COMPOUND INTEREST PROCESSES AND THEIR USE IN BUSINESS 1. Definition of terms.—The compound interest processes are used in many ways in business. A knowledge of them is especially necessary in dealing with the valuation of bonds, the apportionment of bond discounts and bond premiums, and in the dis cussion of the theory of depreciation.
The amount of interest depends upon the "princi pal," the "time" and the "rate." The principal is always a present quantity of money or money's worth which is lent and upon the basis of which interest is earned. The time during which interest is earned is always expressed as a number of time-units. The whole duration of a debt may be taken as a time-unit, for which a rate of interest may be expressed. But the time-units most commonly used are the year, the half-year, the quarter and the month.
The rate of interest always applies to a definite time-unit. The rate may be defined as the relative amount, as compared to the principal, by which the obligation of the debtor increases in the time-unit. Six per cent per annum means that the amount of interest earned during one year is six-hundredths of the principal, or six cents out of every dollar. Six per cent per annum payable semi-annually means that the time unit is the half-year, and that the interest earned during the half-year is three-hundredths of the principal.
2. Simple and compound interest est may accrue by simple interest or compound inter est. Simple interest does not accrue on other un paid interest that is past due. According to the compound interest theory, any interest due but un paid becomes a new principal upon which interest accrues. Simple interest is used in the case of most short time debts. Compound interest, or the allow ance of interest on unpaid interest, is illegal in most jurisdictions.
3. Rate of the time-unit is not al ways the year, the term "time-unit" or "period" will be used in the following discussion, the whole time being referred to as "the number of periods." If the rate of interest is 5 per cent, the interest on $1 for one period is 5 cents, and the "amount" of $1 for one period is $1.05. Evidently the amount of $1,000 for one period is one thousand times as great. The arithmetical operation may be written: 1,000 X $1.05 = $1,050
or $1,000 X 1.05 = $1,050 Evidently the quantity, 1.05, may be regarded as a ratio of increase per period, i.e., at 5 per cent any principal increases so that at the end of one period it is 1.05 times as great as it was at the 4. Rule for finding the amount of any principal.— If the amount of interest mentioned above, namely, $50 is not paid at the end of the first interest period, it would, at compound interest, be added to the orig inal principal and the whole amount, $1,050 becomes a new principal for the second period. If the in terest was paid, we may assume that the recipient could invest it at 5 per cent, or make some other dis position of it just as good, so that the whole principal for the second period is $1,050, any way. This in creases in the ratio 1.05 during the second period so that the amount at the end of the second period is $1,050 X 1.05 = $1,102.50 But since $1,050 equals $1,000 X 1.05, we may write the amount at the end of the second period thus: $1,102.50 = $1,000 X 1.05 X 1.05 = $1,000 X 1.05 It is evident that the amount of any principal may be found by multiplying by the ratio of increase as many times as there are periods. Hence, the following rule: To find the amount of any principal for any num ber of periods, use the ratio of increase as a multiplier of this principal as many times as there are periods.
This rule may be expressed as follows : A=P X (1 i) n (I) In this formula, P stands for any principal, i stands for the rate of interest expressed as a decimal fraction, n the number of periods included in the whole time and A the amount of principal and compound interest at the end of these n periods. Here the ratio of increase for one period is 1 which is also the amount of $1 for one period at the rate i. Of course, in any concrete case, the numeri cal values of P, i and n are supposed to be known, and from these the value of A can be computed. Thus if, in a particular case, P, is $500, n is 10 years, and i is 6 per cent or .06 per annum, then A, which stands for the amount at the end of ten years, is given by the equation, A = $500 X 1.06 from which, by multiplying out the right side, we ascertain that A has the numerical value of $895.42.