PROBLEM I. 688. To find the amount of an annuity in arrears, at simple interest. EXAMPLE. Required the amount of an annual annuity of $500, in arrears for 5 years, at 6 per cent. simple interest. ANALYSIS.-The first payment becomes due at the end of the first year, and, consequently, draws interest to the date of settlement, or four years. The second payment is in arrears three years, and, therefore, is chargeable with interest for that time. For the same reason, the third payment draws interest for two years, and the fourth, for one year. But, 4 + 3 + 2 + 1 years' interest equals 10 years' interest; and hence, we may find the interest on $500 for 10 years at once, instead of on each of the payments separately. The amount is computed in the same manner as annual interest. Hence the RULE.-Multiply the annuity by the number of payments due, and to the product add the interest on the several payments to the end of the term; the sum will be the required amount. The Rule for finding the sum of a series in Arithmetical Progression, also applies. (See Problem III, page 377, Example 4, note.) SOLUTION BY PROGRESSION. $500, First term. 620, Last term, amount of $500 in 4 years. 1120, Sum of the extremes. 5, Number of terms. 2)5600, Product. $2800, Amount of the annuity. EXAMPLES FOR PRACTICE. 1. What is the amount of an annual rent of $800, unpaid for 7 years, at 7 per cent. interest? 2. A gentleman deposits, for the benefit of his son, $200 annually, for 15 years, in a savings bank, which pays 5 per cent. : what will the deposits amount to at the end of the fifteenth year? 3. Required the amount of an annuity of $1500, in arrears for 7 years, at 7% simple interest. ANNUITIES AT COMPOUND INTEREST. 689. An annuity accumulating at compound interest forms a geometrical series, of which the first term (considering the series as ascending) is the first payment; the common ratio is 1 plus the rate of interest expressed decimally, and the last term, the amount of the first payment for the given time. Hence, problems under this head may be solved by the Rules in Geometrical Progression, though, on account of the tediousness of the calculations, Tables are generally employed in practice. PROBLEM II. 690. To find the amount, or Final Value of an annuity, at compound interest for any given time. EXAMPLE. Find the Final Value of an annuity of $1000, running 10 years, at 6% compound interest. SOLUTION.-The Final Value is the sum of the series found by the rule for Problem IV in Geometrical Progression. Hence, Hence the $1000 × 1.0620 - $1000 = 13180.80, Ans. RULE. Find the sum of a Geometrical Series, of which the first term is the annuity, the ratio the amount of a unit for one interval, and the number of terms the number of payments due. NOTE.—To save the labor of involving the ratio, take the amount of $1 for 10 years, found in the Compound Interest Table, page 203. This corresponds with the 10th power of the ratio. EXAMPLES FOR PRACTICE. 1. What will an annuity of $450 amount to in 8 years, at 4 per cent. compound interest? 2. A widow has a life interest in an estate, amounting to $1300 annually: what is the final value of her interest, at 6 per cent. per annum, if her "Expectation of Life" is 15 years? 3. If a person should save $100 per annum, and invest it at 7% compound interest, what would he be worth in 20 years? PROBLEM III. 691. To find the present worth of a certain annuity. RULE.—First find the Amount or Final Value of the annuity by the previous rules; then divide this amount by the amount of $1 for the given time. NOTES.-1. When compound interest is allowed, take the amount of $1 for the divisor from the Table, page 203. 2. The Present Worth of a deferred or reversionary annuity, is a sum which, if put at interest at the present time, will amount to its Final Value at its expiration. EXAMPLES. 1. What is the Present Value of an annuity of $600 for 15 years, allowing 6 per cent. compound interest? SOLUTION.-The Final Value found by Prob. II, is $13965.58; and $13965.58 2.396558, the amount of $1 for 15 years = $5287.34, Ans. 2. A widow is entitled to a pension of $96 per annum for life. Suppose her "Expectation of Life" to be 20 years, what is the present value of the pension, at 6 per cent. compound interest? Ans. $1121.06. 3. What is the Present Value of an annuity of $300, to commence 5 years hence, and continue 10 years, at 5 per cent. compound interest? SOLUTION.-The final value for 10 years = $3773.367. Amount of $1 for 15 years = $2.078928. 4. What is the present worth of a Perpetual Annuity of $1000 per annum, at 5 per cent. ? SOLUTION.-The Present Worth is a sum which will produce $1000 interest per annum, and hence, by dividing $1000 (the interest) by the rate per cent., we obtain the principal, $1000 ÷ .05 = $20,000, Ans. ANNUITY TABLES. 692. By the use of Annuity Tables, constructed upon the principles of the foregoing rules, the solution of problems is rendered very simple. The following Table exhibits the Amount and Present Value of an annuity of $1, at various rates of compound interest, for any number of years from 1 to 50 inclusive. Hence, 693. To find either the Amount or Present Worth of an annuity, we have the following RULE.--Multiply the Amount or Present Value of $1 for the given time, as found in the Table, by the given annuity. 694. To find the Present Value of an annuity deferred or in reversion. RULE.-Find, from the Table, the Present Worth of an annuity of $1 until the annuity commences, and also until it terminates. Then multiply the difference of these values by the annuity. 695. To find the annuity, the time, rate per cent., and Amount or Present Worth being given. RULE.-Divide the Amount or Present Worth, by the Amount or Present Worth of $1 for the given time. NOTE.-When the payments are due semi-annually, quarterly, etc., take the number of years in the Table which is equal the number of intervals, and find the amount at a rate of interest less than the given rate in the ratio of one year to one interval. Thus, the amount of $100 for 5 years, at 6 per cent. per annum, payable semi-annually, is equal to the amount of $100 for 10 years, at 3 per cent. per annum, payable annually. TABLE, SHOWING THE AMOUNT OR FINAL VALUE, AND ALSO THE PRESENT WORTH OF AN ANNUITY OF ONE DOLLAR, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING Fifty, at 4, 5, 6, and 7 PER CENT. COMPOUND INTEREST. Amount of an Annuity of $1. Present Worth of an Annuity of $1. Years. 4 per cent. 5 per cent. 6 per cent. 7 per cent. 4 per cent. 5 per cent. 6 per cent. 7 per cent. 9 10.582 795 10 12.006 107 8.393 838 9.897 468 11.491 316 13.180 795 14.971 643 16.869 941 18.882 138 21.015 066 23.275 970 25.670 528 39.992 727 1.000 000 0.961 538 0.952 381 0.943 396 0.934 579 1.886 095 1.859 410 1.833 393 1.808 017 2.775 091 2.723 248 2.673 012 2.624 314 3.629 895 3.545 951 3.465 106 3.387 209 4.451 822 4.329 477 4.212 364 4.100 195 5.242 137 5.075 692 4.917 324 4.766 537 6.002 055 5.786 373 5.582 381 5.389 286 6.732 745 6.463 213 6.209 744 5.971 295 7.435 332 7.107 822 7.153 291 44.865 177 14.029 160 12.821 185 13.003 166 11.825 779 034 13.210 534 11.986 709 127 13.406 164 12.137 111 074 13.590 721 12.277 674 451 13.764 831 12.409 041 11.026 564 12.577 893 11 13.486 351 14.206 787 12 15.025 805 15.917 127 13 16.626 838 17.712 983 14 18.291 911 19.598 62 15 20.023 588 21.578 564 16 21.824 531 23.657 492 17 23.697 512 25.840 366 18 25.645 413 28.132 385 19 27.671 229 30.539 004 20 29.778 079 33.065 954 21 31.969 202 35.719 252 22 34.247 970 38.505 214 23 36.617 889 41.430 475 24 39.082 604 44.501 999 25 41.645 908 47.727 099 26 44.311 745 51.113 454 59.156 383 68.676 470 15.982 769 14.275 27 47.084 214 54.669 126 63.705 766 74.483 823 16.329 586 14.613 28 49.967 583 58.402 583 68.528 112 80.697 69116.663 063 14.898 29 52.966 286 62.322 712 78.639 798 87.346 529 16.983 715 15.141 30 56.084 938 66.438 848 79.058 186 94.460 786 17.292 033 15.372 31 59.328 335 70.760 790 84.801 677 102.073 041 17.588 494 15.592 811 13.929 086 12.531 814 32 62.701 469 75.298 829 90.889 778 110.218 154 17.873 552 15.802 677 14.084 043 12.646 555 33 66.209 527 80.063 771 97.343 165 118.933 425 18.147 646 16.002 549 14.230 230 12.753 790 34 69.857 909 85.066 959 104.183 755 128.258 765 18.411 198 16.192 204 14.368 141 12.854 009 35 73.652 225 90.320 307 111.434 780 138.236 878 18.664 613 16.374 194 14.498 246 12.947 672 36 77.598 314 95.836 323 119.120 867 148.913 460 18.908 282 16.546 852 14.620 987 13.035 208 37 81.702 246 101.628 139 127.268 119 160.337 400 19.142 579 16.711 287 14.736 780 13.117 017 38 85.970 336 107.709 546 135.904 206 172.561 020 19.367 864 16.867 893 15.846 019 13.193 473 39 90.409 150 114.095 023 145.058 458 185.640 29219.584 485 17.017 041 14.949 075 13.264 928 40 95.025 516 120.799 774 154.761 966 199.635 11219.792 774 17.159 086 15.046 297 13.331 709 41 99.826 536 127.839 763 165.047 684 214.609 570 19.993 052 17.294 368 15.138 016 13.394 120 42 104.819 598 135.231 751 175.950 645 230.632 240 20.185 627 17.423 208 15.224 543 13.452 449 43 110.012 382 142.993 339 187.507 577 247.776 496 20.370 795 17.545 912 15.306 173 13.506 962 44 115.412 877 151.143 006 199.758 032 266.120 851 20.548 841 17.662 773 15.383 182 13.557 908 45 121.029 892 159.700 156 212.743 514 285.749 311 20.720 040 17.774 070 15.455 822 13.605 522 46 126.870 568 168.685 164 226.508 125 306.751 763 20 884 654 17.880 067 15.524 370 13.650 020j 47 132.945 390 178.119 422 241.098 612 329.224 386 ||21.042 936 17.981 016 15.589 028 13.691 608 48 139.263 206 188.025 393 256.564 529 353.270 09321.195 131 18.077 158 15.650 027 13.730 474 49 145.833 734 198.426 663 272.958 401 378.999 000 21.341 472 18.168 722 15.707 572 13.766 799 50 152.667 084 209.347 976 290.335 905 406.528 929 21.482 185 18.255 925 15.761 861 13.800 746 1 |