n years time ; the value at the present time will there fore be v*p. We may also arrive at this result as follows : The same suppositions being still adhered to, the present value of the sum a to be distributed at the end of n years is av*; and each of the (a + b) persons having the same chance of receiving 1, the value of the expectation of each is 7v n =pv".
a + b f
Lemma 2. To find the present value of 1 to be received in n years time, if a specified person, whose age is now x, shall be then living. The sum to be received in this case is called an endowment, and the person on whose life it depends is called the nominee. The probability that the nominee will be alive is to be found, as already intimated, by means of a mortality table. Out of the various tables of this nature that exist, that one must be chosen which, it is believed, most faithfully represents the probabilities of life of the class of persons to which the nominee be longs. Suppose we have reason to believe that Deparcieux s table, above given, is the most suitable in the case before us, that the age of the nominee is 30, and the term of years 10. Then, observing that, according to Deparcieux s table, the number of persons living at the age 30 is 734, while the number at the age 40 is 657, and the difference, or the number who die between the two ages, is 77, we conclude that the chances of any particular nominee of the age of 30 dying before attaining the age of 40 are as 77 to 734, and the chances in favour of his living to the age of 40 are as 657 to 734; or the probability of his living to 40 is - .
Passing now from figures to more general symbols, we will use l x to denote the number given in the mortality table as alive at any age x ; so that, for example, in the above table, ^ = 734, Z 4l) = 657; and in accordance with what we have just explained, the probability of a nominee of the age x living to the age x + n, will therefore be expressed by . Hence, by lemma 1, the value of 1 to be received if the nominee shall be alive at the end of n years, is -y^V. In the particular case supposed above, L x the actual value will be, taking the rate of interest at 3 per cent, ^ x (1 03)- 10 = "666035. We may look at the question from another point of view. Suppose that 734 persons of the age of 30 agree to purchase from an insurance company each an endowment of 1, payable at the end of 10 years, then the probabilities of life being supposed to be correctly given by Deparcieux s table, we see that 657 of those persons will be alive at the end of 10 years, or the engagement of the insurance company to pay 1 to each survivor amounts to the same thing as the engagement to pay 657 at the end of 10 years, and the present value of this sum is 657 (1 03) 10 . The sum that should be paid by each of the 734 persons, so that the company shall neither gain nor lose by the transaction, is CT *T therefore ^(l 03 )" 10 , as before. If we suppose the pro babilities of life to agree with those of the English Table, No. 3, Males, which is printed at the end of this article, the value of the same endowment will be 272,073 304,534 (1-03)- 10 = -664779.
If now we carefully examine the reasoning of the last paragraph, we see that we have made an assumption that must not be allowed to pass without some further justifica tion. We have assumed, in fact, that the lives we are dealing with will die off at the exact rate indicated by the mortality table. This, however, we know, is not neces sarily the case. Even if the mortality table correctly represents in the long run the rate of mortality among the lives we are dealing with, we know that the rate of mortality will, from accidental circumstances, be some times greater and sometimes less than that indicated by the table. If, for example, we have 734 persons under observation all of the age 30, we have no certainty that at the end of 10 years exactly 77 will have died, leaving 657 alive. It is, indeed, within the range of possibility firstly, that the whole 734 persons may die before the age 40; and, secondly, that none of them may die, or that the whole 734 may attain the age of 40. It appears, there fore, as if we had used the word "probability" in the second lemma in a different sense from that we attached to it in the first ; for, in that case we know that if the whole of the (a + b) balls are drawn, a of them will cer tainly be white, and b black. But the cases will be more parallel if we suppose that each of the balls, after being drawn, is replaced in the bag. If this is done, we see it is no longer certain that when (a + b") drawings take place, a of the balls will be white, and b black. It may, under these altered circumstances, possibly happen that the balls drawn at each of the (a + b) drawings will all be white, or on the contrary all black. But when a very large num ber of drawings are made, we can prove that the ratio of white balls drawn to the black will differ very little from the ratio of a to b, and will exactly equal it if the number of drawings is supposed to be indefinitely large. In this case we know that the probability of drawing a white ball is still -, and passing now to the case of lives under d i b observation, we can say, in the same sense, that the pro bability of a person of the age of 30 living for 10 years is, according to Deparcieux s table, -, and that on the average of a very large number of observations, that frac tion will accurately represent the number of persons surviving. We shall, therefore, be justified in basing all our reasonings on the assumption that the lives we are dealing with die precisely at the rate indicated by the figures of the mortality table.
We* are now in a position to show how the value of a life annuity is calculated. The annual payment of the annuity being 1, which is to be made at the end of each year through which the nominee shall live, the annuity consists of a payment of 1 at the end of one year if the nominee is then alive, of the same payment at the end of two years, at the end of three years, &c., under the same condition, and is therefore equal to the sum of a series of endowments. If # is the age of the nominee, the value of the endowment to be received at the end of the ?ith year is, as we have seen in lemma 2, -y^t B , and the total value of the annuity is therefore
By means of this formula, taking the values of l x , /, +1 , l*+v & c -> from the mortality table, and calculating the values of v, v, v 5 , &c., according to the desired rate of interest, or taking their values from the compound interest tables previously described, we can calculate the value of an annuity at any age with any degree of accuracy desired. In practice the calculations would be most readily made by the aid of logarithms.
availing ourselves of the supposition which we have seen to be allowable, that the lives under observation will
die off exactly at the rate indicated by the mortality table.
