The value of a life annuity depends, therefore, upon the manner in which it is presumed a large number of persons, si milarly situated with the buyer, would die off successively. Various Tables of these decrements of life, as they are called, have been constructed, from observations made among different classes of lives. Some make the mortality greater than others; and of course, Tables which give a large mortality, give the value o' the annuity smaller than those which suppose men to live longer. Those who buy an nuities would, therefore, be glad to be rated according to tables of high mor tality or low expectation of life ; while those who sell them would prefer receiv ing the price indicated by tables which give a lower rate of mortality. In insu rances the reverse is the case : the shorter the time which a man is supposed to live, the more must he pay the office, that the latter may at his death have accumulated wherewithal to pay his executors. We now give in Table V. the values of an nuities according to three of the most ce lebrated Tables.
high a mortality at all the younger and middle ages of life, and, consequently, too low a value of the annuity. The second is from the Carlisle Table, formed by Mr. Milne, from observations made at Car lisle. It gives much less mortality than most other Tables, and, therefore, gives higher values of the annuities ; but it has since been proved to represent the actual state of life among the middle classes, in the century now ending, with much greater accuracy than could have been supposed, considering the local character of the observations from which it was de rived. The third table is that con structed by Mr. Finlaison, from the ob servation of the mortality in the govern ment tontines and among the holders of annuities granted by government in re demption of the national debt, and differs from the former two in distinguishing the lives of males from those of females. Most observations hitherto published unite in confirming the fact, that females, on the average, live longer than males, and in the annuities now granted by go vernment, a distinction is made accord ingly. The mean between the values of annuities on male and female lives, according to the Government Tables, agrees pretty nearly with the Carlisle Tables, the rate of interest being the same.
For the materials of Table V. we are indebted to the works of Dr. Price, on Reversionary Payments ; of Mr. Milne, on Annuities and Insurances ; and to Mr. Finlaison's Report to the House if Com mons on Life Annuities; to all of which we refer the reader. The tables are of course very much abridged.
To use the Table V., suppose the value of an annuity of 100/. a-year, on a life aged 35, is required, interest being at 4 per cent., which is nearly the actual value of money. We find in the column marked 4 per cent., opposite to 35, under the Northampton Tables 1440, under the Carlisle 16'0, and under the Government Tables 151 or 16•9, according as the life is male or female. These are the num ber of pounds which ought to buy an annuity of 11., according to these several authorities ; and taking each of them 100 times, we have:— Northampton Table . . 14001. Carlisle Table . . . . 1600/. Government Table (males) 15701. Government Table (females) 16901. We cannot suppose that the annuity could be bought for less than would be required by the Carlisle Tables.
To find the value of an annuity on a life whose age lies between two of those given in the table, the process must be followed which has been already ex plained in treating of annuities certain.
An annuity on two joint lives is one which is payable only so long as both the persons on whose lives it is bought are alive to receive it.
aged 25 and 55, in which the difference of age is 30 years. In the Carlisle Table opposite to 25, the younger, and under 30, the difference, we find 10.3 ; and 8.8 in the Northampton. For the value of an annuity of 100/., the first tables give, therefore, 10301., and the second 880/.
The value of an annuity on the long est of two lives, that is, which is to be payable as long as either of the two shall be alive to receive it, is found by adding together the values of the annuity on the two lives separately considered, and sub tracting the value of the annuity on the joint lives. For the above species of an nuity puts the office and the parties in precisely the same situation as if an an nuity were granted to each party sepa rately, but on condition that one of the annuities should be returned to the office so long as both were alive, that is, during their joint lives. For example, let the ages be 25 and 55 as before, and let the Carlisle Table be chosen, interest being at 4 per cent., we have then :— The preceding table gives the results of the Carlisle and Northampton Tables on the value of this species of annuity, interest being at 4 per cent. The first column shows the age of the younger life, and the horizontal headings are not the age of the elder life, but the excess of the age of the elder life above that of the younger. For example, to know the value of an annuity in two joint lives, The value, therefore, of an annuity of IL per annum on the survivor is 18.61.