a-P and rə in i (5.) pt pt By means of the above five formulæ all the circumstances connected with the simple interest of money are readily determined. But as the rules for the calculation of simple interest are generally given in reference to the rate per cent., instead of the rate per pound, as above, the formulæ may be all changed into those relating to rate per cent., by making r represent the rate per cent, and substituting 100 stead of r throughout all the formulæ; and the student is requested to write in words the rules which the five formulæ contain, by which he will be led to see the advantage of algebraic formulæ. 1. What is the interest of L.560 for 3 years, at 4, per cent. ? Ans. L.75, 12s. 2. What is the amount of L. 420 for 6 years, at 3 per cent. ? Ans. L.495, 12s. 3. What principal laid out at interest for 5 years at 4 per cent. will gain L.60 ? Ans. L.300. 4. What principal laid at interest for 10 years at 3 per cent. will amount to L.607, 10s. ? Ans. L.450. 5. In what time will L.500 amount to L.800 at 4 per cent. ? Ans. 15 years. 6. At what rate per cent. will L.200 amount to L.344 in 18 years? Ans. 4 per cent. COMPOUND INTEREST. 120. In compound interest, the interest is added to the principal at stated intervals or periods, and this amount is made the principal for the next period. Hence if R represent the amount of one pound at the end of one period, since this is the sum laid at interest during the next period, we will evidently have the following proportions to find the amount of L.1 at the end of any number of periods. 1: R=R: RP, amount at the 2d period. 1: R=Rn-1: R", amount at the nth period. From which it appears that the amount of one pound at the end of any number of periods is R raised to the power denoted by the number of periods, and it is plain that the amount of p pounds will be p times the amount of one pound; hence, representing the amount of p pounds by A, and the number of periods by t, we will have F 4=pR", or log. A=log. ptt log. R. A p= log. p=log. A-t log. R. (1.) (2.) R'= t log. R=log. A_log. p. log. A_log. P (3.) (4.) log. R t log. A_log. p (5.) 121. The interest is generally converted into principal yearly, but sometimes half-yearly, and sometimes even quarterly. If r represent the simple interest of L.1 for a year, and n the number of years for which the calculation is to be made, then when the interest is convertible into principal half-yearly, R and t will have the following values: R=l+a, t=2n; and when it is convertible quarterly, R=l+, t=4n ; also R=1+, and t=mn, when the interest is convertible into principal m times per annum. EXERCISES. 1. What will be the amount of L.1000 in ten years, at 5 per cent. compound interest ? Ans. L.1628, 17s. 94d. 2. What principal laid at compound interest will amount to L.700 in eleyen years, at 4 per cent. ? Ans. L.454, 14s. 11d. 3. In what time will L.365 amount to L.400, at 4 per cent. compound interest? Ans. 2 years 122 days. 4. At what rate per cent. compound interest will L.50 amount to L.63, 16s. 31d. in five years ? Ans. 5 per cent. 5. In what time will a sum of money double itself, at 5 per cent. compound interest? Ans. 14.2 years. 6. In what time will a sum of money double itself, at 4 per cent. compound interest. Ans. 17.67 years. 7. In what time at compound interest, reckoning 5 per cent. per annum, will L.10 amount to L.100 ? Ans. 47.19 years. 8. What will be the compound interest of L.100 for twelve years at 4 per cent., if the interest be payable yearly? what if payable half-yearly? and what if payable quarterly? Ans. L.60, 2s. 02d., L.60, 16s. 101d., and L.61, 4s. 5d. ANNUITIES. 122. ANNUITIES signify any interest of money, rents, or pensions, payable from time to time, at particular periods. The most general division of annuities, is into annuities certain, and annuities contingent ; the payment of the latter depending upon some contingency, such, in particular, as the continuance of a life. Annuities have also been divided into annuities in possession, and annuities in reversion, the former meaning such as have commenced, or are to commence immediately, and the latter such as will not commence till some particular future event has happened, or till some given period of time has expired. Annuities may be farther considered as being payable yearly, half-yearly, or quarterly. The present value of an annuity is that sum, which being improved at compound interest, will be sufficient to pay the annuity. The present value of an annuity certain, payable yearly, and the first payment of which is to be made at the end of a year, is computed as follows: Let the annuity be supposed L.l; the present value of the first payment is that sum in hand, which being put to interest, will amount to L.l in a year; in like manner, the present value of the second payment, or of L.) to be received two years hence, is that sum, which being put to interest immediately, will amount to L.1 in two years, and for years or payments; and the sum of the values of all the payments will be the present value of the annuity. 123. Let the interest of L.1 for one year be represented by r, then L.1 will amount to 1+rin a year, and the sum that will amount to one in one year, which call x, will evidently bear the same proportion to L.1, that L.1 bears to 1+r; hence we have the following proportion: 2:1=1:1+r; value of 1st payment. 1tr In the same manner, that sum which in two years will amount to L.1, is evidently that sum which in one year 1 will amount to 1+r Stating the proportion so that the quantity sought may stand last, we have SO on 1 1 1 1 1 1 1 + (1 toje + ... 1+r:l=l: present value of 1st payment. 1+r 1 1+r:= 1+r+ (1+r)2present value of 2d payment. 1+r:l= (1+r)2 : (1+r)ăpresent value of 3d payment. 1+r:l= (1+r)3 + (1+r)® present value of 4th payment. 1+r:l= (1+r)-1(1+r)" present value of nth payment. 124. The present value of an annuity of L.) for n years is therefore the sum of the series. 1 1 1 1 1 + + +...t (1+r) (1+r) This is evidently a geometrical series, in which the first 1 term is and the common ratio is also hence find 1tr ltri ing its sum as in geometrical progression, and putting p for the sum that is the present value of the annuity, we have 1 1 + (1+r)*+ (1+r) l+r (1.) (1+r). 1 t. (1+r) (1+r) + 1+r (1+r)" (2.) (1+r)n+1 р. PItr (1.)–(2.) (1+r)*+1 1+r (3.) 1 rp=l (3.) *(1+r.) (4) (1+r)" (1+r)"-1 (4.);r. (5.) 125. If the annuity is to continue for ever, then n be 1 comes infinite, and also (1+r)"; hence may be con (1+r)" sidered as 0, and therefore we have for the present value of an annuity of L.1, payable for ever, p=, value of a perpetuity of L.1. It is plain, that if the annuity be a pounds instead of one, it will just be a times as great as before; and therefore the present value of an annuity of a pounds, payable for n years, will be (1+r)"--1 p= Х 1+r)" 1 a or r (1+r) and that of a perpetuity of a pounds will be p=; 126. When an annuity is only to commence n years hence, and then continue for t years, it is called a deferred annuity, and it is plain that its present value will be the difference between the present value of an annuity to continue for t + n years, and another to continue for n years ; but we have seen (5) that the present val of an annuity of L.l to continue ttn years, is (1+r)*+n_1 and that +(1+r)c+n the present value of an annuity to continue n years is (1+r)n-1; the difference of these expressions is therefore (1+r) the present value of an annuity commencing n years hence and continuing afterwards for t years; reducing these expressions to a common denominator and subtracting, we have (1+r)t-1 and therefore the present value of an an(1+r)*+n' nuity to commence n years hence, and afterwards to continue for t years, is p= (1+r)1 ; and if the annuity be a r(1+r)+n pounds per annum, instead of one, it is plain that the whole result will be a times as great; therefore the present value of an annuity of a pounds per annum deferred for n years, and then payable t years, is p=@{(1+r).—1} (1+r)*+n If the annuity be payable for ever after n years, then t, and consequently (1+r) become infinite, dividing both numerator and denominator of the above expression by (1+r)', and observing that becomes 0, we have for (1+r) the present value of a perpetuity of a pounds deferred for 1 n years, p= (1+r)n 127. To find the amount of an annuity left unpaid any number of years, at compound interest. Let A be the annuity, then the amount of the first payment which is foreborne for n-1 years will be A(1+rjat; of the second for n—2 years will be A(1+r)->, &c. .. the whole amount=A(1+(1+r)+(1+r) +, &c.) to n terms; or the amount= Art. 96. (3.) = ((1+-) –1). |