UC-NRLF lllillll lull Hill lllil'l"' $B 532 =i^3 '•-.....l-'ilJ '!.>,.„ !'■' .il- w^ LIBRARY OF ALLEN KNIGHT CERTIFIED PUBLIC ACCOUNTANT 502 California Street SAN FRANCISCO. CALIFORNIA GIFT OF OsilSliw- N^WvMD^t^ LOGARITHMS TO 12 PLACES AND THEIR USE IN INTEREST CALCULATIONS By CHARLES E. SPRAGUE Author of "Thb Phii^osophy of Accounts" '•Text-Book of the Accountancy of Investment and "Extended Bond Tabi^es" New York, 1910 publisht by the author s^ ^-7 Copyright, 1910, by Charles E. Sprague. ^ Oid^ \t^^sA3r TRUNK BROS. 18 FRANKFORT ST. NEW YORK PREFACE. The need of a logarithmic table for special cases, where the usual five-figure and seven-place results are insufiBcient, is often felt by the accountant and the actuary. Rough results will answer for approximativ purposes ; but where it is desir- able, for instance, to construct a table of amortization, sinking fund or valuation of a lease at an unusual rate, for a large amount and for a great many years, exactness is desirable and becomes self-proving at the end. It is, of course, a slower process than that for a few places, but as the figures from which all results are obtainable are containd in two pages instead of 200, there is, on the other hand, a great saving in the mechanical labor of turning leaves. It also contains a thoro analysis of the entire doctrin of interest, explaining every process by the use of logarithms, as well as arithmetically and algebraically. CHARLES E. SPRAGUE. Union Dime Savings Bank, New York, January, 1910. 380313 TABLE OF CONTENTS. PART I. — Thk Properties of Logarithms. Page The Nature of Logarithms 1 Multiplication by IvOgarithms 3 Division of Logarithms 5 Tables of Logarithms 6 To Find the Number 8 To Form the Logarithm 16 Less than 12 Places 21 Multiplying Up 22 Signs of the Characteristics 27 Different Bases 28 PART II. — Tables for Obtaining Logarithms and Antilogarithms to 12 Places of Decimals. Table of Factors 30 Table of Interest Ratios 32 Table of Sub-Reciprocals 33 Table of Multiples 34 Logarithmic Paper 36 PART III. — The Doctrin of Interest. Definitions 39 The Amount 41 The Present Worth 43 The Compound Interest and Discount 45 Finding Time or Rate 46 The Annuity 47 Amount of Annuity 48 Present Worth of Annuity 50 Amortization 52 Special Forms of Annuity 53 The Unit of Time 55 Frequency of Payment 57 Coefficients of Frequency 58 Fractional Periods 63 Sinking Funds 65 Interest-Bearing Securities 67 Multiplying Down 72 Computing Amortizations 73 Discounting 75 Intermediate Purchases 76 Intermediate Balances > 77 Short Periods 80 Finding the Income Rate ... 83 Interest Formulas 86 PART I The Properties of Logarithms PART I. THE PROPERTIES OF LOGARITHMS. 1. — If we multiply 5 lO's together, 10 x 10 x 10 x 10 x 10, we may write the result as 100000 or 10^ or the fifth power of ten. The little " ^ " is the exponent of the power. We may form a series of the powers of 10 : 00000 or 10^ 10000 10* 1000 10« 100 10« 10 10^ 1 10° 2. — The following observations may then be made : 1. The number of the zeroes in the first colum is the exponent in the second. 2. Each term in the first colum is one-tenth of the one above it, while in the second colum each exponent is one less than the exponent above it. This leads to the result that 10° = 1, which at first seems paradoxical. 3. If we multiply together any two terms in the first colum, we add the exponents in the second. 3. — Logarithms are auxiliary numbers having relation to a base. When the base is once fixt, every possible number has its logarithm. The customary and most convenient base is 10, because our whole system of numeration is based upon ten. The logarithms are simply exponents and we re-write the above series thus : Thk Properties of Logarithms. The base being 10, 100000 is the number whose logarithm is 5 or contracted, 100000. nl 5 10000. nl 4 1000. nl 3 100. nl 2 10. nl 1 1. nl .1 nl — 1 .01 7ll -2 .001 nl -3 .0001 nl -4 .00001 nl -5 The copula {nl) means "is the number whose logarithm is ;" while {In) means "is the logarithm of the number ." 4. — We have here logarithms of a few numbers, but we need the logarithms of a great many others. All possible numbers must lie between some of the logarithms now ascer- taind. The numbers between 1 and 10 must have their logarithms between and 1; that is, the logarithms must be fractions, and these are exprest decimally to as many places as desired, the difficulty in calculation greatly increasing as the number of places is increast. Similarly, as the numbers of two figures lie between 10 and 100, their logarithms must lie between 1 and 2; that is, they must be 1 -f a decimal « fraction. 5. — We will now illustrate the properties of logarithms, confining our attention to the single-figure numbers 2, 3, 4, 5, 6, 7, 8 and 9, which are as follows, rounded at 12 places : .301029 995 664 In 2 .477 121254 720 In 3 .602 059 991328 In 4 .698 970 004 336 In 5 .778 151250 384 In 6 .845 098 040 014 In 7 .903 089 986 992 In 8 .954 242 509 439 In 9 MuivTlPLICATION BY LOGARITHMS. 6. — The third observation in Art. 1 leads to the following rule : The sum of the logarithms of several numbers is the logarithm of their product. 2 111 .301029 995 664 3 7il .477 121254 720 2x3= (o nl .778 151250 384 2 X 5=~ 2 X 10=^ 7.— In the 2 5 10 2 20 logj nl nl nl nl 7ll irith .301029 995 664 .698 970 004 336 1.000 000 000 000 (See Art. 3) .301029 995 664 1.301029 995 664 ms of 20, 200, 2000, 20.000, 200,000 2,000,000, etc., we shall find the same decimal part .301029 995 664, {_ln 2) preceded by the figures 1, 2, 3, 4, 5, 6, etc., indicating the distance from the units place of their left-hand figure, or the number of zeroes interpolated to hold that position. This is also true of any combination of figures ; the decimal part of the logarithm is the same what- ever their place-value, while the whole number prefixt indi- cates the place-value, being the number of places to the left of units. Thus, if the logarithm of 2 . 378 is .376 211850 283, then 1.376 211850 283 2.376 211850 283 3.376 211850 283 4.376 211850 283 5.376 211850 283 etc. 8. — Where the number is less than unity (a decimal frac- tion) the characteristic or index (the prefixt figure) is negativ, altho the decimal (or mantissa) remains positiv. It is usual to put the minus sign over the characteristic: .2378 {7il) 1.376 211850 283 .02378 {7il) 2.376 211 850 283 .002378 («/) 3.376 211850 283 • etc. etc. 23.78 7ll 237.8 7ll 2,378 nl 23,780 nl 237,800 nl etc. 4 The Properties of Logarithms. Here the position of the left-hand figure of the number again determins the characteristic. 1 indicates that the left-hand figure, 2, is in the Jirsi place to the right of the unit place ; 2 indicates that this figure is in the second place, and so on. The following list of characteristics will show that the left- hand figure of the combination, by its location to the right and left of the unit figure, determins the characteristic. Unit Places 000 000 00 0- 000 000 000 Characteristics 9 876 543 21 1 2 3 45 6 78 9 This principle saves a vast amount of time in the compu- tation of logarithms, and also in their application. 9. — Since division is the converse of multiplication, it may be performd by subtraction as that is by addition. The difference of the logarithms of two numbers is the logarithm of their quotient. Required the quotient of 6 -H 2. 6 nl .778 151250 384 2 '' .301029 995 664 6/2 nl .477 121254 720 In 3 Required the value of >^ or 1 -f- 2 1 nl .000 000 000 000 2 '• .301029 995 664 1/2 ;2/ 1.698 970 004 336 In .^ Required the value of Yi 1 nl .000 000 000 000 3 '' .477 121254 720 1/3 w/ 1.522 878 745 280 /;^ .3333 .522 878 745 280 is called the cologarithm of 3 or the logarithm of the reciprocal of 3. 10. — Powers of numbers are found by multiplication. Let it be required to find the third power of 2, which may be written 2^ or 2x2x2. By the process first shown 2 nl .301029 995 664 2 " .301029 995 664 2_ " .301029 995 664 2 X 2 X 2 = 2^ nl .903 089 986 992 In 8 Roots and Powers. Required the square (2d power) of 3 3 nl .477 121254 720 3 " .477 121254 720 3' nl .954 242 509 440 In 9 In each of the above examples it would have been simpler to multiply the logarithm by the exponent. 2^ nl (.301 029 995 664) x 3 = .903 089 986 992 In 8. 3^ nl (All 121 254 720) x 2 = .954 242 509 440 In 9. Therefore, to "raise" a number to a certain power, we multiply its logarithm by the exponent and then find the number corresponding to the product-logarithm. 11. — The second power is usually called the square, and the third power the cube. 12. — If a certain number is a power of another, we call the latter a root of the former. Thus if 2^ = 32, we may say that the 5th root of 32 is 2. The usual way of expressing this is c- / Y 32 = 2, or 32^ = 2. Using the latter form gives a symmetrical list of ex- ponents and their meanings: a" A positiv exponent denotes a power a~** A negativ exponent denotes the reciprocal of a power; ajr A fractional exponent denotes a root, or the root of a power; a* The exponent ^ denotes the number itself; a° The exponent ^ denotes unity. ^ 13. — As roots are powers with fractional exponents, there- fore roots are found (or extracted) by dividing logarithms insted of multiplying. Thus if it be required to find the 6th root of 64, we take (from Colum A of the Table of Factors) the logarithm of 64, and divide it by 6. 64^ nl (1.806 179 973 984 /6) == .301 029 995 664 In 2. Therefore 2 is the 6th root of the number 64. 14. — Such an exponent as f may require explanation. It signifies the third power of the fourth root or the fourth root of the third power. 15. — Fractional exponents may be represented as decimal, insted of vulgar fractions. Thus we may write 2*^^ insted of 2* or 3* for 3 . In fact, that is what most logarithms are: fractional exponents of 10, exprest decimally. The; Properties of Logarithms. Tabi^es of Logarithms. 16. — The decimal fractions j^ch constitute that part of the logarithm requiring' tU^>ilation are interminate ; their values may be computed to any number of decimal places. If all the logarithms in a certain table are carried to 5 deci- mal places, it is called a 5-place table, and so on. Thus the logarithm of 2 has been computed, with great labor, to 20 places and even further. 2 nl .301029 995 663 981195 21 + In a 4-place table this would be rounded off to .3010; in a 7-place, .3010300; inalO-place, .30102 99957; in a 12-place, .301029 995 664. The terminal decimal is never quite accurate, but is nearer than either the next greater or the next less. 17. — The number of figures in the numbers for which the logarithms are given must also be considered. The tables most in use, like those of Vega, Chambers and Babbage, are of five figures and seven places. A six-figure table would have to contain ten times as many logarithms and occupy ten times the space. A sixth and a seventh figure may be obtaind from them by interpolation. The United States Coast Survey tables (now out of print) are five-figure ten-places. Nine figures may be obtained by simple proportion, but the tenth is, for the most of the work, unreliable. Both of the foregoing systems give auxiliary tables of proportionate parts, or differences. 18. — Peter Gray and Anton Steinhauser have publisht tables of 24 and 20 places respectivly, but the plan for extending the numbers of figures is quite different from the simple interpolation above referd to. They both procede by subdividing the number into factors, and adding together the logarithms of those factors. 19. — All logarithmic calculations end with the ascertain- ment of a number which the problem calld for. The more Tables of Logarithms. 7 decimal places the tables give, the more exact the resulting number, or answer, will be, and the number of figures in the answer can never be more than the number of places in the final logarithm. 20. — I have selected twelve figures as the most useful limit for the accurate computation of interest problems, that being the kind for which the work is specially designd. The logarithms are given to two figures and thirteen places, the extra place insuring the accuracy of the 12th, which would otherwise sometimes be 1, 2 or even 3 units in error, thru the roundings being preponderant in one direction or the other. 21. — The method used is that of factoring, it being pos- sible to construct the logarithm of any number of twelve figures or less (900,000,000,000 in all) by some combina- tion of the 584 logarithms given on the two pages of the Table of Factors. Colum A contains numbers of two figures, 11 to 99, and their logarithms to thirteen places. Colum B contains the logarithms of four-figure numbers 1.001 to 1.099, each beginning with 1.0. . Colum C contains the logarithms of six-figure numbers 1.00001 to 1.00099, each beginning with 1.000. . . Colum D, 1.0000001 to 1.0000099, beginning with one and five zeroes. Colum E, 1.000000001 to 1.000000099, beginning with one and seven zeroes. Colum F, 1.00000000001 to 1.00000000099, beginning with one and nine zeroes. For example, opposit 34 in the table we find : A .531478 917 042,3 In 3.4 B .014 520 538 757,9 In 1.034 C .000147 635 027,3 In 1.00034 D .000 001476 598,7 In 1.0000034 E .000 000 014 766,0 In 1.000000034 F . 000 000 000 147,7 In 1 . 00000000034 By omitting all the prefixt zeroes, the printed table is made very compact, each line containing only 53 figures insted of 78. It will be understood hereafter that C 34, for example, means the number 1.00034, and F 34 means 1.00000000034. The Properties of I^ogarithms. To Find the Number when the Logarithm is Given. 22. — In this process there are two stages : first, to divide the logarithm into a number of partial logarithms among those containd in the T F (Table of Factors) ; second, to multiply together the numbers corresponding to these logarithms. Of course the decimal part only of the logarithm is used and the number has the position of its units figure determind from the characteristic. 23.— Let the logarithm .753 797 472 366,5 be one which has been obtaind as the result of an operation, and the corresponding number be required. Search in Coluni A for the highest logarithm which does not exceed the given one. This is found to be .748 188 027 006,2, which stands op- posit 56. Subtracting from 753 797 472 366,5 A 56 .748 188 027 006,2 we have the remainder 5 609 445 360,3 This is smaller than any logarithm in Colum A. We search for it in Colum B and find opposit 13 pre- cisely the same figures 5 609 445 360.3 These two logarithms added together make the given loga- rithm ; hence the product of their numbers gives the number required. To multiply 56 by 1.013 56 ^ 1013 1013 5 5065 56 e 6078 56 168 r °^ 56728 56728 24. — This process may be greatly simplified as follows, placing the figures of the multiplier in vertical order at the side: 56 56 66 13 X 5 065 168 or 13 X 6 078 56728 56728 , Notice that the first product is moved two colums to the right of the multiplicand. To Find the Number when the Logarithm is Given. 9 25. — We will now take a little larger logarithm and continue the subtraction A 56 B13 C26 D29 E58 G65 + 753 911 659 107,4 748 188 027 006,2 5 723 632 101,2 5 609 445 360,3 114 186 740,9 112 901 888,7 1 284 852,2 1 259 452,2 25 400,0 25 189,1 210,9 208.5 2,4 2,4 There is no colum G; but it is found by simply taking the first two figures from E. It may be either 55 or 56, which may make the thirteenth figure of the result doutful, but probably not the twelfth. See Note 1. See Note 2. See Note 3. See Note 4. 5600 56 168 567280000 113456 3 4 3 6 8 4 2 74928000 * 11.348550 5106847 507 429 97 1.5. 94 70 54 28 3 5 6 7 4 2 9 17 15 2 6, lo The PropertiEvS op Logarithms. Note 1. — The second multiplication jumps its right-hand figure (6) y^z^r places to the right, which may be markt off by four zeroes, or four dots. Note 2. — Having extended the product to include the 13th figure, contraction begins in this multiplicand ; its first figure used being the 7th (markt *) allowing for the carrying from the 8th. Thus the starting point for this multiplication is moved six places back. Note 3. — The multiplicand need no longer be extended, as has been done at successiv stages above, but remains the same to the end. For convenience, dots may be placed in advance under the first figure to be used in multiplication in each line. j Note 4. — The thirteenth figures are added, but only used for carrying to the twelfth. In this example the total of the last colum is 31, but it does not appear, except as contribut- ing 3 to the next colum. The dot below a figure indicates where the contracted multiplication begins, all the figures to the right being ignored, except as to their carrying power. 25. — Another example in which there is no suitable loga- rithm in A and w^e must begin with B. Required the number for log. Oil 253 170 227 To Kind the Number when the Logarithm is Given, ii Formation of Number from Logarithm. Logarithm 1 1 2 5 3 1 7 1 2 7 A — B 26 1 1 1 4 7 3 6 7 7 5 8 1 5 8 9 3 5 1 2 C 24 1 4 2 1 8 1 7 1 5 9 1 1 8 1 2 D 36 1 5 6 3 4 5 7 3 2 7 7 2 3 9 E 63 — — 2 7 3 6 5 3 6 3 4 F 83 3 6 5 2 9 G 67 ^^^'^ 2 9 A — B 26 1 2 6 1 2 6 • C 2 2 5 2 4 4 1 4 1 2 6 2 4 6 2 4 • D 3 3 7 8 7 3 9 6 • 6 1 5 7 4 8 1 2 6 2 4 9 9 3 4 4 8 7 E 6 • 6 1 5 7 5 3 • 3 7 9 F 8 • 8 2 1 3 • 3 1 G 6 • 6 7 1 1 2 6^ 2_ 5 0^ 01 0^ ^ 0^ In this example we illustrate the procedure when B furnishes the first logarithm. It also shows the convenience of using paper ruled for the purpose. 12 The Properties of Logarithms. 26. — In order to set down the partial products without hesitation, remember the numbers 2, 4, 6. In multiplying by B the first figure of the product moves two places to the right. In multiplying by C the first figure of the product movers/our places to the right. In multiplying by D the first figure of the 7nultiplica?td moves six places to the left. 27. — The following rule may now be formulated for this process: Rule. — 1. By successiv subtractions separate the given logarithm into a series of partial logarithms found in the colums of the T F, setting opposit each its letter and number. 2. By successiv multiplications find the product of all the numbers thus found, allowing, in the placing of the partial products, for the prefixt 1 and zeroes. 28. The work may be made to occupy fewer lines by setting down the factors E, F and G as one number at the top, multiplying it by A and incorporating it thereafter as one multiplicand with the preceding figures. The result will not be affected. Let the factors be, as before: A 56 B 13 C 26 D29 E58 F48 G 55. E F G 584855 ^^^ 2924275 350913 5600000327519 B13 56000003275 16800000 9 82 5672800331776 C26 1134560066 340368029 567427525987 1 D29 11348551 5106848 567429171 5 afiXL To Find the Number when the Logarithm is Given. 13 29. — Required the number whose logarithm is .5 or >^. .500 000 000 000.0 A 31 491 361 693 834,3 B 20 8 638 306 165,7 8 600 171 761,9 C 08 38 134 403,8 34 742 168,9 D 78 3 392 234,9 3 387 483,7 E 10 4 751,2 4 342,9 F 94 G 03 408,3 408,2 0,1 The resulting factors A 31 B 20 C 08 D 78 E 10 F 94 G 03 when combined produce the result 3.16227766017. 30. — The multiplication illustrates how zeroes are treated when they occur in the multipliers. 31. — The result is the square root of 10, to 12 places, as may- be demonstrated by multiplying 3.16227766017 by itself. Method by Multipi^es. 32. — In order to facilitate the multiplication of the factors, A, B, C, etc., Mr. A. S. Little, of St. Louis, has devised a Table of Multiples, giving the product of each number from 1 to 9 by every number from 2 to 99. (See page 35.) Thus the multiples of 89 read in one line as follows : 123456789 089 178 267 356 445 534 623 712 801 Then, if it be desired, for example, to multiply 68792341 by 89, we would select from the above table under 6 5 3 4 8 712 7 623 9 801 2 178 3 267 4 356 1 089 6122518349 14 .Thk Properties of I^ogarithms. We have thus multiplied each figure of the multiplicand by both figures of the multiplier, setting down each partial product unhesitatingly. 33. — The work may be made more compact by piling the partial products like bricks, using only three lines: 5 3 4.801,356, 7 12,178,089 6 2 3,2 6 7, 6122518349 34. — Three figures must be set down for each partial product, even if the first be a zero. 35. — To use this method in combining the factors of a number, the letters A, B, C, etc., are written above alternate figure spaces, which is facilitated by the use of paper properly ruled. Then the first partial product under each letter is placed with its middle figure under that letter at the top. 36. — The following is an example of a combination al ready performd in another form : A B C D E F G A 56 1 584855 2 8 0,448 4 4 8,280 2 2 4,2 8 56 327519 B 13 6 5 3 9,0 6 078 2 6,2 091 5672800331778 C26 13 0.052,000, 15 6,208,078 18 2.0 0,0 8 5674275259864 D 29 14 5.116,15 17 405 8,1 2 3.203 5 67429171526 37. — Mr. Little has also suggested a process for verifying a numerical result by using a different set of factors in a second operation. 38. — Required the number corresponding to .305 773 384 163.0 To Find the Number when the Logarithm is Given. 15 The factors are A 20 B 10 C 97 D 21 E 94 F 94 G 33. The number is 2.02195383809. In order to check the result and make sure of perfect ac- curacy, we may solve the problem a second time, using two subtrahends from A. The first subtraction may be of any suitable number ; 11 is found to give the greatest facility. 305 773 384 163,0 A 11 041 392 685 158.2 264 380 699 004,8 A 18 255 272 505 103,3 9 108 193 901,5 B 21 9 025 742 086,9 82 451 814,6 C 18 78 165 972,0 4 285 842,6 D 98 4 256 065,1 29 777,5 E 68 29 532.0 245,5 F 56 243.2 2.3 G 53 2.3 39. — The remainder of the operation may be by either method: A B C D E F G A B C D E F G A 18 685653A18 685653 548522 548522 18 1234175 18 1234175 A 11 18 123418 A 11 1^ 123418 19 8 135759 198 135759 B21 396 2715 B21 021 02 1,15 198 136 18 9 6 3,1 2021580138610 1 68 105 C18 202158014 202 1580138610 16172 6 411 C 18 3 6,0 1 8,0 0,1 2021944023035 0.090.018 D98 18197496 3 6.14 4.0 5 1617555 2021944023034 202196383809 D 98 19 6 9 8,39 0.8 8 2.0 ♦ 196392 202196383809 In this example the first method appears to be preferable, especially in the earlier part. 1 6 The Properties op Logarithms. To Form the Logarithm of a Number. 40. — This consists in two processes : first, the number is separated into a series of factors corresponding to the six colums of the thirteen-place table ; second, the logarithms of these factors are copied from the table and added together. 41. — The factoring is effected by a progressiv division, the divivSor receiving successivly more and more of the figures of the number. 42. — To illustrate this division we will assume a number in which the division will be soon completed. To find the logarithmic factors. A, B, C, etc., of 5.6728. First extend the number to 12 places, 567 280 000 000. The first factor A is always the first two figures of the number itself. A 56)56 7 2 80 000 000 (1 . 013 B 56 72 56 168 168 It will readily be seen that one 56 might have been omitted. A 56)7 280 000 000 (B 13 56 168 168 Turning then to the Table we have only to set down the logarithms of these two factors : A 56 nl 748 188 027 006,2 B 13 nl 5 609 445 360,3 56728 nl 753 797 472 366 5 B 13 may be regarded as an abbreviation of L.013. 43.— We will now give an example where a second divisor, at least, is required. A 56) 7 4 2 9 1 7 1 5 2 6 (B 13 56 182 168 A B 56 728 ) 1 4 To Form the Logarithm op a Number. 17 The second divisor is the product of A and B. It might be obtaind in either of three ways. 1.013 = 56728 multiplication 56 X addition 5Q + 56 + 168 56728 But the easiest way is by subtraction 56742 five figures of the number — 14 the remainder 56728 This is the proper method for forming all divisors after the first ; subtract the remainder from the original number so far as used. 44. — We resume the division, bringing down /our more figures, to the ninth inclusiv. AB)56728 )14 9 171526(C26 113456 357155 340368 ABC)56742 7,5) *1678726(D29 1134855 543871 510685 5 67 42 9,174- 3 3 1 8 6 (E 58 28371 4815 4539 2 7 6 (F 48,7 227 49 45 4 The third divisor A B C is also formd by subtracting from the number 5674291715 * the remainder 16 7 8 7 5674274928 1 8 Thb Properties of Logarithms. As only six figures are needed for the divisor and one for carrying, this is rounded up to 5 6 7 4 2 7,5 The fourth divisor is practically the number itself so far as needed, and this lasts to the end. 45. — The entire process is now repeated, but for greater accuracy in the twelfth figure we will divide out to the thirteenth. . A56) 7 429 17 15 26,0 (B 13 56 A B 56 728) 182 168 149171 113 4 5 6 (C26 357155 340368 A B C 56 742 749 28) [Contracted division begins here] 56 742 92) 1 6 7 8 7 2 6,0 (D 29 113485 5,0 5 4 3 8 7 1,0 5 106 84,7 33 186,3 (E58 2837 1,4 4 8 1 4,9 45 3 9,4 2 7 5,5 (F48 2 2 7.0 485 454 (G55 31 28 It remains only to add together the logarithms : A 56 {nl) 748 188 027 006,2 B 13 " 5 609 445 360,3 C 26 •' 112 901 888,7 D29 " 1 259 452,2 E58 " 25 189,1 F 48 " 208,5 G55 '* 2.4 567 429 171 526 («/) 753 911 659,107 To Form the Logarithm of a Number. 19 46. — The figures in the last colum are only usg^ for car- rying to the twelfth, which otherwise would give Stnsted of 7. 47. — We may now formulate the following rule for finding the logarithm: Rule.— 1. Make the number to 13 figures, by adding cifers or cutting off decimals. 2. Cut off the two left-hand figures by a curve, giving A. 3. Divide the next three figures by A, giving the two figures of B, and a remainder. 4. Form the second divisor A B, by subtracting the re- mainder from the first five figures of the number. 6. Bring down four more figures to the remainder and divide by A B, giving the two figures of C and a remainder. 6. Form the third (and last) divisor A B C by subtracting the remainder from ten figures of the number. 7. Divide the remaining figures by the third divisor. As there are ten figures in the divisor and only eight in the divi- dend, contraction begins immediately. Having obtaind the figures of D, the divisor for E, F and G is simply the number itself contracted. 8. Write down the logarithms of A, B, C, D, E and F, obtaind from the several colums of T F; also that of G, being the first two figures of E. The sum will be the logarithm, the thirteenth figure being used for carrying only. 48. — It is advisable to make all logarithmic computations on paper ruled with thirteen down-lines, every third being darker. A specimen is given on page 36. 49. A few examples for practis are given below with the factors and the solution: 5674 = A 56 B 13 C 21 D 15 E 35 F 42 G 70 log. 5674 = 3.753 889 331 458 38.8586468578 = A 38 B 22 C 58 D 31 E 39 F 02 G 25 log. do. = 1 .589 487 673 453 3.1415926535898 + = A 31 B 13 C 41 D 16 E 33 F 11 G 91 log. do. =.497 149 872 694 (This number is the ratio of the circumference of a circle toils diameter.) 1.02625 = B 26 C 24 D 36 E 63 F 83 log. do. = .011 253 170 127 This number begins with an expression of the form B (1.026), hence no division by A occurs. 1026 is the first divisor. 20 The Properties of I^ogarithms. B 1026) 2 5 2052 C24 4480 4104 B C 102624624) 3 7 6 0.0 3 7 8 7 3,9 D36 6 8 1 2 6,1 6 1574,8 102625) 6 55 1,3 6 1 5 7,5 3 9 3,8 3 7,9 K63 8 5,9 8 2,1 F83 3,8 3,1 7 G70 B 26 C24 D36 E63 F83 G70 Oil 147 360 775,8 104 218 170,0 1563 457,3 27 360,6 360,5 3,0 Oil 253 170 127 This result will be found also in the Table of Interest-Ratios, but even more extended. 21 Logarithms to Less than 12 Places. 60. — The T F may be cut down to any lower number of places. In the example in Art. 45 it may be required to give 9 places only, the tenth being used for carrying. We cut down the original logarithm to ten figures, with a comma after the ninth and it becomes 753 911 659,1 A 56 748 188 027.0 5 723 632,1 B 13 5 609 445,4 114 186,7 C 26 112 901.9 1 284,8 D 29 1 259,5 25,3 B58 25,2 F 24 1 A 56 B 1 5 6 3 168 567280000 C 2 113456 6 3 4036,8 56742749 2,8 D 2 1 1 3 4,9 9 5 10,7 E 5 2 8,4 8 4,5 F 2 1 56742917 1,4 The number is slightly in error in its tenth place, but correct to the ninth. 51. — If a table of factors for 18 or some other number of places should hereafter be prepared, the methods which have been explaind would be applicable. 22 The Properties of Logarithms. Multiplying Up. 52. — Mr. Edward S. Thomas, of Cincinnati, has suggested another method for obtaining the factors of the number in forming its logarithm. 53. — It procedes by multiplication insted of division, the latter operation being notabh^- the more laborious. The num- ber, at first taken as a decimal less than 1, is successivly mul- tiplied up to produce 1.000,000,000,0 and these multipliers are the A, B, C, D, E, F and G, whose logarithms added to- gether make the cologarithm, from which the logarithm is easily obtaind. 54. — A is a number of two figures, a little less than the reciprocal of the number, which will be calld the sub-recipro- cal of its two initial figures. A Table of Sub- Reciprocals is given on page 33. The number multiplied by A will always give a product beginning with 9. B is always the arithmetical complement of the two figures following tlve nine, or the re- mainder obtaind by subtracting those two figures from 99. Multiplication by B will usually give a result beginning with 999. C is the next complement and gives 5 9's, 999,99. D similarly brings 999,999,9 >{^ * ^ * >K H^ ^ * . No further multipli- cation is necessary, when D has been used ; the six figures in the places of the stars are the complements of E, F and G. 55. — To illustrate, let it be required to obtain the loga- rithm to the 12th place of 3.14 159 265 359 0. The object is to multiply .314 159 265 -859 up to 1 .000 000 000 000 0. The first step is to find the sub-reciprocal of 31, or A. Turning to the Table of Sub-reciprocals, opposit 81 we find 31, by which we multiply. Multiplying Up. 23 A 31 99 — 73 = 26 B 26 is therefore the next multipli- er; dropping the last two figures (99—21) C 78 (99—43) D 56 (99-47) (99-03) (100-77) E 52 F 96 G23 .3141592653590 .9424777960770 31415 92653 5 9 .9 738937226129 One 9 has been secared 194778744523 58433623357 999214959400 9 6994504716 799 3 719 6 8 Three nines secured 99999434706 9 3 Five nines 49999718 5999966 .9999999470377 Seven nines 52 9 6 23 A31;^/ .4 913616938343 B26 111473607758 C78 3386176522 D56 24320423 E52 225833 F 96 4 16 9 G 23 _j 1^ colog. 0.5 02850127306 log. 1.4 97149872694 56. — It may happen, in the course of multiplication, that the complement of the figures following the 9 does not suffice to secure two nines more. In this case, another supplementary multiplication must take place. This occurs in the following example, which has alredy been solvd. 24 The Propkrties of I,ogarithms. 57. — Required the logarithm of the number 567 429 171 526. In this example the C multiplication also requires an additional figure. This seldom occurs. .567 429171526 A 17 .397 200 420 068 2 .964 629 591594 2 B 35 28 938 887 747 8 4 823 147 958 .998 391627 300 " 01 998 391 627 3 .999 390 018 927 3 C 60 599 634 0114 .999 989 652 938 7 " 01 9 999 896 5 .999 999 652 835 2 D 03 299 999 9 .999 999 952 835 1 47 164 9 { A 17 230 448 921 378 3 B 35 14 940 349 792 9 01 434 077 479 3 C 60 260 498 547 4 01 4 342 923 1 D 03 130 288 3 B 47 20 411 8 F 16 69 5 G 49 21 .246 088 340 892 7 .753 911659 107 3 As the multiplication by 35 brings only 998 insted of 999, we multiply again by B 01, which brings it up. MuivTiPLYiNG Up. 25 58. — In the next example there is a large defect in B, which requires an additional multiplication by 7. 110 175 A 83 881 400 (83, subreciprocal of 11) 33 052 5 914 452 5 B 85 73156 200 4 572 262 5 B 07 C 87 999 995 469 056,9 D 45 3 999 981,9 499 997,7 992 180 962 5 6 945 266 737,5 999 126 229 237.5 799 300 983,4 69 938 836,0 999 999 969 036,5 30 963,5 A 83 919 078 092 376,1 B 85 35 429 738 184,5 B 07 3 029 4*70 553,6 C 87 377 670 935,8 D 45 1 954 320^8 K 30 13 028,8 F 96 416,9 G 35 1,5 957 916 940 818 042 083 059 182 The number 11075 was purposely selected, very slightly in excess of the highest number in colum B, so as to produce the shortage of 7. 26 The Properties of I^ogarithms. 59. — Little's Table of Multipliers may be used in the multiplication, as in the following example. It will be found •that the logarithm when computed has the same figures as the number itself; a remarkable peculiarity which no other com- bination of figures can possess. . 137 128 857 423 9 A 71 710 715 682 846 4 213 142 355 142 49 756 849 721 3 . 973 614 887 709 7 B 26 23 415 620 818 2 1 820 262 080 078 104 182 2 . 998 928 874 790 1 tt 01 998 928 874 8 . 999 927 803 664 9 C 07 69 994 946 3 . 999 997 798 611 2 D 22 1 981 981 8 198 198 19 815 4 . 999 999 998 606 4 01 393 6 E F G A 71 851 258 348 719 1 B 26 11 147 360 775 8 B 01 434 077 479 3 C 07 30 399 549 8 D 22 955 446 8 E 01 434 3 F 39 169 4 G 36 16 862 871 142 576 1 .137 128 857 423 9 which is the log. of 1.371 288 574 239 27 Signs of the Characteristic. 60 — We have seen (Art. 8) that while the decimal part of the logarithm is always positiv, the characteristic is often negativ and has the minus sign above it. 61. — In adding together several logarithms with different signs, the positivs and the negativs must be added separately; the less sum must be subtracted from the greater, and the remainder has the sign of the greater sum. The carrying from the decimal part counts with the positivs. 34 7il. 1.531478 917 042,3 2900 " 3.462 397 997 899,0 .73 '' 1.863 322 860120,5 .056 ♦' 2.748 188 027 006.2 The sum of the decimals is. . . . 2.605 387 802 068,0 The positivs are 1. and 3 Total + 6 The negativs are 1 , 2 — 3 Sum of the logarithms +3.605 387 802 068,0 The decimal point in the result must follow the fourth figure, as indicated by the characteristic 3. 62. — In subtracting one logarithm from another, when the decimal of the subtrahend is the greater, and a unit^ " borrowed," the unit is considered as one more negativ: '^ut ^CT^'^'y'' * the total characteristic changes its signs from plus to minus or /Tv^^^ from minus to plus. . 290 2 462 397 997 899,0 <^^^^=-^ ^ .0058 3.763 427 993 562,9 Operate on the deci- mals only . 462 397 997 899,0 0.763 427 993 562,9 1.698 970 004 336,1 Negativ from subtrahend 3 Total negativ 4 Sign changed +4 Y^^^"^^^ From minuend ... -h 2 ) 6 . 698 970 004 336,1 In 5 000 000^ y 28 The PrOPERTIBS of lyOGARlTHMS. 63. — To multiply a logarithm having a negativ character- istic (in order to obtain a power of a decimal) , multiply the decimal part and the characteristic separately and add the two together: 2.301029 995 664.0 x 5 Decimal part _ 1 . 505 149 978 320,0 Characteristic 10. 9.505 149 978 320,0 Therefore the 5th power of .02 is .000 000 003 2. 64. — To divide a logarithm having a negativ characteristic, (for the extraction of a root;) if the characteristic is exactly divisible, divide the decimal part and the characteristic separately: 12.690 196 080 028,5 -f- 6 2.115 032 680 004,7 But if the characteristic be not so divisible, add to it a negativ quantity, which will make it divisible, and prefix to the decimal part in compensation an equal quantity positiv. 12.690 196 080 028,5 -f- 5 Add 3 15. Add 3 3.690196 080 028.5 Quotient 3 . 738 039 216 005,7 Different Bases. 65. — Ten is the base of the logarithmic system which we have been explaining; it is the most useful of all systems, because ten is also the base of our numerical system. These are usually calld common or vulgar, or Briggsian logarithms, but decimal logarithms would seem more appropriate. 66. — Any number might form the base of a system of lo- garithms, but the only other in actual use is one known as the "natural" system, having for its base the number 2. 718281828459 -h which is the sum of the series 1 J_ 1 1 1 ■^ "^ i "^ 1x2 "^ 1x2x3 "^ 1x2x3x4 "^ 1x2x3x4x5^^^' This is only used in theoretical inquiries, and is seldom of utility to the accountant. PART II. TABLES FOR OBTAINING Logarithms and Antilogarithms TO 12 PLACES OF decimals 30 TABLE OF FACTORS (V a 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ^26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 A * . * ' 041 392 685 158,2 079 181 246 047,6 113 943 352 306,8 146 128 035 678,2 176 091 259 055,7 204 119 982 655,9 230 448 921 378,3 255 272 505 103,3 278 753 600 952,8 301 029 995 664,0 322 219 294 733,9 342 422 680 822,2 361 727 836 017,6 380 211 241 711,6 397 940 008 672,0 414 973 347 970,8 431 363 764 159,0 447 158 031 342,2 462 397 997 899,0 477 121 254 719,7 491 361 693 834,3 505 149 978 319,9 518 513 939 877,9 531 478 917 042,3 544 068 044 350,3 556 302 500 767,3 568 201 724 067,0 579 783 596 616,8 591 064 607 026,5 602 059 991 328,0 612 783 856 719,7 623 249 290 397,9 633 468 455 579,6 643 452 676 486,2 653 212 513 775,3 662 757 831 681,6 672 097 857 935,7 681 241 237 375,6 690 196 080 028,5 B 1.0* H« 434 077 479,3 867 721 531,2 1 300 933 020,4 1 733 712 809,0 2 166 061 756,5 2 597 980 719,9 3 029 470 553,6 3 460 532 109,5 3 891 166 236,9 4 321 373 782,6 4 751 155 591,0 5 180 512 503,8 5 609 445 360,3 6 037 954 997,3 6 466 042 249,2 6 893 707 947,9 7 320 952 922,7 7 747 778 000,7 8 174 184 006,4 8 600 171 761,9 9 025 742 086,9 9 450 895 798,7 9 875 633 712,2 10 299 956 639,8 10 723 865 391,8 11 147 360 775,8 11 570 443 597,3 11 993 114 659,3 12 415 374 762,4 12 837 224 705,2 13 258 665 283,5 13 679 697 291,2 14 100 321 519,6 14 520 538 757,9 14 940 349 792,9 15 359 755 409,2 15 778 756 389,0 16*197 353 512,4 16 615 547 557,2 17 033 339 298,8 17 450 729 510,5 17 867 718 963,5 18 284 308 426,5 18 700 498 666,2 19 116 290 447,1 19 531 684 531,3 19 946 681 678,8 20 361 282 647,7 20 775 488 193,6 C 1.000+* 4 342 923,1 8 685 802.8 13 028 639,0 17 371 431,8 21 714 181,2 26 056 887,2 30 399 549,8 34 742 168,9 39 084 744,6 43 427 276,9 47 769 765,7 52 112 211,2 56 454 613,2 60 796 971,8 65 139 287,0 69 481 558,7 73 823 7^7,1 78 165 972,0 82 508 113,5 86 850 211,6 91 192 266,3 95 534 277,6 99 876 245,5 104 218 170,0 108 560 051,0 112 901 888,7 117 243 682,9 121 585 433,8 125 927 141,2 130 268 805,2 134 610 425,9 138 952 003,1 143 293 536,9 147 635 027,3 151 976 474,3 156 317 878,0 160 659 238,2 165 000 555,0 169 341 828,4 173 683 058,5 178 024 245,1 182 365 388,3 186 706 488,2 191 047 544,7 195 388 557,7 199 729 527,4 204 070 453,7 208 411 336,6 212 752 176,1 D 1.00000** 43 429,4 86 858,9 130 288,3 173 717,8 217 147,2 260 576,6 304 006,0 347 435,4 390 864,9 434 294,3 477 723,7 521 153,1 564 582,5 608 011,8 651 441,2 694 870,6 738 300,0 729.4 158,7 781 825 868 588,1 912 017,5 955 446,8 998 876,2 1 042 305,5 1 085 734,8 1 129 164,2 1 172 593,5 1 216 022,8 1 259 452,2 302 346 389 433 476 881,5 310,8 740,1 169,4 598,7 520 028,0 563 457,3 606 886,6 650 315,9 693 745,2 737 174,5 780 603,7 824 033,0 867 462,3 910 891,5 1 954 320,8 1 997 750,0 2 041 179,3 2 084 608,5 2 128 037,7 ^H o E F a • 3 1.0'** 1.0%* ^ 434,3 004,3 01 868,6 008,7 02 1 302,9 013,0 03 1 737,2 017,4 04 2 171,5 021,7 05 2 605,8 026,1 06 3 040,1 030,4 07 3 474,4 034,7 08 3-908,7 039,1 09 4 342,9 043,4 10 4 777,2 047,8 11 5 211,5 052,1 12 5 645,8 056,5 13 6 080,1 060,8 14 6 514,4 065,1 15 6 948,7 069,5 16 7 383,0 073,8 17 7 817,3 078,2 18 8 251,6 082,5 086.9 19 20 8 685,9 9 120,2 091,2 21 9 554,5 095,5 22 9 988,8 099,9 23 10 423,1 104,2 24 10 857,4 108,6 25 11 291,7 112,9 26 11 726,0 117,3 27 12 160,2 121,6 28 12 594,5 125,9 130,3 29 30 13 028,8 13 463,1 134,6 31 13 897,4 139,0 32 14 331,7 143,3 33 14 766,0 147,7 34 15 200,3 152,0 35 15 634,6 156,3 36 16 068,9 160,7 37 16 503,2 165,0 38 16 937,5 169,4 173,7 39 40 17 371,8 17 806,1 178,1 41 18 240,4 182,4 42 18 674,7 186,7 43 19 109,0 191,1 44 19 543,3 195,4 45 19 977,5 199,8 46 20 411,8 204,1 47 20 846,1 208,5 48 21 280,4 212,8 49 TABLE OF FACTORS— Continued 31 HI A B C D E F s 50 * • * 1.0** 1.000** 1.00000** 1.0%* 1.0%* 217,1 S 50 698 970 004 336,0 21 189 299 069,9 217 092 972,2 2 171 467,0 21 714,7 51 707 570 176 097,9 21 602 716 028,2 221 433 725,0 2 214 896,2 22 149,0 221,5 51 52 716 003 343 634,8 22 015 739 817,7 225 774 434,3 2 258 325,4 22 583,3 225,8 52 53 724 275 869 600,8 22 428 371 185,5 230 115 100,3 2 301 754,7 23 017,6 230,2 53 54 732 393 759 823,0 22 840 610 876,5 234 455 722,9 2 345 183,9 23 451,9 234,5 54 55 740 362 689 494,2 23 252 459 633,7 238 796 302,1 2 388 613,1 23 886,2 238,9 55 56 748. 188 027 006,2 23 663 918 197,8 243 136 837,9 2 432 042,3 24 320,5 243,2 56 57 755 874 855 672,5 24 074 987 307,4 247 477 330,3 2 475 471,5 24 754,8 247,5 57 58 763 427 993 562,9 24 485 667 699,2 251 817 779,4 2 518 900,7 25 189,1 251,9 58 59 60 770 852 Oil 642,1 24 895 960 107,5 256 158 185,1 2 562 329,9 2 605 759,1 25 623,4 256,2 260,6 59 60 778 151 250 383,6 25 305 865 264,8 260 498 547,4 26 057,7 61 785 329 835 010,8 25 715 383 901,3 264 838 866,3 2 649 188,3 26 492,0 264,9 61 62 792 391 689 498,3 26 124 516 745,5 269 179 141,9 2 692 617,4 26 926,3 269,3 62 63 799 340 549 453,6 26 533 264 523,3 273 519 374.0 2 736 046,6 27 360,6 273,6 63 64 806 179 973 983,9 26 941 627 959,0 277 859 562,8 2 779 475,8 27 794,8 277,9 64 65 812 913 356 642,9 27 349 607 774,8 282 199 708,3 2 822 905,0 28 229,1 282,3 65 66 819 543 935 541,9 27 757 204 690,6 286 539 810,3 2 866 334,1 28 663,4 286,6 66 67 826 074 802 700,8 28 164 419 424,5 290 879 869,0 2 909 763,3 29 097,7 291,0 67 68 832 508 912 706,2 28 571 252 692,5 295 219 884,3 2 953 192,4 29 532,0 295,3 68 69 70 838 849 090 737,3 28 977 705 208,8 299 559 856,2 2 ^996 621,6 29 906,3 299,7 304,0 69 70 ^45 098 040 014,3 29 383 777 685,2 303 899 784,8 3 040 050,7 30 400,6 71 851 258 348 719,1 29 789 470 831,9 308 239 670,0 3 083 479,9 30 834,9 308,3 71 72 857 332 496 431,3 30 194 785 356,8 312 579 511,8 3 126 909,0 31 269,2 312,7 72 73 863 322 860 120,5 30 599 721 966,0 316 919 310,3 3 170 338,1 31 703,5 317,0 73 74 869 231 719 731,0 31 004 281 363,5 321 259 065,4 3 213 767,3 32 137,8 321,4 74 75 875 061 263 391,7 31 408 464 251,6 325 598 777,1 3 257 196,4 32 572,1 325,7 75 76 880 813 592 280,8 31 812 271 330,4 329 938 445,5 3 300 625,5 33 006,4 330,1 76' 77 886 490 725 172,5 32 215 703 298,0 334 278 070,5 3 344 054,6 33 440,7 334,4 77 78 892 094 602 690,5 32 618 760 850,7 338 617 652,2 3 387 483,7 33 875,0 338,7 78 79 80 897 627 091 290,4 33 021 444 682,9 342 957 190,4 3 430 912,9 34 309,3 343,1 347,4 79 80 903 089 986 991,9 33 423 755 486,9 347 296 685,4 3 474 342,0 34 743,6 81 908 485 018 878,6 33 825 693 953,3 351 636 136,9 3 517 771,1 35 177,9 351,8 81 82 913 813 852 383,7 34 227 260 770,6 355 975 545,1 3 561 200,2 35 612,1 356,1 82 83 919 078 092 376,1 34 628 456 625,3 360 314 910,0 3 604 629,2 36 046,4 360,5 83 84 9^4 279 286 061,9 35 029 282 202,4 364 654 231,5 3 648 058,3 36 480,7 364,8 84 85 929 418 925 714,3 35 429 738 184,5 368 993 509,6 3 691 487,4 36 915,0 369,2 85 86 934 498 451 243,6 35 829 825 252,8 373 332 744,4 3 734 916,5 37 349,3 373,5 86 87 939 519 252 618,6 36 229 544 086,3 377 671 935,8 3 778.345,6 37 783,6 377,8 87 88 944 482 672 150,2 36 628 895 362,2 382 Oil 083,8 3 821 774,6 38 217,9 382,2 88 89 90 949 390 006 644,9 37 027 879 755,8 386 350 188,6 3 865 203,7 38 652,2 386,5 390,9 89 90 954 242 509 439,3 37 426 497 940,6 390 689 249,9 3 908 632,7 39 086,5 91 959 041 392 321,1 37 824 750 588,3 395 028 267,9 3 952 061,8 39 520,8 395,2 91 92 963 787 827 345,6 38 222 638 368,7 399 367 242,6 3 995 490,9 39 955,1 399,6 92 93 968 482 948 553,9 38 620 161 949,7 403 706 173,9 4 038 919,9 40 389,4 403,9 93 94 973 127 853 599,7 39 017 321 997,4 408 045 061,8 4 082 348,9 40 823,7 408,2 94 95 977 723 605 288,8 39 414 119 176,1 412 383 906,5 4 125 778,0 41 258,0 412,6 95 96 982 271 233 039,6 39 810 554 148,4 416 722 707,7 4 169 207,0 41 692,3 416,9 96 97 986 771 734 266,2 40 206 627 574,7 421 061 465,6 4 212 636,0 42 126,6 421,3 97 98 991 226 075 692,5 40 602 340 114,1 425 400 180,2 4 256 065,1 42 560,9 425,6 98 99 995 635 194 597,5 40 997 692 423,5 429 738 851,4 4 299 494,1 42 995,2 430,0 99 32 TABLE OF INTEREST RATIOS 1 + i Logarithm 1 + i Logarithm 1.00125 1.0015 1.00175 1.002 1.00225 000 542 529 092 294 000 650 953 629 595 000 759 351 104 737 000 867 721 531 227 000 976 064 922 559 1.01375 1.014 1.01425 1.0145 1.01475 005 930 867 219 212 006 037 954 997 317 006 145 016 376 364 006 252 051 369 365 006 359 059 989 323 1.0025 1.00275 1.003 1.00325 1 . 0035 001 084 381 292 220 001 192 670-653 684 001 300 933 020 418 001 409 168 405 876 001 517 376 823 504 1.015 1.01525 1.0155 1.01575 1.016 006 466 042 249 232 006 572 998 162 075 006 679 927 740 826 006 786 830 998 449 006 893 707 947 900 1.00375 1.004 1 . 00425 1 . 0045 1 . 00475 001 625 558 286 737 001 733 712 809 001 001 841 840 403 709 001 949 941 084 268 002 058 014 864 072 1.01625 1.0165 1.01675 1.017 1.01725 007 000 558 602 125 007 107 382 974 057 007 214 181 076 625 007 320 952 922 745 007 427 698 525 323 1.005 1.00525 1.0055 1.00575 1.006 002 166 061 756 508 002 274 081 774 949 002 382 074 932 761 002 490 041 243 299 002 597 980 719 909 1.0175 1.01775 1.018 1.01825 1.0185 007 534 417 897 258 007 641 111 051 437 007 747 778 000 740 007 854 418 758 035 007 961 033 336 183 1.00625 1.0065 1.00675 1.007 1.00725 002 705 893 375 925 002 813 779 224 673 002 921 638 279 469 003 029 470 553 618 003 137 276 060 415 1.01875 1.019 1.01925 1.0195 1.01975 008 067 621 748 033 . 008 174 184 006 426 008 280 720 124 194 008 387 230 114 159" 008 493 713 989 132 1.0075 1.00775 1.008 1 . 00825 1.0085 003 245 054 813 147 003 352 806 825 089 003 460 532 109 506 003 568 230 679 656 003 675 902 548 784 1.02 1.02025 1.0205 1.02075 1.021 008 600 171 761 918 008 706 603 445 309 008 813 009 052 089 008 919 388 595 035 009 025 742 086 910 1.00875 1.009 1.00925 1.0095 1.00975 003 783 547 730 127 003 891 166 236 911 003 998 758 082 352 004 106 323 279 658 004 213 861 842 026 1.02125 1.0215 1.02175 1.022 1.02225 009 132 069 540 472 009 238 370 968 466 009 344 646 383 631 009 450 895 798 694 009 557 119 226 374 1.01 1.01025 1.0105 1.01075 1.011 004 321 373 782 643 004 428 859 114 686 004 536 317 851 323 004 643 750 005 712 004 751 155 591 001 1.0225 1.02275 1.023 1.02325 1.0235 009 663 316 679 379 009 769 488 170 411 009 875 633 712 160 ' 009 981 753 317 307 010 087 846 998 524 1.01125 1.0115 1.01175 1.012 1.01225 004 858 534 620 329 004 965 887 106 823 005 073 213 063 604 005 180 512 503 780 005 287 785 440 451 1.02375 1.024 1.02425 1.0245 1.02475 010 193 914 768 475 010 299 956 639 812 010 405 972 625 180 010 511 962 737 214 010 617 926 988 539 1.0125 1.01275 1.013 1.01325 1.0135 005 395 031 886 706 005 502 251 855 626 005 609 445 360 280 005 716 612 413 731 005 823 753 029 028 1.025 1 . 02525 1.0255 1.02575 1.026 010 723 865 391 773 010 829 777 959 522 010 «35 664 704 385 • Oil 041 525 638 950 Oil 147 360 775 797 TABLE OF INTEREST RATIOS— Continued TABLE OF SUB-RECIPROCALS (Art. 51) 33 1 + i Logarithm Initial Figures Sub-reciprocal 1.02625 1.0265 1.02675 1.027 1.02725 Oil 253 170 127 497 Oil 358 953 706 611 on 464 711 525 690 Oil 570 443 597 278 Oil 676 149 933 909 10 11 12 13 14 90 83 76 71 66 1.0275 1.02775 1.028 1.02825 1.0285 Oil 781 830 548 107 Oil 887 485 452 387 Oil 993 114 659 257 012 098 718 181 213 012 204 296 030 743 15 16 ' 17 18 19 62 58 55 52 50 1 . 02875 1.029 1 . 02925 1.0295 1.02975 012 309 848 220 326 012 415 374 762 433 012 520 875 669 524 012 626 350 954 050 012 731 800 628 455 20 21 22 23 24 47 45 43 41 40 1.03 1.0305 1.031 1.0315 1.032 012 837 224 705 172 013 047 996 115 232 013 258 665 283 517 013 469 232 309 170 • 013 679 697 291 193 25 26 27 28 29 38 37 35 34 33 1.0325 1.033 1 . 0335 1.034 1.0345 013 890 060 328 439 014 100 321 519 621 014 310 480 963 307 014 520 538 757 924 014 730 495 001 753 30 31 32 33 34 32 31 30 29 28 1.035 1 . 0355 1.036 1.0375 1.038 014 940 349 792 937 015 150 103 229 471 015 359 755 409 214 015 988 105 384 130 016 197 353 512 439 35-36 37 38-39 40 41-42 27 26 25 24 23 1.039 1.04 1.041 1 . 0425 1 . 043 016 615 547 557 177 017 033 339 298 780 017 450 729 510 536 018 076 063 645 795 018 284 308 426 531 43-44 45-46 47-49 50-51 52-54 22 21 20 19 18 1.044 1.045 1.046 1.0475 1.048 018 700 498 666 243 019 116 290 447 073 019 531 684 531 255 020 154 031 638 333 020 361 282 647 708 55-57 58-61 62-65 66-70 71-75 17 16 15 14 13 1.049 1.05 1.055 1.06 1.065 020 775 488 193 558 021 189 299 069 938 023 252 459 633 711 025 305 865 264 770 027 349 607 774 757 76-82 83-89 90 12 11 1 1.07 1.075 1.08 1.09 1.10 029 383 777 685 210 031 408 464 251 624 033 423 755 486 950 037 426 497 940 624 041 392 685 158 225 * •• 34 TABLE OF MULTIPLES 1 2 3 4 5 6 7 8 9 001 002 003 004 005 006 007 008 009 002 004 006 008 010 012 014 016 018 003 006 009 012 015 018 021 024 027 004 008 012 016 020 024 028 032 036 005 010 015 020 025 030 035 040 045 006 012 018 024 030 036 042 048 054 007 014 021 028 035 042 049 056 003 008 016 024 032 040 048 056 064 072 009 018 027 036 045 054 063 072 081 010 020 030 040 050 060 070 080 090 Oil 022 033 044 055 066 077 088 099 012 024 036 048 060 072 084 096 108 013 026 039 052 065 078 091 104 117 014 028 042 056 070 084 098 112 126 015 030 045 060 075 090 105 120 135 016 032 048 064 080 096 112 128 144 017 034 051 068 085 102 119 136 153 018 036 054 072 090 108 126 144 162 019 038 057 076 095 114 133 152 171 020 040 060 080 100 120 140 160 180 021 042 063 084 105 126 147 168 189 022 044 066* 088 110 132 154 176 198 023 046 069 092 115 138 161 184 207 024 048 072 096 120 144 168 192 210 025 050 075 100 125 150 175 200 225 026 052 078 104 130 156 182 208 234 027 054 081 108 135 162 189 216 243 028 056 084 112 140 168 196 224 252 029 058 087 116 145 174 203 232 261 030 060 090 120 150 180 210 240 270 031 062 093 124 155 186 217 248 279 032 064 096 128 160 192 224 256 288 033 066 099 132 165 198 231 264 297 034 068 102 136 170 204 238 272 306 035 070 105 140 175 210 245 280 315 036 072 108 ■ 144 180 216 252 288 324 037 074 111 148 185 222 259 296 333 038 076 114 152 190 228 266 304 342 039 078 117 156 195 234 273 312 351 040 080 120 160 200 240 280 320 360 041 082 123 164 205 246 287 328 369 042 084 126 168 210 252 * 294 336 378 043 086 129 172 215 258 301 344 387 044 088 132 176 220 264 308 352 396 045 090 135 180 225 270 315 360 405 046 092 138 184 230 276 322 368 414 047 094 141 188 235 282 329 376 423 048 096 144 192 240 288 336 384 432 049 098 147 196 245 294 343 392 441 TABLE OF MULTIPLES— Continued 35 1 2 3 4 5 6 7 8 9 050 100 150 200 250 300 350 400 450 051 102 153 204 255 306 357 408 459 052 104 156 208 260 312 364 416 468 053 106 150 212 265 318 371 424 477 054 108 162 216 270 324 378 432 486 055 110 165 220 275 330 385 440 495 056 112 168 224 280 336 392 448 504 057 114 171 228 285 342 399 456 513 058 116 174 232 290 348 406 464 522 1 059 118 177 236 295 354 413 472 531 060 120 180 240 300 360 420 480 540 061 122 183 244 305 306 427 488 549 002 124 186 248 310 372 434 496 558 063 126 189 252 315 378 441 504 567 i 064 128 192 256 320 384 448 512 576 065 130 195 260 325 390 455 ^20 585 066 132 198 264 330 396 462 528 594 067 134 201 268 335 402 469 536 603 i 068 136 204 272 340 408 476 544 612 •! 069 138 207 276 345 414 483 552 621 ' 070 140 210 280 350 420 490 560 630 071 142 213 284 355 426" 497 568 639 1 072 144 216 288 360 432 504^ 576 648 1 073 146 219 292 365 438 511 584 657 074 148 222 296 370 444 518 592 666 075 150 225 300 375 450 525 600 675 076 152 228 304 380 456 532 608 684 077 154 231 308 385 462 539 616 693 078 156 234 312 390 468 546 624 702 079 158 237 316 395 474 553 632 711 080 160 240 320 400 480 500 640 720 081 162 243 324 405 486 567 048 729 082 164 246 328 410 492 574 656 738 083 166 249 332 415 498 581 664 747 084 168 252 336 420 504 588 672 756 085 170 255 340 425 510 595 680 765 086 172 258 344 430 516 602 688 774 087 174 261 348 435 522 609 696 783 088 176 264 352 440 528 616 704 792 089 178 267 356 445 534 623 712 801 090 180 270 360 450 540 630 720 810 091 182 273 364 455 546 637 728 819 092 184 276 368 460 552 644 736 828 093 186 279 372 465 558 651 744 837 094 188 282 376 470 564 658 752 846 095 190 285 380 475 570 665 760 855 096 192 288 384 480 576 672 768 864 097 194 291 388 485 582 679 776 873 098 196 294 392 490 588 686 784 882 099 198 297 396 495 594 693 792 891 36 SPECIMEN OF RULED PAPER RECOMMENDED FOR USE WITH THE FOREGOING TABIvES. 1 1 1 i 1 . i i 1^ ! 1 1 ■ j. i L - — — — __ ! 1 . ' ! 1 i i T i ! 1— i 1 i i r •: i j i j I T - j ' 1 ! i 7"! ■ 1 ■ 1 t ! 1 1 1 : i i i i 1 1 .,J_„_ I ! - i i .-,_,.. j.._.,^-_ ■ r -! PART III. The Doctrin of Interest PART III. THE DOCTRIN OF INTEREST. Interest. 67. — Interest, mathematically considerd, is the increase of an indettedness by lapse of time. The rate of such increase varies with circumstances, * and is subject to bargaining ; the resulting contract, exprest or implied, must embody the fol- lowing terms: Principal. The number of units of value (dollars, pounds, francs, marks, etc.,) originally loand or invested. Interest Rate. The fraction which is added to each unit by the lapse of one unit of time ; usually a small decimal. Frequency. The length of the unit of time, measured in years, months or days. Time. The number of units of time during which the indettedness is to continue. 68. — As each dollar increases just as much as every other dollar, it is best at first to consider the principal as one dollar and when the proper function thereof has been calculated, to multiply it by the number of dollars. 69. — The interest rate is usually spoken of as so much percent per period or term. "6%' per annum" means an increase of . 06 for each term of a year. We will designate the interest rate by the letter i ; as, ? = .06. At the end of one term the increast indettedness is 1 -f /, (1.06), a very important quantity in computation. * For discussion of the causes for higher or lower interest rates, see The Rate of Interest, by Prof. Irving Fisher. 40 The Doctrin op Interest. 70. — Punctual Interest. The usual contract is that the increase shall be paid off in cash at the end of each period, restoring the principal to its original quantity. Let c denote the cash payment ; then 1 + z — ^r = 1 ; and the second term would repeat the same process. The payment of cash for interest must not be regarded as the interest : it is a cancel- lation of part of the increast principal. Many persons, and even courts, have been misled by the old definition of interest, "money paid for the use of money," into treating uncollected or unmatured interest as a nullity, tho secured precisely in the same way as the principal. 71. — But the interest money may not be paid exactly at the end of each term, either in violation of the contract or by a special clause permitting it to run on, or by the det being assigned to a third party at a price which modifies the true interest rate. In this case the question arises : how shall the interest be computed for the following periods ? This gives rise to a distinction between simple and compound interest. 72. — Simple Interest. During the second period, altho the borrower has in his hands an increast principal, 1 + i, he is at simple interest only charged with interest on 1, and has the free use of /, which tho small has an earning power pro- portionate to that of 1. His indettedness at the end of the second term is 1 + 2/, and thereafter 1 + 3/, 1 -f- 4/, etc. After the first period he is not charged with the agreed per- centage of the sum actually employed by him, and this to the detriment of the creditor. For any scientific calculation, simple interest is impossible of application. 73.— Compound Interest. The indettedness at the end of the first period is 1 H- /, and up to this point punctual, simple and compound interest coincide. But in compound interest the fact is recognized that the increast principal, 1 -f z, is all sub- ject to interest during the next period, and that the det increases by geometrical progression, not arithmetical. The increase from 1 to 1 + z is regarded, not as an addition of i to 1, but as a multiplication of 1 by the ratio of increase (1 -f z) . We shall designate the ratio of increase by r when convenient, altho this is merely an abbreviation of 1 H- z, and the two expressions are at all times interchangeable. Thk Amount. 41 74. — For the second period, 1 + / is the actual and equitable principal, and it should be again increast in the ratio 1 -j- i. The total indettedness at the end of the second period is therefore 1 x (1 + x (1 + z) = (1 + 0' = r\ At the end of the third period it will have become r^, and at the end of term No. /, r^. Thb Amount. 75. — The sum to which $1 will have increast at compound interest at ^ (or 100/ per cent.) in / periods, is called the Amount, and will be designated as s. We then have the following equation: 5 = rt= (1 + iy 76. — To find the amount of one dollar, raise the ratio to a power whose exponent is the number of periods. 77. — The logarithm of the ratio of increase is the most important logarithm for interest calculations. If the interest rate does not exceed two figures, the logarithm will be found in full in col. B, TF. For convenience we will designate it by a capital letter L. Thus, if i = .065, L will be found opposit Q^ in B. If i = .065 ; log. r = I, = .027 349 607 774,8. 78. — As powers are found by multiplying the logarithm, L must be multiplied by i. r^ nl tl, 79. — To find the amount, multiply the logarithm of the ratio by the number of periods, and the correspond- ing number will be the amount of $1. 80. — Let the interest rate be 3.5%' per annum, payable annually, what will be the amount of $1 at the end of 100 years? Turning to col. B, TF, we find opposit B 35, (or 1.035) the logarithm .014 940 349 792,9. 1.= .014 940 349 792,9 / = 100 /L = 1 . 494 034 979 29 From the characteristic 1, it appears that the amount will be in the tens of dollars ; and as the decimal part of the logarithm is a little more than that which is opposit 31 we know that the amount is $31 and some cents. Thus a rough idea of the amount may be gaind almost instantly. 42 The Doctrin of Interest. 81. — To obtain a more accurate value and one which will be sufi&ciently near for a large principal, we proceed as follows: 82. — In the first place we can only obtain ten correct figures from lOOL. The final figure 9 is never perfect ; it may be 8 . 51 or 9 . 49 or anywhere between. We must, therefore, use only eleven in the logarithm and finally get ten in the number. (?) nl 494 034 979 29 A 31 491 361 693 83 2 673 285 46 B 06 2 597 980 72 75 304 74 C 17 73 823 79 1 480 95 D 34 1 476 60 4 35 E 10 4 34 F 2 1 31 B 06 186 31186 .... C 1 31186 7 2 18 3 2 3119130162 D 3 9357 4 1248 3119140767 E 10 3 1 3 119 140 798 = 5 (/ 100, z .03) 83. — In order to give accurate results up to twelve figures for one hundred interest terms, we have provided on page 32 a special table of the logarithms of the 150 interest ratios (1 + i) which most frequently occur, calculated to 15 places, which allows two places for loss in multiplication. 43 The Present Worth. 84. — The sum which if now invested at i will in / periods amount to $1 is evidently less than $1. It is in the same pro- portion to 1 as 1 is to s. Designating the present worth by /, we have p \\ \\\\ s ox p =— = 5~^ s or the amount and the present worth are reciprocals of each other. A series of amounts reads 1, r\ r% r% r*, r", etc. A series of present worths reads 1, r"\ r~% r % r"*, r~% etc. Reversing the latter series and connecting it with the former we have a continuous series in geometrical progression: r"% r"*, r~^, r"% r"\ 1, r\ r^ r^ r"*, r^ . Using 1.03 as the ratio, the series becomes f' .86260878 r-^ .88848705 r-« .91514166 r"' .94259591 r' .97087379 r° 1. r* 1.03 r' 1.0609 r^ 1.092727 r* 1.12550881 r"^ 1.15927407 In this series, which might be extended indefinitly upward and downward, every term is a present worth of any which follows it and an amount of each which precedes it. .86260878 is the present worth at 10 interest periods of 1 . 15927407 ; 1.12550881 is the amount at eight periods of .88848705. 85. — If any term be multiplied by 1.03, the product will be the next following term ; if it be divided by 1.03 or (which is the same thing) be multiplied by .97087379, the product will be the next preceding term. 44 I^HB DocTRiN OF Interest. 86. — To find the logarithm of the present worth, subtract the logarithm of the amount (for the same time) from zero. In the preceding example, but using L from the 15 place table s nl /fL = 1.494 034 979 293,7 p=^\ls nl —/L = 2.505 965 020 706,3 A 32 505 149 978 319,9 815 042 386,4 B 01 434 077 479,3 380,964 907,1 C 87 377 671935.8 3 292 971,3 D 75 3 257 196,4 35 774,9 K 82 35 612.1 162,8 F 37 160,7 G 49 2,1 32 3 01 032 3 2 032 . . . . C 8 256256 7 224224 32 0-5 986784. . . D 7 22441907 5 1602993 3206010828900 E 8 256481 2 6412 F 3 962 7 22 G 4 2 9 3206011092779 ,0 32060110928=;^ (1.035)^°° to 12 places. The Compound Interest and Discount. 45 87. — That the amount and the present worth are correct reciprocals may be tested by multiplying them together. Taking a few figures of each we have 31 ■ 19 14 .03 2 06 01 935742 62383 1871 1 1 . 00000 Every pair of reciprocals gives a product of 1. The Compound Interest and Discount. 88. — We have hitherto used the word "interest" abstractly as denoting that force or principle which effects the increase of the amount of an indettedness as time goes on. The interest-increment which is thus added is also frequently called " the Interest," which may be v/ritten with a capital letter. 89. — If we take the original principal away from the amount, we evidently have the Interest. For a single period 2 = 1 + z _ 1 = r — 1. When there are more than one period it is the compound Interest, obtaind in the same way and represented by a capital I = (1 + z)^ — 1 = ?-* — 1 == .S — 1. Thus the compound Interest of $1 at 3% per period for 100 periods is $31.19 — 1.00 = $30.19. For two periods it is 1.0609 — 1 = 0.0609. 90. — In the opposit case of a present worth there is a diminution of the principal. The present worth of $1 at 3 per cent, one period, is .97087379; the Discount is not .03, but .02912621, the true principal being not $1, but .97087379, which X . 03 = . 02912621. Representing the simple Discount by d, we have d —\ — p = i y, p = i/s. 91. — If there be more than one term involvd, it is com- pound Discount, which will be represented by D. Thus, at 3 per cent for 5 periods D = 1 — .86261 = .13739. D is also the present worth of the compound Interest for the same time. .15927 x .86261 = .13739. 46 The Doctrin of Interest. In general D = l — p — \p = 1/s, 92.— Thus we see that the variance from par (|1) is called compound Interest or compound Discount, according as regarded from the past or the future point of view and that their properties are as follows: D ^ 1 — ^t and their relation is D = /I ; or I = ^D. Finding Time or Rate. 93.— 'By time we mean the number of periods, terms or intervals, and by this number the logarithm of the interest- ratio is multiplied to produce the logarithm of the amount. (/ X L) /« 5 / X log (1 + i) = log s 94. — If the amount is known and the rate, but the num- ber of periods unknown, we can transform the above equation into this: log s h 95. — At .03 interest, in how many periods will $1 amount to $2, or how long will it take a sum to double itself ? logs = log2 = .3010299956640 I, = log 1.03= .0128372247052 Using only seven places . 0128372) . 3010300 (23.47 2 5 6 7 4 4 442860 385116 57744 49349 8395 The money will double in 24 periods, as it is not quite doubled at 23. 96.' — How many periods must a det of $1 be deferd to be worth now 30 cents, at 3>^% ? Finding Time or Rate. 47 Lo^ 1.035 = .01494035 Log 1.035-' = 1.98505965 Log- .30 = 1.47712125 98 5 05965) 1.47712125( — .01494035)— .52287875(34.9997 4 4 8 2 10 5 7466825 5976140 1490685 1344631 146054 134463 11591 Practically 35 periods. For convenience in division, the minus sign is made to extend over the entire logarithms. Then, as both divisor and dividend are of the same sign, the quotient is positiv. 97. — If the rate be unknown, the equation/ x log. (1 -|- z) = log. s may again be transformed to log.(l + 0='-^. 98. — 20 periods having elapst and the amount of $1 being now $3.20713547, what is the rate? log. 3.20713547 = .506117303 .506117303/20 = .025305865 In 1.06 1 + 1 = 1.06.-. z= .06 The Annuity. 99. — We have now investigated the two fundamental problems in compound interest : viz. , to find the amount of a present worth, and to find the present worth of an amount. The next question is a more complex one : to find the amount and the present worth of a series of payments. If these pay- ments are irregular as to time, amount and rate of interest, the only way is to make as many separate computations as there are sums and then add them together. But if the sums, times and rate are uniform, we can devise a method for finding the amount or present worth at one operation. 48 The Doctrin of Interest. 100. — Annuity. A series of payments of like amount, made at regular periods, is called an annuity, even though the period be not annual, but a half year, a quarter or any other length of time. Thus, if an agreement is made for the follow- ing payments: On Sept. 9 1904 |5100. On March 9 1905 100. On Sept. 9 1905 100. and on March 9 1906 100. this would be an annuity of $100 per period, terminating after 4 periods. It is required to find on March 9, 1904, assuming the rate of interest as 3% per period : First, what will be the total amount to which the annuity will have accumulated on March 9, 1906 ; second, what is now, on March 9, 1904, the present worth of this series of future sums ? It is evident that the answer to the first question will be greater than ;^400, and that the answer to the second question will be less than $400. Amount of an Annuity. 101. — It is easy, in this case, to find the separate amounts of the payments, for the number of terms is very small, and we have already computed the corresponding values of $1.00. The last $100 will have no accumulation, and will be merely $100. The third $100 will have earned in one period |3.00, and will amount to 103 . The second $100 will amount to 106.09 The first $100 (rounded off at cents) will amount to 109 . 27 and the total amount will be $418.36 102. — If, however, there were 50 terms instead of 4, the work of computing these 50 separate amounts, even by the use of logarithms, would be very tedious. 103. — Let us write down the successiv amounts of $1.00 under one another: a Amounts of $1. 1.00 1.03 1.0609 1.092727 Amoun'T of an Annuity. 49 104. — Now, as we have the right to take any principal we choose and multiply it by the number indicating the value of $1.00, let us assume one dollar and three cents, and multiply each of the above figures by 1.03, setting the products in a second colum: a. b. c. Amounts of %\ . 00 Amounts of %\ . 03 Amounts of |0 . 03 1.00 1.03 1.03 1.0609 1.0609 1.092727 1.092727 1.12550881 105. — Our object in doing this was by subtracting colum a from b to find the amount of an annuity of three cents. Before subtracting, we have the right to throw out any num- bers which are identical in the two colums. Expunging these like quantities, we have left only the following: a. b. c. Annuity of $1 . 00 Annuity of $1 . 03 Annuity of |0 . 03 1.00 1.12550881 less 1.00 1.12550881 Amount 0.12550881 That is, an annuity of three cents will amount, under the conditions assumed, to twelve cents and the decimal 550881. Therefore, an annuity of one cent will amount to one-third of .12550881 or .04183627. An annuity of ^1.00 will amount to 100 times as much, or ;^4. 183627, which agrees exactly with the result obtained by addition, in Article 45. 106.— The number .12550881 ( obtaind by subtracting 1 .00 from 1 . 12550881) is actually the compound Interest for the given rate and time, and the number .03 is the single Interest) the amount of the annuity of ^1.00 is .12550881 -f- .03 = 4.183627. This suggests another way of looking at it. The compound Interest up to any time is really the amount of a smaller annuity, one of three cents instead of a dollar, con- structed on exactly the same plan, and used as a model. 107.— Rule. To find the amount of an annuity of ^1.00 for a given time and rate, divide the compound Interest by a single Interest, both exprest decimally . 50 The Doctrin of Interest. 108. — Let S and P represent the amount and the present worth, not of a single $1.00, but of an annuity of $1, then S = I -f- /. Exprest in symbols the reasoning would be this: Amount of annuity of 1 = r*-^ -f r^-{- r^ + r + 1 (a) Multiplying by r. Amount of annuity oi 1 -\- i = r^ -\- r^'^ . .r* + r^-\-r^ ■{• r {b) Subtracting (a) r^-'^ r^ + ^M- r + 1 Amount of annuity of / — r^ — 1 (^:) = r^ — 1 = I Amount of annuity of 1 = S = 1/i 109. — If the number of periods were 50, insted of 4, the advantage of this process, with the use of logarithms, will be very evident. The rate being .03, the logarithm of the ratio, or L = .012 837 224 705 172 50L == .641861235 258,6 Factors, A 43 B 19 C 50 D 34 E 60 F 10 G 14 5= 4 . 38390601876 - 1 I = 3 . 38390601876 I->. 03 = 112. 796867292 = 8 Compare this with the diflSculty of finding the result by arithmetic for even ten periods. Present Worth of an Annuity. 110. — To find the present worth of an annuity, we can, of course, find the present worth of each payment and add them together ; but it will evidently save a great deal of labor if we can derive the present worth immediately, as we have learnd to do with the amount. 111. — The like course of reasoning will give us the result. Take the four numbers representing the present worths of $1.00 at 4, 3, 2 and 1 periods respectivly, and multiply each by 1.03. a. b. Present Worth of Present Worth of Annuity of $1 . 00 Annuity of $\ . 03 .888487 .915142 • .915142 .942596 .942596 .970874 .970874 1.000000 Present Worth of an Annuity. 51 Canceling all equivalents, we have c, . 888487 Present Worth of Annuity of .03 1.000000 1.000000 less .888487 .111513 Annuity of $1 . 00 = . 111513 -H . 03 = 3 . 71710 This is the same result (rounded up) as that obtaind by adding column a. 112.— But .111513 is the compound discount of $100 for four periods, and we therefore construct this rule: 113. — Rule. To find the present worth of an annuity of $1.00 for a given time and rate, divide the compound Discount for that time and rate by a single interest. Symbolically P = D -f- /. We might give this the form P = S -^ 5, being the present worth of the amount of the annuity. 114. — It may assist in acquiring a clear idea of the work- ing of an annuity, if we analyse a series of annuity payments from the point of view of the purchaser. 115.— He who invests $3.7171 at 3%, in an annuity of 4 periods, expects to receive at each payment, besides 3% on his principal to date, a portion of that principal, and thus to have his entire principal gradually repaid. Principal. His original principal is 3 . 7171 At the end of the first period he receives 1 .00, con- sisting of 3% on 3.7171. .1115 and payment on principal .8885 .8885 leaving new principal 2 . 8286 (or present worth at 3 periods). In the next instalment 1 00 there is interest on 2.8286 .0849 and payment on principal 9151 .9151 leaving new principal 1 .9135 Third instalment 1 .00 Interest .0574 on principal .9426 .9426 ^9709 Last instalment 1 .00 Interest .0291 Principal in full 9709 .9709 52 Thk Doctrin of Interest. Thus the annuitant has received interest in full on the principal outstanding, and has also received the entire original principal. The correctness of the basis on which we have been working is corroborated. 116. — It is usual to form a schedule showing the com- ponents of each instalment in tabular form. Total Instalment Interest Payments Payments on Principal Principal Outstanding 1904 Mar. 9 1904 Sept. 9 1905 Mar. 9 1905 Sept. 9 1906 Mar. 1 1.00 1.00 1.00 1.00 .1115 .0849 .0574 .0291 .8885 .9151 .9426 .9709 3.7171 2.8286 1.9135 0.9709 0.0000 4.00 2829 3.7171 • 117. — The payments on principal are known as amorti- zation, which may be defined as the gradual repayment of a principal sum thru the operation of compound interest. It differs from the ordinary compound interest in this, that the new principal for each period is less than the previous one. 118. — As an example of logarithmic evaluation of an annuity, take an annuity of $1, as before, for 50 periods at the rate of .03 per period. At the beginning of the first period, what is its present worth, or what should be paid in one sum for such annuity ? i= .03 r=1.0S nl .012 837 224 705 172 (to 15 places) 60Iy= .641861235 258,6 As we are discounting, not accumulating, we must take _ the cologarithm — 50 L 1 . 358 138 764 741,4 and find the number. Factors A22 B36 C82 DOS El 2 F73 G23 p = 1.03-5 0= .228 107 079 790 D = l — p = .771892 920 210 D-f-.03 =25.729 764 007 =P This may be proved down to maturity by amortization, the schedule beginning thus: Instalment Payments of Payments on Principal Interest at 3% Principal Outstanding 25.729 764 1 1.00 .771893 .228107 25.501 657 2 1.00 .765 050 .234 950 25.266 707 3 1.00 .757 901 .242 099 25.024 608 etc. etc. etc. etc. 49 1.00 .057 404 .942 596 .970 874 50 1.00 .029 126 .970 874 .000 000 Special Forms of Annuity. 53 119. — It may be notist that each payment on principal, or amortization for one period, is the present worth of the instal- ment at the beginning of its period. From this the instalment of amortization may be calculated at any point independently of any other figures. Thus the payment on principal in the 21st instalment of $1 is the present worth of $1.00 in 30 periods, or . 411987 ; because at the beginning of the 21st period there were 30 instalments yet to come. 120. — It will also be notist that each amortization multi- plied by 1 . 03 becomes the next following, these being a series of present worths ; and that thus they may be derived from one another, upwards or downwards. Speciai. Forms of Annuity. 121. — The annuities heretofore spoken of are payable at the end of each period, and are the kind most frequently occurring. To distinguish them from other varieties they are spoken of as ordinary or immediate annuities. 122. — When the instalment (or rent) of the annuity is payable at the beginning of the period, it is called an annuity due, altho ''prepaid" would seem more natural. It is evi- dent that this is merely a question of dating. The instalments compared with those in Art. 103 are as follows: / Immediate Annuity 4 Periods Annuity Due 4 Periods Immediate Annuity 5 Periods 1 1.00 1.03 1.00 1.03 1.0609 1.03 Amounts of / 1.0609 1.0927 1.0609 ( 1.0927 1.1255 1.0927 1.1255 5.3091 —1 4.1836 4.3091 4.3091 To find the amount of an annuity due, for t periods, find the amount of an immediate annuity for t ■\-\ periods and subtract $1. 54 I'hk Doctrin of Interest. 123. — In finding the present worth: Immediate Annuity 4 Periods Annuity Due 4 Periods Immediate Annuity 3 Periods .888487 .915142 .942596 .915142 .942596 .970874 .915142 .942596 .970874 .970874 1.00 2.828612 +1. 3.828612 3.828612 To find the present "worth of an annuity due for t periods find the present worth of an immediate annuity for f — 1 periods and add $1. 124. — A deferd annuity is one which does not commence to run immediately, but after a certain number of periods, as an annuity of 5 terms, 4 terms deferd, which would begin at the fourth period from now and continue to the ninth inclusiv. Its present worth is r^ -\- r-^ -{- r^ -\- r^ -}- r^ An annuity of the entire nine terms would be worth now 1 + ;-! + r-2 + H + r-* + r^ + r^ + r^ + r^ If from this the value of the four deferd terms be subtracted it will leave the value of the deferd annuity. 125. — To find the present worth of an annuity for m terms, deferd n terms, subtract from the value of w + « terms that for n. 126. — A perpetual annuity, or a perpetuity, is one which never terminates. Its amount is infinity, but its present worth can be calculated at any certain rate of interest. If the rent of the annuity is $1 and the rate is . 05, the value of the annuity is such a sum as will produce $1 at that rate or $200, being $1 / .05. The compound discount is the entire $1, being for an infinit number of terms ; therefore the rule still holds : divide the compound discount by the rate of interest. 127. — Annuities at two successiv rates may occur ; say 5 per cent, for 10 years and then 4 per cent, for 10 more. The second part is evidently a deferd annuity, and therefore its present worth is the same as 20 years at 4% less 10 years at 4% + 10 years at 5% The Unit of Time. 55 128.— In all these examples of annuities it has been assumed that the term or interval between payments is the same length of time as the interest-period. For example, the rate of interest may be so much per year, while the payments are half-yearly or quarterly ; or there may be yearly payments while the desired interest-rate is to be on a half-yearly basis. We shall defer the treatment of these cases until the subject of nominal and effectiv rates has been discust. 129. — There may also be varying annuities, where the instalment changes by some uniform law. These seldom occur in practice. Where the change is simple, as in arithmetical progression, the annuity may be regarded as the sum of several annuities, otherwise the values must be separately calculated for each term. An annuity running for 5 terms, as follows : 13, 18, 23, 28, 33, may be regarded as (1) an annuity of 13 of 5 terms ; (2) an annuity of 5, 4 terms ; (3) an annuity of 5, 3 terms ; (4) an annuity of 5, 2 terms ; (5) a single amount of 5. The Unit of Time. 130. — It makes no difference in the result whether each term is a year, or a month, or a day, so long as the number of terms (/) and the rate per term {t) are ascertaind. But unfor- tunately the habit has been fixt in common speech of stating the rate, not at so much per term, but so much per annum, even when the interest is payable or chargeable semi-annually (which is the prevalent custom), or quarterly, or monthly, 131. — When we refer hereafter to a nominal rate per annum, we shall write "per cent." in full, using for actual rates per period the symbol % or the decimal. The letters a, s, q, or m, will stand for "payable annually," "semi- annually," "quarterly," or ** monthly." 132. — The following phrases need interpretation into more exact language: (a) ** Six per cent, per annum, payable annually," means what it says : six per cent, per term, the term being a year. {F) "Six per cent, per annum, payable semi-annually," means three per cent, each half year ; which is more than six per cent, per year. 56 Tun DocTRiN OF Interest. (c) "Six per cent, per annum, payable quarterly,'' means one-and-one-half per cent, per term of three months. (d) ** Six per cent, per annum, payable monthly," means one-half per cent, per month. 133.— In cases (^), (c) and (d), the ''6" is fictitious. The ratios which must be used are 1.03, 1.015 and 1.005, not 1.06 at all. *' Six per cent." is known as the nominal rate, but the effectiv rate for the entire year is different. Taking up the above four cases: (a) Here the nominal and the effectiv rate are identi- cal; .06. (3) Here the effectiv rate is .03 per half year; for the year .0609. (c) Here the effectiv rate is . 015 per quarter ; for the year .06136355. (d) Here the effectiv rate is .005 per month; for the year .06167781. 134. — Thus the words ** six per cent, per annum " have four different meanings, according to the qualifying phrase used, or understood. Let / represent the nominal rate "per annum," i being the rate per term, and k the effectiv rate per year. Then in (a), where r = 1.06 and / = 1, 1 + y^ = 1 -l-y= 1.06 In (5), where r = 1.03, and / = 2, 1 + ^ = ^-^^ (1 + y2jy= (1.03)^= 1.0609 In (c) , where r = 1 . 015 and t = 4, l-{.k = r'=(l + }ijy= (1.015)*= 1.06136355 || In (flO, where ?- = 1 . 005 and if == 12, l-\-k = r^'= (1 + i^y)^^= 1.005)^^= 1.06167781 1| These values may be ascertained by logarithms or by arithmetic. 135. — Case (3) furnishes an arithmetical solution which is very convenient. Expanding (1 •\-j/^y by the binomial theorem we have 1 +/ +/ V4. To the nominal rate the quarter of its square is to be added to give the effectiv rate if compounded The Unit of Tims. 57 at half periods. Thus at 6% for 7, .06^ = .0036, .0036/4 = .0009; .06 + .0009 = .0609. At 8%, .08^ = .0064; .0064/4=1 .0016. k= .0816. 136. — The rate k being =/ -{-j^/4, we may factor this, making ity(l +y/4). 1 +y/4 is thus a multiplier, reducing the nominal rate payable semi-annually to an effectiv annual rate. For six per cent, this multiplier would be 1.015, (. 0609 / . 06) ; for five, 1 . 0125 ; for four, 1 . 01 ; for 3%, 1 . 0075 ; for 2%, 1.005. The same reasoning applies to a nominal half- yearly rate, payable quarterly. If 3% is i for the half-year, 3 (1.0075) is y for the half year with quarterly payments, or 3.0225. 137. — But the annual rate given may be the effectiv rate (z) and the question be, what rate (y) will be equivalent for the case of more frequent payments, giving k as the nominal rate per annum for that frequency. Case {a) is the same as before. Case (3) 1 +y =: (1 + 0^= (1 + y2k) For i = 6%, l-}-J=:(1.06y/-=l. 02956301 ; and /^ = 2j = . 05912602. That is^ to produce 6% payable annually, we must invest at 5.912602% per annum, payable semi-annually, or 2.956301% per period of six months. (c) 1 + y == (1 + t)H =a + H^) For / = 6%, l^ k = 1.05869538, payable quarterly. (d) 1 +y =: (1 + i)^ = (1 + iW For i = 6%; k = .058269, payable monthly. 138. — In annuity calculations the period or interval between cash payments is to be considerd as well as the fre- quency of compounding the interest. Here, also, the terms are reduced to the **per annum" standard. An annuity of $50 per half year is usually spoken of as an annuity of $100, payable semi-annually. What the actual value of the yearly revenue is, depends upon the rate of interest assumed in the problem. 139. — If ^a represents the instalment or "rent" of the annuity for each half-year, and i the rate of interest for the half-year, the equivalent of these two cash payments for the 58 Ths Doctrin of Interest. year will be Yza ■}- }4a (1 -^ i) = a + }4ai = a (1 -f K^')- If y is the nominal rate per annum or 2i, then the annual effectiv payment is a (1 -\- j/^) and 1 +y/4isa multiplier for transforming a yearly annuity into a half-yearly one. This is the same multiplier which was alredy found to transform a yearly nominal rate of interest, compounded semi-annually into its corresponding ejffectiv rate. This multiplier, 1 +y/4, will be found important in practis. It may be called the co-efl5cient of double frequency, or C^^^ The ^^^ represents the ratio of the frequency of compounding to that of payment. 140. — If the rate of interest is 3%" per half year (6 per cent. , s) and the annuity payment $1 per annum, to find the amount of the annuity for four years, we may reduce the interest to the annual standard, the cash being alredy there. The annual equivalent of the rate is .0609 (6 x 1.015). Twice the logarithm of 1 . 03 . 012837224705 is log. 1.0609 = .025674449410. The first step is to find the amount, for which purpose the logarithm is multipHed by 4, . 1026977976400. This is also 8 times the logarithm of 1.03, so that we gained nothing by squaring 1.03. From either view the amount is 1 .26677008 and the compound interest is .26677008. This is next to be divided by the rate of interest, which is not . 03, nor .06, but .0609. .0609). 26677008(4. 3804601, amount of annuity. 2436 2317 1827 4900 4872 2808 2436 372 365 The Unit of Time. 59 141. — We may test this result as follows: End of first year ; cash 1.00000 third half year ; interest .03 on 1.00. . .03 ♦• second year; " .03 on 1.03.. .0309 " •• " cash 1 .00000 Total 2 . 0609 End of fifth half year ; interest .03 on 2.061 . . .06183 '• third year; interest .03 on 2.123. . .06369 cash _1^ Total 3.18642 End of seventh half year ; interest . 03 on 3 . 1 86 .09558 •• fourth year ; interest .03 on 3.282 .09846 cash J_^ Total 4.38046 142. — We may simplify this method a little further. Had we made the instalment 60 cents each half year, the compound interest would have been half as much, or . 13338504. This would have been divided by .03, giving 4.446168. It would have been the same had we divided the compound interest of $1 by .06. But we did divide it by .0609, which is .06 X 1.015, the latter being the coeflficient of double fre- quency. We might, therefore, have divided the amount of the annuity when payable semi-annually by the C^^^ 4.446168/1.015 = 4.38046 143. — Therefore, an annuity payable annually is trans- formd as to its amount into one payable half-yearly by multi- plying it by the C^^). 144. — The present worth of the annuity is subject to the same law ; when the annual payment is divided into two equal sums its present worth is increast in the ratio of 1 +y/4 or 1 -j- z/2. In the case given above the logarithm . 1026977976400 would have been changed to its cologarithm 1 . 8973022023600 the number of which would be the present worth . 789409234 The compound Discount would be .210590766 and the rate or divisor as before .0609 giving the present worth of the annuity as 3 . 45797645 6o The Doctrin op Interest. 145. — The correctness of this may be demonstrated as follows : Amount invested in annuity 3.45797645 Half-year's interest on 3.457976 + .10373929 3.56171574 Half-year's interest on 3.561716— . 10685147 3.66856721 Annual instalment 1.00000000 2 . 66856721 Half -year's interest on 2.668567+ . 08005702 2 . 74862423 Half-year's interest on 2.748624+ .08245873 2.83108296 Annual instalment 1.. 00000000 1.83108296 Half-year's interest on 1.831083— .05493249 1.88601545 Half-year's interest on 1.88 6 015+ . 5 658046 • 1 . 94259591 Annual instalment i 1.00000000 .94259591 Half-year's interest on .942596— .02827788 , . 97087379 Half-year's interest on '.,9 708738+ .02912621 1 . 00000000 Ivast instalment 1.0000000 146. — Had the payments been half-yearly, each being 60 cents, the compound discount would have been 105295383 and we should have divided by .03, giving 3.5098461 Dividing by the C^^) i oi5, we should again have the value 3.457976 + 147. — The conclusion is that there are two ways of calcu- lating the amount or present worth of an annuity where the interest compounds with twice the frequency of the cash payments. (1) Procede as if both were at the greater interval, tak- ing care to use the effectiv rate of interest in dividing. (2) Procede as if both were at the smaller interval, the instalment being half as much and divide the result by the C^^). The Unit of Time. 6i 148. — Where the interest-period is greater than the pay- ment-period, or the payments are made twice as frequently as the interest is compounded, the solution is less easy because it depends on evolution. 149. — Half of the instalment is paid when only half the interest-period has elapst. It may be considered as earning interest for the other half -period, but the rate must be taken effectivly. Thus, if the interest for a period is .03, the ratio for the half-period is the square root of 1.03, or 1.01488916. The half-instalment paid at the half- period becomes at the end of the full period 50 X 1 .01488916 = .50744458 The other half-instalment is only .50 and the total of both is 1.00744458 This is the effectiv instalment, insted of the nominal instal- ment, $1. It is a coejB&cient of frequency as to payments, and may be represented by C^^), meaning that the interest is com- pounded only half as often as a payment is made. 150. — If the period of compounding is the half year at .03 per period (a nominal rate per cent, of .06), the effectiv rate per quarter is 1.01488916 and the G^^) is 1.00744458, being half the square root of 1 + half the nominal rate. If the nominal rate is 3 . 8 per cent, (s), take first the logarithm of 1.019 008 174 184 006,4 and divide by 2... 004 087 092 003,2 The number corresponding to this is ^ 1.009 455 299 halving the decimal part . 1 . 004 727 649 is the C^^^ for a rate of 3 .8 per annum (.y), payments quarterly, or$l (^). 151. — The coefficient of frequency (C^^O has to be com- puted at the commencement of each problem by the above method. 152. — The computation of the amount, or of the present worth, as the case may be, then goes on just as if the payments took place at the same times as the compoundings. When completed, the result is multiplied by O^K 153. — When the interest-period is semi-annual and the instalments are paid quarterly, it is better to ignore the ' * per 62 The Doctrin of Interest. annum" rate and treat of the periods (half years) and half periods (quarters) , after the commencement, 154.— An annuity of $2 per annum, payable quarterly, interest to be compounded semi-annually, for 2 years at 3i^ per cent, per annum, would be stated as an annuity of $1 per period, payable by half-periods interest at lA^ per period, and continuing for four periods. The present worth of this annuity, omitting the condition ''payable by half -periods," would be 3.81698703, which x 1.00472765 is 3.8350324, the present worth when the annuity is paid at the quarters or half-periods. Tested as follows: Present worth 3 . 8350324 Interest at .019 .0728656 3.9078980 First and Second Instalments, with interest on the first 1.0047276 -|- 2 . 9031704 Interest at .019 .055 1602 2.9583306 Third and Fourth Instalments, as before. . . 1 .0047277 1.9536029 Interest at .019 .0371185 1 .9907214 Fifth and Sixth Instalments 1.0017276 .9859938 Interest at .019 .0187339 1 . 0047277 Seventh and Eighth Instalments 3 .0047277 The C<^) being almost exactly 1.00472765, it is taken alternately as 1.0047276 and 1.0047277. 155. — Values of C^^) for all ordinary rates are found by taking half the decimal part of the figures under ' ' Square Root " in Table VI. of the Text Book of the Accountancy of Investment, Part III, the ** 1" remaining where it is. 156. — To find the amount or the present worth of an annuity where half of each instalment is collected midway of the period, procede as if the entire instalment were collected at the end and then multiply the result by C^^\ being 1 + ^ (^rr^'-i). Fractionai. Periods. 63 157. — In some theoretical computations interest is con- ceived as compounding momently or continuously. Interest at 6% per annum, when compounded momently, gives an equiv- alent effectiv rate of .061837. This is obtained by multiply- ing the rate .06 by the constant quantity .4342944819 (or as many figures as required); considering this as logarithm, its number will be the ratio sought, 1.061836546539. If .06 is the effectiv rate and it is desired to find the nominal rate, multiply the logarithm of the ratio (L) by the constant quan- tity 2 . 302585092994, or so much as required and the result will be the nominal rate. log. 1.06 = .0253058652648. This X 2.30585092994 = .0583689 + . These constants depend on the Naperian logarithms. Fractionai. Periods. 158. — We have hitherto treated only of entire periods, but it is quite usual that the number of periods should be a mixt number, sometimes a fraction only. 159. — A det is due in one year from now, at six per cent, annually ; but the dettor has the privilege of paying at the half year ; what interest should he then pay ? There are two answers to this question, depending on whether it is to be con- siderd legally or equitably — by simple interest or by compound interest. 160. — Legally, the rate is .03 per half year, the law not recognizing the justice of compound interest. Equitably, that is not the true proportion in which the interest should be divided. The creditor gets, not six per cent, annually, but six per cent, semi-annually, which we have seen to be more profitable. 161. — The compound interest for a half term is at the rate of .02956301 only, not .03. Compound interest for several periods is greater than simple interest ; conversely, for part of a period the compound interest is the lesser. 162.— If the det spoken of is $1,000,000 and is discharged at midyear by a payment of $1,030,000, the creditor has the 64 The Doctrin of Interest. use for six months of $30,000, at some rate from which the dettor has no benefit, besides the use of the $1,000,000 to which he is entitled. 163. — If interest were not a constant force, but a periodi- cal incident, there would be no such thing as interest between the periodical dates ; one would have to pay a full period or nothing. 164. — The result of this inconsistency is that, conven- tionally, when interest is calculated on a certain number of terms and a fraction of a term, the interest compounds for the integral terms, but remains simple during the fraction of a term. 165. — For four-and-a-half years on the conventional inter- est plan at six per cent, annually, the compound interest must be calculated for four years ; amount 1.26247696 then this must be multiplied by 1.03 (the con- ventional ratio for the half year) producing 1.30035127 This number is exactly midway, arithmetically, between the amount at four years 1.26247696 and that at five years 1.33822558 This plan of dividing the difference in proportion to the time elapst is generally used where the even periodical values can be obtaind from tables, especially in case of valuation of bonds, as will be shown hereafter. 166. — In scientific interest, the ^ forms part of the num- ber of terms. The log. 1 . 06 025 305 865 264,8 being multiplied by 4.5 gives 113 876 389 191,6 the number for which is 1 .29 979 957 070 This result might have been obtaind by multiplying the 4 year amount 1 . 26 247 696 by the inconvenient number 1 . 02 956 301 which is the effectiv ratio. 167. — When annuities are to be sumd or valued, it is necessary to get the value for the entire terms first and then multiply by the effectiv rate for scientific interest ; for conven- tional interest either multiply by the conventional rate, or "split the difference," according to time elapst. It is impos- sible to value or sum the annuity in one operation by a fractional multiplier, for the reason that these processes depend entirely on a uniform ratio. Sinking Funds. 65 168. — It is the universal custom in actual business to treat parts of terms by simple interest, not by compound ; conven- tionally, not scientifically. Sinking Funds. 169. — We have hitherto assumed the periodical instal- ment, or rent of an annuity, to be 1. When this is some other number, the amount or present worth of $1 is multiplied by that other number ; that is, the amounts (or present worths) are directly proportionate to the rent. But sometimes we have given the amount or the present worth as a fixt sum and wish to find an instalment which will produce that amount or ex- tinguish that present worth. 170. — We have seen that the amount of an annuity of $1 at 3% for 50 periods is $112.79687. If the amount were $1000 insted of $112.79687, it is evident that each instalment must be increast as many times as $112 . 79687 is containd in $1000. The quotient is 8.8655. Therefore, under the same conditions where $1 amounts to $112.79687, $8.8655 will amount to $1000. If the growth of the two annuities be com- pared it will be seen that at any point the one which is to accumulate to $1000 is 8 . 8655 times as large as the one which accumulates to $112.79687. Instalments Instalments of of $1 $8.8655 8 . 8655 . 2660 8 . 8655 1 . 0000 . 0300 1 2 . 0300 . 0609 1 3 . 0909 . 0927H- 1 17.9970 . 5399 8.8655 27 . 4024 .8220 8 . 8656 4 . 1836 37 . 0899 etc. etc. Therefore, to find the instalment which contributed each period, will amount to a given sum S, divide S by the amount of an annuity of $1. 66 The Doctrin of Interest. 171. — Where an annuity is so constructed that it shall accumulate to a certain amount at a certain time, it is called a sinking fund. Frequently the uniform periodical contribu- tion is itself calld the sinking fund, and is found in the fore- going manner. 172. — Where the present worth is the quantity given, the process of finding the uniform contribution which will gradu- ally extinguish or amortize that present worth by the aid of compound interest is similarly performd. The fixt quantity is the present worth of an annuity of x dollars ; the given present worth divided by P, the present worth of $1 gives the instal- ment, x^ necessary to amortize it. 173. — It is required to find what annual payment will clear off $1000 in 50 periods, allowing .03 interest. We have alredy calculated that a payment of $1 per period will pay off $25 . 729764, with interest. $1000 is 38 . 8655 times $25.729764 ; therefore the contribution must be $38 . 8655 per period, which will, by forming a schedule, be found to amortize the $1000. 174. — As a provision for liquidating indettedness, or for replacing vanishing assets, sinking fund and amortization are two different applications of the same principle. Formerly, the terms were used interchangeably, but more recently they are distinguisht as follows: 175. — The sinking fund permits the det to stand till maturity, but in the meantime provides a fund which at maturity pays off the entire det, the Interest on the original sum being paid separately. 176. — The amortization plan accumulates nothing, \iU\. gradually reduces the det, applying to this reduc- tion all the excess of the contribution over the Interest. 177. — The two operations which we have performd show that the sums necessary to be set aside for a det of $1000 dur- ing 50 periods at . 03 are, Sinking Fund $ 8.8655 Amortization 38.8655 The difference is the $30 per period, which on the sinking fund plan is required to pay the current interest, so that actually the two methods of con- tribution come to the same thing. Interest-bearing Securities. 67 178. — The number of terms necessary for a certain contri- bution per period to amount to a certain principal may be found, but first the amount of a single dollar must be found. The amount of the annuity is I/z, the total compound interest divided by the rate of interest. Multiplying that amount by the rate gives, therefore, the compound interest. Adding to this $1 we have the amount of a single dollar, ^ or S X z -h 1. We then proceed as shown in Art. 93. Similarly the present worth of the annuity being D/t, /> = 1 — P X z, and / may be deduced therefrom. 179. — The rate of interest oi an annuity cannot be ascer- taind by any direct formula, as it involvs the solution of equations of higher degrees. 180. — A special method for finding the income-rate of securities by gradual approximation will be given hereafter. (Art. 231). Interest- BEARING Securities. 181. — A bond (which is the most usual form of interest- bearing security) is a complex promise to pay: 1. A certain sum of money at a future time ; this is known as the principal, the par or the capital. 2. Certain smaller sums, proportionate to the principal, and at various 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. 182. — These various sums are never worth their face or par until the stipulated times arrive, but are always at a dis- count. The principal is never worth its face until its maturity; the coupons are never worth their face until the maturity of each. Yet while both principal and coupons are at a discount, the aggregate may easily be worth more than the par, and it is the aggregate, principal and coupons which is the subject of the valuation. 183. —If the bond is sold at par, the coupon and the in- terest are equivalent. Take a five per cent. (5) bond for $10,000, due in 5 years, at par. Its value consists of 68 Thb Doctrin of Interest. EXAMPLE 1 1. The present worth of $1000 at 10 periods at .025 781 . 1984 2. The present worth of an annuity of |25 per period, 10 periods 218.8016 Aggregate 1000.0000 EXAMPLE 2 But if the coupons were $30 each, the bond being ** six per cent," the principal would still be valued at : 781 . 1984 while the coupons would be worth 262 . 5619 Aggregate 1043.7603 EXAMPLE 3 If the bond were a "four per cent." bond, the coupons being $20 each, the valuations would be, principal 781 .1984 coupons 175.0413 Aggregate 956.2397 All the above calculations may be made by logarithms, commencing with the logarithm (L) of 1 . 025. 184. — From these computations we may draw the follow- ing inferences : 1. If the coupon rate is the same as the income rate, the bond is at par. 2. If the coupon rate is greater than the income rate, the bond is worth more than par. 3. If the coupon rate is less than the income rate, the bond is worth less than par. 185. — Rule I. Any bond may be valued so as to earn a given interest rate by adding together 1. Present worth of the principal ; 2. Present worth of the annuity, consisting of all the coupons. 186. — Representing the coupon rate or the proportion which the coupon bears to the principal, by c and the value of the bond for $1 by V 1 — /-* t r^ is the only quantity which requires logarithms for its com- putation, which always begins with L, the logarithm of r. /L is the logarithm of r^ and subtracted from zero is the loga- rithm of r^. In the above example Interest-bearing Securities. 69 I, or log. (1 + or log. 1 . 025 = . 010 723 865 391,8 log. 1.025^° = /L = __.107 238 653 918 log. 1025-^ « = 1.892 761 346 082 1.892 761 346 082 /« .781198 401727 Substituting the above value for r-^ will give the results in Examples 2 and 3. 187. — In the second and third case the correctness of the figures may be proved by forming a schedule of amortization, which, starting with the present value, will bring the value, up or down, to par at maturity. Six Per Cent. Bond, Net Income 025. Coupons Interest at .025 Amortization 1043.7603 30. 25.0940 3.9060 1039.8543 30. 25.9964 4.0036 1035.8507 30. 25.8902 4.1038 1031.7469 30. 25.7937 4.2063 1027.5406 30. 25.6885 4.3115 1023.2291 30. 25.5808 4.4192 1018.8099 30. 25.4702 4.5298 1014.2801 30. 25.3570 4.6430 1009.6371 30. 25.2410 4.7590 1004.8781 30. 25.1219 4.8781 1000.0000 300. 356.2397 43.7603 Four Per Cent. Bond, Net Income .025. Coupons Interest at .025 Amortization 956.2397 20. 23.9060 3.9060 960.1457 20. 24.0036 4.0036 964.1493 20. 24.1038 4.1038 968.2531 20. 24 2063 4.2063 972.4594 20. 24.3115 4.3115 976.7709 20. 24.4192 4.4192 981.1901 20. 24.5298 4.5298 985.7199 20. 24 6430 4.6430 990.3629 20. 24.7590 4.7590 995.1219 20. 24.8781 4.8781 1000.0000 200. 243.7603 43.7603 70 Thk Doctrin of Interest In the six per cent, example the amortization is subtracted ; in the four per cent, example the amortization of discount (called also accumulation or accretion) is added. The figures in the two amortization colums are identical. 188. — This is the most natural method of valuation, and for one who only occasionally employs it, perhaps the safest. There are other methods which in practis are briefer. 189. — The excess over par in the second example (six per cent, coupons), $43.7603, is known as premium. In the third example (four per cent, coupons), the value is less than par and the difference is known as discount, a word which has several meanings. When I have occasion to speak of both premiums and discounts I shall use the word variance ; that is, variance from par. 190. — The difference between the coupon-rate, or cash- rate, and the interest-rate, or income-rate, is the sole cause of the variance. This difference will be called the interest- difference. . 025 being assumed as the interest-rate, and the coupon- rate .03 or .02, the interest- difference is .005. 191. — Where the coupon is .03, the .025 may be con- sidered as interest on $1, and each .005 is a future benefit or extra profit, which should be paid for. Reduced to present values, these benefits are the present worth of an annuity of .005 per period. 192. — The present worth of an annuity of .005 for 10 periods is .0437603, or on $1000, $43.7603, the same variance as found by the previous process. 193. — Rule II. The variance is the present worth of an annuity of the interest-difference. When the coupon-rate is the greater, the variance is added to the par ; when the coupon- rate is the less, the variance is subtracted from par. 194. — Representing the variance by Q, the second rule may be exprest as follows: 1 — ^' or ^^ (1 — r-') Intkrbst-bearing Securities. 71 195. — Multiplying both numerator and denominator by 200 will not alter the value of the fraction, hence it will be the same thing if we use the nominal annual rates. Insted , .03— .025 6 — 5 , . , . of ^r^ we may use — = — which is easier. In the above example, No. 2, the variance would be obtaind thus : Q = 5-=-^ (1 — .781198401727 o V = 1 + ?-=^ (1— .78119840172/ 5 = 1+ i (. 218801598273) o = 14- .0 43760319655 = 1.0 437603 + In the third example, where c = 4 Q = ^^^ (.2188016) = — i (.2188016) o V = l— .0 437603=. 9 562397 The results may be carried to 11 or 12 decimals if desired. 196.— A third method (suggested by Mr. Arthur S. Little) is based upon the value of a perpetual bond. This, as there is no redemption, is merely a perpetual annuity, or perpetuity of the ' ' coupon. ' ' The value of such a perpetuity is c/i. A six per cent, (s) bond to pay five per cent. (5) is .03/.025 = 6/5 = 1.20, and this value is perpetual, there being no redemption. But if it is known that the variance (.20) will vanish 10 years from now, the value of the bond is now lessend by the present worth of that variance. 197. —Rule III. The terminant value is the perpetuity value, minus the present worth of the perpetuity variance. v= ^_(£_i)(^.) In Example 2, the perpetuity value is 6/5 = 1 . 20 The present worth of the vanishing quan- tity .20 is 1562397 Remainder 1.0437603 72 The Doctrin of Interest. Here the first step is to obtain the perpetuity value by simple arithmetic. The variance is, of course, .20. Then r-t==.781 198 401 727 is found as usual and multiplied by .20, giving 156239680+ In Example 3, the perpetuity is 80 but the variance is still .20, and its present worth is .15623968 which is added, making 95623968 because minus a minus is plus. 198. — Multiplying down. Whichever of these three methods has been employd for ascertaining the value of the bond at a certain date, if the successiv values for each period are expected to be required (and they usually are) , it is pre- ferable to find them by schedules of amortization rather than resort to independent logarithmic calculation for each. A thoro test of the correctness of all the intermediate values is the fact that the series reduces to par at maturity. In this test, insted of a formal schedule in colums, all the figures may be brought into a single colum, so that no marginal computa- tion may be needed. The amount of each amortization is not exprest, but implied, in the following example of such a single- col um schedule : A four per cent, (j) bond for 4 years to net three 96/100 (5). The r is 1.0198: 1.0198 nl .008 515 007 6315 X8 _.068 120 061 052 Subtract from zero : 1 .931 879 938 95 (A85 B05 C67 D93 E75 F27) Present worth of |1 854 830 361 69 Compound discount 145 169 638 31 Divide by .0198 7.331 799 914.75 Multiply by interest difference 000 2 Premium .001 466 359 98 Value 1 .001 466 359 98 .0198 .01 010 014 663 60 .009 009 013 197 24 .0008 .000 801 173 09 1.021 295 393 91 Coupon 02 1.001295 393 91 MuivTiPLYiNG Down. 73 We save a number of figures by adding and subtracting at the same time, putting a circle round the coupon to indicate subtraction : Resuming 1 . 001 295 393 91 .01 10 012 953 94 .009 9 011658 55 7)801 036 32 & 1.001121042 72 The operation may be still further abridgd by amortizing the premium only, but subtracting the interest-difference only, not the entire coupon : .001 121041 72(1) 11 210 43 ^0 089 38 (2) 896 83 943 239 36 (2) 9 432 39 ^^8 489 15 (2) 754 59 761 915 49 (3) 7 619 15 6 857 24 (2) 609 53 577 001 41 (4) 5 770 01 5 193 01 (V) 46160 388 426 03 (5) 3 884 26 ^^3 495 83 (2) 310 74 196 116 86 (6) 1 961 17 1 765 05 (2) 156 89 Error in decimals 03 .000 000 000 00(7) The (2) is two places further to the right than in the first procedure. 199. — Computing Amortizations. It may sometimes be advisable to find and verify at first the instalments of amorti- zation, leaving this series of amounts to stand, not filling in the remaining colums of the schedule, until required. The obtaining of the amortization colum is remarkably easy, as shown in Art. 119. 74 The Doctrin of Interest. 200. — Starting with the premium or discount at t periods, as above explaind, it is next amortized to the extent of p that is the present worth of the interest-difference. This mul- tiplied by r gives the next amortization, — ^ and so on down. 201. — In the last example, the premium at 4 years was found to be 001 466 359 98 The next amortization is simply the present worth at 8 periods of . 0002. The present worth of $1 has been found to be .854 830 36169; this x .0002 = .000170 966 072 238 Three figures may be dropped from this with safety, leaving the decimals as much extended as in the previous operation. 1.000 000 170 966 072 (1) .01 1709 661 .009 1538 695 .0008 136 773 174 351 210 (2) 1 743 512 1 569 161 139 481 177 803 355 (3) 1 778 034 1 600 230 142 243 181 323 862 (4) 1 813 239 1 631 915 145 059 184 914 075 (5) 1 849 141 1 664 227 147 931 188 575 374 (6) 1 885 754 1 697 178 150 860 192 309 166 (7) 1 923 092 1 730 782 153 847 . 000 196 116 887 (8) These are the eight instalments of amortization, which, added together, should equal the total premium. Computing Amortization. 75 .000 170 966 07 174 351 20 177 803 35 181 323 86 184 914 07 188 575 37 192 309 17 196 116 89 .001 466 359 98 But if this test were not available, the last amortization would have to be multiplied up by 1 .0198, which should produce .000200. .000 196 116 887 1 961 169 1 765 052 156 893 .000 200 000 001 202. — Small discrepancies in the last figure are to be ex- pected and disregarded ; therefore, the decimals should be carried beyond the figures which are to be utilized. 203. — Discounting. A series of values in reverse order, beginning from maturity, may be obtaind (without using logarithms) by division, the interest-ratio being the divisor. The entire amount to be receivd on the above bond is: princi- pal $1, coupon .02, total $1.02. This should be divided by 1.0198. 1.02 -V- 1.0198 = 1.000 196 116 89 which is the value 1 period before maturity. The coupon .02 must be added before the second discounting process. 1.020 196 116 89 ^ 1.0198 = 1.000 388 426 03 .02 1.020 388 426 03 1.020 388 426 08 -f- 1.0198 = 1.000 597 001 41 This is a laborious process, even if, insted of dividing, we mul- tiply, by the reciprocal of 1 . 0198, . 980584428. 76 The Doctrin of Interest. 1.02 -|- coupon .980 584 428 .019 611 689 1.000 196 117 .02 1.020 196 117 .980 584 428 .019 611 689 098 058 88 253 5 884 98 10 7 1.000 388 427 Table of Multiples 1. .980 584 428 2. 1.961168 857 3. 2.941753 285 4. 3.922 337 713 5. 4.902 922 142 6. 5.883 506 570 7. 6.864 090 998 8. 7.844 675 427 9. 8.825 259 855 204. — Intermediate Purchases. It happens very often (perhap? in a majority of cases), that bonds are not purchast on the very day when the interest is payable. In the preced- ing examples it was supposed that exactly 8 or 10 periods would elapse from the purchase of the bond till its maturity ; but the purchase may have been a month, or several months, or months and days, after the beginning of the period. 205. — We saw (Art. 167) that an annuity cannot be valued by the usual formula when the number of terms is a mixt number. We must derive it from the next regular term or interpolate it between the two nearest. It was also explaind that this interpolation maybe done in two ways: one by simple proportion, conventionally, or by compound interest, scientifi- cally. In the present state of knowledge the conventional or non-scientific method is establisht by usage, altho it works injustice to the buyer. The difference is usually not very large. 206. — Let us suppose that in the above Examples, in Art. 182 the purchase had been made 9}^ periods before maturity, that is, 4 years 9 months. Intermediatf Purchases. 77 The value at 10 periods (Ex. 2) is. . . 1043.7603 the value at 9 periods is 1039.8543 the difference, or amortization, is 3 . 9060 As half the time has elapst, we assume that half the amortization has taken effect 1.9530 and this we subtract from the 10- period value 1043.7603 making 1041.8073 which is exactly half way between the 9-period and the 10-period values. Besides this, however, the purchaser must pay one-half of the current coupon, or $15.00, as "accrued interest," the entire cost being 1056.8073. This is called the flat price, and formerly this was the form in which securities were quoted at the Exchanges ; now, however, the quotations are understood as so much "and interest," meaning that the accrued interest is made a separate item in the bill. 207.— Had the 10-year value been multiplied by 1.0125 (half the interest rate) the same flat value would have been obtaind. 1043.7603 X 10125 = 1056.8073 208-— To apply the scientific plan, the 1043.7603 would have been multiplied by 1.01242284, giving 1056.7267 insted of 1056.8073 a difference of .0806 in favor of the purchaser, but he could not claim it under the law and the customs of the market. 209. — As the usual period is divided into six months, and as the odd days are considerd as thirtieths of a month, the amortization for each day is 1/180 of that for a half year. If 2 months, 17 days had past after the interest-date, then 1.9530 must be multiplied by 77/180, giving .8359 as the proportion- ate amortization for 77 days. 210. — Thus the value of a bond at any date from its issue to its maturity, at some given rate of interest, may be calcu- lated by first valuing it at at two consecutiv interest dates, according to the rules given, and then "splitting the differ- ence" by dividing it into 180ths. 78 The Doctrin of Interest. 211. — Even if the unit of time employed in the coupon- payments and the interest-compoundings be different, the rules given in the section entitled " The Unit of Time " will enable these to be allowed for, if Rule I (Art. 185) is used for valuing. 212. — Intermediate Balances. — When the regular interest-periods do not coincide with the date of the balance sheet, it becomes necessary to adjust the valuations for that purpose in the manner just described for purchases at odd times. 213. — If the interest-dates are May 1 and November 1, and the dates for balancing are January 1 and July 1, the bond must, on January 1, have been amortized to the extent of one- third of that from November to May, conventionally. 214. — A six per cent, (s) bond for $1000, to yield five per cent. (^) due Nov. 1, 1925, is worth on November 1, 1920 1043.7603 on May 1, 1921 1039.8543 the amortization for 6 months is, therefore 3.9060 For 2 months it is one-third 1 . 3020 and this subtracted from 1043.7603 leave the value on Jan. 1, 1921 1042.4583 Applying the same method between May 1 and Nov. 1, 1921 1039.8543 1035.8507 3) 4.0036 1.3345 1039.8543 we have the balance, July 1, 1921 . . 1038.5198 If we take the January value 1042 . 4583 and multiply down ; x .025 26.0615 1068.5198 — Coupon 30. we also get the July result 1038.5198 Thus we have the choice of interpolating each balance- value, or having obtaind one, of multiplying down to maturity, Intermediate Bai^ances. 79 which can be done on the conventional, but not on the scientific plan. The resulting schedule would be as follows: Date Collected Interest at .025 Amortization Value Jan. 1,1921 .... Value at 5 per cent. basis 1042.4583 Julyl, " .... 30.00 26.0615 3.9385 1038.5198 Jan. 1, 1922.... 30.00 25.9630 4.0370 1034.4828 Julyl. - .... 30.00 25 8621 4.1379 1030.3449 Jan. 1, 1923.... 30.00 25.7586 4.2414 1026.1035 Julyl, " .... 30.00 25.6526 4.3474 1021.7561 Jan. 1, 1924.... 30.00 25.5439 4.4561 1017.3000 Julyl, - .... 30.00 25.4325 4.5675 1012.7325 Jan. 1, 1925.... 30.00 25.3183 4.6817 1008.0508 Julyl, - .... 30.00 25.2012 4.7988 1003.2520 Nov. 1, " .... 20.00 16.7480 3.2520 1000.0000 The last period is of only 4 months, July 1 to Nov. 1, Yz of the half year. The cash collected is, therefore, considerd as only $20, yz of $30 ; each previous coupon had included Yi of the following half year, and this must now be squared up. In the colum " Interest at .025 " the procedure is peculiar. The 16,7480 is composed of two parts : 1. Yi of .025 on the $1000 par $16.6667 2. .025 infullon the $3.2520 .0813 $16.7480 215. — To explain why interest in full for the half-year is reckond on the premium, go back to the normal schedule in Art. 187, and it will be seen that the premium on May 1 was 4.8781. Now, on the conventional plan, based on simple interest, this 4 . 8781 should not vary during the period ; there- fore the interest ought to be : Yz of .025 of 4 8781 = .0813 which is the same as .025 of ^ of 4.8781 (3.520) = .0813 Hence in the last or broken period the variance from par must be treated as having earnd interest during the entire period, while the par itself has only earnd interest for the actual time, as four months. 216.— It will also be notist that 3.2520 is ^ of 4.8781, so that if we had calculated 4.8781 by discounting, it would have been sufi&cient confirmation of the preceding values to take Yz of 4.8781 and compare it with 3.2520. 8o The Doctrin of Interest. 217. — It must be rememberd that the periods introduced for balancing purposes are artificial, and that, strictly speaking, amortization takes place only at the dates when interest becomes due. The charging of part of the coupon, tho not yet collected, is fictitious, but in each period until the last, this borrowing is compensated for by a fresh loan. 218.— Short periods, terminal or initial. — It happens sometimes (altho it should be avoided) that the bond does not mature at the end of an interest period, but at some previous date. This gives rise to a fractional period, not an artificial one like those establisht for balancing purposes (Art. 212), but an actual one, which must be taken into account in the valuation. 219. — We will take the case of a six per cent. (5) bond for $1000. issued Jan. 1, 1921, and payable Nov. 1, 1925, interest payments January and July 1, and valued to pay five per cent. (5). There are 9 full periods and a short period of 4 months, or ^ of a period. The coupon for this short period would be $20 insted of $30, as in such cases the last coupon is always proportional to its time. The interest ratio is also reduced for that period to 1 + ^ of .025, or 1.016^ by the conventional plan. 220. — Using the first method of evaluation, the following are the components of the value : An annuity of 9 terms of |30 each at . 025 $239 . 1260 The present worth of $1020 at .025 for 9 terms and .016^ for 1 term. $1 for 9 terms .8007284 Divided by 1 . 016% .7876017 Multiply by 1020 803.3537 v 1042.4797 It will be observd that this differs slightly from the value obtaind in Art. 209 for the same length of time, but with interest May and November 1042 . 4583 The $1020 referd to is composed of the principal and the last or partial coupon. 221. — To divide by a number like 1 .016^, it is easier first to multiply both divisor and dividend by 3, converting each into a whole number. 1.0116^3 X3 -=3.05 .8007284 X3 =2.40218508 2 . 40218508 -- 3 . 05 = . 78760167 Short Periods, TerminaIv or Initiai. 8i 222. — To illustrate the case of an initial short period, suppose that the above bond had been issued on October 1, 1920, 3 months earlier than has been assumed ; issued Oct. 1, 1920, interest January and July, principal maturing Nov. 1, 1925. There is then a preliminary coupon for 3 months, $15, to be discounted at 1 . 0125 ; 9 coupons of $30 each forming an annuity ; one coupon of $20 and the principal, $1000, dis- counted for 9 periods at 1.025, and one period at 1.016^. The value on Jan. 1, 1921, obtaind as before, is 1042.4797. The simplest way will probably be to add to this the initial coupon $20, and discount back the entire 1062.4797 by divid- ing by 1.015, giving 1046.7780. 223. — We may have two other complications : the bond may be purchast within one of the odd periods ; the balancing period may be at still another date. 224 — If the above bond were bought on Dec. 1, 1920, the price would be between 1046.7780, the October value, and 1042.4797, the January value, a three months' interval. The difference is 4 . 2983 ; and either Yz of this (1.4328) may be added to the January value or ^ (2.8655) subtracted from the initial value. 1042.4797 + 1.4328 = 1043.9125 1046 . 7780 — 2 . 8655 = 1043 . 9125 225. — The adjustment of values to balancing dates pre- sents no special difi&culty, being performd by simple proportion. 226. — Cash payments on principal. — Bonds at the same rates of coupon and of interest, tho at different dates of matu- rity, may be combined into one schedule. This may be done even if the interest-dates are different, but it is practically better in that case to keep the schedules distinct. 227. — We must commence with an aggregate value, made up of the separate values of the groups of bonds maturing on the same day. 228. — $2000 six per cent. (5) bonds, maturing as follows : $1000 on Nov. 1, 1923, $1000 on Nov. 1, 1925 ; interest .025, interest payments May and November. Required, value on Nov. 1, 1920. The value of the first bond is 1018.8099 The value of the second 1043.7603 Aggregate value 2062.5702 82 The Doctrin of Interest. This value, multiplied down in the usual manner, gives the following schedule. At the date of the maturity of bond No. 1, the cash colum must contain not only the coupon $60, but the $1000 payable on principal. Date Cash Collections Interest at .025 Payments on Principal Investment Value 1920 Nov. 1 2062.5702 1921 May 1 60.00 51.5643 8.4357 2054.1345 " Nov. 1 60.00 51.3534 8.6466 2045.4879 1922 May 1 60.00 51.1372 8.8628 2036.6251 •• Nov. 1 1060.00 50.9156 1009.0844 1027.5407 1923 May 1 30.00 25.6885 4.3116 1023.2291 •♦ Nov. 1 30.00 25.5808 4.4192 1018.8099 etc. etc. etc. etc. 229. — The remainder of the schedule continues as in Article 187. 230. — For intermediate balances (Article 214), the interest requires adjustment at the date of partial payment. We now assume that the above group of bonds is to be valued on January 1 and July 1 of each year. The values of the bonds under consideration for January 1, 1921, would be, by inter- polation 1017.3000 and 1042.4583 Afir2're2'ate. . 20.^9 7.^ft.^ Dates Cash Collections Interest at .025 Payments on Principal Investment Value 1921 Jan. 1 2059.7583 - July 1 60.00 51.4940 8.5060 2051.2523 1922 Jan. 1 60.00 51.2813 8.7187 2042.5336 '• July 1 60.00 51.0633 8.9367 2033.5969 1923 Jan. 1 1050.00 42.5066 1007.4934 1026.1035 - July 1 30.00 25.6526 4.3474 1021.7561 etc. etc. etc. etc. ^31. — The entries under January 1, 1923, are peculiar. The $1000 paid off was only in possession for 4 months, Yi of a period; therefore, $20 is the appropriate sum to be con- siderd as paid with it, and if it was kept in a separate account, that is all which would be allocated to it. The other $1000 is on interest during the full period and $30 is charged to it. Short Periods, Terminai, or Initial. 83 Cash entries : For bond No. 1 par 1000 Gross interest thereon, .03, 4 months. . 20 Gross interest on No. 2, .03, 6 months._^ 30 1050 Interest entries : Bond No. 1, Yz of period at .025 16.6667 Bond No. 2, full period, (Art. 210), at .025 on 1033.5969 25.8399 42.5066 Applied on principal : 1050.00-42.5066== 1007.4934 232. — While this procedure may be applied where the successiv partial payments on principal are of the most irregu- lar amounts and intervals, their chief utility is in what are known as serial bonds, where regular payments of principal are made, usually annually. Each bond or group of bonds, as it is dropt from the total, carries with it the appropriate cash and interest entries exactly as exemplified above. To Find thb Income Rate. 233. — When the cash-rate, the time and the price of the bond are known, it is very desirable to know what is the income-rate, for, of course, every one wishes to get the highest income, security being equal. 234. — There is no positiv, direct method of doing this beyond three periods, as an equation of a higher degree is not directly soluble. There are only methods of approximation and trial. 235. — When printed tables are accessible it is easy to make a rough approximation by observing between what values the given price lies. The smaller the interval between the rates of the table, the closer is the approximation, and an addi- tional decimal may be obtaind by proportion. With an ex- tended table at close intervals, the result is suflSciently accu- rate for commercial purposes. 236. — There is a method devised by the author which will produce even greater accuracy, up to 12 places, at the expense of considerable labor. It appears in the later editions of his " Text- Book of the Accountancy of Investment," from which it is here quoted. $4 The Doctrin of Interest. To Find the Income Rate. 1. — Given a bond on which there is a premium or discount Q, cash-rate c payable in n periods, what is the income-rate, /? 2. — Every premium or discount is the present worth of an annuity of n terms, each instalment of which is the difference of rates ; or it is the difference of rates X such an annuity of $1 (Art. 193). Writing P for the present worth of an annuity of $1 (Art. 108): Q = P X {c — i). If instead of the pre- mium on $1 we use that on $100, we have 100 Q = P X (100^ — 100/). It will not affect the value of the right-hand side if we halve one factor and double the other. 100 Q = ^ P X (200^ — 2000. 200^ is the rate per cent, per annum as conventionally termed. Thus we pay 4 per cent, per annum, meaning .02 per period. So also of 200/ for i. 3. — It is evident that if we divide lOOQ, the premium on $100, by %P (which we will hereafter call the trial-divisor), we shall find the difference of rates. But as the annuity de- pends on the unknown rate, this does not help us at all. 4. — Let us assume the rate of income per annum to be any rate whatever, and calculate the trial- divisor at that rate. Then there is this property : If the assumed rate is too large, the quotient or difference of rates will be too small, and yet will be nearer the truth, and vice versa. From this approximate difference of rates we derive a new rate and proceed with this as a trial-rate. The result of this trial will give a new rate still nearer and so on. We may slightly modify any rate to make it more easy to work. If we select our first trial-rate near the true rate, fewer successiv approximations will be necessary. 5. — Fortunately for our purpose, any table of bond values will readily give the trial- divisor, by taking the difference between the values at the same income-rate of two successiv $100 bonds, say a 3% and a 4%, a 5% and a 6%, always 1% apart. 6. — For example, a 6% bond for $100 (semi-annual) for 50 years is sold at 133 . 00, what is the income rate ? To Find the Income Rate. 85 7. — With so large a premium as 33, the income-rate is evidently much less than 6. Let us assume 4. Then from any bond table we find on the 4% line the value of a 5% bond to be 121.55 and that of a 4% bond to be 100 00 The first trial-divisor is therefore 21 . 55 33.00 -f- 21. 55 = 1.531. the difference of rates. 6 — 1.531 = 4.469, the new trial rate. Taking 4.45 as more convenient, the new trial-divisor is 19.98. 33.00-^-19.98 =- 1.651. 6 — 1.651 = 4.349. We find that 20.315 is the trial-divisor for 4.35. 33.00 -> 20. 315 = 1.6244. 6 — 1.6244 = 4.3756. Next using 4.37, trial-divisor 20.25 : 33.00 -f- 20.25 = 4.37, almost exactly, so that 4. 37 has reproduced itself. The value of the bond at 4.37, as computed by logarithms, is 133.0069, an error of less than one cent. 8. — It will be noticed in the foregoing example that the results always swing to the opposit side of the true rate ; that is, if the trial-rate is too large the next rate is too small, and the true rate is between them. The successiv rates were 4. . .4.469. . .4.349. . .4.3756. . .4.37. 4.37 lies between any pair of these. This is always the case with bonds above par. With bonds below par it is different. The true rate always lies beyond the approximation. 9. — As an example of a bond below, take a 3% bond pay- able in 25 years. If purchased at 88. 25, what is the income- rate ? The following may be the steps, the dividend being always 11.75, the discount. Trial-rates 3.70 3.725 3.7265 Trial- Divisor 16 . 2190 16 . 175 16 . 17245 Result 3.7244 3.7264 3.7265 As 3 . 7265 reproduces itself, it must be correct to the 4th decimal ; and the value of a 3% bond for 25 years to yield 3.7265 is found by logarithms to be 88.25015. 86 The Doctrin of Interest. 237. — Since the publication of the above method some simplifications have been suggested, referring particularly to the employment of Sprague's Extended Bond Tables. Insted of obtaining a divisor as shown in par. 5, we may first multiply Q itself by the interest-difference and then use the entire variance at the trial-rate (Q'') as the divisor, giving the same result. Thus in par. 7, where the first assumed rate is 4% and the interest difference 2, finding from the Table that the premium is 430.983.52 33 X 2 -^43.098352 = 1.531 insted of 33-^^21.55 = 1.531 238. — The valuable suggestion has further been made by Mr. E. S. Thomas that by using the two nearest income rates and interpolating, five decimals may be obtaind at once. In the example where Q = 33.00, 200^ = 6, select 4.35 as 200i^ and 4.40 as 200/2. From the Tables we find opposit 4.35, 335,201.06 and opposit 4.40, 322,371.36, the interest-differences being 1.65 and 1.60. 4 . 35% 33 . 00 X 1 . 65 -^ 33 . 520106 = 1 . 624398 4 . 40% 33 . 00 X 1 . 60 -^- 32 . 237176 = 1 . 637861 4 . 35 -f 1 . 624398 = 5 . 974398 = 6 . — 025602 4.40 + 1.637861 = 6.037861 = 6. + .037861 Difference in error . 063463 239. — If the same operation be performd on a third rate, 4.45 : 4.45% 33.00 X 1.55 -h 30.974371 = 1.651333 4.40% 33.00 X 1.60 -f- 32.237176 = 1.637861 4.45+1.651333 = 6.101333 4.40 + 1.637861 = 6037861 Difference .063472 which differs so slightly from .063463 that further differenc- ing may be neglected. To Find the Income Rate. 87 240. — By proportion, we may now ascertain at what rate the coupon will become 6. From 4.35 to 4.40, an extent of .05, it varies by .06347. Therefore .06347 : .025602 : : .05 : .0201686 Add 4.35 Rate required 4.3701686 or safely 4.37017 241. — In the other example (par. 9), where Q = 11.75, it appears from the Tables, (p. 10) , that the nearest rates are 3.70% and 3.75%; the corresponding values being 88.646703 and 87.900397 and the discounts 11.353297 and 12.099603. 3 . 70% — 11 . 75 X . 70 -f- 11 . 353297 = — . 72445916 3.75% — 11 75 X .75 -^ 12.099603 = — .72832968 3 . 80% — 11 . 75 X . 80 -f- 12 . 837828 = — . 73221109 Diff. 3.70 — .72445916 = 2.97554084 3.75— .72832968 = 3.02167032 -04612948 3 . 80 — . 73221109 -= 3 . 06778891 04611859 3 — 2.97554084 -= .02445916 .04612 : .02446916 : : .05 : .02650169 3.70 3.72650169 or safely 3.7265 242. — The process may be still further abridged by mak- ing the interest-difference the standard of comparison insted of the coupon, giving the same result. Thus in 238, insted of 4.35 + 1.624398 = 5.974398 = 6. - .025602 4.40 + 1.637861 =-6.037861 = 6. + -037861 .063463 might have been written 1.65 — 1.624398 = + .025602 1.60 — 1.637861 = — .037861 And in 241 .70 — .72445916 = — .02445916 •75 -- .72832968 = + .02167032 .80 — -73221109 = + .06778891 .063463 .04612948 .04611859 88 INTEREST FORMULAS. i = Rate of Interest, or the Interest on Unity for 1 period. r~ 1 + /, or the Ratio of Increase. / =^ Number of Periods of Time. r*^ = (1 + 0* = Amount = s. rt^-= (i + /)-t = -^^^^ Present Worth = /. r^ — 1 = Compound Interest = I. 1 — r-^ = Compound Discount = D. Amount of Annuity — l-^-r-^-r^-^-r^ + r^-^ = l/t = S. Present Worth of Annuity = 1 + r^ + r^ + r^ . . r^-^= B/i = P. y = Nominal Rate Per Annum. Coefficient of Double Fre- quency = C(^) = 1 + V*- Coefficient of Half Frequency = c(^^) = i + y2 (y"rT7— 1) Sinking Fund = 1/S = 2*/I. Amortization = 1/ P =:=^ /*/ D = //I + t. c = Coupon Rate of Bond, or Cash Rate. V • = Value of Bond at c to Earn i — r'^ -\- cV. or = 1 + {c—i) P. or= f-(f-l)P. UE ON THE LAST DATE -^\oP^t ^^^^i---°^ ;-lfr/^^.te.e-^h ^iA:;t?S\*'"^^ ^^.602^ LIBRARY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. THIS BOOK IS DUE BEFORE CLOSIMr xrvtn ON LAST DATE STAmIeD BEUW^**^ JJBRARY USb y REC'D LD OCT 4 '64-8 PM l-^i^^ LD 62A-50m-2,'64 (E34948l0)9412A , General Library University of California Berkeley ■'..■:;;|i|l«i«' 1 m mKKkm,.