Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/939

647

NNUITIES. As an addition to the article , we beg to insert here an expeditious method of calculating the values of annuities on single or joint lives, from any tables or bills of mortality, with sufficient accuracy for all practical purposes.

We must begin by determining the mean complement of life, according to the average number of deaths during a certain period, which must vary according to the nature of the proposed calculation; being shorter as the rate of interest is higher; and as the number of lives concerned is greater; but not requiring to be very accurately defined. If the rate of interest be $$r$$, we must find the time in which the number of deaths is expressed by the fraction $$\frac{3}{r+3}$$ of the whole number of survivors at the given age, for a single life: for two lives, the fraction must be $$\frac{3}{r+5}$$, and for three, $$\frac{3}{r+7}$$; and, in each of these cases, the time determined from the age of the oldest life must be employed for finding the complements of both the others.

Having thus calculated the complements for each of the ages, we may, in most instances, save ourselves the trouble of further computation, by employing tables of the value of annuities on one and two lives, according to Demoivre’s hypothesis. For this purpose, we have only to subtract the complement from 86, and we obtain an equivalent life on this hypothesis. If we take, for example, the age of 20, the number of survivors in the Northampton tables ts 5132; and, for a single life, at 3 and at 6 per cent. we must find the time at which they are reduced $$\frac{3}{6}$$ and $$\frac{3}{9}$$ respectively; that is, to about 2566 and 3421: now at 54 and 43, the numbers are 2530 and 3404; and $$\frac{5132 \times 34}{5132 - 2530} = 67{\cdot}07$$, and $$\frac{5132 \times 23}{5132 - 3404} = 68{\cdot}3$$; whence the equivalent ages in Demoivre’s tables are 18·93 and 17·7, giving 18·62 and 12·43, for the value of the annuity; while Dr Price’s table, deduced from the actual decrements at all ages, gives 18·64 and 12·40.

The utility of this mode of calculation will be still further illustrated by a comparison of the very different values of lives, as indicated by different tables. Taking, for example, the age of 30, and the interest at 5 per cent. we may find the value of the annuity, by this approximation, in different situations, for which correct tables have been published by Dr Price, and may thence infer how much nearer it approaches to the truth than the generality of the results approach to each other:

According to the bills of mortality of London for 1815, out of 9472 survivors at 30, 5573 lived to 50, and this is near enough to $$\frac{3}{8}$$ for our purpose: hence the complement is 48·58, and the value of an annuity at 5 per cent. 12·16 years’ purchase. Where the age is much greater, the approximation is somewhat less accurate, though not often materially erroneous; thus, at 70, the values, according to the Northampton tables, at 3 and 6 per cent, are 6·23 and 5·35, instead of 6.73 and 5.72 respectively.

In the values of joint lives, there is more difference, according to the different tables employed, than in those of single lives: thus, at 30, the value of an annuity, at 4 per cent. on a single life, differs at Northampton, and in Sweden, in the proportion of 14·78 to 16·00, or of 12 to 13; but, for two joint lives at 30, in that of 11·31 to 12·96, or of 7 to 8; and for three lives, the disproportion would be still greater.

In the absence of Demoivre’s tables, or for cases to which they do not extend, it becomes necessary to calculate the value of the annuity for each particular instance. Calling then the complement, as already determined, $$a$$, the number of survivors after $$x$$ years will be represented by $$a-x$$, and the present value of any sum to be paid to each of them by $$av^x-xv^x$$, $$v$$ being the present value of a unit payable at the end of a year: and if we suppose such payments to be made continually, their whole present value may be found by multiplying this expression by the fluxion of $$x$$, and finding the fluent, which will be $$-pv^x (a-x-p)$$, $$p$$ being $$=-\frac{1}{\operatorname{HL} v}$$, or the reciprocal of the hyperbolical logarithm of the amount of a unit after a year. When $$x$$ vanishes, this fluent becomes $$-p (a-p)$$, and when $$x = a$$, $$p^2 v^a$$; the difference, divided by $$a$$, gives the present value of the annuity, $$p - \frac{p^2}{a} + \frac{p^2 v^a}{a}$$; from which, when the annuity is supposed to become due and to be paid periodically, we must subtract in all cases half a payment; that is, ½ for yearly payments, and ¼ for quarterly; and if, at the same time, we choose to assume that money is capable of being improved by laying out the interest more frequently than once