Note 1. When the annuity is payable half-yearly, and the value received for it can be improved at an interest payable half-yearly; or, when the annuity is payable quarterly, and the money arising from it can be improved at an interest also payable quarterly; and so on; the above theorems will answer if t denote the number of times payment is made; r the interest of £1 in one of those times, and A the sum, or part of the annuity payable each time. 2. When the annuity is not payable at the same intervals at which the interest is convertible into principal; the amount of any annuity (A) payable yearly for t years is = Ax[(1+")" —1] ÷ and universally the amount of any annuity (A), payable u times in a year; for t years, each payment being the uth Α nt partof the annuity is== [(1+)"~1]÷[(1+5)*—1] : X น in which formulæ n is the number of payments of interest in a year, each payment being the nth part of r. 3. In calculating the value of any periodical income, the first payment is always considered to be made at the end of the first period from the time of the valuation, unless the contrary be expressly stated. When the first payment is made immediately, then Ꭱ -1 M=Ax XR. EXAMPLES. 1. What will an annuity of £60, payable yearly, amount to in 30 years, reckoning compound interest at the rate of 4 per cent.? Here A-60, R-104, and t=30; consequently m=60x (1·04)30-1 ⚫04. =60×56-0849377 £3365 1 11. Ans. = 2. To what sum will an annuity of £60, receivable in equal halfyearly payments, amount in 30 years, when money yields an interest of 2 per cent. every half year? m=30x Here A-30, R=1-02, and t-60; consequently (1.02)60 ⚫02 30x114-051539=£3421 10 11. Ans. 3. To what sum will an annuity of £60, receivable in equal quar. terly payments, amount in 30 years, at the rate of 4 per cent. compound interest, when the interest is payable yearly? (1·04)001 Here m=15x 15×227-67734 1·041 £3415 3 2. Ans. EXERCISES. 1. To what sum will an annuity of £20 for 20 years amount, when each payment is improved at compound interest, at the rate of 3 per cent., from the time it becomes due until the expiration of the term? 2. To what sum will an annuity of £100 for 25 years amount, improved at the rate of 5 per cent. compound interest? 3. What annuity, improved at the rate of 3 per cent. compound interest, will, at the end of 20 years, amount to £1413:19:8? 4. In what number of years will an annuity of £20 amount to £890: 09 at the rate of 5 per cent. compound interest? 5. Find the amount of an annuity of £20 for 15 years, improv ed at the rate of 4 per cent. compound interest; both the interest and the annuity being payable half-yearly. 6. Find the amount of an annuity of £100 for 25 years, improved at the rate of 5 per cent. compound interest; the annuity being payable half-yearly, and the interest quarterly. 7. A seaman, who was engaged to serve at 30 per month, after 6 years' service, received his wages, with compound interest on the monthly payments, at the rate of 4 per cent.; what sum did he then receive, supposing the interest to have been con verted into principal yearly? 8. A servant, after 7 years' service, received his wages, being 5/ per week, with compound interest on the weekly payments, at the rate of 5 per cent. per annum: what sum did he receive, supposing the interest to have been converted into principal every half year. II. On the Present Value of Annuities. The present value of an annuity is such a sum, as, being improv ed at compound interest, will just be sufficient for the payment of the annuity as it becomes due. The present value (p) of any annuity (A) to continue (t) years, may be found as follows: Let r denote the simple interest of £1 for a year, and R, the amount of £1 at the end of a year: it is plain, that at the same rate of interest, the present value of any annuity must bear the same proportion to its amount at the end of a given term that £1 does to its amount at the end of the same term. Now the amount of £1 at the end of t years is Rt, and the amount of any annuity A at the Rt. 1 end of the same term is Ax Rt 1 ; therefore Rt: Ax When the annuity is to continue for ever, it is called a perpetui. ty; and in that case, the quantities R 1 and Rt in the 4th theorem, may evidently be considered as the same, and therefore VII. p=A÷r VIII. A=pr IX. r=A÷p Note 1. When the annuity is payable at the same intervals at which the interest is convertible into principal, the above theorems will answer, if t denote the number of times payment is made, r the simple interest of £1 in one of those times, and A the sum paid each time. 2. When the annuity is not payable at the same intervals at which the interest is convertible into principal, the present value of any annuity (A) payable yearly for. t years is 1 = AX ent value of any annuity (A) payable u times in a year for t years, each payment being the uth part of the annuity is A : which expression, in 1 these formulæ n is the number of payments of interest in a year, each payment being the nth part of r. 3. The present value of an annuity of £1 is commonly called the number of years purchase that any annuity is worth for that time and rate of interest. 1 EXAMPLES. 1. Find the present value of an annuity of £120 for 50 years, at the rate of 4 per cent. compound interest. Here A=120, R=1·04, and t=50; consequently p = 120 × (1·04)501 £2577 17 24. 120 × 6·10668335 → ·284267334 2. Find the present value of an annuity of £120 for 50 years, re ceivable in equal half-yearly payments, when money yields an interest of 2 per cent. every half year. Here A-60, R=1·02, and t=100; consequently 3. Find the present value of an annuity of £120 for 50 years, re ceivable in equal quarterly payments, at the rate of 4 per cent. compound interest. when the interest is payable yearly. 1. Find the present value of an annuity of £30, to continue 35 years, at the rate of 34 per cent. compound interest. 2. What annuity, to continue 20 years, may be purchased for £260: 3:24, reckoning compound interest at the rate of 4 per cent. ? 3. An annuity of £25 is purchased for £416: 11: 63; how many years should it continue, reckoning interest at the rate of 4 per cent. ? 4. The present value of an annuity of £80 is £800, at the rate of 5 per cent. find its duration. 5. Find the present value of an annuity of £25 for 50 years, receivable in equal half-yearly payments, when money yields an interest of 2 per cent. every half year? 6. Find the present value of an annuity of £40 for 50 years, receivable in equal quarterly payments, at the rate of 4 per cent. compound interest, when the interest is payable yearly. 7. Find the value of a freehold estate of £250-a-year, allowing the purchaser interest at the rate of 5 per cent. for his money. 8. Find the yearly rent of an estate, bought for £4000, at the rate of 74 per cent. interest. 9. An estate of £70 a-year is bought for £2000, what rate of interest does the buyer receive? 3 III. On Deferred Annuities. A deferred or reversionary annuity, is an annuity to be entered on at the end of a given term. The present value (p) of any annuity (A) to be entered on at the end of (d) years, and then to continue (1) years, is evidently equal to the excess of its present value for (d+t) years, above its present = [Log. A — Log. (A—pr R2 ]÷Log. R XII. t= XIII. 4= [ Log. (1— — 1) — Log, (pr÷A) ] ÷ Log R When the annuity is to continue for ever after its commencement, Rt 1 being in that case the same with Rt, therefore XIV. p = A÷÷÷rRa XV. A=pr Rd XVI. d = (Log. A Log. pr) Log. R Note. Universally, the present value of any deferred annuity for t years after d years, payable u times in a year, and the interest being payable n times in a year, is = |