of The following Table will very much facilitate calculations compound interest on any sum, for any number of years, at various rates of interest. The Amounts of 17. in any Number of Years. Yrs. 3 34 4 41 5 1 4 6 1.3382 1.4185 1.5036 1'0300 10350 1:0400 1.0450 10500 1.0600 12 15 The use of this Table, which contains all the powers, R, to the 20th power, or the amounts of 17, is chiefly to calculate the interest, or the amount of any principal sum, for any time, not more than 20 years. For example, let it be required to find, to how much 5231. will amount in 15 years, at the rate of 5 per cent per annum compound interest. In the table, on the line 15, and in the column 5 per cent. Note 1. When the rate of interest is to be any other time than a year; as suppose to year, &c; the rules are still the same; S 2 determined to a a year, or but then will express express that time, and R must be taken the amount for that time also. 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 17. 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 logarithms 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 payments. 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 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; 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: a as R: 1 :: a: the present value of a due 1 year hence. R be the present values of a, due at the end of 3, 4, 5, &c, years respectively. Consequently the sum of all these, or 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 having both for its first term and common ratio, and the R number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be X 1 1 1 When the annuity is a perpetuity; n being infinite, R 1 is also infinite, and therefore the quantity becomes = 0, R" X also = 0; consequently the expres 1 R" sion becomes barely v = ; that is, any annuity divided R- -1 by the interest of 17. for 1 year, gives the value of.the perpetuity. 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 per petuity petuity at 4 per cent: And 100a ÷ 3 = 334a is the value of the perpetuity at 3 per cent and so on. Again, because the amount of 17. in n years, is_R", 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 R" 1, so is a to m; that is, m = R" 1 R 1 x a. Hence, the several cases relating to Annuities in Arrear, will be resolved by the following equations: R" 1 × a = vR"; R" - log. × VR"; MR-m+a a log. R 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 any But the amount and present value of any annuity for number of years, up to 21, will be most readily found by the two following tables. TABLE TABLE I. The Amount of an Annuity of 17. at Compound Interest. Yrs. at 3 perc. 34 per c. 4 per c. 4 per c. 5 per c. 6 per c. 7 8 4.3746 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 2.0300 2:0350 2.0400 2.0450 2.0500 2.0600 3 3:0909 3.10652 3.1216 3.1370 3.1525 8.1836 4 4.1836 4.2149 4.2465 4-2782 4:3101 5 5:3091 5.3625 5.4163 5.4707 5.5256 5.6371 6 6.4684 6.5502 6.6330 6.7169 68019 6.9753 7.6625 7.7794 7.8983 80192 81420 8.3938 8.8923 90517 92142 9-3800 9.5191 9.8975 9 10 1591 10-3685 10 5828 10.8021 11.0266 11.4913 10 11.4639 11-7314 12-0061 | 12-288212-5779 13.1808 11 12.8078 13-1420 13.4864 13 8412 14-2008 14.9716 12 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 1620-1569 20.9710 21.8245 22.7193 23 6575 25-6725 17′′ 21-7616|22.7050 23.6975 24-7417 25-8404 28-2129 18 23-4144 24.4997 25.6454 26.8551|28-1324 30-9057 19 25 1199 26-3572 27-6712 29.0636 30.5390 33.7600 20 26 8704 28-2797 29-7781 31.3714 33.0660 | 36·7856 21 28.6765 30·2695 |31·9692|33·7831|35·7193|39.9927 TABLE II. The Present Value of an Annuity of 17. Yrs. at3 perc.3 per c. 4 per c. 42 per c. 5 per c. 6 per c. 2 3 1 0.9709 0.9662 09615 0.9569 0.9524 0.9434 6 7 8 9 10 4.2124 4:9173 5.5824 6.2098 7:0197 68740 6.7327 6.5959 6-4632 7.7861 7:6077 7.4353 7.2688 7.1078 6.8017 8.5302 8.3166 81109 7.9127 7·7217 7.3601 11 9.2520 90116 87605 8.5289 8.3054 7.8869 12 9.9540 9-6633 93851 9.1186 8.8633 8.3838 13 10.6350 10-3027 9:9857 9:6829 9.3936 8.8527 14 11-2961 10-9205 10-5631 10-2228 9.8986 9.2950 15 119379 11.5174 11.1184 10.7396 10.3797 9.7123 16 12:5611 12-0941 | 11-6523 11 2340 10 8378 10.1059 17 13-1601 12.6513 12.1657 11-7072|11-2741 10-4773 18 13 7535 13.1897 12:6593 12-1600 11.6896 10.8276 1914-3238 13.7098 13 1339 12.5933 12.0853 11∙1581 20 14 8775 14.2124 13.5903 13-0079 12-4622 11·4699 21 15-4150 14.698014.0292 |13·4047 12.8212 11 7641 |