54 MATHEMATICAL THEOEY OF INVESTMENT EXAMPLES 1. Compute the exact interest on $43,729.29 from April 1 to May 18 at 5%. The time is 47 days. From the table we find Interest on $40000 = $257.534 Interest on 3000 = 19.315 Interest on 700 = 4.507 Interest on 20 = .129 Interest on 9 = .058 Interest on ____-W =.001 Interest on $43729.29 = $281.54 The exact interest on the same principal and for the same time at 6% would be $281.54 + $56.31 == $337.85. 2. Find the simple interest at 6% on $23,738.42 from February 1,1908, to March 30, 1908. 3. Find the simple interest at 6% on $38,472.27 from March 21 to April 11 of the same year. 22. Compound interest. Interest, -whether it be in the form of money paid for the use of money that has been loaned, or whether it be in the form of dividends on capital profitably in- vested, should be paid promptly when due. When paid, it may itself be put at interest, so that it is, in effect, added to the prin- cipal to form a new principal. This process may go on to any length. In case the interest is not paid by the borrower, it may still be considered as added to the principal at stated intervals. When the interest is added to the principal at stated intervals throughout a given time, the difference between the original principal and the sum due at the end of the time is called com- pound interest. The total amount due is called the compound amount. The interest is usually added to the principal at the end of each successive year or half year or quarter year. We say the interest is " compounded, or converted into principal, annu- ally, semiannually, or quarterly, or m times a year," or that in- terest is "payable annually, or semiannually, or quarterly, or m times a year." The time elapsing between two successive con- versions of interest into principal is called the conversion interval. INTEREST 55 In all theoretical work in compound interest the principal is assumed to be a unit of money, without regard to the particular currency to which the unit belongs. Thus, we speak of the "com- pound amount on 1" for a given time, and the formulas obtained will apply equally well whether the unit be a dollar, a pound, a mark, or a franc. The compound amount on any principal is then found by multiplying the given principal by the compound amount on a unit principal, since the amount is proportional to the principal employed. PROBLEM. To find the compound amount on 1 for n years with interest at rate i payable annually. At the end of the first year the principal becomes 1 + i, and at the end of the second it is l+i+i(l+i)=(l+iy; at the end of the third year it is (l+zy+z(l+zy=(l+,y; and so on. At the end of the wth year it will be the nth term of the geometrical progression 1+{, (l+O2, (l+O3, -, or, denoting the amount by s, »=(i+»y1. (i) The compound amount of any principal, P, is S=P (!+»)". (2) It should be noted that, while the compound amount is propor- tional to the principal, it is not proportional to the rate of interest or to the time, as is simple interest. ILLUSTRATIVE EXAMPLE. Find the compound amount on $175.50 for 3 years at 5%. Solution. The compound amount of 1 for 3 years is (1.05)3, or 1.157625. Multiplying this amount by 176.60, we find that the compound amount of $176.50 is $203.16. 56 MATHEMATICAL THEORY OF INVESTMENT PROBLEM. To find the compound amount on 1 compounded m times a year for n years. ' For reasons that will appear later, we use the letter j for the rate instead of i. We assume the rate for the mth part of a year to be the with part of the rate for a year. The rate for one in- f\ terval of" time will then be "- The problem is exactly like the first problem, except that the rate per interval of time is ^- and the number of intervals is mn. m The required expression is, then, (i \mn s= l+ii) (3) Similarly, the compound amount of any principal, P, with interest convertible m times a year at rate j is (i \m'n s=-pl+ii) (4) The foregoing definitions and formulas apply only when the time contains an exact number of conversion intervals. Conse- quently, compound interest can have no meaning when the time does not contain an exact number of conversion intervals. To obviate this difficulty we assume that formulas (1) and (3), viz. s = (1 + i)" and s == (l + ^-)", \ m/ hold for all values of n when n is expressed in years. This assumption is of course equivalent to a definition. According to the definition the compound amount of 1 for 1 year and 6 months at 5%, convertible annually, would be (l.OSy^^l.OS)3, and not (1.05)fl+^5V ' ' as it is frequently given. Similarly, the compound amount of 1 for 6 months, at rate .05, convertible annually, would be (1.05)^, which is less than the amount, at simple interest, of the same principal at the same rate and for the same time. INTEREST 57 However, in practice the compound amount is usually com- puted for the integral number of conversion intervals, and the amount of simple interest is then figured on this amount for the fractional part of the interval remaining. For example, the com- pound amount on 1 for 5 years and 6 months at 6%, payable annually, would be given as (1.06)5 (1.03) =$1.38. The expression (1+i)", or its equivalent (l+--) , is some- times called an accumulation factor (see § 26). PROBLEM. To find the rate of interest when the principal, the compound amount, and the time are given. (a) If interest is payable annually, the problem is to find i from the formula (2), viz. S=P(l+f)". Dividing through by P and extracting the nth root of both sides of the resulting equation, we find i+-^€
and, finally, i=^-l. (5) (b) If interest is convertible m times a year, the relation con- necting P, S, n, j, and m is, by (4), / i V"" S=p{l+3-\ . \ m/ Solving this equation for j, we find /""^ 1^ ^ g=m(^-^-lj. (6) The two expressions -\|— and "!"!_ are easily found by loga- rithms when the values of <S', P, n, and m are known. 58 MATHEMATICAL THEORY OF INVESTMENT ILLUSTRATIVE EXAMPLE. In 5 years $1000, placed at interest convert- ib