Thus, suppose that I, persons of the- age x buy each an annuity of 1. Then the number of persons who will survive to the age x+ 1, and claim the first payment of the annuity, will be l, +l . The value of 1 to be paid at the end of a year is v, and therefore the present sum that will be required to provide for all the payments at the end of the first year will be If+iU. The number of persons who will survive two years, so as to claim the second year s payment, will be l a+2 , and multiplying this into the value of 1 payable at the end of two years, we get l x+ ^ as the present sum necessary to provide for the payments at the end of the second year. Proceeding in this way, the total sum that will be required to provide the annuities to the l x persons, will be l x+1 v + l x+z v 2 + l a+3 iP + . . . Hence the value of an annuity on a nominee of the age x, or the sum that will on the average be required to provide for such an annuity,
will be7 ... .
which is at once seen to be the same as (1) formula under another shape.
If we suppose money to bear no interest, or make v= 1 in the formula for the value of an annuity, we shall obtain a quantity which is called the " expectation of life," or the " average duration of life," being the average number of years which persons of the given age will one with another live. Denoting this by e x , and making v=lin the formula above given, we get
[ math ]
As in the formula for the annuity, no payment is made on account of the year .in which the nominee dies, this formula gives the average number of complete years that persons of the given age will live according to the mortality table, and makes no allowance for the portion of the year in which death occurs. The expectation thus found is called the curtate expectation; and in order to obtain the complete expectation of life, which is denoted by e x , half a year must be added to it.
The first writer who is known to have attempted to obtain, on correct mathematical principles, the value of a life annuity, was Johan De-Wit, Grand Pensionary of Holland and West Friesland. All our exact knowledge of his writings on the subject is derived from two papers contributed by Mr Frederick Hendriks to the Assurance Magazine, vol. ii. p. 222, and vol. iii. p. 93. The former of these contains a translation of De Wit s report upon the value of life annuities, which was prepared in consequence of the resolution passed by the States General, on the 25th April 1671, to negotiate funds by life annuities, and which was distributed to the members- on the 30th July 1671. The latter contains the translation of a number of letters addressed by De Wit to Burgomaster Johan Hudde, bearing dates from September 1 670 to October 1671. The existence of De Wit s report was well known among his contemporaries, and Mr Hendriks has collected a number of extracts from various authors referring to it; but the report is not contained in any collection of his works extant, and had been entirely lost for 180 years, until Mr Hendriks conceived the happy idea of searching for it among the state archives of Holland, when it was found together with the letters to Hudde. It is a document of extreme interest, and (notwithstanding some inaccuracies in the reasoning) of very great merit, more especially con sidering that it was the very first document on the subject that was ever written; and Mr Hendriks s papers will well repay a careful perusal.
for life annuities to be granted to nominees of any age } in the constant proportion of double the rate of interest allowed on stock; that is to say, if the towns were borrow ing money at 6 per cent., they would be willing to grant a life annuity at 12 per cent.; if at 5 per cent., the annuity granted was 10 per cent.; and so on. De AVit states that " annuities have been sold, even in the present century, first at six years purchase, then at seven and eight; and that the majority of all life annuities now current at the country s expense were obtained at nine years purchase; but that the price had been increased in the course of a few years from eleven years purchase to twelve, and from twelve to fourteen. He also states that the rate of interest had been successively reduced from 6-| to 5 per cent., and then to 4 per cent. The principal object of his report is to prove that, taking interest at 4 per cent., a life annuity was worth at least sixteen years purchase; and, in fact, that an annuitant purchasing an annuity for the life of a young and healthy nominee at sixteen years purchase, made an excellent bargain. It may be mentioned that he argues that it is more to the advantage, both of the country and of the private investor, that the public loans should be raised by way of grant of life annuities rather than per petual annuities. It appears conclusively from De Wit s correspondence with Hudde, that the rate of mortality assumed as the basis of his calculations was deduced from careful examination of the mortality that had actually pre vailed among the nominees on whose lives annuities had been granted in former years. De Wit appears to have come to the conclusion that the probability of death is the same in any half-year from the age of 3 to 53 inclusive; that in the next ten years, from 53 to 63, the probability is greater in the ratio of 3 to 2 ; that in the next ten years, from 63 to 73, it is greater in the ratio of 2 to 1 ; and in the next seven years, from 73 to 80, it is greater in the ratio of 3 to 1 ; and he places the limit of human life at 80. If a mortality table of the usual form is deduced from these suppositions, out of 212 persons alive at the age of 3, 2 will die every year up to 53, 3 in each of the ten years from 53 to 63, 4 in each of the next ten years from 63 to 73, and 6 in each of the next seven years from 73 to 80, when all will be dead. This is the conclusion we have drawn from a careful study of the report; but, in consequence of the inaccuracies above mentioned, some doubt exists as to De Wit s real meaning; and Mr Hen driks s conclusion is somewhat different from ours (see his note, Ass. Mag. vol. ii. p. 246). The method of calculation employed by De Wit differs much from that described above, and a short account of it may interest the reader. Suppose that it were desired to apply it to deduce the value of an annuity according to Deparcieux s mortality table given above, then we assume that annuities are bought on tha lives of 1000 nominees each 3 years of age. Of these nominees, 30 will die before attaining the age of 4, and no annuity payment will be made in respect of them; 22 will die between the ages of 4 and 5, so that the holders of the annuities on their lives will receive payment for 1 year; 18 attain the age of 5 and die before 6, so that the annuities on their lives are payable for 2 years. Reasoning in^the same way, we see -that the annuities on 15 of the nominees will be payable for 3 years; on 13, for 4 years; on 12, for 5 years; on 10, for 6 years; and so on. Proceeding thus to the extremity of the table, 2 nominees attain the age of 93, 1 of whom dies before the age of 94, so that 90 annuity payments will be made in respect of him; and the last survivor dies between the ages of 94 and 95, so that the annuity on his life will be payable for 91 years. Having previously calculated a table of the values of annuities certain for every number
of years up to 91, the value of all the annuities on the