TABLE IV. Showing the present worth of £1 annuity for any number of years, from 1 to 40. 9.71225 15 9.10589 16 15 11.41839 10.73954 10.37965 10.03759 16 11.65229 11.23401 10.83777 10.46216 17 12.16567 11.70719 11.27407 10.86461 10.47726 17 18 12.65929 12.15999 11.68958 11.24607 10.8276 18 19 13.13394 12.59329 12.08532 11.60765 11.15811 19 20 13.59032 13.00793 12.46221 11.95034 11.4699220 21 14.02916 13.40472 12.82115 12.27524 11.76407 21 22 14.45111 13.78442 13.163 12.58317 12.04158 22 23 14.85684 14.14777 13.48857 12.87504 12.30338 23 24 15.24696 14.49548 13.79864 13.15170 12.55035|24| 25 15.62208 14.82821 14.09394 13.41391 12.78335 25 26 15.98277 15.14661 14.37518 13.66250 13.00316 26 27 16.32959 15.45130 14.64303 13.89810 13.21053 27 28 16.66306 15.74287 14.89813 14.12142 13.4061628 29 16.98371 16.02189 15.14107 14.33310 13.59072 29 30 17.29203 16.28889 15.37245 14.53375 13.76483 30 31 17.58849 16.54439 15.59281 14.72393 13.92908 31 32 17.87355 16.78889 15.80268 14.90420 14.08404 32 33 18.14764 17.02286 16.00255 15.07507 14.23023 33 34 18.41126 17.24676 16.1929 15.23703 14.36814 34 35 18.66461 17.46101 16.37419 15.39055 14.49825 35 36 18.90828 17.66604 16.54685 15.53607 14.62098|36| 37 19.14258 17.86224 16.71129 15.67400 14.73678 37 38 19.36786 18.04999 16.86789 15.80474 14.84602|38| 39 19.58448 18.22965 17.01704 15.92866 14.94907 39 40 19.79277 18.40158 17.15909 16.04612 14.92640 40 TABLE V. Rate Half yearly Quarterly p. ct.payments. payments. 3 The construction of this table, is from an algebraic theorem, given by the learn1.007445 1.011181 ed A. De Moivre, in his trea31.008675 1.013031 tise of Annuities on lives, 4 1.009902 1.014877 42 1.0111261.016720 thus: 1.0123481.018559 5 5 1.013567 1.020395 6 which may be in words, For half yearly payments take a unit from the ratio, and from the square root of the ratio; half the quotient 1.0172041.025880 of the first remainder divid 1.014781 1.022257 61.015993 1.024055 7 ed by the latter, will be the tabular number. For quarterly payments use the 4th root as above, and take one quarter of the quotient. CASE 1. The annuity, time, and rate of interest given, to find the amount. RULE. From the ratio involved to the time take a unit, or one, for the dividend; which divide by the ratio less one; and multiply the quotient by the annuity, for the amount or answer. Or, by Table III. Multiply the number under the rate, and opposite to the time, by the annuity, and the product will be the amount for yearly payments. If the payments be half yearly or quarterly, the amount for the given time, found as above, multiplied by the proper number in Table V. will be the true amount. EXAMPLES. 1. What will an annuity of 50L. per annum, payable yearly, amount to in 4 years at 5 per cent.? 1.05×1.05×1.05×1.05—1—21550625 1.05-1.05).21550625 4.310125 50 Ans. L. 215.506250-215L. 10s. 1d. 2qrs. 2. What will an annuity of 30L. per annum, payable yearly, amount to in 4 years, at 5 per cent. per annum, and what would be the respective amounts, if the payments were to be half yearly or quarterly? Ans. Amount for yearly payments is L. 129.30375 L. 130.9004 L. 131.7035 3. If a salary of 35L. per annum to be paid yearly, be omitted for 6 years at 5 per cent. what is the Ans. 241L. 1s. 7d. 2.5+qrs. amount? CASE 2. The annuity, time, and rate given, to find the present worth: RULE. Divide the annuity by the ratio involved to the time, and subtract the quotient from the annuity; divide the remainder by the ratio less one, and the quotient will be the present worth: Or, by Table IV. Multiply the number under the rate, and opposite the time by the annuity, and the product will be the present worth. When the payments are half yearly or quarterly, multiply the present worth so found, by the proper number in Table V. EXAMPLES. 1. What is the present worth of a pension of 30L. per annum for 5 years, at 4 per cent.? Number from Table IV. 4.45182 Ans. 133L. 11s. 1d. X30 annuity. L. 133.55460 Or, 133L. 11s. 1.104d. 2. What is the present worth of 20L. a year for 6 years, payable either yearly, half yearly, or quarterly, computing at 5 per cent. per annum? L. 3. What is the yearly rent of 50L. to continue 5 years, worth in ready money, at 5 per cent.? Ans. 216L. 9s. 10d. 2.24qrs. ANNUITIES TAKEN IN REVERSION. AT COMPOUND INTEREST. Annuities taken in reversion, are certain sums of money payable yearly for a limited period, but not to commence till after the expiration of a certain time. CASE 1. The annuity, time of reversion, time of continuance, and rate given, to find the present worth of the annuity in reversion: RULE. Divide the annuity by the ratio involved to the time of continuance, and subtract the quotient from the annuity for a dividend; multiply the ratio involved to the time of reversion by the ratio, less one, for a divisor; the quotient of this division will be the present worth. Or, Take two numbers under the given rate in Table IV. viz. that opposite the sum of the two given times, and that against the time of reversion, and multiply their difference by the annuity of the present worth. When the payments are half yearly or quarterly, use Table V. EXAMPLES. 1. What is the present worth of a reversion of a lease of 40L. per annum, to continue for six years, but not to commence till the end of 2 years, allowing 6 per cent. to the purchaser? 40 annuity. Ratio involved to Lo1.4185191)40.000000000000(28.19842 the time. 11.80158 1.06x1.06 x.06.067416) 11.80158(175.056+L. Ans. Or by Table IV. First, the sum of the two given times is 8 years, and the time of reversion 2 years; therefore, Take for 8 years 6.20979 for 2 do. 1.83839 Difference 4.37640 x40 annuity. L. 175.05600 Ans. as before. 2. A person owns a farm which he proposes to let for 8 years, at 100 dollars per annum; but cannot give possession till after the expiration of two years; what is the present worth of such a lease, allowing 4 per cent. for present payment? Ans. 622.48dols. 3. What is the present worth of a reversion of a lease of 60L. per annum, to continue 7 years, but not to commence till the end of 3 years, allowing 5 per cent. to the purchaser? Ans. 299L. 18s. 2.112d. PERPETUITIES, AT COMPOUND INTEREST. Perpetuities are such annuities as continue for ever. CASE 1. The annuity, and rate given, to find the present worth. RULE. Divide the annuity by the ratio less one, for the present worth. Note. For perpetual half yearly, or quarterly payments, Table V. is to be applied as in similar cases of temporary annuities. EXAMPLES. 1. What is an estate of 140L. per annum, to continue for ever, worth in present money, allowing 4 per cent. to the purchaser? L. 1.04-1.04)140.00 L. 3500. |