Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year; it may be determined in the following manner: 1st, For any time which is some aliquot part of a year :Find the amount of 17 for 1 year, as before; then that root of it which is denoted by the aliquot part, will be the amount of 11. This amount being multiplied by the principal sum, will produce the amount of the given sum as required. 2d, When the time is not an aliquot part of a year :Reduce the time into days, and take the 365th root of the amount of 11 for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before. And in these calculations, the operation by loga. rithms will be very useful. OF ANNUITIES. ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual pay ments. Annuities are divided into those that are in Possession, and those in Reversion: the former meaning such as have commenced; and the latter such as will not begin till some particular event has happened, or till after some certain time has elapsed. When an annuity is forborn for some years, or the payments not made for that time, the annuity is said to be in Arrears. An annuity may also be for a certain number of years; or it may be without any limit, and then it is called a Perpetuity. The Amount of an annuity, forborn for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due. The Present Worth or Value of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off, or paid all at once. Let a the annuity, pension, or yearly rent; n = the number of years forborn, or lent for ; R= m = the amount of the annuity; v = its value, or its present worth. Now, 1 being the present value of the sum R, by proportion the present value of any other sum a, is thus found: α R2 the present value of a due 1 year hence. is the present value of a due 2 years hence; for R: 1 :: : α a α α a So also be the present values of a, due at the years respectively. Consequently the a R a α a 1 R R2 R3 R4 R5 &c. will + + + + &c. = + + + &c.) X R3 R4 R3 R4 a continued to n terms, will be the present value of all the n years' annuities. And the value of the perpetuity, is the sum of the series to infinity. But this series, it is evident, is a geometrical progression, 1 R having number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be 1 1 1 but for its first term and common ratio, and the When the annuity is a perpetuity; n being infinite, R" 1 is also infinite, and therefore the quantity becomes = 0, the interest of 17 for 1 year, gives the value of the perpetui. ty. So, if the rate of interest be 5 per cent. Then 100a 5 = 20a is the value of the perpetuity at 5 per cent.: Also 100a ÷ 4 25a is the value of the perpetuity at 4 per cent. : And 100a ÷ 3 = 331a is the value of the perpetuity at 3 per cent. and so on. Again, because the amount of 1l in n years, is R2, its increase in that time will be R" - 1; but its interest for one single year, or the annuity answering to that increase, is R - 1; therefore, as R - 1 is to Rn - 1, so is a to m; that Rn 1 is, m = R X a. Hence, the several cases relating to Annuities in Arrear, will be resolved by the following equations: In this last theorem, r denotes the present value of an annuity in reversion, after p years, or not commencing till after the first p years, being found by taking the difference R" 1 a between the two values years and p years. RP 1 α for n But the amount and present value of any annuity for any number of years, up to 21, will be most readily found by the two following tables. TABLE I. The Amount of an Annuity of 11 at Compound Interest. Yrs. at 3 perc. 31 per c. 4 per c. 41 perc. 5 per c. 6 per C. 2-0300 2.0350 2.0400 2.0450 2.0500 2.0600 3 4 5 3-0909 3.1062 3.1216 31370 31525 3.1836 4:3746 5.5256 5.6371 6 7 6.4684 6.5502 6.6330 6.7169 68019 6.9753 8 12 S-8923 9.0517 9.2142 9-3800 9.5491 9.8975 9 10-1591 10-3685 10-5828 10-8021 11:0266 11:4913 10 11-4639 11.7314 12.0061 12:2882 12-5779 13-1808 11 12-8078 13.1420 13-4864 13 8412 14-2068 14.9716 14.1920 14-6020 15.0258 15-4640 15-9171 16-8699) 13 15.6178 16-1130 16-6268 17-1599 17.7130 18 8821 14 17.0863 17.6770 18-2919 18.9321 19:5986 21 0151 15 18-5989 19-2957 20-3236 20 7841 21-5786 23-2760 20-1569 20-9710 21 8245 22-7193 23-6575 25-6725 21-7616 22.7050 23-6975 24 7417 25-8404 28-2129 23-4144 24-4997 25 6454 26 8551 28 1324 30-9057 25-1169 26-3572 27-6712 29 0636 30-5390 33-7600 26-8704 28.2797 29-7781 31-3714 33-0660 36-7856 28-6765 30-2695 31 9692 33-7831 35 7193 39.9927 16 17 18 19 20 21 TABLE II. The Present Value of an Annuity of 17. Yrs. at 3 per c. 3 per c. 4 per c. 4 per c. 5 per c. 9-9540 9.6633 9.3851 1310-6350 103027 9.9857 14 11-2961 10 9205 10-5631 10-2228 15 16 17 18 19 20 21 8.8527 9.8986 9.2950 11-9379 11-5174 11·1184 10.7396 10:3797 9.7123 12:56 11 12-0941 11·6523 11·2340 10.8378 10-1059 13-1661 12-6513 12:1657 11.7072 11-2741 10-4773 13-7535 13-1897 12-6593 12-1600 11.6896 10.8276 14-3238 13-7098 13-1339 12.5933 12·0853 11.1581 14-8775 14-2124 13:5903 13-0079 12:4622 11-4699 15-4150 14-6980 14:0292 13:4047 12-8212 11-7641] To find the Amount of any Annuity forborn a certain number of years. TAKE out the amount of 17 from the first table, for the proposed rate and time; then multiply it by the given annuity; and the product will be the amount, for the same number of years, and rate of interest. And the converse to find the rate of time. Exam. To find how much an annuity of 501 will amount to in 20 years, at 3 per cent. compound interest. On the line of 20 years, and in the column of 31 per cent. stands 28.2797, which is the amount of an annuity of 17 for the 20 years. Then 28-2797 X 50, gives 1413·9857 = 14137 19s 8d for the answer required. To find the Present Value of any Annuity for any number of years.-Proceed here by the 2d table, in the same manner as above for the 1st table, and the present worth required will be found. Exam. 1. To find the present value of an annuity of 501, which is to continue 20 years, at 31 per cent.-By the table, the present value of 1l for the given rate and time, is 14.2124; therefore 14.2124 × 50710-621 or 710l 12s 4d is the present value required. Exam 2. To find the present value of an annuity of 201, to commence 10 years hence, and then to continue for 11 years longer, or to terminate 21 years hence, at 4 per cent. interest. In such cases as this, we have to find the difference between the present values of two equal annuities, for the two given times; which therefore will be done by subtracting the tabular value of the one period from that of the other, and then multiplying by the given annuity. Thus, tabular value for 21 years 14.0292 |