64 MATHEMATICAL THEORY OF INVESTMENT If we -remember that we have, by formula (6), , l+i=eS, and that the fundamental relation between nominal and effective rates of interest may take the form / -\m ^-(^y' the formulas for compound interest, viz. (i\mi> ,=(l+0n=l^) and S = P(\ + i)" = p(l + ^Y", are expressible in the simple forms s = e»< (8) andS = Pe^. (9) These forms justify the name "compound interest law," which was applied to the relation v = a* at the close of § 18, for if we put 0s equal to the constant a, (8) becomes S ^ CE y where the variables are n and s instead of x and y, as in § 18. Formulas (8) and (9) are only two of many cases in which a simplifi- cation is effected by means of the introduction of the force of interest. EXAMPLES 1. By means of the exponential series, find the effective rate when the nominal rate is .06 and the interest is compounded momently. 2. Find the force of interest corresponding to the effective rate .06. 3. If the force of interest is .06, find the compound amount of $1259 for 3 years and 6 months. 25. Computation of compound interest. The formulas S=P(l+i')" I A"" =?!+"-) \ m/ =Pe"5 are admirably adapted to computation by logarithms, but in order to secure accurate results for compound amounts on principals up to $10,000, one would need at least six-place logarithms. INTEREST 65 Compound amounts play such an important part in practical affairs that compound-interest tables have been constructed for all the rates in common use and for times up to one hundred years. Such a table gives the amount for unit principal, but the amount for any principal is easily found by multiplication. Table III gives the amount of 1 at compound interest at rates U, H, 2, 2,L, 3, U, 4, U, 5, and 6% for times up to one hun- dred years. ^The solutions of the examples that follow will show how to use the table most expeditiously. A little care in noting the number of decimal places that are required will often save a considerable amount of work. For example, if the principal does not exceed $100, four-place tables will give results accurate to the cent, but if the principal is as large as $100,000, accurate results cannot be obtained without tables carried out to at least seven places. Eight-place tables would be still better. If logarithms are not used in performing the multiplication, it is well to begin with the left-hand figure of the multiplier, so that unnecessary figures can be dropped out as the multiplica- tion proceeds. Indeed, it is not necessary to write down multi- plicand and multiplier. ILLUSTRATIVE EXAMPLE. Required the compound amount of $236.41 for 10 years with interest payable annually at 5%. The work may be arranged as follows: Using 6 places, we find from the table that the amount of $1 for 10 years at 5% is 1.62889. By multiplication we find Amount of $200.00 for 10 years at 5% = $325.778 Amount of 30.00 for 10 years at 5% = 48.866 Amount of 6.00 for 10 years at 5% = 9.773 Amount of .40 for 10 years at 5% = .651 Amount of .01 for 10 years at 5% = 016 Amount of $236.41 for 10 years at 5% = $385.084 By utilizing the properties of a power of a base the range of the table may be considerably extended. If, for example, we wished to find the amount of $236.41 for 75 years at 5%, with only a fifty-year table at hand, we could write s = $236.41 x (1.05)60 x (1.05)26 = $236.41 x 11.46740 x 3.38635. 66 MATHEMATICAL THEORY OF INVESTMENT The value of this product may be found by means of a table of six-place logarithms. If interest is convertible oftener than once a year, we can use the table for the rate ^ and for mn years. For example, to find the amount of 1 for 10 years at 5%, convertible semiannually, we have to look up the amount of 1 for 20 years at 2^%, con- vertible annually, since, by the formula, we have / ()K\10X2 s=(l-f-—) =(1.025)20. If the time does not contain the conversion interval an exact number of times, it will ordinarily be necessary to use logarithms or straight-out multiplication to reach the result. If, for example, we wish to find the amount of $250 for 5 years and 6 months at 5% when interest is payable annually, we must find the value of the product ^ ^ ^^ If a table of logarithms is not at hand, the value of (1.05)^ may be found by the binomial theorem, or we may take the value of (1.05)" from the table and find the value of (1.05)*, either by the binomial formula or by simple root extraction. MISCELLANEOUS EXAMPLES 1. Find the compound amounts in the following examples: (a) $239.54 at 5%, convertible annually, for 6 years. (b) $250 at 5%, convertible annually, for 3 years and 6 months. (c) $300 at 5%, convertible annually, for 120 years. (d) $1000 at 6%, convertible quarterly, for 20 years. 2. How long will it take for a sum to double itself at 5%, convertible annually ? 3. At a certain university which had 4000 students in 1910 the attend- ance has been increasing at the rate of 10% each year over the previous year's attendance. If the rate of increase should be kept up, what would be the attendance in 1920 ? 4. A merchant starting out with a capital of $5000 finds that after 8 years his capital is $10,000. What has been the annual rate of in- crease if the rate of increase is supposed to have been uniform through the 8-year period? "