What annuity may be purchased for 1,000l. to continue for 20 years, in half yearly payments, allowing interest at 5 per cent.? 1 This is, in fact, 40 payments, at an interest of 2 per (1+); the present value of the second payment cent. Now, by Table III. 17. will purchase, under such conditions, an annuity of ⚫039836 +2; the third payment, 1 ANNUITIES. i ever. Mult. by 1 1000 (1 + r) "'+2 ; 1 (1 + r) "'+n Consequently, 8. A perpetuity is an annuity that is to continue for Now, in the foregoing article, we have seen that n denotes the number of years, or the number of pay- (1 + r) ments, and we have introduced no condition that ought to limit its value; therefore, the first two formulæ will still apply to this case, by making n infinite, that is, in the case where the annuity is to continue for ever; but to render this transformation the more perspicuous, it will be better to change them into the following form, viz. ANNUITIES. TABLE I. Showing the amount of an annuity of 11. for any number of years, not exceeding fifty; and for the different rates of interest from 2 to 7 per Cent. 12 13 14 15 16 18 19 23 24 13-41208 13.79555 | 14·19202 14.60196 15.02580 15:46403 15.91712 16.86994 14-68033 15-14044 15.61779 16.11303 16.62683 17.15991 17.71298 18-88213 20·14064 15.97393 16-51895 17.08632 17.67698 18.29191 18-93210 19.59863 21.01506 22-55048 17-29341 17-93192 18-59891 19.29568 20-02358 20.78405 21.57856 23.27596 18-63928 19-38022 20.15688 20-97102 21.82453 22-71933 23-65749 25-67252 17 20.01207 20.86473 21-76158 22-70501 23.69751 24-74170 25.84036 28.21287 21.41231 22.38634 23.41443 24-49969 25.64541 26.85508 28.13238 30-90565 22.84055 23.94600 25.11686 26-35718 27-67122 29.06356 30.53900 33.75999 20 24.29736 25.54465 26.87037 28-27968 29-77807 31-37142 33-06595 36.78559 21 25-78331 27-18327 28-67648 30-26947 31.96920 33.78313 35.71925 39-99272 22 27.29898 28-86285 30.53678 32.32890 34-24796 36.30337 38.50521 43·39229 28.84496 30.58442 32-45288 34-46041 36-61788 38.93702 41-43047 46-99582 30-42186 32-34903 34-42647 36.66652 39-08260 41.68919 4450199 50.81557 17.88845 25.12902 27.88805 30-84021 33-99903 37-37896 40.99549 44.86517 49.00573 53-43614 58.17667 25 30 31 32 63.24903 68.67647 74-48382 80-69769 87-34652 32.03029 34-15776 36-45926 38.94985 41.64590 44.56521 47-72709 54-86451 26 33-67090 36.01170 38-55304 41-31310 44.31174 47.57065 51.11345 59.15638 27 35.34432 37.91200 40-70963 43-75906 47.08421 50-71132 54.66912 63·70576 28 37-05121 39.85980 42.93092 46-29062 49.96758 53.99333 58.40258 68.52811 29 38.79223 41.85629 45.21885 48.91079 52.96628 57-42303 62.32271 73.63979 40.56807 43.90270 47.57541 51-62267 56-08493 61.00706 66:43884 79-05818 94.46078 42-37944 46.00027 50-00267 54-42947 59-32833 64-75238 70-76078 84-80167 102-07304 44.22702 48.15027 52.50275 57.33450 62.70146 68.66624 75-29882 90.88977 110-21815 33 46.11157 50.35403 55.07784 60-34121 66.20952 72-75620 80-06377 97-34316 118-93342 34 48.03380 52.61288 57-73017 63.45315 69-85790 77·03025 85-06695 104.18375 128-25876 35 49.99447 54.92820 60.46201 66.67401 73.65222 81.49661 90-32030 111·43477 138-23687 36 51-99436 57.30141 63-27594 70.00760 77.59831 86-16396 95.83632 119-12086 148-91345 37 54-03425 59.73394 66-17422 73-45786 81-70224 91-04134 101-62813 127-26811 160-33740 38 56.11493 62.22729 69-15944 77.02889 85.97033 96-13820 107-70954 135.90420 172.56102 39 58.23723 64.78297 72.23423 80-72490 90-40914 101.46442 114-09502 145 05845 185-64029 40 60-40198 67.40255 75.40125 84.55027 95.02551 107-03032 120-79977 154-76196 199-63511 41 62.61002 70.08761 78-66329 88.50953 99.82653 112.84668 127-83976165-04768 214.60956 42 64.86222 72.83980 82-02319 92-60737 104.81959 118-92478 135-23175 175.95054 230-63223 43 67.15946 75-66080 85.48389 96.84862 110-01238 125-27640142-99333 187-50757 247-77649 69.50265 78.55282 89-04840 101-23833 115-41287 131-91384 151 14300 199-75803 266-12085 71-89271 81.51613 92.71986 105-78167 121-02939 138-84996 159-70015212-74351 285-74931 74.33056 84.55403 96.50145 110-48403 126.87056 146-09821 168.68516 226-50812 306-75176 47 76-81717 87.66788 100-39650 115-35097 132.94539 153-67263 178 11942 241-09861 329-22438 79-35351 90-85958 104-40839 120-38825 139-26320 161.58790 188-02539 256.56452 353-27009 49 81.94058 94-13107 108-54064 125-60184 145·83373 169·85935 198·42666272-95840 378-99899 50 84-57940 97-48434 112-79686 130-99791 152-60708 178.50302 209-34799290-33590 406-52892 44 45 46 48 N NES. TABLE II. Showing the present value of an annuity of 11. per annum, for any number of years, not exceeding fifty, and at different rates of interest, from 2 to 7 per Cent. AN NUITIES. 6 5.60143 5.50812 5.41719 5.32855 5-24213 5.15787 5·07569 4.91732 7 6.47199 6.34939 6-23028 6.11454 6·00205 5.89270 5.78637 5.58238 8 9 7.32548 7.17013 7:01969 6.87395 6.73274 6.59588 6.46321 6.20979 98039 1.94156 97560 1.92742 97887 1-91346 96618 1.89969 96153 1-88609 95693 1.87266 95238 1.85941 94339 -93457 1.83339 1-80801 2.62431 3.38721 2.88388 2.85602 2.82861 2.80163 2.77509 2.74896 2.72324 2.67301 4.10019 10 12 13 8.98258 8.75206 8.53020 8.31660 8.11089 7.91271 7-72173 7.36008 11 9.78684 9.51420 9.25262 9.00155 8.76047 8.52891 8.30641 7.88687 10.57534 10-25776 9.95400 9.66333 9-38507 9.11858 8.86325 8.38384 11.34837 10.98318 10.63495 10-30273 9.98564 9-68285 9-39357 8.85268 6.51523 7.49867 7.94268 12.10624 11.69091 11.29607 10.92052 10-56312] 10-22282 9.89864 9.29498 37 38 40 41 42 43 44 45 47 48 49 50 ANNUITIES. TABLE III. Showing the annuity that 11. will purchase for any number of years, not exceeding fifty; AN NUTME AN MIES: § II. Of life annuities. 10. Life annuities are of that class which we have fe annui- called contingent annuities, and, indeed, they form the principal part of them; for, although an annuity may be made to depend upon certain other contingencies beside the duration of life, yet such is seldom the case, and it will be unnecessary here to enter upon any such speculation. By a life annuity is to be understood, the payments which depend upon the continuance of any given life, or lives, and they may be distinguished into two principal classes; viz. those to commence immediately, and those which are to commence at some future period, or reversionar life annuities. The value of a life annuity is, properly, that sum which will be sufficient, when improved at interest, to pay the annuity without loss; if, therefore, we were certain as to the duration of the life on which the annuity depends, this doctrine would be immediately reduced to principles in every respect the same as those we have just examined; and on the contrary, without some data derived from tables of mortality, it would be impossible to establish any principle of computation whatever. But numerous tables of his kind have been kept in different places, and from these we may deduce such information, as to render the calculation at least approximatively correct; for, although with regard to any one life the result may be very different from the actual value, yet where many lives are concerned, these results correct each other, and approach so much the nearer to a medium value. nciples 11. In order to apprize the reader, in some measure, compu- of the principles upon which the doctrine of life annuities are made to depend; we may take the following example:-Observations show that, according to the mean probability of human life, the expectation of a life, aged 10, is nearly forty years; that is to say, of any number of lives all of this age, they will, one with another, enjoy 40 years of existence, or which is the same, taking a specific number as 100, the sum of all their ages before they become extinct, will be 40 x 100 =4000; and, in a similar manner, the expectation of a life at any other age is computed, from tables such as those to which we have just alluded. ectation fe. AN that at which they were granted, will have the advantage of accumulating longer at compound interest, than NUITIES. would have been contemplated in computing for an annuity certain for 40 years; consequently, upon the whole, a less sum will purchase an annuity upon a life whose expectation of existence is 40 years, than would purchase an annuity certain for the same period. In general it may be assumed, that one-half nearly of the payments on a certain number of life annuitants will be made after the expiration of a term of years equal to the expectation of the lives, and that this half having a longer time for accumulation than that indicated by the expectation, the value of such annuities must be less than the value of annuities to be paid regularly every year for a time equal to the expectation. The proper deduction arising from this consideration, or rather the correct method of computing such annuities, will form the subject of a subsequent article; but let us first offer a few remarks relative to the tables to which we have alluded: such are our Tables IV. and V. 12. The nature of these tables will be readily com- Explanaprehended, without being particularly described; it Table IV. will be sufficient to observe, that the second column and V. shows the number of persons supposed to be living of any given age, and the third the number that will die in the course of the following year, and which, therefore, deducted from the first, will show the number living at the beginning of the succeeding year. Thus we see, that in Table V. of 11,650 children born, 3,000 will die before the expiration of the first year; of the number, 8,650 which live to the age of one year, 1,367 will die before they attain the age of two years; and so on for any other age. In this table, the whole number of lives are supposed to become extinct in 97 years; in Table IV. the duration of life is limited to 95 years. tion of of life. 13. The next succeeding tables, viz. VI. and VII. ex- Table of exhibit the expectation of life for the several ages there pectations specified; it is formed from the preceding ones, upon the principles we have already referred to, that is, by computing the whole number of years that all the several lives of any given age will amount to, and dividing that sum by the number living at that age; or, more simply, by dividing the sum of all the living in the table at: the age whose expectation is required, and at all greater ages, by the number living of the proposed age, adding or 5, to the quotient; the result will be the expectation sought. The expectation is, of course, different according to the tables of mortality from which it is deduced, and unfortunately these differ very essentially from each other; we have selected those of De Parcieux and Dr.' Price; the former is generally considered as offering the best medium results, but that of Dr. Price is, notwithstanding, more generally consulted in the valuation of annuities in this country. It must not, however, be understood from what has, been above stated, that the value of an annuity upon a life aged 10, is the same as that of an annuity certain for 40 years; we shall see hereafter, that supposing the annuity to be 17 and interest allowed at 4 per cent. the value of such a life annuity is only 17. 10s. 6d. whereas it will be found by the tables given for annuities certain, that its value for 40 years is 197. 16s. The principal reason for this, is the difference between the value of forty payments of an annuity to be made every year regularly one after the other, till in 40 years they are all made; and the value of the same number of payments to be made at greater distances of time, and not to be all made till the end of 70 or 80 years. Or it may, perhaps, be more intelligibly illustrated thus: suppose a person to grant a number, say 100 of such annuities, upon as many lives, each aged 10; of these lives, some will fall very soon, others will live to 50, and others to a greater age; of those that live to 50, the exact value will have been paid: but of those that fall early, the difference between their actual value, and On the value of life annuities. 14. The computation of the value of life annuities is, The docas we have already remarked, dependent on the doc- trine of protrine of probabilities; it will, therefore, be proper to babilities. make a few such remarks on the latter subject as will be sufficient to explain the method of proceeding in the case in question. For this we may observe, that if there are a ways all possible in which a thing may happen, and a ways, in which it may take place in a certain manner, then the |
