45B7 S74 UC-NRLF AMORTIZATIO N A Guide to the Ready Computation of the Investment Value of Bonds by the use of the Extended Bond Tables By Charles E. Sprague second edition New York, 1910 pubusht by the author T\ :)age 7. is is the aper or I tyle of ru ooks may ing reconfim be orderdd GEOIGE H/RJES CO. 35 V^EST 31si STREET NEW YORK, N. Y. ended in from AMORTIZATION A GuiDK TO THE Ready Computation of the Investment Value of Bonds by the use OF THE Extended Bond Tables By Charles E. Spragub second edition New York, 1910 publisht by the author < Copyright, 1908, by Charles E. Sprague PREFACE. TDELIEVING that the principle of amortization was the only true method of valuing securities held for investment, and that ulti- mately it would be the one ofiScially adopted, I commenced in 1898 to make the necessary computations for placing the securities of this savings bank on that basis. I found that the existing tables of bond values were useless for that purpose, as their decimals are not suflBciently extended, giving accurate values to $100 only. I succeeded at the beginning of 1901 in inaugurating the system of amortization, after an enormous amount of labor, but with conscientious accuracy. Realizing the need for extended tables which would greatly facilitate the work, and utilizing the experience and some of the results already obtained, I then took up the still greater task of making all the calculations for a set of tables which would have saved me nine-tenths of my labor had they been in existence in 1898. These tables were issued in 1905, and are extended to eight places of decimals. As amortization has now been enacted into law, I have prepared this little treatise as a companion to the Tables, hoping it may be useful to all officers of financial institutions and may aid them in their labors. I have explained the theory in my book entitled "Text-Book of the Accountancy of Investment, ' ' but even those who have studied the theory are benefited by having at hand a simple set of rules. I am satisfied that by the use of the Extended Tables, guided by these rules, scientific amortization, within the conditions laid down by the Banking Department, will be much easier of operation than the so- called •* simple " but very inaccurate method prescribed as an altprnativ. CHARIvES E. SPRAGUE. Union Dime Savings Bank, New York, June, 1908. 247038 CONTENTS. PAGE Introductory 1 What is Investment Vai.ue ? 1 Prewminary Steps 3 Determining the Basis 3 1. For Even Half- Years 4 2. For Broken Times 4 3. For Serial or Various Maturities C 4. Bonds at Unusual Rates 6 The Inaugurai, Vai^ue 7 1. For J. & J. Bonds 7 2. For Bonds at other Dates 7 3. For Serial Bonds 8 4. Combining Several Purchases.. 10 Continuing the Amortization 10 1. For J. & J. Bonds 11 Schedule of Amortization 12 2. For Bonds at other Dates 13 3. Serial Bonds 18 New Purchases 19 Intermediate Rates 19 OuARTERi,Y Bonds 21 Annuai, Bonds 21 Residues 21 Accrued Interest 22 The Proper Basis of Bond Accounts vi^hen held for Invest- ment 23 AMORTIZATION The Banking Law of the State of New York as amended, paragraph 20, makes it the duty of each Savings Bank to report semi-annually to the Superintendent of Banks, among other things, " the estimated invest- ment value of all stock and bond investments" "in the manner pre- scribed by the Superintendent of Banks." In accordance with this provision, the Superintendent has Qune, 1908) issued a circular in which he recognizes as admissible two methods of ascertaining investment values, designated as the scientific and the simple, or pro rata, method, respectivly. If we were compelled to compute each separate value by arithmetic, the pro rata method would in most cases be easier; but where the values at all ordinary rates and times have been exactly calculated and tabu- lated, as they are in Sprague's Bond Vai.uks, it is quite as easy to use the "scientific" method. The scientific method gives far more satisfactory results and equalizes the rate of interest during the life of the bond. If the "simple " method be applied to a seven-year 7% bond bought on a 4% basis, it will be found that the income on the investment value, instead of being a uniform 4%, varies from 3.73% to 4.36%. It does not^ therefore, "result in giving the purchaser the exact periodical income upon the basis of which the bond was purchased," even for this short maturity. This treatise is intended to furnish rules of procedure by which the scientific investment-value may most readily be ascertained from tables of extended bond-values; the "simple" or "pro rata" values will be disregarded While very little of the theory of the subject need be introduced here, a definition of the terms used will be proper. What is Investment Value? The income-basis being that rate of compound interest which, ap- plied to the original cost, will produce at the proper times all the pay- ments provided for in the security, whether designated as principal or as interest: the investment value is that value which, upon the original income basis at the time of investment, will amortize or gradually extinguish the premium or discount so as to bring the security to par at maturity. The successiv values of bonds under these definitions are obtain- able in two ways : 1. By discounting to the present date at true compound interest, on the income basis, every future payment of principal or interest. 2 Amortization 2. By applying to the value reported at the next previous report the amount known as the amortization. The amortization is the difference between the nominal and the effectiv interest or net income. The nominal interest is a constant quantity being a fixt percent- age of the par value. The effectiv interest is a fixt percentage of the investment value, but as this constantly varies, approaching to par, the effective interest also varies and the amortization, whether of premiums or discounts, constantly increases. • Thus, if a 6% bond for $10,000., interest semi-annual, was pur- chased at some time in the past on a 4% basis and now has one year to run to maturity, the nominal interest is |300. each half year ; its effectiv interest is 2% each half year on whatever the investment value may be at the time. The income basis is 4%. The investment value is composed of the following values : 1. Present worth of $ 300. at 4% >^ year from now . . $ 294.12 2. Present worth of 300. at •• (cpd.) 1 year from now . . 288.35 3. Present worth of 10,000. at " " 1 year from now . . 9,611.69 ToTAi, $10,194.16 This result is corroborated by various tables of bond values. At six months before maturity the corresponding figures would be: 1. Present worth of § 300. at 6 months $ 294.12 2. Present worth of 10,C00. at 6 months 9,803.92 New Investment Vai,ue... $10,098.04 To proceed by amortization, we begin with the value at one year $10,194.16 The interest for 6 months on this investment is $203.88 The nominal interest is 3% of $10,000 $300.00 The effectiv interest is 2% of $10,194.16 203.88 The amortization is the difference $ 96.12 Subtracting this from the investment value 96.12 We obtain a new investment value (as above) $10,098.04 The nominal interest for the next year is $300.00 The effectiv interest, 2% of $10,098.04, is 201.96 The amortization is .T7T7TT7 98.04 Leaving THE Par $10,000.00 Thus the results by compound discount and by gradual amortiza- tion are precisely the same; and in either case the effectiv rate is actually received and the amount calculated to maturity is exactly what it should be. Sprague's Extended Bqnd Tabi^es (2nd Edition, New York, 1907) are accurately calculated on precisely the above principle. All ,values are given on the assumption of half-yearly conversion, which corresponds with the serai-annual periods covered by the reports of Amortization 8 savings banks. Their use in computing schedules of amortization (which was the original purpose in constructing them) will save many tedious hours of computation, and their extreme minuteness (to the nearest cent on one million dollars), as well as their wide range of incomes {l%% to 10%), make them the only tables which will fully meet this need. PreIvIminary Steps. As the reports to the Banking Department show the condition at the opening of business on the first day of January and the first day of July, we shall assume that amortization is uniformly to be effected up to the close of business on the last days of December and of June, altho the interest may be payable at some other dates. An exact list of all the holdings of bonds must be prepared, giving all the data necessary for the future account on the investment value basis. This may be made up in the tabular form, or a paper or ticket may be devoted to each lot of bonds. The following are the data to be obtained from the books as to each lot of bonds. Where bonds of a certain issue were purchased in several lots at different times and prices, they must be treated separately at this stage. 1. Thenar or principal sum. 2. The date when it will become due. 3. The rate of interest stipulated to be paid. 4. The dates of interest payments. 5. The date of purchase. 6. The price per $100 paid and the total cost, exclusive of accrued interest. From the above data is to be ascertained the income-basis upon which the purchase was made, and to which all subsequent computa- tions must conform. Determining the BAvSis. As the entire computation of investment values depends upon the basis of income, the first step is to ascertain what was the basis at which each lot of bonds was purchased. If, at the time 6f each purchase a record was made^ showing the basis at the cost price, then the basis may be at once inserted in the descriptiv ticket and much labor saved. But this is not always the case. It must be remembered that after finding the basis, it is not neces- sary to continue the successiv values down to the time of installing the plan of amortization. On the inaugural date^ as we shall call it, a fresh start is made with the inaugural value, and the values listed until they reach par at maturity. The bond-tables contain in the left-hand column of each page the basis of income from 2.50% to 5.00%, between which almost all savings 4 Amortization bank investments are found. Rates above and below these are given in the second edition. These are given one-twentieth of one per cent, apart, 2.50, 2.55, 2.60, 2.65, 2.70, etc. Intermediate values, one hundredth of one per cent, apart (as 2.51, 2.52, 2.53, 2.54) are found by takmg so many fifths of the difference and then correcting by means of the blue pages. We will first explain the method of determining the basis, finding the inaugural value and completing the amortization to maturity, on the assumption that the tabular values (2.05, 2.10, 2.15, 2.20, etc.) are the ones which apply or else are considered near enough for practical purposes. 1. For Even Hai^f-years. The income basis depends upon three things : the cash rate, or coupon rate (as a 3% or a 6% bond), the price (exclusi v of accrued interest), and the time the bond has to run. Let us take as an illustration a 6% bond for $20,000 which was purchased when it still had just 42 years to run, at 132.88. We turn to that portion of the tables devoted to 5% bonds; find the column headed 42 years, and run down the column until we come to the nearest value to 132.88. We find that, at the income basis, 3.50, the value of $100 would be 132.877107, hence 3.50 is practically the exact income basis of the bond, and its present investment values must be computed on that basis. If, at the inaugural date the bond still has 12 years to run, then turning to the 12 year column on page 83 we find, opposit the net income basis 3.50, the investment value of $1,000,0C0 as 1,145,955.14, and con- sequently the inaugural value for $20,000 par will be $22,919.10. This value will furnish 3^% income on the amount remaining invested and will reduce the investment to par on the day of maturity. Where a number of bond accounts are reduced at the same time to the investment value, the result will be substantially accurate if computed on the tabular values, to the nearest 1/20 of one per cent. We will here- after explain the more minute calculations, by which accuracy to the 1/100 of one per cent, may be attained. 2. For Broken Times. In the foregoing examples, it has been supposed that the purchase was made on an interest date, so that the search is confined to a single column. It is, however, very frequently the case that, besides the years and half years, there are odd months and days in the time which the bond has to run. In this case, the time and hence the value must lie between two columns of the table, and the difference of the values is di- vided proportionately to the difference of the times. On page 8 of the Banking Department circular, authority is given for treating the inaugural value as of the precedmg January or July first aud for disregarding any terminal portion of a period, thus reducing the computation to even half-years. All inaugural values are then found in Amortization 5 the tables. To extract them is far easikr and at The same time FAR MORE accurate THAN THE " SIMPI^E " METHOD. But if absolute accuracy is desired, the basis may be found as follows : If the given value lies between two values, and no others of the adjoining columns, the income-basis of that line is the basis sought, within 1/20 of 1%. For example, a 6% bond having 18 years, 9 months and 6 days to run, bought at 140 and interest; what is the basis? The time may be better exprest as 18>^ years, 3 months and 6 days. Each day is computed as the 30th of a month and each month as the 6th of a half year. We must look down both the 18^ year column and the 19 year column, to see if we can find a pair of values between which 140 lies. We find the pair: iot/ in ^ 18;^ yr. 19 yr. 3.15 1 397 278 83 1 405 147 76 and this is the only pair which embraces 140; and, therefore, 3.15 is the income-basis within the limits of 1/20%; no other tabular value will do. But if there are two pairs between either of which the value for the given time might lie, or if there are no such pairs, we shall have to interpolate according to time. If the price had been 120 and interest and the time 8}4 years, 3 months, 6 days, we should have to choose between these pairs : S'A yr. 9 yr. 3.30 1 198 706 06 1 208 761 50 3.35 1 194 614 08 1 204 439 71 Either pair has 120 within its limits. The difference between 1208 76150 and , 1 198 706 06 is the amortization for 6 months 10 055 44 three months will be half as much 5 027 72 6 days will be 1/30 of 10,055-44 or 335 18 3 months, 6 days, will be 5 362 90 This added to 1 198 706 06 gives the investment value at 3.30% 1 204 068 96 1 204 439 71 1 194 614 08 9 825 63 By the same process applied to the rate 3 35 / o.^J ^9 5 240 33 1 194 614 08 1 199 854 41 we find the much nearer approximate, hence 3.35 is the proper basis. It is true even to the 1/100 of 1%, as 3.30 appears by inspection to be wider of the mark. If advantage be taken of the permission given by the Banking Department, page 8, all purchases and all redemptions will be treated as of June 30th and December 31st. This will be easier, but will give a less accurate result. 6 Amortization 3. For Seriai. or Various Maturities. Bonds are often issued in series. For example: $30,000, of which $1,000 is payable after one year, another $1,000 after two years, and the last $1,000, 30 years from date. In offering such bonds for sale, they are often listed as of "average maturity — 15)4 years." This is entirely delusiv, and frequently causes the buyer to believe that he is getting a more favorable basis than will be realized. The only correct valuation of a series is the sum of all its separate values. If we assume that the $30,000 above referred to was a series of 5% bonds bought on an alleged 3.50% basis, the true value would be 35,005.00 whereas the value for the *' average time " would be 35,348.22 In computing the basis of a series, the basis corresponding to the average time may be taken as a point of departure, but it will be found that it is invariably too high. A series of 30 5% bonds maturing at 1 to 30 years from the date of purchase at 116.68; what basis ? looking in the 15)4 year column (average date) the nearest value to 116.68 is 1 165 200 00 which is at a 3.60 basis. If, however, we take off on an adding machine the values for 30 years at 3.60 1 255 549 38 for 29 years at 3.60 1 250 705 95 and so on to one year, we shall find that the total is 34 531 390 28 and the average price 115.10 The basis 3.55 will give the result 116.06, but 3.50 will give 116.68, and 3.50 is the true basis on separate maturities. 4. Bonds at Unusuai. Rates. All the usual rates (2, 3, 3>^, 4, 4)4, 5, 6 and 7 per cent bonds) are provided for in the tables. Infrequently, bonds of different rates occur, as 3.60, 3.65, 3^, 4Xi 5^. These may accurately and readily be de- rived from the tabular values by "splitting the difference " : 4)( bonds are midway between 4's and 4j^'s of the same income basis : 5>^'s between 5 and 6. For 3.60 bonds, to the value of a 3^ add 1/5 the differ- ence between S^ and 4. For example, a 3% bond of 25 years at a 3 . 55 basis is worth 909 350 85 and a 4% bond on the same basis is worth 1 074 167 48 The difference is 164 816 63 We should, therefore, expect a 5% bond to be worth 1 238 984 11 which is found to be true on turning to page 88. The dif- ference for y2% should be half of 164 816 63, or 82 408 31,5 and the value of a 4yi/c bond on a 3.55 basis is 1 156 575 79,5 A 3% bond is worth 909 350 85 4- 82 408 31.5 And a 3>^% bond is worth 991 759 16,5 Adding to this .15 of 164 816 63 + 24 722 49.4 5 we have the value of a 3 65 bond 1 016 481 66 Amortization ' 7 The Inaugural Valuk. Assuming that the entire system of amortization is to be inaugu- rated as of a certain date, which we call the inaugural date, and that the income basis has been ascertained, it remains to compute the value on the inaugural date. In our illustrations we shall assume the 1st of January, 1909, as the inaugural date. The process of finding the value is just the reverse of that for finding the basis, but it is done with greater nicety, to the last decimal. The first step is to compute the time, that is, the number of years, months and days from January 1st, 1909, to the day of maturity, and this time is also to be entered on the list of data. All the computations should, if possible, be made on blank books rather than on loose pieces of paper. Paper ruled like the inside cover of this book is recommended, to be made up into manila-covered books of convenient size, say about 7 x 10 inches, paged consecutively from book to book, for reference. Each inaugural value will be computed on this book, either followed by the successiv investment values down to maturity or by space enough to contain them if that work is postponed. 1. For J. AND J. Bonds. If the income-basis is one of those given in the body of the tables, it is only necessary to turn the appropriate column and multiply the value there found by the amount at par, to as many figures as required. 2. For Bonds at other Dates. Assuming that advantage is not taken of the authority on page 8 of the Banking Department Circular, the procedure is as follows : Copy the values from the column earlier and the column later than the time given, and find the difference. This difference is the amortization (or "run-off") for C months; for one month it must be ]/(, of this difference and for other numbers of months, various fractions as usually computed. For F. and A. bonds Ye of the difference ; For M. and S. >^ ; For A. and O. Yz ; For M. and N. 2^, and For J. and D. 5/6 of the difference are to be added to the shorter time value, for bonds above par, and subtracted from the shorter time value for bonds below par. If among our 5% bonds we have six different lots, $10,000 each, all on a 3.80 basis, and maturing as follows : (a) January 1st, 1919, J. and J., 10 years. (b) February 1st, 1919, F. and A., 10 years, 1 month. (c) March Ist, 1919, M. and S., 10 years, 2 months. (d) April 1st, 1919, A. and O., 10 years, 3 months. (e) May 1st, 1919, M. and N., 10 years, 4 months. (f) June 1st, 1919, J. and D., 10 years, 5 months : 8 Amortization Turning to page 82, we find that the value of lot (a) is $10,990.62. The others must be a little larger, lying between 10,990 . 6200 and 10;^ years 11.031.0304 Difference for six months 40 .4104 Of this difference we take the following fractions: Ye is 6.7351 yi is 13.4701 yi is 20.2052 2^ is 26 . 9403 5/6 is 33.6753 Values rounded off to the nearest cent : of the February bond 10,990 6200 + 6 . 7351 = 10,997 . 36 March bond 10,990.0200 + 13.4701 = 11,004.09 April bond 10,990.6200 + 20.2052 = 11,010.83 May bond 10,990.6200 + 26.9403 = 11,017.56 June bond 10,990.6200 + 33.6753 = 11,024.30 Bonds having interest payable on other days of the month are similarly prorated by months and days, each interest-day being treated as the 180th part of a half year. It may be noted that in May and June it would have been easier to find Yi and Ye than % ^^^ ^/^I ^^^ ^^^ same result would have been ob- tained by subtracting Y or Ye from the greater value as in adding % or 5/6 to the less value. 11,031.0304 — 13.4701 = 11,017.56 11,031.0304 — 6.7351 = 11,024.29 3. Seriai. Bonds. The inaugural value of a series of bonds is, of course, the sum of all the values at the proper times and basis. As already shown, the readiest method is to transcribe these values on the adding-machine. The case of a J. and J. bond is easiest. Suppose on the inaugural date we have a series of 5% bonds, of |10,000 each, maturing July 1st, 1915 to 1919, incl., and that the basis is 3.60. The values are as follows for 11,000,000 : Maturing July 1st, 1915 6>^ yrs. 1 080 495 95 1916 7>4 yrs. 1 091 305 38 1917 8>^ yrs. 1 101 735 92 1918 9K yrs. 1 111 800 87 1919 10>^ yrs. 1 121 513 03 5 506 851 15 The value of the five bonds will, therefore, be $55,068 .51. This is the easiest way of finding the inaugural value ; but with a view to the future values, it is recommended to commence at yi year and continue the addition to 10>^ years, the longest time, with a sub-total after each value. Amortization 9 K yr 1 006 876 23 lYz yrs 1 020 266 08 2 027 142 31 2K yrs 1 033 186 61 3 060 328 92 3K yrs 1 045 654 27 4 105 983 19 4^2 yrs 1 057 684 92 5 163 668 11 hYz yrs 1 069 293 89 6 232 962 00 6>^ yrs 1 080 495 95 7 313 457 95 lyi yrs 1 091 305 38 8 404 763 33 ^Yz yrs 1 101 735 92 9 506 499 25 9>^ yrs 1 111 800 87 10 618 300 12 10% yrs. ..... . 1 121 513 03 11 739 813 15 By subtracting from this total all the values preceding that for 6>^ years we have the inaugural value as before: 11 739 813 15 6 232 962 00 5 506 851 15 If the bonds are not J. and J., but some other date, it will be neces- sary to take off the values for 1 year, 2 years, 3 years, etc. This will cause no loss of time, as those values would anyhow have to be taken off for future amortization : 1 yr 1 013 630 87 2 yrs 1 026 783 97 2 0^0 414 84 3 yrs ] 039 476 04 3 079 890 88 4 yrs 1 051 723 25 4 131 614 13 5 yrs 1 063 541 18 5 195 155 31 6 yrs 1 074 944 88 6 270 100 19 7 yrs 1 085 948 87 7 356 049 06 8 yrs 1 096 567 17 8 452 616 23 9 yrs 1 106 813 28 9 559 429 51 10 yrs 1 116 700 26 10 676 129 77 5 195 155 31 6 480 974 46 10 Amortization The difference between 5 506 851 15 and 5 480 974 46 which is 25 876 69 measures 6 months added to the life of the bonds. If they matured in Feb. instead of July, ye of 25 876 69 being. . . 4 312 78 must be added to 5 480 974 46 giving the value 5 485 287 24 and similarly for March 5 489 6U0 02 " April 5 493 912 80 ♦• May 5 498 225 59 ♦• June 5 502 538 37 Combining Severai, Purchases. When a certain security has been invested in at various times, vari- ous quantities, various prices, and various income bases, the basis of each lot must be separately ascertained, as has been explained. But it is not necessary to carry as many accounts as there are lots ; a combined value may be ascertained and an equated income basis, upon which the amorti- zation will proceed, giving a joint result substantially the same as the sum of the separate values, and with less labor. We have |22,000 5% bonds of a certain issue, due January 1st, 1920, purchased as follows : $10,000 Jan. 1, 1900, for $12,992.00, a 3% basis. 5,000 Jan. 1, 1903, for 5,955.00, a 3>^% basis. 7,000 Jan. 1, 1905, for 7,340.00, a 4.55% basis. $22,000 cost $26,287.00 Each parcel is to be reduced to its value, at 3%, 3>^% and 4.55% respectivly, on the inaugural date, January 1st, 1908. $10,000 on a 3% basis = $12,003.04 5.000 on a 3J^% basis = 5,729.77 7,000 on a 4.55% basis ^ 7,288.82 Investment value Jan. 1, 1908 $25,021.63 At the nearest basis 3.58%, the bonds are valued at $25,025.64, which we adopt as the inaugural value, and this will amortize the bonds to par at maturity. The basis is actually 3.5818, or $25,021.63. It is, therefore, unnecessary to treat each lot of bonds separately in the scientific method of amortization. The remarks of the Superintendent on page 10 of his circular seem to refer to the pro rata method. Continuing the Amortization. Having establisht the inaugural value, we must find successiv in- vestment-values all the way down to par, and the differences between these values are the amortization or the accretion as the case may be. The first question which arises is this: shall we (1) calculate these on each bond all the way to maturity, making a complete schedule? or (2) shall we only continue each class of bonds for one period and per- form this work every half year ? Amortization 11 If there is enough time, I would recommend plan (1), filling up a complete schedule to maturity. If the premium or discount has then exactly disappeared it is the strongest evidence that all the successiv values are correct. It is evidently easier to keep at work on one process than to shift from one to another, and, therefore, the plan of finishing one before taking up another is labor saving. If, however, the time allowed for transforming the accounts is very short, it may be necessary to compute only one, or a very few, of the values at present, deferring the completion of the schedule. In this case, space should be left in the calculation-books for continuing the computa- tions to maturity. There are two general methods for amortization (or accretion): 1. By transcription ; 2. By multiplication. The former consists in deriving each value independently from the tables; the latter in deriving each value from its predecessor, which is multiplied by the income- rate, the cash received being then subtracted. We will exemplify both of these methods and show when they are applicable, respectively. 1. On J. AND J. Bonds. Here the values for $1., 110., $100., $1,000., $10,000., $100,000. or $1,000,000. are at tabular values; that is, they are found in the columns of the book of tables and require only to be multiplied by such number as will correspond to the principal in question. Thus, we have determined that 3.70 is the true income basis for $30,000. 5% bonds due January 1st, 1914, the inaugural value being $31,765 .43 Write down the values for five years, four years, etc., of $10,000, pointing off from the table, but retaining the mills; opposit each value of $10,000, multiply it by 3, to cents; the carrying figure from the mills being added in : 5yrs. 10,588.47,7 X 3 31,705.43 AYi yrs. 10,534.36,4 81,603.09 4yrs. 10,479.24,9 31,437.75 3>^ yrs. 10,423.11,6 31,269.36 3 yrs. 10,365.94,3 31,097.83 2>^yrs. 10,307.71,3 30,923.14 2 yrs. 10,248.40,6 30,745.22 l>^yrs. 10,188.00,1 30,564.00 1 yr. 10,126.48,0 30,379.44 >^yr. 10,063.81,9 30,191.46 The figures on the right are reliable as the successiv investment- values; the only possible error would be from having incorrectly set down the first column or from having made an error in multiplying by 3. Both of these may be effectually checkt by the use of an adding- and-listing machine. Let a list of the right-hand column be made, with total; also a list of values of $1,000,000. on which they are based, with total, 12 Amortization which is 10,332,556.86 Multiplying by 3 30,997,670.58 which indicates that the total of the other column should be 309,976 . 71 In fact it is 309,976.72 and the trifling discrepancy may be ignored. Small differences in the cents are usually caused thru the suppression of decimals. This, in such a case, is of course the easiest way of obtaining the values required. In making up the schedule the values occupy the right- hand column; the column preceding it is derived by subtraction; the one before that again by subtraction from the interest column, which is constant. ScHKDuivE OF Amortization. Date Interest Net Income Amortization Investment Value 1909 Jan. 1 July 1 1910 Jan. 1 July 1 1911 Jan. 1 July 1 1912 Jan. 1 July 1 1913 Jan. 1 July 1 1914 Jan. 1 750 66 750 00 750 00 750 00 750 00 750 00 750 00 750 00 750 00 750 00 587*66 584 66 581 61 578 47 575 31 572 08 568 78 565 44 562 02 558 54 162 '34 165 34 168 39 171 53 174 69 177 92 181 22 184 56 187 98 191 46 31.765 43 31,603 09 31,437 75 31,269 36 31,097 83 30,923 14 30,745 22 30,564 00 30,379 44 30,191 46 30,000 00 7,500 00 5,734 57 1,765 43 The totals furnish several valuable checks on the accuracy of the schedule. The total amortization must equal the inaugural premium, or the total accretion must equal the inaugural discount : 31,765.43 — 30,000 = 1765.43. The total amortization equals the total interest less the total net income : 7500 — 5734.57 = 1765.43. The several amortizations bear the following relation to one an- other: each is the product of the preceding by 1 .0185; or each equals the 70%. preceding -{- six months interest at 3. 162.34 X 1.0185 1.62 1.30 .08 165.34 1.65 1.32 .08 168.39 etc. Amortization 13 The amortization of the above premiums by multiplication would be as follows : Inaugural value (retaining the mills) 31,765.43,1 X 1.0185: viz: 1 31,765.43,1 .01 317.65,4 .008 254.12,3 .0005 15.88.3 32,353.09,1 — 750 750.00 31,603.09,1 316.03,1 252.82,4 15.80,2 32.187.74,8 750.00 31,437.74,8 and so on to maturity. If the mills figure had not been retained there would have been some inaccuracy in the last figure. 2. Intkrest-days other than January 1st and Jui.y 1st. The successiv values for intermediate dates are computed in the same way as the inaugural value by transcription of the tabular values and interpolation by simple proportion. The following procedure will facilitate and check these operations. We will illustrate it by the follow- ing problems : 100,000 4%'s on a 3.50 basis, and 100,000 4%'s on a 4.50 basis, payable February 1st, 1912; 3 yrs., 1 mo. Set down the values for even years and half years, beginning with S)4 years, leaving spaces between the lines and on each side. It is as well to set down the extended figures for the full |1, 000,000. First for the 3.50 basis : 1 016 336 60 1 014 122 49 1 Oil 869 64 1 009 577 36 1 007 244 96 1 004 871 75 1 002 457 00 1 000 000 00 14 Amortization These amounts, exclusiv of the first, are totaled on the machine for the purpose of proving : ^ q^q ^^^ go But the millions may be neglected, leaving 50 143 20 Next subtract each value from the next above and place the differ- ence on the left of the lower; total these differences 1 016 336 60 2 214 11 1 014 122 49 2 252 85 1 Oil 869 64 2 292 28 1 009 577 36 2 332 40 1 007 244 96 2 373 21 1 004 871 75 2 414 75 1 002 457 00 2 457 00 1 000 000 00 16 336 60 50 143 20 The column of difference aggregates the same as the previous pre- mium, which indicates that no error has been made. As this is an F. A. Bond we take ye of each diflference and place it above the adjoining value and add it: 1 016 336 60 + 369 02 2 214 11 1 014 122 49 1014 491 51 375 47 2 252 85 1 Oil 669 64 1 012 245 11 382 05 2 292 28 1 009 577 36 1 009 959 41 388 73 2 332 40 1 007 244 96 1 007 633 69 395 54 2 373 21 1 004 871 75 1 005 267 29 402 46 « 2 414 75 1 002 457 00 1 002 859 46 409 50 2 457 00 1 000 000 00 1 000 409 50 6) 16 336 60 52 865 97 52 865 97 Errors of multiplication or subtraction are detected by adding Ye of the first column to the total of the second, which should produce the third. Amortization 15 The bond on a 4 . 50 basis could give the following results, the yi being subtracted downward, not added: 2 139 43 2 187 56 2 236 78 2 287 11 2 338 57 2 391 18 2 444 99 6) 16 025 62 983 974 38 — 356 57 986 113 81 364 59 988 301 37 372 80 990 538 15 381 18 992 825 26 389 76 995 J 63 83 398 53 997 555 01 407 50 1 000 000 00 6 950 497 43 2 670 94 6 947 826 49 985 757 24 987 936 78 . 990 165 35 992 444 07 994 774 07 997 156 48 999 592 50 6 947 826 49 The effects of adjustment for intermediate months may be sum- marized as follows: ^- -^^ ^ 1 Above par add to 1 i r t, . ♦• j^ 3 ^ I Below par subtract from j^^^"^^^''^^^^^^^ ^^°^«- A. O. % Midway between, add or subtract. M. N. >^ I Above par subtract from 1 i r i J ^ y^ \ Below par add to j ^^^^^ ^^^^ l°°g^^ ^^°^«- A schedule formed from the last example would read : SCHEDUI.E OF Accretion. Date Interest Net Income Accretion Interest Value 1909 Jan. 1 98 575 72 July 1 2 666 6o 2 217 96 217 96 98 793 68 1910 Jan. 1 2 000 00 2 222 85 222 85 99 016 53 July 1 2 000 00 2 227 88 227 88 99 244 41 1911 Jan. 1 2 000 00 2 233 00 233 00 99 477 41 July 1 2 000 00 2 238 24 238 24 99 715 65 1912 Jan. 1 2 000 00 2 243 60 243 60 99 959 25 Feb. 1 333 33 374 08 40 75 1 00 000 00 12 333 33 13 757 61 1 424 28 The foregoing problems might have been workt by multiplication, the only difficulty being at the fractional period at the end, one month. In exemplifying this we will make the further simplification of indi- 16 Amortization eating subtraction by a line drawn around the figures of cash interest, and performing the subtraction and the addition at the same operation. At the rate 3 .50 it is easier to take 1% 01 divide it by 2.... 005 and divide that by 2 .0025 .0175 than to multiply .01 by 7 007 and by 5 0005 $100,000 4% bonds 3.50% basis, payable February 1st, 1909. F. & A. Inaugural value January 1st, 1009 101 449 15 6 mouths' interest at .0175 .01 1014 49 .005 507 25 .0025 /^'^^ ^^ Subtract 6 months' cash interest \2j Value July 1, 1909 101 224 51 Coutmue the operation 1 012 25 506 12 253 06 2' © January 1, 1910 100 995 94 1 009 96 504 98 ^252 49 2' & July 1, 1910 100 763 37 1 007 63 503 82 251 91 2' © January 1, 1911 100 526 73 1 005 27 502 63 251 32 2' © July 1, 1911 100 285 95 1 002 86 501 43 ^250 71 2' © January 1, 1912 100 040 95 We have brought the value down to the last full period. For the broken period it is evident that $40.95 is the amortization, and this is corroborated by the fact that it is exactly ^ of the last 6 months' pre- mium in the 4% table, 3 . 50 basis, $245 . 70. It might be obtained by mul- tiplication, with this peculiarity that ^ of the par plus the entire pre- mium is multiplied by .0175 and yi of the cash interest is subtracted. Amortization 17 16 666 67 40 95 16 707 62 100 040 95 167 08 83 54 41 77 100 333 34 333 34 100 000 00 But this process is unnecessary. If the last January or July pre- mium isy^, y^, Yz, ^ or 5/6 of the last tabular premium, the work may be considered accurate. The amortization of the value 98,575 72 at 4.50 basis would be as follows : we retain the mills. 98 575 72,4 .02 1 971 51,4 .002 197 15,1 .0005 49 28.8 i of .002) © 98 793 67,7 1 975 87,4 197 58,7 49 39.7 99 016 53,5 1 980 33,1 198 03,3 49 50,8 © 99 244 40,7 1 984 88,8 198 48,9 ^^ 49 62,2 © 99 477 40.6 1 989 54,8 198 95.5 ^^ 49 73,9 © 99 715 64.8 1 994 31,3 199 43.1 ^^ 49 85,8 © 99 959 25,0 100,000 — 99,959,25 = 40.75 100,000 — 99,755.50 = 244.50 244.50-^6 = 40.75 18 Amortization 3. Seriai, Bonds. If the serial bonds are of the J. & J. maturity they may be re- computed at each half-year by the method outlined under ''Inaugural Values." Two complete lists are made up on the adding machine, one of the even years, the other of the intervening half years, beginnmg at Yz year and ending with the longest term. Subtotals are inserted after every value. If the series has commenced to mature, the total is given at sight; if some of the shorter maturities have not yet arrived, the total of those earlier bonds must be subtracted from the total which embraces all now outstanding. In short, each successiv value is computed exactly as the inaugural vahie was. But, except for J. & J. bonds, it will usually be found easier to multiply down, and even with J. & J. it may be done. We take as an example a series of three bonds maturing March 1st, 1911 to 1913; 6% bonds, 3.80 basis. We take the full extent of the figures in the tables. The inaugural value January 1st, 1909, is 3 194 386 49 Until the bonds begin to mature we proceed as usual, mul- tiplying by .019 .01 31 943 86 .009 28 749 48 3 255 079 83 6 months' interest earned on $3,000,000 (9) July 1st, 1909 3 165 079 83 31 650 80 28 485 72 © Jan. 1, 1910 3 135 216 35 31 352 16 28 216 95 © July 1, 1910 .3 104 785 46 31 047 85 27 943 07 © Jan. 1, 1911 3 073 776 38 Here a complication arises: $1,000,000 of the principal is only invested for 2 months, 3^ of a period. We therefore compute it for interest as 333 333 33 and the remainder at full value 2 073 776 38 .019 on 2 407 109 71 24 071 10 21 663 99 3 119 511 47 We must subtract principal 1 000 000 Interest earned thereon 10 000 " on 2.000.000 60 1070 000 00 Julyl, 1911 2 049 511 47 During the half year July— Dec. we proceed as usual 20 495 11 18 445 60 © Jan. 1, 1912 2 028 452 18 AmormzaTion 19 Brought forward 2 028 452 18 This being the half-year in which a payment is made we proceed as follows : 333 333 33 1 028 452 18 Interest on 1 361 785 51 13 617 86 12 256 07 1 000 000 2 054 326 11 10 000 1 040 000 00 30 000 Julyl, 1912 1 014 326 11 10 143 26 9 128 93 ® 3^ 1 003 598 30 Instead of working this down, we verify its accuracy by observing that the premium is exactly y^ of §10,794.90, the tabular premium for 6 months. New Purchases. Future purchases will doubtless be made for the most part on some agreed basis. If not, their basis will be found precisely as in earlier purchases. The new purchase, if not on any exact basis, requires to be squared up by the next following balancing-period, so that the amortization may thereafter proceed with regularity. On long bonds, this squaring-up process may sometimes involve a considerable adjustment, enough to make a perceptible jar; while the same adjustment spread over many years, as in inaugurating the system, would be imperceptible. It is, therefore, in the new purchases that the most exactness is re- quired and it may be thought desirable to compute them to a higher degree of nicety, in respect to using a basis correct to the nearest hundredth of 1% ( .0001 ) and in other particulars. Intermediate Rates. The tables give values 5/100 of 1% apart, as 2.50, 2.55, 2.60, 2.65. The intermediate values, at 2.51, 2.52, 2.53, 2.54, — 2.56, 2.57, 2.58, 2.59, may be obtained approximately by taking the values at two tabular rates and dividing their difference by 5. Thus a 6% bond for 25 years at the basis 4.60 is worth 1 206 716 65 and at 4 . 65 1 198 321 45 The interval is 8 395 20 one-fifth is 1 679 04 and if this is subtracted successivly from 1 206 716 65 we have the values for 4.61 1 205 037 61 4.62 1 203 358 57 4.03 1 201 679 53 4.64 1 200 000 49 and again 4 .65 1 198 321 45 20 Amortization For a small lot of bonds, these values would be accurate to the nearest cent, but on $20,000 par the error would be perceptible. All these values are too large; the corrections to be subtracted are found in the blue pages; in this case, p. 178. According to that page the "difference" for a G% bond 25 years, basis between 4.60 and 4. 05, isG.G4andthe next below it is 6.56. The difference between these two differences is called the "sub-difference." Our corrections, according to the rules, on pages 122 — 123 are as follows : For 4.61, the difference 6.64 For 4.62, 1^ times the difference less 1/10 the sub-difference 9.95 For 4.63, 1]4. times the difference less 1/5 the sub-difference 9.94 For 4.64, the difference less 1/5 the sub- difference 6.62 Subtracting these from the approximate values we have the exact values : 1 205 037 61 — 6.64 = 1 205 030 97 (4.61) 1203 358 57 — 9.95 -= 1203 348 62 (4.62) 1 201 679 53 — 9.94 = 1 201 669 59 (4.63) 1200 000 49 — 6.62 = 1199 993 87 (4.64) If the approximate values were used in multiplying down there would always be an excess at maturity and this would be an increasing quantity, thus vitiating the test afforded by reaching par at maturity. But it will be found that the corrected values will multiply down with accuracy. The values at intermediate rates should be workt out first, before interpolating for broken times. In serial bonds at intermediate rates the adjustment may be made in a lump on the aggregate. Thus, if a series of 6% bonds, three in num- ber, due in 18, 19 and 20 years, respectively, at a 4% basis, is worth 3 792 849 62 and at a 4.05 basis 3 760 788 11 a 4.03 basis would be approximately 3 779 012 71 To correct this take off the differences and sub-differences opposit 4 .00 onpagel77 18 yrs 4.56 .04 19yrs 4.97 .05 20 yrs 5.39 .05 14.92 li lYz times 14.92, less 1/5 of .14 22.35 3 778 990 36 Amortization 21 QuARTERiyY Bonds. A quarterly bond of a certain bond-rate is worth more at a certain basis than a semi-annual bond at the same basis. If it is decided to ignore the advantage of collecting half the interest in advance, as permitted by the Superintendent, no very great error can arise and the bond will work out like any semi-annual one. If, however, it is desired to make the values perfectly exact, a set of multipliers will be found on page VII (p. 184 first edition), by which the premiums or discounts are to be multiplied, increasing the value of the bond in either case. It will be the simplest way to calculate each successiv premium by multiplication, and for this purpose a little table may be constructed of the multiple in question, from 1 to 9 (Problems and Studies p. 20, 21). For example: for a 3.90 basis on a 5%, the table will be as follows, all figures below the second line being formed by ad- dition of the top line : 1 022 052 1 2 044 104 2 3 066 156 .S 4 088 208 4 5 110 260 5 C 132 312 6 7 154 361 7 8 176 416 8 9 198 468 9 220 521 Bonds at a lower income-basis than 2.50 are entered in the table (Additional Volume, included in second edition) on a basis of quarterly collection. These are almost entirely U. S. bonds, which are all quarterly, Annuai, Bonds. These do not occur so frequently as quarterly bonds, but multi- pliers for reducing the premiums or discounts to the semi-annual stand- ard are given in the second edition. Having establisht the annual values, the amortization at the half- year may be considered as one-half of the yearly, unless great accuracy is required. Residues. Where great accuracy is intended, it often happens that, having establisht an income-basis as close as possible, it is found that there is still a residue of difference which should affect the rate of income, but it is undesirable to extend it beyond the second decimal of 1%. There are three ways of disposing of such residues : 1. By taking it all at once out of the first period's amortization, or adding it, as the case may be. 2J Amortization 2. To distribute it over the list of values by dividing into equal partSy to be added to or subtracted from each value. 3. To distribute it still more accurately over the list in proportion to the premiums or discounts. Thus, if a certain purchase of $1,000,000 5% 5 year bonds is made at 104.50, and we find that the basis of 4% is 104.49129., which is nearer than the 3 . 99% basis. This leaves a residue of |8 . 71 to be disposed of. 1. Put it all into the first amortization, making it $418.88, instead of 1410 . 1 7. All the subsequent values are regular. 2. Divide $8.71 into 9 parts of 87 cents and one of 88 cents, which are added to the amortization. 3. Divide $8.71 into -^siris proportionate to the several amortiza- tions. These will vary from 80 cents for the first, to 95 for the last. This will make a still closer approximation. . AccRUKD Interest. That portion of the half year's interest which has been earned and accrued {''■grown on") is exactly as much part of the det secured by the obligation as the principal itself. It is a frequent practis to ignore the earnings until the date of collection ; this, while convenient, is not strictly correct. It is not the cash received, but the earning-power which makes the asset ; the collection of the cash is an incident . If all the items of interest were payable, and were punctually paid, on the last days of December and June, the result would be the same ; but when, as usually happens, there is a single balancing period for bonds of all sorts of dates, a correct balance-sheet cannot be given with- out adjusting everything— interest and amortization down to that date. It has been attempted to facilitate the calculations by amortizing to the next previous coupon-date only ; but this would be no saving of labor, for the accrued interest for the unexpired term would have to be computed at the effectiv rate, resulting in practical amortization. AMORTIZATION ' 23 The Proper Basis of Bond AccoirNXs When Held for Investment. By Charles E. SpraguE, Ph.D. From The Annals of the American Academy of Political and Social Science, September, 1907, entitled: " Bonds as Investment Securities." A bond is a complex promise to pay; 1. A certain sum of money at a future time ; this is known as the principal, or par. 2. Certain smaller sums, proportionate to the principal, and at vari- ous earlier times. These are usually known as the interest, but as they do not necessarily correspond to the true rate of interest, it will be better to speak of them as the coupons. The sale of a bond is the transfer of the right to receive these vari- ous sums at the stipulated times. They are never worth their face, or par, until these times arrive, but are always at discount. The principal is never worth its face until its maturity ; the coupons are never worth their face until their maturities. Yet while both principal and coupons are always at a discount, the aggregate may easily be worth more than the principal alone ; and it is the aggregate, principal and coupons, which is the subject of the bargain. The purchaser, in fixing the price which he is willing to pay, is guided by several considerations : 1. The amount of the principal. 2. The amount of each coupon. 3. The length of time to which the principal is deferred. 4. The number of coupons. 5. The times of their payments. 6. The rate of interest which can be earned upon securities of a similar grade. He discounts the principal and each coupon at compound interest, at such rate and for the times which they respectivly have to run, and the sum of these partial present-worths is the value of the bond. If he can buy at a price below this value he will receive a higher rate of interest than he anticipated ; if he is required to pay more than his price, he refuses to buy. As he cashes each coupon, he receives what he paid for it plus interest at the uniform rate ; thenceforward he earns interest on a diminisht investment so far as coupons are concerned, but on an increast investment as to principal. If each coupon is less than the total earning during its period there is an increase in the total investment ; if it is greater, there is a surplus which operates to reduce the investment or to amortize the premium. 24 Amortization We have then two fixt points in the history of the bond : the origi- nal cost or money invested, and the principal sum or par, or money to be received at maturity. Between these two points there is a gradual change ; if bought below par, the bond must rise to par ; if above, it must sink to par ; these changes being the effect of interest earned and coupons paid. At any intermediate moment there is an investment-value which can be calculated, and which is just as true as the original cost and the par. In fact these latter are merely cases of investment- value ; the in vestment- value at the date of purchase is cost ; the investment- value at the date of maturity is par. The gradual change in the investment is ignored by some invest- ors, who either use the original cost all thru, or the par. In the former case they suppose that the investment value remains at its original figure until the very day of maturity and is instantly reduced to par, by a loss of all the premium or a sudden gain of the discount. Those who use par as the investment-value assume also that there is this sudden change of value, but that it took place at the instant of purchase. These treatments are manifestly fictitious and unreal and only resorted to because the labor of computing intermediate values is shunned. Experience would tell us, if theory did not, that there is no such violent change. In any complete system of bookkeeping (popularly called double entry) the accounts representing assets and those representing profits and losses are mutually dependent. You cannot arbitrarily change a value without affecting and distorting the general profit-and-loss account. A year's actual gain might be swept away, on paper, by the investment, perhaps a very advantageous one, in a security at a premium. The disappearance of premium being regarded as a consumption of capital, instead of a return requiring reinvestment, the entire coupon is looked upon as income and the impairment of capital becomes actual. In case of a sale, the true profit or loss is unknown, the proceeds being compared either with a value which has passed into history or one which is yet to be realized — not with a value which is adjusted to the present. The error in these faulty methods of accountancy arises from the assump- tion that interest is only earned when specifically collected in cash — that the coupon is exactly the measure of the interest earned. When the bond is at par this is true : the coupon and the interest are co-extensiv. But if there is any premium or discount we must disre- gard the distinction between principal and interest and consider that the original investment goes on increasing at compound interest, period by period, but diminisht by the coupons and the final redemption. In other words, we must think of the coupons and the principal as merely instal- ments, the periodic instalments and the final one, but all of the same nature. A familiar instance of interest earnings not represented by specific cash payment but by accretion is the discount of a note. If a three- months' note for $1000 is discounted at six per cent, the investment is 1985, which by accretion becomes |1000. Altho interest is not mentioned, Amortization 26 the purchaser earns $15, or more than 6 per cent, on his investment of $985. If the note were payable three years from now instead of three months, he would expect to earn compound interest and would pay, per- haps, |837.48. His earnings on this investment, compounded semi- annually, would bring the investment up to ^1000 in three years. This note would be equivalent to a bond without coupons ; no interest is stip- ulated for, but interest is actually earned. If coupons were added, the bond would simply be worth so much more, according to their value. I therefore regard the cost and the par value, while correct at the beginning and at the end of the period of ownership, as entirely incorrect during the interim. The true standard is the present worth, compound- discounted, of all recipiendSy or sums of cash to be received, whether called coupons or principal. These three values resemble three tenses in grammar : the cost is the past, what was paid ; the par is future, what will be received ; the investment value is the present. There is a fourth value, which may be considered as in the potential mood ; what might be obtained on sale, at the present time. This is the market value and is a matter of judgment, opinion and inference. Some bonds are bought and sold so frequently that there is a current quotation which is fairly reliable ; other issues, in which dealings are rarer, are valued by analogy with those whose con- ditions are nearest like them. It may be observed here that the market value depends solely on the rate of interest which prevails on the particular grade of security. This has sometimes been doubted ; even the courts have sometimes assumed that there could be a depreciation, regardless of interest-rate, for the "badness" of the investment; or a premium paid, regardless of interest-rate, for the "excellence" of the security. It is necessary to analyze this view, which I regard as essentially unsound. No one buys a bond by reason of admiration, as he would buy a painting or a statue. He is dealing solely in earnings, that is to say, in interest. He is impelled by no other motiv than that of receiving his money back with the increment which shall accru in the meantime. If an investment is offered him which is superlativly "good," but which returns only what he originally invested without any increment, he will certainly refuse it. The rate of interest, however, is affected by the risk of loss. Every rate of interest may be regarded as composed of two parts : one, a com- pensation for the use of the capital ; the other, a premium of insurance against chances of loss. Thus an interest-rate of 5 per cent, per annum may be conceived as 3% riskless interest or compensation for use of capital ; + 2% premium for insurance against loss. Another and safer investment, where the chance of loss is twice as remote would rate at 4 per cent. : 3% riskless interest ; , -f- 1% premium of insurance. • <26 Amortization The chance of loss may be very remote, it may be imaginary, but it is worth insuring against. Similarly, the chance of any one house being destroyed by fire is remote, but men willingly pay a part of the income of the house to secure themselves against it. The loss feared in the case of investment is not merely direct loss, or failure to return, but losses by delay, by difficulty of collection, by expense of litigation, by the very feeling of suspense which acts as a penalty. An opinion that the loss is possible is exactly as potent as the reality, in producing a loading of the rate on account of risk-insurance, provided this opinion is suffi- ciently widespread. A lowering of the grade of security means an increase of the insurance-premium and hence of the rate of interest. This may happen by deterioration of the physical property which underlies the investment, by bad management, by accidental loss of custom, and in various ways, preventable or non-preventable. The other element in the interest-rate, the value of the use of cap- ital, also fluctuates, as in times of capital famin or capital glut, in new countries as against old countries, and it is difficult to decide how much of the rate is due to this source and how much to insurance. But taking the rate as a whole, the question is, does the price of a bond ever vary except thru the interest-rate ? We may test this by experiment. Taking some railroad company which has fallen into misfortune and whose S}4 per cent, bonds, once at a premium, are now below par, so that their present market price is equivalent to discounting all the recipiends at 4.50, If this depreciation is not entirely a consequence of this high rate of discount, if there is an intrinsic depreciation, it must apply to all obligations of the road. But if the same road now puts out bonds bearing 5 per cent, interest under the same mortgage it will invariably be found that these will sell at a premium, on approximately the same (4>^) basis. We may deduce the following conclusions : 1. There is no sanctity in par ; it is merely a convenient round sum to be received in the future. 2. There is no necessary identity between the size of the coupons, or periodical instalments, and the rate of interest. 3. All the recipiends (coupons and par) must be sold below par ; their aggregate may amount to more or less than par or to exactly par. 4. The rate of interest is affected by the degree of belief in the certainly and punctuality of the payments ; and this rate determins the price. It may further be stated that no investment is so insecure that, theoretically, it will not be discounted at some rate. A $1000 bond secured by something which must be annihilated at the end of five years, but bearing 30 per cent, semi-annual coupons, would doubtless find pur- chasers at better than $400, and would be an advantageous purchase. Amortization 27 The market value is of absolutely no importance to an investor who does not contemplate changing the investment, but will hold it to maturity. The ups and downs of the market do not in the least affect the value to him ; if he were to record these fluctuations it would be merely to substitute an undulating zig-zag for the natural and logical curve of the investment value, since in either case the point of final rest is par at maturity. Such a case is that of a trustee who, under the decisions of the courts of New York, is bound to keep his trust intact, carrying the investments at their investment value and re-investing all in excess. But to the investor who has the privilege of selling and replacing his investments, acquaintance with market values is highly advanta- geous. It is his guide to the advisability of making such changes and of forecasting the future. It is his duty, therefore, to watch the fluctuations of the market and, in a perfect legitimate way, seek to improve his in- come, without impairing the factor of safety. A large investor will not endeavor to have all his investments at the same grade ; he will probably have at the same time some capital out at high rates and some at lower. The money at high rates is not quite so secure, not quite so available, and requires more effort for its collection. That at lower rates is nearer to absolute freedom from risk and from labor ; it almost automatically collects its own income. On some of the high-interest investments the security may have improved in the course of time ; the credit of the municipality or the revenue of the corporation may have so risen that the 4 per cent, bond, which was bought at par, is now selling at a pre- mium which, if a further investment were made, would yield only S}4 per cent. If the bond has still ten years to run, he may sell at a profit of 4 per cent, and thus have |104 to re-invest in some other 4 per cent, security. Altho the market price is of great utility, I do not admit that it can be introduced into the accountancy of investment. It is not an act nor a fact of the business ; it is a statement of what might be done. When the bond just mentioned has gone up to 104, the owner has not gained a penny. He merely has an opportunity presented ; if he lets it pass, the opportunity has not had the slightest effect on his financial status. Unless the accounts are kept on the investment-value basis, he cannot even tell whether a certain price would result in a loss or a gain. If his books are kept on the basis of par, every sale above par will appear as a gain, tho it may be a losing bargain ; while a comparison with origi- nal coat will be equally delusiv. Where liquidation, entire or partial, is a possible contingency, as in a savings bank or an insurance company, market values are an appro- priate basis for an estimate of solvency. It must be remembered, how- ever, that solvency for going on and solvency for winding up are different matters and that in a going concern, going insolvency is primarily to be considered. 28 Amortization My conclusions as to the proper basis of accountancy for an investor are as follows : 1. Neither original cost nor ultimate par is a proper permanent basis, but the bond should enter at cost, which is a fact, and should go out at par, which is another fact. 2. During the interim the reduction from cost to par should take place gradually by the processes of amortization and accumulation at the basis-rate of true interests 3. Information should be obtained of the fluctuations in market value, but these should not be carried into the accounts as actualities. 4. A list of market values should accompany the balance-sheet of any concern which may be subject to liquidation for the purpose of showing its ability to liquidate. Address t wn-H E M REMITTAN( '.e ElOOKS ATH Text Book of the Accountancy "three parts i la. The Accountancy <)n a large yaluation of lb. Problems and above, contai fliflicult cases ic. Tables of Compound i>i decimals, Complete ixtended Eight places 2a. [Not isold separa 2b. [Soldj separately] The Philosophy of A(*:ounts Amortizatipn 12-piace Logfarithms ON MAT CHARLES ISi! of 1 one ; sold s of Investme:lscoant, Si^kingr ih two work 1 14 above ST., NEW Price f4.oo the book-ketping of securities annuities, sinking funds anid the Price $2.oo uppleraentai-y volume to the solutions ; elabcjration of spe( ially Price I1.50 etc.: Calculated to 8 places Price $1.00 Price I J 0.00 Price Price Price 50 Price l3. I3-00 00 dents 00 13. -^T^T-VFT'^TT THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. DEC 2 1932 ''' S t932 *'0«' 6 1934 APR 30 1938 MAY 14 19C0 "•^i 28 192 J ]a- fi-b^ '/( DfC ^61940M LD 21-50m-8,-32 PAMPHLET BINDER Syracuse, N. Y. Stockton, Colif.