CHAPTER V INTEREST 19. Simple interest. Broadly speaking, interest is defined as the income on capital profitably invested. In a more restricted sense it is the sum paid for the use of money that is loaned. The sum invested, whether it be in the form of money loaned or capital invested in a business enterprise, is called the principal. Interest is computed as so many hundredths of the principal earned in a given unit of time. The unit of time is almost in- variably one year. The rate of interest* is that fraction which expresses the ratio of the interest earned in the unit of time to the principal. It may also be defined as the fraction which ex- presses the amount paid for the use of a unit of principal for a unit of time. Simple interest is interest which is proportional to the time. The sum of the principal and the interest is called the amount. If we use the symbols -P, J, i, n, and A, to denote the princi- pal, the interest, the rate, the time, and the amount, respectively, the fundamental formulas for simple interest are I=Pni (1) and A=P+I, or A=P(l+ni). (2) All problems that can arise in simple interest may be solved by means of formulas (1) and (2). « Care must be taken to note that the rate of interest is always a traction. We use the expression 6% meaning that the rate is .06. 51 52 MATHEMATICAL THEORY OF INVESTMENT ILLUSTRATIVE EXAMPLE. Let it be required to find the principal which will yield $500 in interest at 5% in 2 years and 6 months. By formula (1) J ' ' 500 = P x .05 x 2.5; whence p_ 500
.05 x 2.5 = $4000. 20. Ordinary and exact interest. In practice the greater num- ber of problems that occur in simple interest are problems in- volving short periods of time. For reasons of convenience it is customary in ordinary transactions to compute the interest on the basis of 360 instead of 365 days in a year. Interest com- puted on the basis of 360 days in a year is called ordinary interest. Interest computed on the basis of 365 days in a year is called exact interest. To find the relation between, the two kinds of interest, let I denote ordinary and I' exact interest. If d denote the time expressed in days, ^Im- (1) and ^^ <2) To find the relation between I and J', divide equation (1) bv equation (2) and obtain _[^365^ 1. I' 360 72' whence I=II+^L. ^\ 72 v - Translating this formula into words, we obtain as the rule for deriving ordinary from exact interest: Ordinary interest is equal to the exact interest increased by one seventy-second of itself. Solving formula (3) for I', I'=72! 73 ' r=i-^ (4) INTEREST .33 From formula (4) we have the rule for deriving exact interest from ordinary interest: Exact interest is equal to ordinary interest diminished by one seventy-third of itself. ILLUSTRATIVE EXAMPLE. Find the exact interest on $5278.17 from May 11, 1911, to June 25, 1911, at 6%. The time is 45 days, which is ^ of 360 days. Hence, the ordinary interest is $5278.17 x .06 x ^ = $39.586. The exact interest is then $39.586 - g39-586 = $39.044. i3 21. Computation of simple interest. The computation of simple interest, however the work may be arranged, depends upon one of the formulas J= Pm, I=pi^' and I'-pi^ If a large amount of such work is to be done, an interest table is almost a necessity. Elaborate tables giving the interest on all sums up to $1000, for times less than one year and at rates from 3 to 8 or 9%, are published for the use of bankers and others who wish to obtain results quickly. For ordinary purposes a much simpler table is sufficient. From Table II we may find the exact interest at 5%, on all amounts from $1 to $100,000, for times up to 365 days, with a small amount of labor. A table like Table I, which gives the number of the day of the year, counting forward from January 1, is a useful aid in finding the time between two dates. Since simple interest is proportional to the rate, we may find the interest at a rate differing from 5% by increasing or dimin- ishing the interest at 5% by the proper number of fifths of itself. Thus, to find the interest at 6% we add to the interest at 5% one fifth of itself.