the Present Value of the sum of the several payments supposed due at the Equated time. Thus, if P, p, be due at the end of times T, t, respectively, r the rate of interest, and a be the equated time, from which simple equation with respect to x, we get P P + 1+ Tr 1 + tr COR. If the quantities multiplied by r be neglected, since r is generally a very small fraction, we have 408. To find the Amount of an annuity or pension left unpaid any number of years, allowing simple interest upon each sum or pension from the time it becomes due. Let A be the annuity; then at the end of the first year A becomes due, and at the end of the second year the interest of the first annuity is rA (Art. 396); at the end of this year the principal becomes 2A, therefore the interest due at the end of the third year is 2r A; in the same manner, the interest due at the end of the fourth year is 3rA; &c. Hence the whole interest at the end of n years is and the sum of the annuities is n A, therefore the whole amount MnA+n. 409. Required the Present Value of an annuity which is to continue a certain number of years, allowing simple interest for the money. Let P be the present value; then if P, and the annuity, at the same rate of interest, amount to the same sum, they are upon the whole of equal value. The amount of P in n years is P+ Pnr (Art. 396); and the amount of the annuity in the same From this equation, any three of the four quantities P, A, n, r being given, the other may be found. n A 410. COR. Let n be infinite, then P = an infinite quan tity; therefore for a finite annuity to continue for ever, an infinite sum ought, according to this calculation, to be paid; a conclusion which shews the necessity of estimating the Present Value of an annuity upon different principles. 411. To find the Amount of an annuity in any number of years, at compound interest. Let A be the annuity, or sum due at the end of the first year; then 1 R A RA, its amount at the end of the second year; therefore A+RA is the sum due at the end of the second year; in the same manner, 1 : R :: (1 + R) × A : (R + R2) × A, the amount of the two payments at the end of the third year; and (1 + R + R2) × A is the whole sum due at the end of the third year; in the same manner, is the sum due at the end of n years, that is, the amount required Rn × A. 412. COR. 1. From this equation, any three of the quan tities being given, the fourth may be found. COR. 2. If interest be paid q times per annum, and be each pay ment per 1£, the amount of the annuity in n years, reckoning compound interest, will be A COR. 3. If the annuity (4) be payable m times per annum, each of the payments being and Р be the annual rate of interest, the amount in n years will be m and if the interest also be payable q times a year, each payment of interest for every 1£ being, this amount becomes r 413. To find the Present Value of an annuity to be paid for n years, allowing compound interest. Let P be the Present Value, ▲ the annuity; then since PR" is 414. COR. 1. Any three of the quantities P, A, R, n being given, the fourth may be found. If the number of years be infinite, RTM is 415. COR. 2. infinite, and R- vanishes; therefore P = Ex. If the annual rent of a freehold estate be 1£, what is its value, allowing 5 per cent. per ann. compound interest? 1, R − 1 = 0·05; therefore the Present Value 1 £20, or 20 years purchase. P = COR. 3. The Present Value of an Annuity of A£ payable m times per annum for n years, each of the payments being and rate of interest, will be A , Р the annual m and if the interest also be payable q times a year, each payment of interest for every £1 being, the Present Value will be COR. 4. If the annuity is to continue for ever, this Present Value 416. The Present Value of an annuity, to commence at the expiration of p years, and to continue q years, is the difference between its present value for p+q years, and its present value for p years, COR. If the annuity commences after p years, and continues for AR-1 R-1' Ex. What is the Present Value of an annuity of 1£, for 14 years, to commence at the expiration of 7 years, allowing 5 per cent. per ann. compound interest? hence the value of the annuity for fourteen years after the expiration of 7 is 7-03£, or 7 years purchase, nearly. The preceding Article contains the whole Theory of the RENEWAL OF LEASES. 417. To determine the fine which ought to be paid for renewing any number of years lapsed in a lease. Let P + q be the number of years for which the lease was originally granted; p the number lapsed; and A the clear annual value of the estate, after deducting reserved rent (if any), taxes, and all other fixed annual charges. Then it is clear that the lessee has to purchase an annuity of A£ to commence at the expiration of q years, and to continue p years, the Present Value of which is the Present Value for p+q years Present Value for q years A {R--R-P+). R 1 -- - Ex. In a lease of 21 years 7 years lapsed are to be renewed, the reserved rent is 10£, and the estate is really worth 150£ a year, what fine ought to be paid for the renewal, reckoning interest at 5 per cent.? In this case the lessee has to pay for an annuity of 140£ to commence at the end of 14 years and to continue 7 years; therefore the fine required is |