The Journal of Accountancy Official Organ of the American Institute of Accountants Vol. 28 NOVEMBER, 1919 No. 5 Introduction to Actuarial Science* By H. A. FINNEY In the more comprehensive meaning of the term, actuarial science includes an expert knowledge of the principles of compound interest as well as the laws of insurance probabilities. Public accountants, however, are usually interested only in the interest phases of actuarial science, leaving the application of the laws of insurance probabilities to the actuary, who ascertains the measurement of risks and establishes tables of rates. This discussion of actuarial science will, therefore, be restricted to the phases thereof which deal with compound interest. There seems to be a more or less prevalent belief that the exacting of compound interest is illegal, and that this illegality makes the mathematics of compound interest an impractical matter of purely academic importance. While it may be illegal in some jurisdictions for a creditor to charge the debtor interest on unpaid interest, there can be no legal restriction against the collection and reinvestment of the interest. In fact, the mathematical theory of investment is based on the assumption that all interest accretions become themselves a part of the investment, being converted at periodical intervals into interest-earning principal. The compound interest basis is the only scientific one where the accumulation or reduction of an investment extends over a series of periods. It is with compound interest, therefore, and not with simple interest, that actuarial science deals. Interest is the increase in investment or indebtedness caused by the use of money or credit. The rapidity and extent of the increase depend on the factors of interest, which are rate, frequency and time. *Copyright, 1919, by Harry Anson Finney. The rate is usually expressed in terms of percentage and measures the fraction by which the investment is increased at each date of conversion of interest. Thus, a rate of 5% per period indicates that the interest each period will be .05 of the investment at the beginning of the period; or, stated in another way, the investment at the end of the period will be 1.05 times the investment at the beginning of the period. Expressing the decimal interest rate by the symbol i (for instance, .05=i), the interest earned during any period may be computed by multiplying the investment at the beginning of the period by i; and the increased investment at the end of each period may be computed by multiplying the investment at the beginning of the period by 1+i. The symbol 1+i is called the ratio of increase, because it measures the ratio existing between the investment at the beginning and the investment at the end of each period. The frequency is the length of the period in years, months or days between the dates of interest conversions. It is evident that the frequency of compounding will materially affect the rapidity with which an investment increases. For instance, an investment of $1.00 for one year will amount to more if the loan is at 172% per period of 3 months than if at 6% per period of 12 months. Increase during one year in investment of $1.00 at 12% per period of 3 months : 1.00 Original investment Multiply by 1.015* Ratio of increase 1.015 Investment end of 3 months 1.061364 Increase during one year in investment of $1.00 at 6% per period of 12 months : 1.00 Original investment Multiply by 1.06 Ratio of increase 1.06 Investment end of 12 months. One frequently sees interest tables in which the interest rates are stated as a certain per cent. per annum. It is better to express the rate as a certain per cent. per period, because, when the compounding occurs more frequently than once a year, the effective rate earned during a year is really greater than the nominal rate. Thus, in the foregoing illustration, the loan is made at 112% per period of three months. The customary statement, however, is that the rate is 6% a year compounded quarterly. The nominal rate is 6%, but since the investment increases during the year from 1.00 to 1.061364, the effective rate per annum is 6.1364%. The time is the total number of periods over which the investment extends. It is customary in commercial parlance to state the time as a certain number of years, but as a matter of principle the time is a certain number of periods which may be of any duration, and the rate should, therefore, be stated as the rate per period. For instance, if money is lent at 6% per annum for 41/2 years, compounded semi-annually, the time (represented by the symbol n) is 9 periods, and the rate (represented by i) is 3% per period. Each dollar of an investment increases in the same ratio as every other dollar; therefore, in compound interest computations, it is customary to compute the required value on the basis of a principal of $1.00, and to multiply by the number of dollars in the principal. THE AMOUNT OF 1. Since interest increases the investment, the fundamental problem in interest is the computation of the amount to which an investment will increase in a given time. It has already been noted that the increase depends upon i (the rate per period) and n (the number of periods). During each period the investment increases in the ratio of 1+i. At 6% per period, an investment of 1 will amount, at the end of 1 period, to 1.06, or 1+i; during the next period the investment will increase to 1.06, the investment at the beginning of the period, multiplied by 1.06, the ratio of increase, or to 1.1236, which is (1+i)*; at the end of the third period the investment will amount to 1.1236, the investment at the beginning of the period, multiplied by 1.06, or 1.191016, which is (1+i). Or, stated generally, the investment will amount, at the end of n periods to (1+i)n. This means that the ratio of increase is raised to the nth power, or that the amount is a product obtained by using the ratio of increase as a factor as many times as there are periods. Representing the amount of 1 by a, the formula is a=(1+i)n When compound interest tables are available, the amount can be determined by reference to them. A table of amounts appears as follows: = 8 1.266770 1.316809 1.368569 1.422101 1.477455 When interest tables are not available, the amount is easily computed by logarithms. To illustrate: what is the amount of 1 at 6% compound interest for 4 years, compounded semi-annually? a=1.03 .012837 The product is .102696, which is the logarithm of 1.266764, or 1.03'. An interest table states this amount as 1.266770 When neither an interest table nor a table of logarithms is available, the amount may be computed by repeated multiplications. The required amount is the product obtained by using the ratio of increase as a factor as many times as there are periods. Thus, 1.03' may be computed as follows: 1.266770 8 This work can be materially reduced by recognizing the principle that the multiplication of any two powers of a number results in a power represented by an exponent equal to the sum of the exponents of the powers multiplied. For instance, 1.0609 =1.03. Now since 1.0609 contains 1.03 twice as a factor, the product of 1.0609 multiplied by 1.0609 will contain 1.03 four times as a factor. Thus 1.0609 x 1.0609=1.125509=1.03 The eighth power can be obtained thus: 1.125509 The 4th power multiplied by 1.125509 4 1.266770 8 The seventh power can be obtained by multiplying any powers, the sum of whose exponents is 7. |