ANNUITIES. AN ANNUITY is a sum of money payable every year, for a certain number of years, or for ever. When the debtor keeps the annuity in his own hands, beyond the time of payment, it is said to be in arrears. The sum of all the annuities for the time they have been forborn, together with the interest due upon each, is called the amount. If an annuity be to be bought off, or paid all at once, at the beginning of the first year, the price, which ought to be given for it, is called the present worth. To find the Amount of an Annuity at Simple Interest. RULE.* 1. Find the sum of the natural series of numbers 1, 2, 3, &c. to the number of years less one. 2. Multiply * DEMONSTRATION. Whatever the time is, there is due upon the first year's annuity, as many years' interest as the whole number of years less one; and gradually one less upon every succeeding year to the last but one ; upon which there is due only one year's interest, and none upon the last; therefore in the whole there is due as many years' interest of the annuity, as the sum of the series 1, 2, 3, 4, &c. to the number of years less one. Consequently one year's interest, multiplied by this sum, must be the whole interest due; to which if all the annuities be added, the sum is plainly the amount. Q. E. D. Let r be the ratio, n the annuity, t the time, and a the amount. Then will the following theorems give the solutions of all the 1 2. Multiply this sum by one year's interest of the annuity, and the product will be the whole interest due upon the annuity. 3. To this product add the product of the annuity and time, and the sum will be the amount sought. NOTE. When the annuity is to be paid half-yearly or quarterly; then take, in the former case, the ratio, half the annuity, and twice the number of years; and, in the latter case, the ratio, the annuity, and 4 times the number of years, and proceed as before. EXAMPLES. 1. What is the amount of an annuity of 50l. for 7 years, allowing simple interest at 5 per cent. ? 1+2+3+4+5+6=21=3×7 21. 10s. 1 year's interest of 50l. cannot be found equal to the amount, the problem is impossible in whole years. 1 NOTE. Some writers look upon this method of finding the amount of an annuity as a species of compound interest; the annuity itself, they say, being properly the simple interest, and the capital, whence it arises, the principal. 2. If a pension of 600l. per annum be forborn 5 years, what will it amount to, allowing 4 per cent. simple interest ? Ans. 3240l. 3. What will an annuity of 250l. amount to in 7 years, to be paid by half-yearly payments, at 6 per cent. per annum, simple interest ? : Ans. 20911. 58. : To find the present Worth of an Annuity at Simple Interest. Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these will be the present worth required. EXAMPLES. * The reason of this rule is manifest from the nature of discount, for all the annuities may be considered separately, as so many sin gle and independent debts, due after 1, 2, 3, &c. years; so that the present worth of each being found, their sum must be the present worth of the whole. The estimation, however, of annuities at simple interest is highly unreasonable and absurd. One instance only will be sufficient to shew the truth of this assertion. The price of an annuity of 50l. to continue 40 years, discounting at 5 per cent. will, by either of the rules, amount to a 'sum, of which one year's interest only exceeds the annuity. Would it not therefore be highly ridiculous to give, for an annuity to continue only 40 years, a sum, which would yield a greater yearly interest for ever ? It is most equitable to allow compound interest. EXAMPLES. 1. What is the present worth of an annuity of 100l. to continue 5 years, at 6 per cent. per annum, simple interest ? 106 : 100 :: 100: 94°3396 = present worth for 1 year. 2. What is the present worth of an annuity or pension of 500l. to continue 4 years, at 5 per cent. per annum, simple interest ? Ans. 17821. 58. 7d. To find the Amount of an Annuity at Compound Interest. RULE.* : 1. Make I the first term of a geometrical progression, and the amount of 11. for one year, at the given rate per cent. the ratio. 2. Carry The other two theorems for the time and rate cannot be given in general terms. * DEMONSTRATION. It is plain, that upon the first year's annuity, there will be due as many years' compound interest, as the given number of years less one, and gradually one year's interest less : 2. Carry the series to as many terms as the number of years, and find its sum. 3.- Multiply the sum thus found by the given annuity, and the product will be the amount sought. EXAMPLES. less upon every succeeding year to that preceding the last, which has but one year's interest, and the last bears no interest. Letr, therefore, rate, or amount of 11. for 1 year; then the series of amounts of Il. annuity, for several years, from the first to the last, is 1, r, ,r3, &c. to r-1. And the sum of this, according to the rule in geometrical progression, will be r-I I , = amount of sl. annuity for years. And all annuities are proportional to amount of any given annuity n. Q. E. D. Letr = rate, or amount of Il. for one year, and the other And from these equations all the cases relating to annuities, or pensions in arrears, may be conveniently exhibited in logarithmic terms, thus : HI. I. Log.n+Log. rt-1-Log.r-1=Log. a. II. Log.a-Log.r-1+Log.r-1=Log.n. 1 |