crease indefinitely, or interest to be convertible momently, the above amount becomes e *, where e is the base of the
natural (or Napierian) logarithms.In consequence of the above-mentioned practice as to half-yearly interest, the values given in Smart s tables for the odd half-years, though theoretically correct, are prac tically useless, and they have been superseded by the other tables above mentioned. It is important, however, always to bear in mind that when interest is thus payable half- yearly or quarterly, the true rate of interest exceeds the nominal. From want of attention to this point, the sub ject has become involved in much confusion, not to say error, in the works of Milne and some other writers.
It is easily seen from the above formula that the amount of 1 in mn years, at the rate of interest , is the same as tTb that of 1 in n years, at the rate of interest i convertible m times a year; and a similar property holds good of present values. Hence, the tables calculated at the rate of interest may be used to find the amounts and present values m at the rate i convertible m times a year ; for example, the tables calculated for interest 2 per cent, will give the results for 4 per cent, payable half-yearly. For this reason it would be an improvement, as remarked above, to use the word "terms" in the headings of the tables instead of " years."
We pass on now to the consideration of the theory of life annuities. This is based upon a knowledge of the rate of mortality among mankind in general, or among the particular class of persons on whose lives the annuities depend. If a simple mathematical law could be discovered which the mortality followed, then a mathematical formula could be given for the value of a life annuity, in the same way as we gave above the formula for the value of an annuity certain. In the early stage of the science, De- moivre propounded the very simple law of mortality which bears his name, and which is to the effect, that out of 86 children born alive 1 will die every year until the last dies between the ages of 85 and 86. The mortality, as determined by this law, agreed sufficiently well at the middle ages of life with the mortality deduced from the best observations of his time ; but, as observations became more exact, the approximation was found to be not suffi ciently close. This was particularly the case when it \vas desired to obtain the value of joint life, contingent, or other complicated benefits. Demoivre s law is now, accord ingly, entirely a thing of the past, and does not call for any further notice from us. Assuming that law to hold, it is easy to obtain the formula for the value of an annuity, immediate, deferred, or temporary ; but such formulas are entirely devoid of practical utility. Those who are curious on the subject may consult the paper by Charlon, Ass. Mag., xv. 141. In vol. vi. p. 181, will be found an in vestigation by Gray of the formula for the value of an annuity when the mortality table is supposed to follow a somewhat more complicated law. No simple formula, however, has yet been discovered that will represent the rate of mortality with sufficient accuracy ; and those which satisfy this condition are too complicated for general use.
The rate of mortality at each age is, therefore, in practice usually determined by a series of figures deduced from observation; and the value of an annuity at any age is found from these numbers by means of a series of arith metical calculations. Without entering here on a descrip tion of the manner of making these observations and de ducing the rate of mortality, and of the construction of " Mortality Tables," we append, for the sake of illustration, one of the earliest tables of this kind, namely, that of Deparcieux, given in his Essai sur les Probcibilites de la Duree de ia Vie Humaine, 1746.
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Number Number Number Age. Number living. dying in the next Age. Number living. dying in the next Age. Number living. dying in the next vear. year. year. X I, d, X I* d* X lx - d, 3 1000 30 34 702 8 65 395 15 4 970 22 35 694 8 66 380 16 5 948 18 36 686 8 67 364 17 6 930 15 37 678 7 68 347 18 7 915 13 38 671 7 69 329 19 8 902 12 39 664 7 70 310 19 9 890 10 40 657 7 71 291 20 10 880 8 41 650 7 72 271 20 11 872 6 42 643 7 73 251 20 12 866 6 43 636 7 74 231 20 13 860 6 44 629 7 75 211 19 14 854 6 45 622 7 76 192 19 15 848 6 46 615 8 77 173 19 16 842 7 47 607 8 78 154 18 17 835 7 48 599 9 79 136 18 18 828 7 49 590 9 80 118 17 19 821 7 50 581 10 81 101 16 20 814 8 51 571 11 82 85 14 21 806 8 52 560 11 83 71 12 22 798 8 53 549 11 84 59 11 23 790 8 54 538 12 85 48 10 24 782 8 55 526 12 86 38 9 25 774 8 56 514 12 87 29 7 26 766 8 57 502 13 88 22 6 27 758 8 58 489 13 89 16 5 28 750 8 59 476 13 80 11 4 29 742 8 60 463 13 91 7 3 30 734 8 61 450 13 92 4 2 31 726 8 62 437 14 93 2 1 32 718 8 63 423 14 94 1 1 33 710 8 64 409 14 95
It is to be understood from this table that the mortality among the persons observed was such that out of every 1000 children alive at the age of 3, 30 died before attain ing the age of 4, leaving 970 alive at 4 ; 22 died between; the ages of 4 and 5, leaving 948 alive at the age of 5 ; and so on, until one person is left alive at the age of 94, who died before attaining the age of 95.
For the purpose of explaining more fully the method of finding the value of a life annuity, it will be convenient, in the first instance, to establish the two following lemmas.
a future time in the event of the happening of a given contingency. Suppose that the sum of 1 is to be re ceived in n years time, provided that a certain event shall then happen (or shall have then happened), the probability of which is p. We have seen that the value of 1 to be certainly received in n years time is v n . In order to introduce the idea of probability into the problem, suppose that p = - - , so that there are a cases favourable to the happening of the assumed event, and b unfavourable, the total number of possible cases, all of which are equally probable, being (a + b). We may sup pose, for instance, that there are (a + b) balls in a bag, of which a are Avhite and b black ; and that 1 is to be received if a white ball is drawn. In order to determine the value of the chance of receiving 1 in consequence of a white ball being drawn, suppose that (a + b) per sons draw each one ball, and that every one who draws a white ball receives 1 ; then the total sum to be received is a, and the value of the expectation of all the (a + b) persons who draw is also a. But it is clear that each of the persons has the same chance of drawing a white ball, therefore the value of the expectation of each of them is r = >. This is the value of the chance of
receiving 1 immediately before the drawing is made ia