable half yearly, and are redeemable by drawings as follows : $10,000 on 1 April 1912 at 101. $10,000 on 1 April 1913 at 102. &c. &e. &e. $10,000 on 1 April 1921 at 110. What should be the issue price to yield 5% compounded half yearly? The price of the bonds=Ai+A,. where A1=price disregarding the premiums on redemption and A,=cash value of the premiums payable on redemption. Makeham’s formula is appropriate to find A. Ai=K + ai (CF) 4 where C =100,000 K=present value of 10 annual payments of $10,000 each =present value of 20 semi-annual payments of $4,938.30 each (see table, page 32) = $4,938.30 ay at 214%............ = $76,984 PCR = + of $2018... See a = 18,413 4 Rt ea os A, Uc ALUM CR aas Nel Pee See eaaris lattes ioe uriienie men ogra oa $95,397 Az,=100 (v?+20!+3v8+....+908+ 100°) at 2%4% v? Ao=100 (vt+2v8+ 33+ . . +9v?°-+ 10v??) .. (1—v?) Ap =100 (v? + 04+ 08+. . +0?°) — 1000 v” = 49.383 a9] — 1000 v” 86 Interest and Bond Values. yee 769.84 — 580.86 .048186 so that the full price 41 +A2=$99,319. = $3,922 16. A loan of $5,000. was made 5 years ago to be repaid in 25 years by equal half yearly instalments of $100. each with interest at the rate of 414% per annum on the capital out- standing in each half year. Just after the 10th half yearly payment the lender offers his security for sale. How much should be given by a purchaser so that he may realise 5%, pay- able half yearly on his investment while replacing his capital at 4% compounded half yearly? There is now $4,000 of the loan outstanding. The next half yearly payment will be $100 capital and $90 interest. The following half yearly payment will be $100 capital and $87.75 interest. The following half yearly payment will be $100 capital and $85.50 interest. Or the future payments of interest will decrease by $2.25 each half year. If we denote the payments made by the borrower as Ly e—Y, £—2y, 4 —3y,.. 2 X= O0Y where x = $190.00 and y=$2.25, and also denote the price paid by the purchaser as P, and write 7 for .025, then we have available for the sinking fund x—j P,x—-y—j P,x—2y—j P, &c. in successive half years, so that (x—7P) (1.02)99+ (x—y—j7P) (1.02)%8+ ..+(«-—389y—7iP) =P or (x—jP) (1.02)#+ (x—y—jP) (1.02)+.... .... +(x—39 y—jP) (1.02) =P (1.02) ea (—7F)) (1.02) —4 2 (1:02)? ye 1-02) 80 ee —y (1.02) —(x—39y—jP)=PX.02 FP) CLO) yes ig 2 — Oy a =e pa* (1.02)? — y sgo] 24—x+40y j (1.02)*°-+ .02 —7 Some Problems. 87 _ 190 X 2.20804 —2.25 60.40198—190+90 025 X 2.20804 — .005 one $3,657 . 80. ~ 050201 17. A loan of $18,323. is to be repaid in five years by an an- nuity calculated at3% toinclude principal and interest at 5%. If the first payment of the annuity is $4,120.26 and the pay- ments are to increase in geometrical progression, prove that their common ratio is 1.03 and draw up a schedule shewing the repayment of the loan. The annual interest demanded is $916.15. If the successive payments be P, Pr, Pr’, Pr’, Pr’, we have (P—916.15) (1.03)4+ (Pr— 916.15) (1.03) + (Pr*—916.15) (1.03)?+ (Pr? — 916.15) (1.03) + (Pr*—916.15) = 18,323. Or P (1.03)*+ Pr (1.03)'+ Pr? (1.03)?-+ Pr? (1.03) +Pr4 = 916.15 s5] at 8%+18.323 = 4,864+18,323 =23,187 But P =4,120.26 nd Beet 8s = §.6275=5 X 1.1255 =5 (1.03)4 4,120.26 wir 1 OS The schedule illustrating the repayment is as follows: 3% on The Annuity payment contains & Capital re- Capital Capital § Annuity jpaid. This} 5% on 2% on repaid still > | Payment. comes Capital | Capital | A Capita to date. outstanding, from re-| outstand-| repaid. | Repayment investm t. ing. —— ee —— OO 1 |$4,120.26)$ 0.00'$916.15)$ 0.00/$3,204.11/$ 3,204.11 $15,118.89 ee | 4,243.87| 96.12) 755.95) 64.08} 3,423.84) 6,627.95) 11,695.05 bo —————$————— | — | | 3 | 4,371.19] 198.84) 584.75) 1382.56} 3,653.88] 10,281.83) 8,041.17 —-. | oJ Ss | es |S SE | 4,502.33] 308.45) 402.06) 205.64} 3,894.63) 14,176.46; 4,146.54 -_——————— | | | | fF 4,637.40) 425.29} 207.33] 283.53) 4,146.54) 18,323.00 0.00 We | $916.15 on 88 Interest and Bond Values. 18. A Timber berth of 30,000 acres of pine estimated to pro- duce 120 million feet of lumber is bought for $800,000. and a ground rent of $8 per square mile. Logging may begin after 3 years at the rate of 6 million feet per annum, and at a cost of $8 per thousand feet to the mill. The stumpage dues are $3 per thousand. The mill charges $3.50 per thousand. At what price should the lumber be sold in order to make a net profit of 10% per annum on the proprietor’s outlay? It is to be assumed that lumber prices will rise 2% per annum; also that $500,000 can be raised at once by the issue of 5% bonds at par, the bonds to be redeemed at the rate of $50,000 a year, at the ends of the years sixth to fifteenth after issue. Assuming that each year’s cut is got out, put through the mill and marketed during 12 months, the final proceeds will come in 23 years after the purchase. At that time the accum- ulations of the outgo at the yield rate will amount to Original outlay, $300,000 (1+7)........... = $2,686,290 (srOund Tent, pot O Seq ae oe ee re eee = 29,829 Stumpage, logging and mill costs, $87,000 saj. = 4,982,925 $7,699,044 plus the payments for interest and redemption of bonds as follows: For coupons during the first six years, $25,000 a6). Clot) tae ee a ulate coe Wesera ieee $974,957 For capital redemption from the end of the 6th to the end of the 15th year inclusive, $0; 000% sipi Cl thet) oes eo eerie ae ae Se ee 1,708,165 For coupons from the end of the 7th to the end of the 15th year inclusive: $22,500 (1-+7)*+20,000 (1+72)5+.... +2,500 (1-+7)8 = $2,500 {9 (1+1)%+8 (1+2)8+7 (1+1)4+ .. 2 $2 (1+4)9+(1+7)8 } = $2,500 10 Sealer (sygj— S3]) 4 BAe te og 819,075 $3,502,197 Some Problems. 89 The total outgo for all purposes at the yield rate will amount to $11,201,241. If $p per thousand is the first price realised, the accumulated income on the same basis is $6,000 {p (1+4)+ p (1.02) (1+2)8+......p (1.02) (1+2)+ +p (1.02)} (1-+7)?°— (1.02)?° (1+72) — (1.02) = $6,000 a p = $393,116 p = $6,000 p, butz=.10 Reet 53-45 393,116 The first year’s cut should sell for $28.50 per M. The second year’s cut should sell for $29.10 per M. The last year’s cut should sell for $41.50 per M. 19. A provincial government makes an issue of 314% bonds to obtain the money necessary toreplantanareaof 12,500 acres with white pine. The initial expenses together with the pur- chase price of the property amount to $75,000. During each of the first five years $1000 will be spentin road making. Four per cent of the area must be set aside for roads, fire lines &c. Planting is done at the rate of 1000 acresa year at $10 an acre. Salaries and expenses amount to $5,000 a year. After 30 years, 50 cents per acre can be secured on each area from thinnings. This can be repeated at the end of each 10 year period and the proceeds will increase at the rate of 2% per annum. At the end of 60 years the final harvest will produce $300 per acre. Assuming that bonds will be sold at par as the money may be needed, what will be the maximum issue out- standing ? When will this maximum occur? When can the bonds be retired ? What will the forest be worth when the last of the issue has been paid off ? 90 Interest and Bond Values. The total.area is........ 12,500 acres. Road allowance........ 500 VeavViIng eit ts ae 12,000 acres available for planting at the rate of 1000 acres a year for 12 years, and costing $10,000 each year. No income can come from the forest until the end of the 30th year when the proceeds from thinnings will begin and con- tinue as follows :— End of 30th year—$500 a year for 12 years. End of 40th year—$500 (1.02)!° a year for 12 years. End of 50th year—$500 (1.02)?° a year for 12 years. At the end of the 60th year the final harvest will produce $300,000 a year for 12 years. The income from thinnings will never be enough to pay in- terest on the bonds outstanding at the time. At the end of the 60th year the debt on the property will amount to:— Purchase price at 344% interest for 60 years, | 1 DOO Let 4) Or a ite Acre siete. png ee ae a 590,857 Cost of road making, $1,000 sz (1-+7)*® = 35,570 Cost of planting, $10,000 sj (1+72)*%........ = 761,286 Cost of maintenance, $5,000 s@j............ = 982,584 $2,370,297 less the proceeds from sales of thinnings, $500 { si (1+2)8+- (1.02) say (1+1)8 HE (L.02)7 sitters awe ee ae = 33,997 $2,336,300 And this will be the maximum issue outstanding. The harvest from the first 1000 acres will now produce $300,000 leaving $2,036,300 outstanding. The whole issue will be redeemed in » years more where $300,000 szj + $500 (1.02) sy (1+1)""? — $5,000 sq = $2,036,300 (1-+7)”, Some Problems. 91 where 295,000 {(1+7)” —1}+500 (1.02)? (1—v?) (1+7)"= 2,036,300 (1+1)” 3, where 295,000 {(1+7)"—1}+49 (1+7)"=71,271 (14+1)", where (1+7)” = Ea 1.318, or where 7 is alittle more 223,778 than 8. Nine years after the first harvest when the 10th harvest has just been sold, the debts should be all paid off and there should be a surplus of $295,000 s9j + $500 (1.02)? szj (1+7)7— $2,036,300 (1+7)° = $285,361. And the value of the harvests to come is $295,000 ay = $560,410. So that the total value when the bonds are all paid off will be $845,771. INTEREST AND BOND VALUES. EXERCISES. 1. What rate of interest compounded half yearly is the equi- valent of 5% compounded (i) yearly (ii) quarterly ? 2. Prove that 1=d+d’?+d'+ etc. 62 6° SA Clatstats atcls GEC and find similar expansions for.d in terms of 7 and in terms of 6, also for v in terms of 7 and in terms of 6, also for 6 in terms 4 and in terms of d. 3. Find (1+)! and v™ at (i) 8% (ii) 5%. 4. For how long a time should $100 be left to accumulate at 5% in order that it may amount to double the accumulated value of another $100 deposited at the same time at 3% ? 5. Fill in the 27 blanks in the following schedule: The present value of $1000 a year for 20 years at 4% First Taverest The $1000 payable. Payment Compounded- Yearly Half-yearly Quarterly Yearly At once + Half-yearly | Quarterly RIES | ESD | eS NS | A SSS _ Yearly Aivearien 4 z hence less EN \ Quarterly Yearly Three = ||————-_——— years ‘| Half-yearly hence |\Pamnctergtemncc ir Quarterly Exercises 93 6. An annuity of $1000. a year for 25 years, first payment one year hence, is to be altered so as to become payable (i) quarterly in advance, (ii) yearly in advance, (iii) half yearly, first payment 3 months hence. Find the equivalent payments in each case at 4%. 7. What payment should be made in cash by the annuitant to obtain each of the conversions of the previous question and still receive $1000 a year? 8. Aman deposited $100 on the Ist January, 1890, and $100 every six months thereafter, at 4% per annum compounded half-yearly, making his last deposit on the Ist January, 1910. What sum was standing to his credit immediately after his last deposit? To what will this accumulate at the same rate of interest by the Ist July, 1915? 9. Explain the following equation in words: ox ae) hipaa where— is the value of a perpetuity of 4 4 1 per period at 7 per period. 10. Make verbal and self-explanatory statements of the following formulae. (i) w=d; (ii) t—-d=d1; (iii) (1+2)"=1-+7 sz; (iv) 1=2a;j;+0"; (Vv) a= — si; (vi) (1—v”) as =i (vii) {(1-+2)"-1} sai. To what do formulae (iii) to (vii) reduce when n=1? 11. If x=a;] and y=aa, find 7 in terms of x, and y. 12. How would you express the present value of 1 due n years hence at a rate 7 compounded m timesa year; andalso the accumulated amount of 1 deposited years ago at the same rate? What do these expressions become when is infinite? 13. What is the present value at 3% of an annuity to run for 25 years; the payments being $100., $103., &c., increasing 3% per annum? 14. If the payments of the annuity in the previous question were to be discounted at 5% instead of at 3%, by how much would the value differ from $ “3 ay at 2% ? 94 Interest and Bond Values. 15. Find, at 344% interest, the value as at the Ist January, 1910, of: (i) Twenty annual payments of $1000 each, the last to be made on the Ist January, 1930. (ii) Twenty annual payments of $1000 each, the last to be made on the Ist January 1929. (iii) A Bond for $10,000 bearing annual coupons for inter- est at 4% and due on the Ist January, 1925. 16. A certain property producing a fixed income to per- petuity is left in equal shares to four hospitals, A, B, C, and D. A,B, and C are each in succession to enjoy the whole in- come for a time and the final reversion is to be to D. Assum- ing interest at 4% compounded yearly, for what length of time should A, B, and C each enjoy the income before the property goes absolutely to D ? 17. What do the following functions (1 +1)”, v0", ag, Sap = ss become when 7 is zero and n= (i) zero (ii) infinity? 18. A man possessing a certain sum invested at rate 1 spends 15% of his interest the first year, 314 times his interest the second year, 4 % times his interest the third year, and soon. At the end of the 16th year he has nothing left. Shew that in the 8th year he spent as much as he had left at the end of that year, and that his money was invested at 4%. 19. A bridge costs $20,000 which can be raised by an issue of 6% annuity bonds at par to run for 15 years. The bridge will cost $150 a year in repairs and must be replaced at the end of 15 years. What annual sum should be included in the taxes to provide for this bridge? a 20. Twelve level railway crossings in and near a city, each cost $1,750 a year to guard and maintain. They could all be abolished by lowering the tracks and building bridges. How much could the railway company afford to spend upon this work? Assume that 1% of the outlay would be the annual cost of repairs and 50% of the outlay would have to be spent again for reconstruction after 25 years. Interest at 5% per annum. Exercises. - 95 21. An apartment house costs $250,000 to build and $25,000 a year to maintain. Every five years it must be renovated at a cost of $7,000. Assuming that the life of the house under these conditions will be 80 years, what should be the annual rent-roll to return the builder 8% upon his invest- ment and replace his capital by a sinking fund at 5%? All income and outgo in half-yearly amounts. : 22. A cottage hospital costs $100,000 to build and $20,000 a year to maintain. Assuming that it must be rebuilt at the same cost at the end of 50 years and that the maintenance charges will always be the same, what sum at 5% per annum compounded yearly will provide for this hospital to perpetuity? 23. A factory employs 20 girls each at $500 a year to doa certain work. A machine requiring 5 such girls could do the same work in the same time. Assuming that the life of the machine is ten years and that the cost of running it and keeping it in repair is $750 a year, what must be the maximum price of the machine in order that it may effect a saving of $3000 a year. Assume that the depreciation fund can be in- vested at 4% per annum compounded quarterly and that the capital invested must earn 8% compounded quarterly. 24. Amandeposited $100 everysix months at8% compound- ed half-yearly. Every four years he drew out the amount at his credit and invested in 5% bonds with half yearly coupons at par. The coupons of these bonds have always been de- posited with the half-yearly $100. This process has been going on for 24 years and the man has just made a new purchase of bonds. How much does he now hold in bonds? 25. How would the result of the last question have been affected if the bonds had all been bought at 105? 26. Twenty years ago $10,000 in 414% bonds with half- yearly coupons was deposited with a trust company to ac- cumulate as follows:—The coupons to be deposited in a sav- ings account at 38% compounded half-yearly. At the end of every fifth year the accumulated amount in the savings account to be invested in 4% bonds at par. Assuming that the Trust Company charged 5% of income, to what amount has the fund now grown? 96 Interest and Bond Values. 27. If a 30-year annuity is worth 20 years purchase, what should be paid on the same basis for a 40-year annuity? 28. Find the value of 20 future half-yearly payments as follows: $100. at the end of thirty months, $105. at the end of thirty-six months, $110. at the end of forty-two months, and so on, increasing by $5 each six months. Assume 5% compounded half-yearly. 29. A man borrows $5,000 at 5% and agrees to repay the loan by an annuity covering principal and interest in ten years. What annual payment will he make? How much of the fifth payment will be a return of capital? What capital will be out- standing after the fifth payment ? If the borrower then wishes to repay the balance and it is agreed to discount future pay- ments on a 4% interest basis, what will be the redemption price? 30. An investor, calculating prices to yield him 44% com- pounded half-yearly, buys $100,000 of 5% 40-year bonds with half-yearly coupons for $114,365.30. Find agj and v® at 2%. What should that investor bid for a similar bond with yearly coupons? 31. A corporation borrows $50,000 at 414% and agrees to repay the loan at the endof30years. Assumingthat it must accumulate a sinking fund at 8%, what annual charge will this loan impose upon the corporation? Had the lender been willing to accept repayment in the form of an annuity covering principal and interest at 5% what difference would it have made in the annual charge? 32. A30-year 4% bond with half-yearly coupons is bought at 95. Find the investment yield on the following suppositions: (i) that the purchaser debits the account with interest each half year at the yield rate and credits it with the coupons as they are paid. (ii) that the purchaser provides for the shortage in the in- terest each half year from another source, allowing 6% on all items so transferred. 33. A 20-year 6% bond with half-yearly coupons is bought at 134. Find the investment yield on the following suppositions: Exercises. 97 (i) that the purchaser writes his investment down each half year by the excess interest in the coupon. (ii) that the purchaser sets up a sinking fund at 4% com- pounded half-yearly to replace the premium at which he bought the bond. 34. Bonds to the amount of $1,000 are bought to yield 444% interest convertible half-yearly, reckoning from the next in- terest date. They bear 4% payable semi-annually, and the price paid for them is $966.50 and accrued interest of $15.00. What are the entries required to record this purchase; also the entries when the first payment is received, assuming that the company amortizes its bonds? 35. Certain 6 per cent. bonds maturing February 1, 1934, interest payable semi-annually, contain an option giving the right to the issuing corporation to redeem them at 110 on or after February 1, 1919. Compute the value of these bonds as at February 1, 1909, on a 5 per cent. basis. 36. A dies, leaving an estate of $44,000 in cash, from which a tax of 1% is to be deducted; the balance is to be invested in 5% bonds, then quoted at 132, and the income is to be divided equally among three children. What will be the annual in- come of each of the children on the supposition that the trustee sets up a sinking fund to replace the premium on re- demption 30 years hence,and that the sinking fund will earn 4%? Allinterest compounded and payable half-yearly. 37. Find an expression for the value of a bond due nm years hence and bearing interest at the nominal rate g, payable p times a year, in order to pay the purchaser interest at the nominal rate 7, convertible m times a year. What does the expression become when m=p? 38. A loan of $200,000 at 5% payable half-yearly is to be repaid as follows: $5,000 at the end of 5 years, $6,000 at the end of 6 years, $7,000 at the end of 7 years, and soon. The issue price is 9244. What rate of interest is the borrower paying? 98 Interest and Bond Values. 39. A purchases from B a piece of property worth $10,000, agreeing to pay for it in equal instalments at the end of each year for ten years, including interest at the rate of 5% per an- num. There is a tax of 1% on the property and by agree- ment A is to pay at the end ofeach year only his share of the tax, reckoning as his share 1% of the principal paid up to and including the instalment then due, but in place of a varying amount from year to year he desires to pay a level extra amount with each of the ten instalments. Find (a) a general formula for the extra. (0) A’s total an- nual payment, assuming 5% interest per annum throughout. 40. The value of an annuity for 30 years of which the pay- ments are successively 30, 29, 28, etc., is 225. Determine the interest yield. 41. (a) Determine an expression for the amount which should be paid a lender ¢ years hence for an immediate ad- vance of 1 made upon the condition that the lender is to receive interest at the rate 7 per annum for m years (n>12), though he can make re-investments only at the rate 12, (j7>1). (b) From (a) obtain the present value of an annuity of 1 per annum for 1 years, the remunerative rate being 7 per annum, and the reproductive rate 7 per annum. (c) Discuss the redemption of such an annuity after ¢ years upon application by the borrower. 42. Investigate a convenient formula for ascertaining approximately the true rate of interest yielded by debentures terminable at the end of m years, issued at a premium and redeemable at par. Apply the formula so obtained to determine the rate of in- terest yielded by a terminable 6% debenture, repayable at par at the end of 20 years, purchased at 120. 43. It is desired to raise $100,000 by an issue of debentures, $5,000 is to be set aside each year to pay interest and provide for the redemption of the debentures—the sum to be appor- tioned as follows: (1.) Interest at 4% is to be paid at the end of each year on the debentures then outstanding. Exercises. 99 (2.) The balance of the $5,000 is to be invested to yield 3% to provide for triennial drawings of the debentures at a premium of 5%, the first drawing to be at the end of the third year from the date of issue. Find the number of years necessary to pay off the loan. 44, A foreign government loan of $1,000,000 at 5% with half yearly coupons is to be redeemed at 110 by the operation of an accumulative sinking fund in 36 years. What semi-annual sum should be set aside for the service of the loan ? (i) When redemptions are made each half year. (ii) When redemptions are made at intervals of 4 years from a fund which accumulates in the meantime at 4% compounded half-yearly. 45. Find a ready approximation to the period in which a 5% loan with yearly coupons will be redeemed by an accumulative sinking fund of 114% allowing for quinquennial redemptions at 110 from the sinking fund which can be invested during each quinquennium at 4% compounded half-yearly. 46. A loan of $10,000 at 5% payable half-yearly was made 15 yearsago. For the first 5 years the borrower paid $350 each half year. For the next five years he paid $325 each half year, and for the past five years he has paid only $300 each half year. How much must he pay each half year for the next five years to extinguish the debt? 47. What payment made half-yearly in advance for 7 years will secure a quarterly annuity of 1 per annum the first in- stalment of which will fall due at the commencement of the (r+1)th year and the last three months before the end of the (r+n)th year? Interest at the nominal rate 7 convertible quarterly throughout. 48. By paying a certain rate 1 per annum in quarterly in- stalments the effective rate becomes (1.017) +. How would you approximate to the value of 7? 49, Find the value of an annuity payable annually whose several payments are 1, 2, 3, 4, etc. when the annuity is to run (i) for years; (ii) forever. 50. If a debt bearing interest at rate 1 compounded yearly can be discharged, principal and interest, by » annual instal- 100 Interest and Bond Values. ments, in how much less time would the same debt be dis- charged by the same annual instalments payable half-yearly, (i) When the interest on the debt remains payable yearly ? (ii) When the interest on the debt is to be payable half yearly ? 51. Prove that the sum of the first x terms of the 4% values of sq] is equal to aut 52. Shew that ifa@ be the value of an annuity of 1 per annum payable at the end of each year, then the value of the same annuity of 1 per annum, but payable at the end of each quarter is @ (1+387) approximately. 53. A perpetuity of $x per annum, the first payment of which is due at the end of 7 years, is to be purchased by an- e e e e e . P nual instalments, commencing at P and diminishing by — n each year, so that the last instalment, ahs will be payable n at the beginning of the mth year. Find the value of P. 54. An annuity of $1000 a year payable half yearly for 20 years, the first payment to be made 20 years from now, is to be purchased by ten annual instalments commencing at once and increasing by5%perannum. Find the amount of the first instalment assuming interest at 4% compounded half yearly. 55. A man holds $10,000 in 5% municipal debentures re- deemable at par in 7 years and standing in his books at $10,097. He is offered conversion into 4% inscribed stock at the rate of 111 for every 100 of his debentures. Assuming that he may . count upon realising the stock at parin 7 years time, and in the meanwhile is able to invest small sums at 4%, or borrow them at 6%, what difference in the rate yielded by the invest- ment would the conversion make? All rates are payable half-yearly. 56. Aloan of $10,000 is to be repaid in 15 years by uniform semi-annual payments which include interest at 5% for the first half of the time, and at 414% for the second half, and also sinking fund payments which will improve at 4%. Find the semi-annual payments needed to repay this loan. INTEREST AND BOND VALUES. EXAMINATION PAPERS. 1: 1. Distinguish carefully between interest and discount. When a banker discounts three months bills at 8%, what effective annual rate of interest is he earning? What is the fundamental assumption underlying the idea of simple interest? 2. Define the symbols sjj and az. $100 deposited 20 years ago has grown at interest to $235. The interest was compounded twice a year. What was the rate? How much should be set aside at the beginning of each year for 10 years to amount to $1,000 at the end of the 10th year? 3. Find a7 at 4% and sgq at 5%. If a thirty-year 7% bond with yearly coupons sells at a premium of ~, and a forty-year 7% bond with yearly coupons sells at a premium of g, show that a-seventy-year 7% bond with yearly coupons will sell at a premium of p+v°°g or g+u0"p where v is taken at the investment rate. 4, Find the price of a 20-year 6% bond with half-yearly coupons to net the investor 5%. What would the yield be if the bond were bought at 121}? 5. A corporation which is issuing 5% twenty-year bonds can sell them at 110, but can accumulate the sinking fund for their redemption at only 4%. The corporation could borrow the same sum at 5% by the issue of twenty-year annuity bonds. Find the annual charge upon the corporation in each case. 102 Interest and Bond Values. Il. 1. Clearly distinguish between nominal and effective rates of interest. | What rates compounded half yearly are equivalent to 4% compounded (i) yearly, (ii) quarterly? 2. Answer the following questions by reference to Tables. (i) How long will it take for a sum of money accumulating by interest at 10% compounded quarterly to amount to double the sum to which it would have accumulated at 3% com- pounded half. yearly? (ii) Ground rents of $1,000 a year payable quarterly for 10 more years are sold for $6,000. What rate of interest does this represent? (iii) Ten years ago a man began making deposits of $100 every six months into a trust fund. He has just made his twenty-first deposit, and is informed that there is $2,500 to his credit at 4% compounded half yearly. What should be to his credit, and what rate would the $2,500 represent? 3. A 6% Bond is bought at 120. Show that this price will yield the investor 5% if and only if the bond is a perpetual bond. What should have been the price to yield 5% had the bond been a 10 year bond? Show that the price of the 10 year bond is less than 120 by the value of 20 due 10 years hence at the yield rate. 4, A corporation can raise money by issuing a twenty year 6% Bond at 973 for which it must provide a sinking fund that will accumulate at 4%. On the other hand the Corpor- ation could raise the same sum by selling 20 year 7% annuity bonds at par. All rates are compounded half yearly. Which method is the cheaper? 5. A trustee invests $25,000 in 6% bonds at par 22,500 in 64% bonds at 1124 and 18,600 in 5% bonds at 93. All the bonds run for 20 years. What sinking fund to accu- mulate at 4% must the trustee set up, and what rate of interest will the life tenant obtain? Examinations Papers. 103 ITT, 1. Compare the simple interest and the true interest on $1,000 for 73 days at 5% compounded half-yearly. 2. Make verbal explanatory statements of the following formulae: ae ee e PROVOST RS OL Be Mirra (ii) 1 =tay7-+v" (iil) Az = Spi +1 3. Show that the present value of a geometrical series of emia bapayinenits are xv", XV", Rao. Cate so xv"—* is equal to | (i) 3; a; where da; is at rate j such that (1+7)y =1+1. (ii) v x sq where sy is at rate j such that y=(1+2) (1+)) where 2 is the effective periodic rate of interest. 4. Show that the accumulated value of 1 per annum deposited in p instalments each year for ” years at a nominal rate 7 compounded g times a year is equal to Seer (i+2) —1 q ee AC] 5. State and prove Makeham’s theorem for the value of a loan. A mortgage of $1,000 at 6% is to be repaid in 10 years by quarterly payments of $25 on account of capital; interest to date being payable with each payment of capital. Find the value of this mortgage to net the purchaser 8% com- pounded quarterly. 6. A municipality issues a 20-year annuity bond for $10,000 at 5%. What can a purchaser afford to bid for this bond so as to net 45% on his whole investment for the whole time though he can replace his capital at only 4%? All rates to be compounded yearly. 104 Interest and Bond Values. 7. B mortgages his house for $1,500 to A and agrees to pay interest half-yearly at the rate of 8% per annum, also to pay each half-year such a sum of money as will, if deposited in the bank, wipe out his loan in 15 years. A borrows $1,000 of the $1,500 at 7% by pledging his security to a mortgage corporation. What rate of interest does A make and what rate does B actually pay for what money he got while he had it? The Bank allows 3%. 8. Four and a-half per cent. semi-annual coupon bonds due on the Ist January, 1933, are bought on the 2nd of March, 1913, at a price of 94% flat, i.e., including the interest adjustment to date. What rate of interest will this invest- ment yield? IV. 1. Define the standard interest symbols, Sreyet nj? Snj° Make out a table expressing each of the symbols 7, v, d in terms of each of the remaining ones and deduce the relations you give without the use of algebra. Also without algebra show that (i) L=taq+e", (ii) 1+¢sq=(14+7)", (ili) ae = sti, n| 1, 0; d, Gal, Say @ 2. (a) Find the half yearly payment Sk accumulated at 6% compounded half-yearly is equivalent to $1,000 a year. (Use simple interest for the fractions of a year). ; (b) What is the present value of an annual payment of $1,000 a year, to run for 20 years, the first payment being one year hence and the rate of interest being 6% compounded half yearly? 3. A 20-year $1,000 bond with annual coupons at 5% is bought to yield 4%. Assuming that the elements of capital in the coupons can be accumulated at only 3%, find Gee price of the bond. 4, A town wishes to issue $80,000 worth of straight term bonds with annual coupons at 5%, in such a way that the annual charge it has to meet is constant from year to year, Examination Papers. - 105 and so that the entire issue will be redeemed by the end of 30 years. Show how this may be carried out and calculate the face value of the bonds maturing at the end of the Ist, 15th, and 30th years. 5. (a) State and prove Makeham’s formula for the price of an interest bearing security. (b) An issue of bonds of nominal value $100,000, and bearing interest at 5% per annum payable half yearly, is to be redeemed at 105 by ten annual drawings of $10,000, the first drawing to be made one year from the date of issue. Find the price to give a yield of 6% compounded half yearly. Na 1. Define the symbols 2, d, v, and from your definitions deduce and explain without algebra the equations (i) w=d (ii) ~—d=di What nominal rate of interest compounded half-yearly is the equivalent of a discount rate of 8% compounded quarterly? 2. Find the present value of an annuity of 1 per annum for nm years, payable m times a year, at a nominal rate of interest 4, compounded h times a year. What does your expression become when m and h are both infinitely great? 3. A man buys a twenty-year four per cent. bond with half-yearly coupons at 87%, and also buys a twenty-year five per cent. semi-annual annuity bond at 983. Each is nomi- nally a $1,000 bond. Find the yield of each investment on the assumption that the shortage in interest of the straight- term bond is made up from the repayments of capital under the annuity bond, that 6% is allowed on all items so trans- ferred and that the balance of the capital repayments made under the annuity bond are invested at 4%. 4. A borrows .$2,000 at 9% from B. The loan is to be repaid by 20 equal semi-annual payments of $100 each. What interest will this yield B, who must replace his capital by investments at 5%? Immediately after the fifth semi- 106 Interest and Bona Values. annual payment, A wants to pay off his Cebt. What should B claim as the redemption price? Would it nake any differ- ence if the loan were to be paid off at B’s request :1stead of A’s? 5. A corporation issues straight term bonds which it can sell at rate 7 and which are redeemable in m periods by a sinking fund that will accumulate at rate 1. Show that if 4 be 7 it will be more economical to issue them at a discount. ae 1. Explain in words the truth of the following statements (1) l1—v=d, (ii) va=d, Gii) 1i—d = di. Give Pa definitions of the symbols oF and s= and n| show why a; — s =, 2. Find the present value of an annuity of 1 per annum to run for m years. The annuity is payable p times a year. The rate of interest is 7 compounded m times a year. What does this become when p=m= =? 3. A straight term $1,000.00 bond for 20 years at 44% with half yearly coupons is bought for $967.95. Find the yield on the assumption that the investor writes up his pur- chase by the difference between the interest due and the coupon each half year. What will the investment be standing at in his books five years after purchase? 4, What will an investor give for a 12 year annuity bond of $10,000 at 5% so as to make 6% on his investment for the whole time, although he must reinvest repayments of capital at 4%? All rates are yearly. 5. Sketch a solution to the following problem. A foreign government loan of $750,000 at 5% is to be redeemed at 105 during 20 years by quinquennial drawings. The sinking fund will accumulate for each five year period at 4%. What annual sum should be set aside to serve the loan? All rates are yearly. Examination Papers. 107 VLE 1. Show by general reasoning: i—d=di; { (1-+1)"—-1}sS =i. 2. A Company buys a 45% bond maturing in 10 years, on the assumption that interest is payable semi-annually, the price being 98. It subsequently appears that the interest is payable annually. How much should the seller return to the company so that the effective rate under the bond will remain as before. 3. Show by general reasoning that (v"+-jamq)(1+%)-—j=v""'+jaz-7, the »v and a functions being taken at rate 7. An n-period bond for 1 with coupons at rate j is bought for 1+. Assuming that the purchaser writes down his investment each period by the excess interest in the coupon, show that the investment yield 7 will be obtained on solving for 7 the equation 1+p=v0"+ jaz. 4. A government is issuing a loan of $1,000,000 bearing interest at 5% and repayable by drawings of $200,000 at the end of every 5 years. A syndicate takes up the entire issue at a price of $1,045,000. What rate of interest does it realize on the investment? 5. A loan is to be repaid by an annuity of $1,000 a year for 5 years. Find the amount of the loan, assuming that the lender is to receive 5% interest over the whole term, and can replace his capital at the end of the term by a sinking fund accumulating at only 3%. Construct a schedule showing the division of the yearly payments into principal and interest, and discuss the redemp- tion price just after the third payment has been made. we ere. le, SAS ON RS Bae ae Re ONES NESE SRR Ga ota nee, CROC) Ee) alla aL ee eee ; Sy es >. tS > mon 4 Opeth “as Sot ia Oh Ate cae > ae ‘% i a ; <. J wy oe, Cre eS et \ TORE Fe te Pits ORS, yy, ha i a | eS Gdn athe i ae — « & 7 ‘ i? os SO pig eae “ar a 4 ~ i Faye. Se ‘ ; “t r ze a ~ os fe Ee i 1 2 INTEREST TABLES. ‘ This short eollecnen will serve the student in the sobitige” ee of typical problems; but the tables given herein must not be regarded as a substitute for such tables as those of Mr. Archer or Colonel Oakes. 1 oe oR ae ow OO Ub WN eK 1% I‘O1000 I‘O2010 1°03030 1°04060 I*‘O5101 1°06152 1°07214 1°08 286 1°09369 1*10462 I°11567 1°12683 1°13809 1°14947 1°16097 1°17258 1°18430 I°19Q615 1°20811 1°22019 1°23239 1°24472 1°25716 1°26973 1°28243 1°29526 1°30821 1°32129 1°33450 1.34785 1°36133 1°37494 1°38869 1°40258 1°41660 1°43077 1°44508 1°45953 1°47412 1°48886 1°59375 1°51879 1°53398 1.54932 1°56481 1°58046 1°59626 1°61223 1°62835 1°64463 TABLE I. Amount of 1: viz., (1+7)”. 147% I°OI250 1°02516 1°03797 1°05095 1°06408 1°07738 1°09085, 1°10449 1°11829 1713227 1°14642 © 1°16075 1°17526 1°18995 120483 1*21989 1°23514 1°25058 1*26621 1°28204 1*29806 1°31429 1°33072 1°34735 1°36419 1°38125 1°39851 1°41599 1°43369 I°45161 1*46976 1°48813 1°50673 1°52557 1°54464 1°56394 1°58349 1°60329 1°62333 1°64362 1°66416 1°68497 1°70603 1°72735 1°74895 1°77081 1°79294 1°81535 1°83805 1°86102 13% *01500 03023 "04.568 061 36 07728 °09344 *10984 *12649 "14339 *16054 "17795 *19562 °21355 °23176 °25023 *26899 *28802 "30734 "32695 *346086 1°36706 1°38756 1°40838 1°42950 1°45995 1°47271 1°49480 1°51722 1°53998 1°56308 1°58653 1°61032 1°63448 1°65900 1°68388 I°70914 1°73478 1°76080 1°78721 1°81402 =x = SS eH =x = = eS et = = = et et -— = = eS et 1°84123 1°86885 1°89688 1°92533 1°9542! 1°98353 2°01328 204348 2°07413 2°10524 127 1*01750 1°03531 1°05342 1°07186 1'09062 I*10970 I°I2912 1°14888 1°16899 1°18944 1°21026 1°23144 1°25299 1°27492 1°29723 1°31993 1°34 303 1°36653 1°39045 1°41478 1°43954 1°46473 1°49036 1°51644 1°54298 1°56998 1°59746 1°62541 1°65386 1°68280 yieas 1°74221 1°77270 1°80372 1°83529 1°86741 I °(Q0009 1°93334 1°96717 "00160 03663 07227 "10853 "14543 "18298 *22118 "26005 ‘29960 "33984 2°38079 N NHN NNNNND WN xX + =~ S = = Se I I I I I I I I I I I I I I I I I I I I I I I I I 27 *02000 "04040 06121 08243 °10408 *12616 *14869 *17166 “19509 *21899 "24337 *26824 °29361 "31948 "34587 °37279 "40024 42825 *45681 "48595 "51567 "54598 "57690 "60844 "64061 67342 *70689 °74102 "77584 *81136 84759 "88454 "92223 *96068 "99989 2°03989 2°08069 2°12230 2°16474 2°20804. 2°25220 2°29724 2°34319 2 2 2 *39005 "43785 48661 2°53634 2 2 2 *58707 63881 69159 2% 1'02250 1°0455! 1°06903 1°09308 1°11768 1°14283 1°16854 1°19483 I°22171 1°24920 1°27731 1°30605 1°33544 1°36548 1°39621 1*42762 1°45974 1°49259 t'S2017 1°56051 1°59562 1°63152 1°66823 1°70577 1°74415 1°78339 1°82352 1°86454 1*90650 1°94939 1°99325 2°03810 2°08396 2°1 3085 2°17879 2°22782 2°27794 2°32920 2°38160 2°43519 2°48998 2°54601 2°60329 2°66186 2372570 2°78300 2°84561 2°90964. 2°97511 3°04205 n SS ee Oe ee. es Ll OO ONO UhWN TABLE I. (continued). Amount of 1: vts., (1+72)”. n 23% 3% 33% 47% 43% 5% I I*°02500 | 1°03000 | 1°03500 | I°04000 | 1°04500 | 1°05000 I 2 1°05063 | 106090 | 1°07123 | 1°08160 | 1°09203 | 1°10250 2 3 1.07689 | 1°09273 | 1°10872 | 1°12486 | 1°14117 | 1°15763 3 4 T°IO38r | 1°12551 | 1°14752 | 1°16986 | 1°19252 | I'21551 4 5 I°13141 | 1°15927 | 1°18769 | 1°21665 | 1°24618 | 1°27628 5 6 1°15969 | 1°19405 | 1°22926 | 1°26532 | 1°30226 | 1°3q40I0 6 7 1°18869 | 1°22987 | 1°27228 | 1°31593 | 1°36086 | 1°40710 8 1°21840 | 1°26677 | 1°31681 | 1°36857 | 1°42210 | 1°47746 8 9 1°24886 | 1°30477 | 1'36290 | 1°42331 | 1°48610 | 1°55133 9 10 1°28008 | 1°34392 | 1*41060 | 1°48024 | 1°55297 | 1°62889 | 10 If 1°31209 | 1°38423 | 1°45997 | 1°53945 | 1°62285 | 1°71034 | II 12 1°34489 | 1°42576 | 1r°51107 | 1°60103 | 1°69588 | 1°79586 12 13 1°37851 | 1°46853 | 1°56396 | 1°66507 | 1°77220 | 1°88565 13 14 I1°41297 | 1°51259 | 1°61869 | 1°73168 | 1°85194 | 1°97993 le 15 | 144830 | 1°55797 | 1°67535 | 1°80094 | 1°93528 | 2°07893 | 15 16 1°48451 | 1°60471 | 1°73399 | 1°87298 | 2°02237 | 2°18287 16 17 1°52162 | 1°65285 | 1°79468 | 1°94790 | 2°11338 | 2°29202 17 18 1°55966 | 1°70243 | 1°85749 | 2°02582 | 2°20848 | 2°'40662 18 19 1°59865 | 1°75351 | 1°92250 | 2°10685 | 2°30786 | 2°52695 19 20 1°63862 | 1°80611 | 1°98979 | 2°I9112 | 2°4117I | 2°65330 | 20 21 1°67958 | 1°86029 | 2°05943 | 2°27877 | 2°52024 | 2°78596 | 21 22 1°72157 | I°QI610 | 2°13151 | 2°36992 | 2°63365 | 2°92526 22 23 1°76461 | 1°97359 | 2°20611 | 2°46472 | 2°75217 | 3°07152 23 24 1°80873 | 2°03279 | 2°28333 | 2°56330 | 2°87601 | 3°22510 | 24 25 | 1785394 | 2°09378 | 2°36324 | 2°66584 | 3°00543 | 3°38635 | 25 26 1°90029 | 2°15659 | 2°44596 | 2°77247 | 3°14068 | 3°55567 | 26 27 1°94780 |. 2°22129 | 2°53157 | 2°88337 | 3°28201 | 3°73346 23 28 1°99650 | 2.28793 | 2°62017 | 2°99870 | 3°42970 | 3°92013 | 28 29 2°04641 | 2°35657 | 2°71188 | 3°11865 | 3°58404 | 4°11614 29 30 | 2°09757 | 2°42726 | 2°80679 | 3°24340 3°74532 | 4°32194 | 30 ay 2°15cor | 2°50008 | 290503 | 3°37313 | 3°91386 | 4°53804 | 31 32 2°20376 | 2°57508 | 3°00671 | 3°50806 | 4°08998 | 4°76494 32 33 | 2°25885 | 2°65234 | 3°11194 | 3°64838 | 4°27403 | 5°00319 | 33 34 | 2°31532 | 2°73191 | 3°22086 | 3°79432 | 4°46636 | 5°25335 | 34 35 | 2°37321 | 2°81386 | 3°33359 | 3°94609 | 4°66735 | 5°51602 | 35 36 | 2°43254 | 2°89828 | 3°45027 | 4°10393 | 4°87738 | 5°79182 | 36 37 | 2°49335 | 2°98523 | 3°57103 | 4°26809 | 5°09686 | 6°08141 | 37 38 2°55508 | 3°07478 | 3°69601 | 4°43881 | 5°32622 | 6°38548 | 38 39 | 2°61957 | 3°16703 | 3°82537 | 4°61637 | 5°56590 | 6°70475 | 39 2°68506 | 3°26204 | 3°95926 | 4‘80102 | 5°81636 | 7°03999 | 40 2°75219 | 3°35999 | 4°09783 | 4°99306 | 6°07810 | 7°39199 | 41 2*82100 | 3°46070 | 4'24126 | 5°19278 | 6°35162 | 7°76159 | 42 2°89152 | 3°56452 | 4°38970 | 5°40050 | 6°63744 | 8°14967 | 43 2°96381 | 3°67145 | 4°54334 | 5°61652 | 693612 | 8°55715 | 44 3°03790 | 3°78160 | 4°70236 | 5°84118 | 7°24825 | 8°9g8501 45 3°11385 | 3°89504 | 4°86694 | 6°07482 | 7°57442 | 9°43426 | 46 319170 | 4°01190 | 5.03728 | 6°31782 | 7°91527 | 9°90597 | 47 3°27149 | 4°13225 | 5°21359 | 6°57053 | 8°27146 | 10°40127 | 48 3°35328 | 4°25622 | 5°39606 | 6°83335 | 8°64367 | 10°92133 | 49 3°43711 | 4°38391 | 5°58493 | 7°10668 | 9°03264 |11°46740 | 50 RR RR A A A aAbAAA HAAAH FH CO ON A NHWNH OC eS es Td TABLE II. Present Value Ofal ai vise ON. n Lond CO OND MNPWN | Oe Be | ee | CWO OND ONL WN iS) ~ NHN NHN NN Cm OO Oh wW bd 29 30 TABLE II. (conz¢nued). Present Value of 1: vtg., uv”. ——— | | | | | | 16 | °67362 ; °62317 | °57671 | °53391 | “49447 | ‘45811 16 17 65720 | ‘60502 | °55720 | °51337 | °47318 | *43630 17 18 | *64117 | *58739 | °53836 | °*49363 | *45280 | “41552 | 18 19 | ‘62553 | ‘57029 | ‘52016 | *47464 | *43330 | ‘39573 | 19 20 | ‘61027 | °55368 | °50257 | °45639 | °41464 | °37689 | 20 21 | °59539 | °53755 | ‘48557 | ‘43883 | °39679 | °35894 | 21 22 *58086 "52189 *46915 *42196 *37970 *34185 22 23 | °56670 | *50669 | °45329 | “49573 | °36335 | °32557 | 23 24 | ‘55288 | ‘49193 | °43796 | “39012 | °34770 | ‘*31007 | 24 25 °53939 "47761 "4.2315 °37512 °33273 °29530 25 26 52623 *46369 "40884 *36069 *31840 "28124 26 27 | °51340 | “45019 | ‘39501 | °34682 | *30469 | ‘26785 | 27 28 *50088 *43708 *38165 33348 *29157 *25509 28 - 29 *48866 42435 *36875 *32065 *27902 °24295 29 30 "47674 ‘41199 35628 *30832 *26700 *23138 30 3! "46511 | *39999 | °34423 | ‘29646 | ‘25550 | °22036 | 31 32 ‘45377 | ‘38834 | °33259 | °28506 | ‘24450 | ‘20987 | 32 33 | °44270 | °37703 | °32134 | °27409 | ‘23397 | ‘19987 | 33 34 | “43191 | *36604 | *31048 | °26355 | ‘22390 | ‘19035 | 34 35 | °42137 | °35538 | *29998 | °25342 | ‘21425 | ‘18129 | 35 36 ‘41109 *34503 *28983 *24.367 *20503 17266 36 37. | ‘40107 | ‘33498 | ‘28003 |. °23430 | ‘19620 | ‘16444 | 37 38 *39128 *32523 *27056 *22529 18775 "15661 38 39 | °38174 | °31575 | ‘26141 | ‘21662 | ‘17967 | ‘14915 | 39 40 | *37243 | °30656 | ‘25257 | *20829 | ‘17193 | "14205 | 40 41 "36335 | *29763 | ‘24403 | ‘20028 | *16453 | °13528 | 41 42 "35448 | *28896 | °23578 | ‘19257 | ‘15744 | ‘12884 | 42 43 *34584 *28054 *22781 *18517 *15066 *12270 43 44 "33740 ‘29237 ‘22010 *17805 °14417 "11686 44 45 *32917 *26444 *21266 *17120 *13796 ‘11130 45 46 “32115 *25074 *20547 "16461 "13202 *10600 46 47 tat ant *24926 "19852 °15828 | . 12634 "10095 47 48 *30567 *24200 ‘1g181 "15219 *12090 "09614 48 49 | °29822 | *23495 | °18532 | 14634 | ‘11569 | ‘og156 | 49 50 *29094 ‘22811 *17905 ‘14071 ‘11071 *08720 50> _ OO ON OO HANBWN 1) | Pe COMO ONO Ubwhd Nv oN NOR 23 24 25 26 27 28 29 30 ay 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Amount of 1 per Period: vtz., sz}. 1% I "00000 201000 3°03010 4.°06040 510101 6°15202 721354 8°28567 9°36853 10°46221 11°56683 12°68250 13°80933 14°94742 16°09690 17°25786 18°43044 19°61475 20°81089 22°01900 23°23919 24°47159 25°71630 26°97346 28°24320 29°52563 30°82089 32°12910 33°45039 34°78489 36°13274 37°49407 38°86901 40°25770 41 °66028 43°07688 44°50765 45°95272 47°41225 48°88637 50°37524 51°87899 53°39778 54°93176 56°48107 58°04588 59°62634 61°22261 62°83483 64°46318 13% I*O0000 2.01250 3°03765 4°07503 5°12657 6°19065 7°26804 8°35889 9°46337 10°58167 11°71394 12°56036 14°O2112 15°19638 16°38633 17°59116 18°81105 20°04619 21°29677 22°56208 23°84502 25°14308 20°45737 27°78808 29°13544 30°49963 31 °88087 33°27938 34°69538 36°12907 37°58068 39°05044 40°53957 42°04530 43°57087 ep 46°67945 48°26294 49 86623 51°48956 53°13318 54°79734 56°48231 58°18834 59°91569 61°66464 63°43545 65°22839 67°04374 68°88179 TABLE Ili; 13% I *00000 2*O1500 3°04523 4°09090 515227 6°22955 732299 8°43284 9°55933 10°70272 11 °86326 13°O4121 14°23683 15°45038 16°68214 17°93237 19°20136 20.48938 21°79672 23°12367 24°47952 25°83758 27°22514 28°63352 30°06302 31°51397 32°98668 34°48148 35°99870 37°53868 39°10176 40°68829 42°29861 43°93309 45'59209 47°27597 48°98511 50°71989 52°48068 54°26789 56°o08191 57°92314 59°79199 61°68887 63°61420 65°50841 67°55194 69°56522 71°60870 73°68283 I*00000 2°O01750 3°05281 4°10623 517809 6°26871 7°37841 8°50753 9°65641 10°82540 12°01484 13°22510 14°45654 ES 79953 16°98445 18°28168 19°60161 20°94463 ZSR1T Ey 23°70161 25°11639 26°55593 28°02065 29°51102 31°02746 32°57044 34°14042 35°73788 37° 36329 39°O17I5 40°69995 42°41220 44°15441 45°92712 47°7 3084, 49°56613 51°43354 5333362 55°26696 57°23413 59°23573 61°27236 63°34462 65°45315 67°59858 69°78156 72°00274 74°26278 76'56238 78°90222 2% I 00000 2*02000 3°06040 4°12161 5.20404 6°30812 7°43428 8°58297 9°75463 10°94972 12°16872 13°41209 14°68033 15°97394 17 °29342 18°63929 20°01 207 21°41231 22°84056 24°29737 25°78332 27°29898 28°84496 30°42186 32°03030 3367091 3534432 3705121 38°79223 40°56808 42°37944 44°22703 46°11157 48°03380 49°99448 5199437 5403425 56°11494 58°23724 60°40198 62°61002 64°86222 67°15947 69°50266 71°89271 74°33056 76°81718 19°35352 81°94059 84°57940 24% I'00000 2°02250 306801 4°13704 5°23012 6°34780 7°49062 8°65916 9°85399 I1°07571 12°3249I 13°60222 14°90827 16°24371 17°60919 19°00540 20°43302 21°89276 23°38535 2491152 26.47203 28°06765 29°69917 31°36740 33°07317 34°81732 36°60071 38°42422 40°28877 42°19526 44°14466 46°13791 48°17602 50°25998 52°39083 5456962 56°79744 59°07538 61°40457 63°78618 66°22137 68°71135 71°25735 7386064 76°52251 79°24426 82°02726 84°87287 87°78251 90°75762 con AO Uh w& bd Ne) a CR RR RR EY ST A ETRE BES ES RRR ES Amount of 1 per Pertod: vts., sy}. n 23% 3% 33% 4% 437% 5% % I 1*0000 I ‘o0co I*0000 1‘0000 I*0000 I*0000 I 2 2°0250 2°0300 2°0350 2°0400 2°0450 2°0500 2 3 | 370756 | 3°0909 | 3°1062 | 3°1216 | = 31370} = 3°1525| 3 4 | 41525 | 41836) 4°2149] 4°2465] 4°2782] 4°3101) 4 5) {2 -5°2563 ;|) 5°3991 | 593025} 574163 | —°5:4797,| | Gs 5250 Ras 6 6°3877 6°4684 6°5502 6°6330 6°7169 6°8019 6 7 75474 7°6625| 7°7794| 778983] S8o192| 81420} 7 8 8°7361 8°8923 9°0517 9°2142 9°3800 9°5491 8 9 9°9545 IO°I59I | 10°3685 | 10°5828] 10°8021] 11°0266 aps fe) 11*2034 11°4639 | 11°7314.|. 12°0061 | .12°2882.| 212°5770' ac II 12°4835 12°8078 | 13°1420] 13°4864| 13°8412| 14°2068] 11 12 13°7956 14°1920 | 14°6020| 15°0258| 15°4640] I5°9171 |] 12 13 15°1404 15"6178 |. 16°1136 | 16°6268 |. 17°5500.|- 17 Fi4a te es 14 16°5190 17°0863 | 17°6770] 18°2919| 18°9321 | 19°5986]| 14 15 17°9319 18°5989 | 19'2957 | 20°0236| 20°7841| 21°5786|] 15 16 19°3802 | 20°1569| 20°9710] 21°8245| 22°7193| 23°6575| 16 17 20°8647 | 21°7616| 22°7050| 23°6975| 24°7417| 25°8404| 17 18 22°3863 |. 23°4144 | 24"'4997 | 25°6454 1. 26°8551:| _28°1324 |} a8 19 | 23°9460 | 25°1169| 26°3572| 27°6712| 29°0636| 30°5390| 19 20 25°5447 26°8704 | 28°2797| 29°7781 | 31°3714| 33°0660] 20 21 2771833" 1. 28°6765 | )30°2695)| «389092199 33°763T 957 1Ot eee 22 28°8629 | 30°5368 | 32°3289| 34°2480| 36°3034] 38°5052| 22 23 | 30°5844 | 32°4529 | 34°4604 | 36°6179 | 38°9370] 41°4305| 23 24 32°3490 | 34°4265 | 36°6665]| 39°0826| 41°6892| 44°5020|] 24 25 | 34°1578 | 36°4593| 38°9499| 41°6459| 44°5652) 47°7271 | 25 26 | 3670117 | 38°5530| 41°3131 | 44°3117| 47°5706| 51°1135| 26 27 | 37°9120 | 40°7096| 43°7591 | 47°0842| 50°7113| 54°6691 | 27 28 | 39°8598 | 42°9309| 46°2906| 49°9676| 53°9933 |~ 58°4026| 28 29 | 41°8563 | 45°2189) 48°9108| 52°9663| 57°4230| 62°3227| 29 39 | 43°9027 | 47°5754| §1°6227| 56°0849| 61°0071 | 66°4388 | 30 31 | 46°0003 | 50°0027| 54°4295| §9°3283] 64°7524| 70°7608| 31 32 | 48°1503 | 5§2°5028| 57°3345 |- 62°7015| 68°6662} 75°2988| 32 33 | 50°3540 | 55°0778| 60°3412| 66°2095| 72°7562| 80°0638| 33 34. | 52°6129 | 57°7302| 63°4532| 69°8579| 77°0303| 85°0670}] 34 35 54°9282 | 60°4621 | 66°6740| 73°6522| 81°4966| 90°3203]| 35 36 | 57°3014 | 63°2759| 70°0076| 77°5983| 8671640} 95°8363} 36 37 | 59°7339 | 661742 | 73°4579| 81°7022) 91°0413 | tor’6281 | 37 38 | 62°2273 | 69°15904| 77°0289] 85°9703| 96°1382 | 107°7095| 38 39 | 64°7830 | 72°2342 | 80°7249| go*4ogi | 101°4644 | 114°0950] 39 49 | 6774026 | 75°4013 | 84°5503| 95°0255 | 107°0303 | 120°7998| 40 4I 70°0876 | 78°6633| 88°5095 | 99°8265 | 112°8467 | 127°8398 | 41 42 72°8398 | 82°0232| 92°6074 | 104°8196 | 118°9248 | 135°2318 | 42 43 75°6608 | 85°4839| 96°8486 | 110°0124 | 125°2764 | 142°9933] 43 44 | 78°5523 | 89°0484 | 101°2383 | 115°4129 | 131°9138 | I51°1430| 44 Aso} Bree Oy 92°7199 | 105°7817 | 121°0294 | 138°8500 | 159°7002 | 45 46 | 84°5540 | 96°5015 | 110°4840 | 126°8706 | 146°0982 | 168°6852 | 46 47 | 87°6679 | 100°3965 | 115°3510 | 132°9454 | 153°6726 | 178°1194 | 47 48 | g0°8596 | 104°4084 | 120°3883 | 139°2632 | 161°5879 | 188°0254 | 48 49 | 94°1311 | 108°5406 | 125°6018 | 145°8337 | 169°8594 | 198°4267 | 49 50 | 97°4843 | 112°7969 | 130°9979 | 152°6671 | 178°5030 | 209°3480 | 50 TABLE III. (continued). ——— | | 0’99010 1°97040 2°94099 3°90197 4°85343 5°79548 6°72819 7°65168 8°56602 9°47130 10°36763 11°25508 12°13374 13°00370 13°86505 14°71 787 15°50225 16°39827 17°22601 1804555 18°85698 19°66038 20°45582 21° 24339 22°02316 22°79520 23°55961 24°31644 25 06579 25°80771 26°54229 27°26959 27°98969 28°70267 29°40858 30°10751 39°7995! 31°48466 32°16303 32°83469 33°49969 34°15811 34°81001 35°45545 36°09451 36°72724 37°35379 37°97396 38°58808 39°19612 9°34553 10°21780 II ‘07931 I1°93018 12°77955 13°60055 14°42029 15°22992 16°02955 16°81931 17°59932 18°36969 19°13056 19°88204 20°62423 21°35727 22°08125 22°79630 23°50252 24.°20002 24°88891 25°56929 26°24127 26°90496 27°56046 28°20786 28°84727 29°47878 30°10250 30°71852 31°32093 31°92784 32° 2132 33°10748 33°68640 34°25817 34°82288 35°38062 35°93148 36°47554 37°01288 TABLE IV. Present Value of 1 per Period: vtz., az}. 10°O7112 10°90751 11°73153 12°54338 13°34323 14°13126 14°90765 15°67256 16°42617 17°16864 17°QO014. 18°62083 19°33086 20°03041 20°71961 21°39863 22°06762 22°72672 23°37608 24°01584 24°64615 25°26714 25°87896 26°48173 27°07560 27°66068 28°23713 28°80505 29°36458 29°91585 30°45896 30°99405 31°52123 32°04062 32°55234 33°05649 33°55319 3404255 34°52468 34°99969 9°92749 10°73955 11°53764 12°32201 1 3°09288 13°85050 14°59508 15°32686 16°04606 16°75288 17°44755 18°13027 18°80125 19°46069 20°10878 20°74573 21°37173 21°98695 22°59160 23°18585 23°76988 24°34386 24°99797 25°46238 26°00725 26°54275 27°06904 27°58628 28°09463 28°59423 29°08524. 29°50780 30°04.207 30°50817 30°96626 31°41647 31°85894 32°29380 Raeoa le 33°14121 2% 0°98039 1°94156 2°88388 3°80773 4°71346 5°60143 6°47199 7°32548 8°16224 8°98258 9°78685 10°57534" 11°34837 12°10625 12°84926 13°57771 14°29187 14°99203 15°67846 16°35143 I7°O1I21 17°65805, 18°29220 18°91 393 19°52346 20°12104 20°70690 21°28127 21°84438 22°39646 22°93779 23°46833 23°98856 24°49859 24°99862 25°48884 2596945 26°44064 26°90259 27°35548 27-79949 28°23479 28°66156 29°07996 29°49016 29°89231 30°28658 30°67312 31°05208 31°42361 23% 097800 1°93447 2°86990 3°78474 4°67945 5°55448 6°41025 7°24718 8°06571 8°86622 9°64911 10°41478 11°16360 11*89594 12°61217 13°31 263 13°99768 14°66766 15°32290 1596371 16°59043 17°20335 17°80279 18°38904 18'96238 19°52311 20°07150 20°60783 21°13235 21°64533 22°14702 22°63767 23°11753 23°58683 24°04580 24°49467 24°93366 25°36299 25°78288 26°19352 26°59513 26°95790 27° 37293 27°74771 28°11512 28°47444 28°82586 29°16955 2950507 29°83440 | + OO ON A ONXHWD & TABLE IV. (continued). Present Value of 1 per Pertod: viz., ay}. n 237% 3% 33% 47% 43% 5% ” et | ee fa | | | \O ~J \O ~J e) \O ~J ~I oo Oo’ = wy io) ie) “J ~JI ~JI aS w On ioe) ~I LS) ON co ioe) ~sI _ ie) ~T co 14 I1*6909 | I1°2961 | 10°9205 | 10°5631 | 10'2228 98986 14 15 | 12°3814 | 11°9379 | 11°5174 | I1°1184 | 10°7395 | 10°3797 | 15 16 | 13°0550 | 12°5611 | 12°0941 | 11°6523 | 11°2340 | 10°8378 16 17 13°7122'} 33°1661-*) +12°6513 |) 12*1657