Calculations for the amounts of $1 at rates of interest in general use for any number of years or pe riods may be found in published tables known as com pound interest tables.
5. Present converse of finding the amount of a given principal at compound interest is to find the "present worth" or "present value" of a sum of money payable at a future date, interest being taken into consideration.
The present worth at compound interest at any given rate of any given sum, payable at any price fu ture date, may be defined as that present principal which, invested at the given rate and accumulated at compound interest, will amount to the given sum at the given future date.
Thus, to make use of foregoing illustrations, the present worth, at 5 per cent compound interest, of $1,102.50 receivable at the end of two periods is $1,000, because $1,000 invested at 5 per cent com pound interest will amount to $1,102.50 in two periods.
This is evident since $1,000 X 1.05 2 = $1,102.50, $1,000 = $1,102.50 1.05 and since $1,000 X 1.05 = $1,050, $1,000 = $1,050 1.05; and since $500 X 1.06 1° =-- $89.5.42, $500 = $895.42 =1.06 1°.
6. Rule and formula for finding the present worth. the above cases $1,102.50, $1,050 and $895.42 are quantities of money receivable at future dates, while $1,Q00, $1,000 and $500 are their respective present worths at the rate named. The divisor in each case is the ratio of increase used as many times as there are periods in the whole time or the amount of $1 at the given rate for the given time. These three pairs of equations express the following general prin ciple: The present worth is obtained by dividing the given sum by the amount of $1 at the given rate for the given time, or by multiplying the given sum by the present worth of $1 at the given rate for the given time.
A formula for computing present worth may be obtained from Formula I by solving for P, as fol lows: P — A or A X (II) 0 +On inn Since (1 i) n represents the amount of $1 at rate i for n periods, the 1 stands for the pres ent worth at the rate i of $1 payable at the end of n periods. Hence the second part of the rule. In
practice, a tremendous amount of labor can be saved by using published present worth tables in which are computed the present worths of $1 for various lengths of time at various common rates of interest.
7. Annuities defined and diff erentiated.—An an nuity is a series of equal sums of money payable or re ceivable at equal intervals of time. A life annuity is an annuity, the number of payments of which depends upon the continuance of a designated person's life. , This is distinguished from an annuity certain, the number of whose periods is definitely fixed. Thus, if a man were to buy the right to receive $1,000 at the end of each year for ten years, he would have pur chased an annuity certain. The chief arithmetical problems connected with an annuity certain are the computation of its present worth and the computation of its amount at compound interest at the end of the annuity period.
8. Amount of an annuity.—Consider an annuity consisting of five instalments of $100 each, none of which have yet been paid, and let us find its amount at 5 per cent compound interest on the date of the last instalment. The first instalment, coming at the end of the first period may be considered as a single principal which accumulates at 5 per cent compound interest for the remaining four periods. It would amount to $100 X The second, accumulating during three periods, would amount to $100 X 1.05 the third to $100 X 1.05 the fourth to $100 X 1.05 the fifth to just $100. Therefore, designating the amount of the whole annuity by "A," we have