This principle may be applied when it is desired to determine an amount beyond the scope of an interest table. Thus, if an interest table extends only to 20 periods, the amount for, say, 75 periods, can be computed by multiplication of the amounts for periods within the scope of the table, as follows: 20th x 20th x 20th x 15th=75th Or the principle may be applied when an amount must be computed with no table available. For instance, (1+i)" may be computed thus: (1+i) x (1+i) =(1+i)' 32 THE COMPOUND INTEREST Since the interest increases the investment, the difference between 1 and the amount of 1 is the compound interest. Representing the compound interest by the symbol I, I=a-1 Illustration required the compound interest on 1 at 6% per annum for 4 years, compounded semi-annually? The amount of 1.03 has already been computed as 1.266770. Then 1.266770-1-.266770 the compound interest. THE RATE When the investment, the amount and the time are known, the rate can be computed by logarithms. Illustration: if $80.00 invested at an unknown rate, compounded annually will amount to $107.20 in six years, what is the rate? 80 x (1+i)=107.20 Hence (1+i)=107.20÷80-1.34 Since 1.34 is the 6th power of 1+i, it is necessary to extract the 6th root, which is accomplished as follows: log. 1.34 .127105 log. V1.34.127105÷6-021184 .021184 is almost the exact logarithm of 1.05, or 1+i Hence i-.05 the rate. THE TIME When the investment, amount and rate are known, the time can be computed by logarithms. Illustration: for how many years must $2,000.00 remain at 5% interest compounded annually to produce $5,054.00? The present value of a sum due at a fixed future date is a smaller sum which, with interest, will amount to the future sum. When the time is more than one period, the smaller sum will accumulate at compound interest, increasing each period in the ratio of increase of 1+i. Representing the present value by the symbol p, the accumulated sum at the end of 1 period will be px (1+i) That is, the present value of 1 due n periods hence at a given rate may be computed by dividing 1 by the amount of 1 for n periods at the same rate. Illustration: what is the present value of 1 due four periods hence at 3%? The table of amounts of 1 at 3% shows that 1.03-1.125509. Then 11.125509-.888487, which is the present value of 1 at 3% for 4 periods as shown in the following illustrative table: When a table of present values is not available the value of p can be computed by dividing 1 by the value of a for the same time and rate, shown by a table of amounts of 1. In the absence of any table, the value of p can be computed in one of the following ways: Required: the present value of 1 due 4 periods hence at 3%. Compute the value of a, which in this case is 1.03', and divide 1 by the value of a. 1.03-1.0609 1.03-1.0609'-1.125509 11.125509-888487 Or, use the ratio of increase as a divisor as many times as there are periods, using 1 as the first dividend, and each quotient as the dividend in the succeeding division; thus: 11.03 .970874 present value of 1 due 1 period hence Or, divide 1 by the ratio of increase, and raise the quotient to the nth power; thus: .970874 x .970874-942596 present value of 1 due 2 periods hence .942596 x .942596-.888487 66 1 4 COMPOUND DISCOUNT The present value of 1 is the sum which will accumulate to 1 in a given time at a given rate. The difference between 1 and the present value of 1 is the compound discount, which will be represented by the symbol D. The compound discount can be computed by the formula: D=1-p Illustration: required the compound discount on 1 due 4 periods hence at 3%. The value of p for 4 periods at 3% was computed above. It is .888487. Therefore 1-888487-.111513, the compound discount. This is the formula to use when one has a table of present values. If one has only a table of amounts of 1, it is better to use the formula: This formula requires explanation. The compound discount is really the compound interest earned on the present value. It is the interest which increases p to 1. For instance, if one lends .888487 for 4 periods at 3%, the interest will be .111513, increasing the investment to 1. Now the interest earned is proportionate to the sum invested. If the investment is 1, the compound interest will be I. If the investment is half of 1, the compound interest will be 1⁄2 of I. If the investment is .888487, the interest will be I x .888487. That is, if the investment is p the earning of D will be I x p. But p is 1÷a. To illustrate the application of this formula: required the compound discount on 1 for 4 periods at 3%. Given: 1.03= 1.125509. Then .125509÷1.125509-111513. ANNUITIES A series of equal payments, due at regular intervals, is an annuity. The intervals may be periods of any length, as a month, a quarter, a half year or a year. The periodical payments of an annuity are called rents. Since the rents may be invested when received, there is the problem of determining the amount to which the rents will accumulate. On the other hand it may be desired to compute the investment which, with interest accumulations, will permit the withdrawal of rents of stated amounts at stated intervals. The sum so invested is the present value of the annuity. These are the two fundamental annuity problems. The amount of an annuity of 1 will be represented by the symbol A; the present value of an annuity of 1, by the symbol P. AMOUNT OF AN ANNUITY To illustrate the methods of computing the amount to which the rents of an annuity will accumulate, let it be assumed that a contract requires the following payments: Required the amount to which these payments will accumulate at December 31, 1919, if each rent is invested immediately at 3% per annum. Clearly the amount of the annuity will be the sum of the four $300 rents plus the interest on each rent, as follows: Although the amount of an annuity may be computed by determining the amount of each rent and the sum of these amounts, it is unnecessary to resort to this labor. The following short method may be used: To find the amount of an annuity of 1 for a given number of periods at a given rate, divide the compound interest on 1 for the number of periods at the given rate by the simple interest rate. Or, in symbols: A=1÷i |