To find the Amount of an Annuity at Compound Interest. RULE. * 1. Make 1 the first term of a geometrical progression, and the amount of 11. for one year, at the given rate per cent. the ratio. 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 upon every succeeding year to that preceding the last, which has but one year's interest, and the last bears no inter est. Let r therefore rate, or amount of 11. for 1 year; then the series of amounts of 11. annuity, for several years, from the first to the last, is 1, r, r2, r3, &c. tort-1. And the sum of this, according to the rule in geometrical progression, will be amount of 11. annuity fort years. And all annuities are propor tional to their amounts, therefore 1 : pt- 1 amount of any given annuity n. Q. E. D. :: n : Xn = Let r rate, or amount of 11. 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: I. Log.n+Log.r'— \— Log. r—\ —Log, a. II. Log, a.—Log. —¡ + Log. r—1=Log. n. 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. 1. What is the amount of an annuity of 40l. to continue 5 years, allowing 5 per cent. compound interest? 2 3 1+105+1'05 +1°05 +1°05|=5 ̊52563125 5'52563125 40 221 02525 20 0'505 12 6'06 Ans. 2211. 6d. 2. If 501. yearly rent, or annuity, be forborn 7 years, what will it amount to, at 4 per cent. per annum, compound interest ? Ans. 3941. 18s. 34d. To find the present value of Annuities at Compound Interest. RULE.* 1. Divide the annuity by the ratio, or the amount of 11. for one year, and the quotient will be the present worth of the first years annuity. *The reason of this rule is evident from the nature of the question, and what was said on the same subject in the purchasing of annuities at simple interest. = Let present worth of the annuity, and the other letters as before, then as the amount = r-l -xn, and as the present worth or principal of this, according to the principles of compound interest, is the amount divided by r', therefore 2. Divide the annuity by the square of the ratio, and the quotient will be the present worth of the annuity for the second year. 3. Find, in like manner, the present worth of each year by itself, and the sum of all these will be the value of the annuity sought. And from these theorems all the cases, where the purchase of annuities is concerned, may be exhibited in logarithmic terms, as follows. Lett express the number of half years or quarters, n the half year's or quarter's payment, and the sum of one pound and or year's interest, then all the preceding rules are applicable to half-yearly and quarterly payments, the same as to whole years. The amount of an annuity may also be found for years and parts of a year thus: 1. Find the amount for the whole years as before. 2. Find the interest of that amount for the given parts of a year. 3. Add this interest to the former account, and it will give the whole amount required. The present worth of an annuity for years and parts of a year may be found thus: 1. Find the present worth for the whole years as before. 2. Find the present worth of this present worth, discounting for the given parts of a year, and it will be the whole present worth required. EXAMPLES. 1. What is the present worth of an annuity of 401. to continue 5 years, discounting at 5 per cent. per annum, compound interest? ratio = 2 [year. 1'05)40'00000(38'095 present worth for first = ratio= 1'1025)40 ̊00000(36°281= rauo3=1'157525)40'00000(34 556= do. for 2d year. do. for 3d year. do. for 4th year. ratio= do. for 5th year. 173 173 1731. 3s. 5d. = '=1′215506)40 ̊00000(32*899 = ratio=1'276218)40 ̊00000(31 ̊342= whole present worth of the annuity required. 2. What is the present worth of an annuity of 211. 10s. 944. to continue 7 years, at 6 per cent. per annum, comAns. 1201. 5s. pound interest? 3. What is 70l. per annum, to continue 59 years, worth in present money, at the rate of 5 per cent. per annum ? Ans. 1321 30211. To find the present worth of a Freehold Estate, or an Annuity to continue forever, at Compound Interest. RULE.* As the rate per cent. is to 100l. so is the yearly rent to the value required. * The reason of this rule is obvious; for since a year's interest of the price, which is given for it, is the annuity, there can neither more nor less be made of that price than of the annuity, whether it be employed at simple or compound interest. The same thing may be shown thus: the present worth of an annuity to continue forever is -+ + + &c.ad infinitum, r r2 n n n as has been shown before; but the sum of this series, by the rules of geometrical progression, is EXAMPLES. 1. An estate brings in yearly 791. 4s. what would it sell for, allowing the purchaser 41 per cent. compound interest for his money? 4 5 100 :: 79°2 : 100 4*5)7920 ̊0(17601. the answer. 4.5 342 315 270 270 2. What is the price of a perpetual annuity of 401. discounting at 5 per cent. compound interest? Ans. 8001. 3. What is a freehold estate of 751. a year worth, allowing the buyer 6 per cent. compound interest for his money? Ans. 12501. n which is the rule. The following theorems show all the varieties of this rule. n n =p. II. r~1 xp=n. III. +1=r, or -=r—1. |1= p 1 The price of a freehold estate, or an annuity to continue forever, at simple interest, would be expressed by + 1+r 1+2r series is infinite, or greater than any assignable number, which sufficiently shows the absurdity of using simple interest in these cases. |