TABLE, SHOWING THE AMOUNT OF AN ANNUITY OF ONE DOLLAR PER ANNUM, IMPROVED AT COMPOUND INTEREST FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY. 32 52.502 759 57.334 502 33 55.077 841 60.341 210 34 57.730 177 35 60.462 082 36 37 38 90.889 778 110.218 154 80.063 771 97 343 165 118.933 425 85.066 959 104.183 755 128.258 765 90.320 307 111.434 780 138.236 878 62.701 469 66.209 527 63.453 152 69.857 909 66.674 013 73 652 225 63 271 944 70.007 603 77.598 314 95.836 323 119.120 867 148.913 460 66.174 223 73.457 869 81.702 246 101.628 139 127.268 119 160.337 400 69.159 449 77.028 895 85.970 336 107.709 546 135.904 206 172.561 020 39 72.234 233 80.724 906 90.409 150 114.095 023 145.058 458 185.640 292 40 75.401 260 84.550 278 95.025 516 120.799 774 154.761 966 199.635 112 41 78.663 298 88.509 537 99.826 536 127.839 763 165.047 684 214.609 570 42 82.023 196 92.607 371 104.819 598, 135.231 751 175.950 645 230.632 240 43 85.483 892 96.848 629 110.012 382 142.993 339 187.507 577 247.776 496 44 89.048 409 101.238 331 115.412 877 151 143 006 199.758 032 266.120 851 45 92.719 861 105.781 673 121.029 392 159.700 156 212.743 514 285.749 311 46 96.501 457 110.484 031 126.870 568 168.685 164 226.508 125 306.751 763 47 100.396 501 115.350 973 132.945 390 178.119 422 241.098 612 329.224 386 48 104.408 396 120.388 297 139.263 206 188.025 393 256.564 529 353.270 093 49 108.540 648 125.601 846 145.833 734 198.426 663 272.958 401 378.999 000 50 112.796 867 130.999 910 152.667 084 209.347 976 290.335 905 406.528 929) 70.760 790 84.801 677 102.073 041 TABLE, SHOWING THE PRESENT WORTH OF AN ANNUITY OF ONE DOLLAR PER ANNUM, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY. Years. 1234∞∞ 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 6.463 213 6.209 744 5.971 295 12345678 5 4.579 707 4.515 052 6 5.417 191 5.328 553 7 6.230 283 6.114 544 8 7.019 692 6.873 956 6.732 745 9 7.786 109 7.607 687 7.435 332 10 8.530 203 8.316 605 8.110 896 7.721 735 7.360 087 11 9.252 624 9.001 551 8.760 477 8.306 414 12 9.954 004 9.663 334 9.385 074 8.863 252 13 10.634 955 10.302 738 9.985 648 9.393 573 8.357 635 13 14 11.296 073 10.920 520 10.563 123 9.898 641 9.294 984 8.745 452 14 15 11.937 935 11.517 411 11.118 387 10.379 658 9.712 249 9.107 898 15 16 12.561 102 12.094 117 11.652 296 10.837 770 10.105 895 9.446 632 16 17 13.166 118 12.651 321 12.165 669 11.274 066 10.477 260 9.763 206 17 18 13.753 513 13.189 682 12.659 297 11.689 587 10.827 603 10.059 070 18 19 14.323 799 13.709 837 13.133 939 12.085 321 11.158 116 10.335 578 19 20 14.877 475 14.212 403 13.590 326 12.462 210 11.469 421 10.593 997 20 21 15.415 024 14.697 160 12.821 153 11.764 077 10.835 527 21 22 15.936 917 15.167 115 13.163 003 12.041 582 11.061 241 22 23 16.443 608 15.620 410 14.856 842 13.488 574 12.303 379 11.272 187 23 24 16.935 542 16.058 368 15.246 963 13.798 642 12.550 358 11.469 334 24 25 17.413 148 16.481 515 15.622 080 14.093 945 12.783 356 11.653 583 25 26 17.876 842 16.890 769 14.275 185 13.003 166 11.825 779 26 27 18.327 031 17.285 586 14.643 034 13.210 534 11.986 709 27 28 18.764 108 17.667 063 14.898 127 13.406 164 12.137 111 28 29 19.188 455 18.035 715 15.141 074 13.590 721 12.277 674 29 30 19.600 441 18.392 033 15.372 451 13.764 831 12.409 041 30 31 20.000 428 18.736 494 15.592 811 13.929 086 12.531 814 31 32 20.338 766 19.068 552 15.802 677 14.084 043 12.646 555 32 33 20.765 792 19 390 646 16.002 549 14 230 230 12.753 790 33 34 21.131 837 19.700 198 16.192 204 14.368 141 12.854 009 34 35 21.487 220 20.000 613 16.374 194 14.498 246 12.947 672 35 36 21.832 252 20.290 852 14.620 987 13.035 208 36 37 22.167 235 20.570 287 14.736 780 13.117 017 37 38 22.492 462 20.841 893 14.846 019 13.193 473 38 39 22.808 215 21.102 041 14.949 075 1.264 928 39 40 23.114 772 21.355 086 15.046 297 13.331 709 40 41 23.412 400 21.599 368 15.138 016 13.394 120 41 42 23.701 359 21.834 208 15.224 543 13.452 449 42 43 23.981 902 22.062 912 15.306 173 13.506 962 43 44 24.254 274 22.282 773 15.383 182 13.557 908 44 45 24.518 713 22.495 070 15.455 832 13.605 522 45 46 24.775 449 22.700 067 15.524 370 13.650 020 46 47 25.024 708 22.899 016 15.589 48 25.266 707 23.091 49 25.501 657 23.276 50 25 729 764 23.455 791 20.548 158 15.650 027 13.730 474 48 574. To find the amount of an annuity, at compound interest, forborne, or in arrears, for any number of years. Ex. 1. What will an annuity of $ 60, unpaid, or in arrears, 4 years, amount to, at 6 per cent. compound interest? Ans. $262.476. The amounts of the several payments form a geometrical series, of which the annuity is the first term, the amount of $1 for one year the ratio, the years the number of the terms, and the amount required is the sum of the series. Hence, RULE.- Find the sum of the series as in geometrical progression. Or, Multiply the amount of $1 for the given time, found in the table, by the annuity, and the product will be the required amount. NOTE. The amount of an annuity at simple interest corresponds to the sum of an arithmetical series, of which the annuity is the first term, the interest on the annuity for one term the common difference, and the time in years the number of terms. 2. What will an annuity of $500 amount to for 5 years, at 6 per cent. compound interest? Ans. $2818.546. 3. What is the amount of an annuity of $80, unpaid, or in arrears, for 9 years, at 5 per cent. compound interest? Ans. $882.125. 4. What is the amount of an annuity of $1000, forborne for 15 years, at 3 per cent. compound interest? Ans. $19295.68. 5. What will an annuity of $30, payable semiannually, amount to, in arrears for 3 years, at 7 per cent. compound interest? 6. Suppose a salary of $ 600 per year, payable quarterly, to remain unpaid 5 years; to what sum will it amount, at 6 per cent. compound interest? Ans. $3875.63. 575. To find the present worth of an annuity, at compound interest. Ex. 1. What is the present worth of an annuity of $ 60, to be continued 4 years, at 6 per cent. compound interest? Ans. $207.906. OPERATION. = The present worth re$3.465106 X 60 $207.906+. quired evidently may be obtained by finding the amount of the given annuity, by the last articles, and then finding in the usual way the present worth of that amount. A more expeditious method, however, is to find, in the table, the present worth of an annuity of $1 for the given time and rate, and take that sum as many times as there are dollars in the given annuity, as in the operation. RULE.-Multiply the present worth of an annuity of $1 for the given time and rate by the number denoting the given annuity. 2. What is the present worth of an annuity of $100, for 9 years, at 6 per cent.? Ans. $ 680.169. 3. What is the present worth of an annuity of $ 200, for 7 years, at 5 per cent.? Ans. $1157.27. 4. Required the present worth of an annuity of $500, to continue 40 years, at 7 per cent. 5. A gentleman wishes to purchase an annuity, which shall afford him, at 6 per cent. compound interest, $ 500 a year, for ten years. What sum must he deposit in the annuity office to produce it? Ans. $3680.04. 6. If a widow be entitled to $160 a year, payable semiannually, from a fund, for 8 years, what is its value at present, at 6 per cent. compound interest? Ans. $1004.88. 576. To find the present worth of an annuity in perpetuity. Ex. 1. What is the present worth of a perpetual lease, which yields an income of $ 600, the rate of interest being that of 6 per cent.? Ans. $10000. OPERATION. $600.06 RULE. = $ 10000. The question is evidently the same as one requiring what principal in one year, at 6 per cent. interest, will yield $ 600. - Divide the given annuity by the number denoting the interest of $1 for one year. NOTE. - When the annuity is payable quarterly, semiannually, or in any other periods less than a whole year, the annuity must be increased by the interest which may thus accrue on the parts of the annuity payable before the end of the year, before dividing by the interest of $1 for one year. 2. A ground rent in the city of Philadelphia yields an annual income of $963, at 6 per cent. interest. value of the estate? What is the Ans. $16050. 3. What is the present value of a perpetual lease, yielding an income of $ 6335, interest being at 7 per cent. Ans. $90500. 4. What sum should be paid for a perpetual annuity of $1200, payable semi-annually, interest being at 5 per cent.? Ans. $24000. 577. sion. To find the present worth of an annuity in rever Ex. 1. What is the present worth of an annuity of $ 300, to commence in 3 years, and to continue 5 years, allowing compound interest at 6 per cent.? Ans. $1061.03. The present worth of an annuity of $ 1, at 6 per cent., com. mencing at once, = and continuing till the termination of the annuity, or for 3 + 5 — 8 years, is $ 6.209744; and the present worth of the same annuity up to the time of the commencement of the reversion is $2.673012. The difference of these present worths multiplied by the number of dollars in the given annuity is the present worth of the reversion. RULE. Find the present worth of an annuity of $ 1, commencing immediately and continuing till the reversion COMMENCES, and also till the reversion TERMINATES; and multiply the difference of these present worths by the number of dollars in the given annuity. The result will be the present worth required. 2. The reversion of a lease of $350 per annum, to continue 11 years, which commences 9 years hence, is to be sold. What is its worth, allowing the purchaser 6 per cent. per annum for his ready money? Ans. $1633.70. 3. A father presents to his daughter, for 8 years, a rental of $70 per annum, payable yearly, and its reversion for the 12 years succeeding to his son. What is the present value of the gift to his son, allowing 4 per cent. compound interest? Ans. $480.03. 4. What is the present worth of the reversion of a perpetuity of $240 per annum, payable yearly, but not to come into possession till the expiration of 100 years, compound interest being allowed at 6 per cent.? Ans. $11.78. 578. To find the annuity, the present worth, time, and rate being given. |