ANNUITIES, any income of a. certain yearly amount, payable at particular peri ods, which may be either yearly, half yearly, quarterly, monthly, weekly, or at any other intervals. They are usually dis tinguished into annuities certain, and con tingent annuities, or such as are for an un certain period, being determinable by some future event, such as the failure of a life or lives.
The present value of an annuity is that sum, which, if improved at compound in terest, would be sufficient to pay the an nuity- ; the present value of an ainvity certain, pay able yearly-, and of whillOrthe first payment is to be made at the end of a year, may therefore be calculated in the following manner.
Suppose a person has 100/. due to him a twelve month hence, and he wishes to have the value of the same advanced im mediately, the sum which ought to be giv en as an equivalent thereto,allowing 5 per cent. interest, is 95L 4s. 9-td.for this is the sum, which, put out to interest, at the rate of 5 per cent will, at the end of the year, amount to 100/. So also, if a person has 100/. due to him at the end of two years, and he wishes to have the value of the. same advanced immediately, the sum which ought to be given as an equivalent thereto is 901. 14r. Old. for this is the sum, which, put out at the same rate of interest, will, at the end of two years, amount to 100/. In like manner, if a person has 100L due to hint at the end of three years, and he wishes to have the same advanced im mediately, the sum which ought to be given as an equivalent thereto is 86/. 78. M. for this is the sum which, at the same rate of interest, will at the end of three years amount to 100L And if these three values are added together, they will make 272/. 6a. 6d. being the sum which ought
to be paid down for an annuity of 100/. for three years; as this sum improved at the given rate of interest is just sufficient to make the three yearly payments.
As the amount or present worth of IL for any given term is usually adopted as the foundation of calculations relating to annuities, let r represent the amount of IL in one year ; that is, one pound in creased by ayear's interest; then •n, or f• raised to the power whose exponent is any given number of years, will be the amount of IL in those years; its increase in the same time is ro — 1; now the interest for a single year, or the annui ty corresponding with the increase, is r — ; therefore as r —1 is to •n —1, so is a (any given annuity) to a its amount ; hence we have 7€ X rn— 1 a EXAMPLE.—To what sum trill an an nuity of 501. amount in 6 years, at 5 per cent. per annum, compound interest ? 50 x 3401. 19a. ld. .05 In this manner the amount of an annuity for any number of years, at any given rate of interest, may be found. But when the term of years is considerable, it will be more convenient to work by logarithms, by which the labour of all calculations re lating to compound interest is greatly abridged. There is, however, little occa sion in general to calculate the amount or present worth of annuities, except for par ticular rates of interest, as the following tables, and others of a similar nature, for different rates of interest, which are given inmost books on compound interest, save much time and labour in common prac tice, and are therefore in general use.