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/642

 age, greater than that of any proposed life by the number of years in the term, and the product be divided by the number in the same table, against the present age of that life; the quotient will be the present value of L. 1, to be received at the expiration of the term, provided that the life survive it.

And if, in this manner, the value be determined of L. 1, to be received upon any proposed life, surviving each of the years in its greatest possible continuance, according to the table of mortality adapted to it; that is, according to the Carlisle table, upon its surviving every age greater than its present, to that of 104 years inclusive; then, the sum of all these values will evidently be the present value of an annuity on the proposed life.

36. If 5642 lives at 30 years of age be proposed, and 5075 at the age of 40; since each of the 5642 younger lives may be combined with every one of the 5075 that are 10 years older, the number of different pairs, or different combinations of two lives differing in age by 10 years, that may be formed out of the proposed lives, is 5642 times 5075.

But at the expiration of 15 years, the survivors of the lives now 30 and 40 years of age, being then of the respective ages of 45 and 55, will be reduced to the numbers of 4727 and 4073 respectively; and the number of pairs, or combinations of two, differing in age by 10 years, that can be formed out of them, will be reduced from 5642 × 5075 to 4727 × 4073.

So that L. 1 to be paid at the expiration of 15 years for each of these 5642 × 5075 pairs or combinations of two, now existing, which may survive the term, will be of the same value in present money, as 4727 times L. 4073, to be received certainly at the same time.

Now let A be any one of these lives of 30 years of age, and B any one of those aged 40; and from what has been advanced it will be evident, that the present value of L. 1 to be received upon the two lives in this particular combination jointly surviving the term, will be the same as that of the sum L. $$\frac{4727 \times 4073}{5642 \times 5075}$$ to be then received certainly.

But, when the rate of interest is 5 per cent. L. 1 to be received certainly at the expiration of 15 years, is equivalent to L. 0·481017 in present money (Table I).

Therefore, at that rate of interest, and according to the Carlisle table of mortality; the present value of L. 1 to be received upon A and B now aged 30 and 40 years respectively, jointly surviving the term of 15 years, will be $$\frac{4727 \times 4073 \times \text{L. } 0{\cdot}481017}{5642 \times 5075.}$$.

37. Hence it is sufficiently evident, how the present value of L. 1 to be received upon the same two lives jointly surviving any other year may be found. And if that value for each year from this time until the eldest life attain to the limit of the table of mortality be calculated, the sum of all these will be the present value of an annuity of L. 1 dependent upon their joint continuance.

In this manner, it is obvious that the value of an annuity on the joint continuance of any other two lives might be determined.

38. If, besides the 5642 lives at 30 years of age, and the 5075 at 40 (mentioned in No. 36), there be also proposed 3643 at 60 years of age; each of these 3643 at 60, may be combined with every one of the 5642 × 5075 different combinations of a life of 30, with one of 40 years of age; and, therefore, out of these three classes of lives 5642 × 5075 × 3643 different combinations may be formed; each containing a life of 30 years of age, another of 40, and a third of 60.

But at the expiration of 14 years, the numbers of lives in these three classes will, according to the table of mortality, be reduced to 4727, 4073, and 1675 respectively; the respective ages of the survivors in the several classes being then 45, 55, and 75 years; and the number of different combinations of three lives (each of a different class from either of the other two), that can be formed out of them, will be reduced to 4727 × 4073 × 1675.

Hence, by reasoning as in No. 36, it will be found, that if A, B, and C be three such lives, now aged 30, 40, and 60 years, the present value of L. 1 to be received upon these three jointly surviving the term of 15 years from this time, will be $$\frac{4727 \times 4073 \times 1675}{5642 \times 5075 \times 3643} \times \text{L. } 0{\cdot}481017$$: interest being reckoned at 5 per cent.

Thus it is shown, how the present value of an annuity dependent upon the joint continuance of these three lives might be calculated, that being the sum of the present values thus determined, of the rents for all the years which, according to the table of mortality, the eldest life can survive.

39. But it is easy to see, that the same method of reasoning may be used in the case of four, five, or six lives, and so on without limit. Whence, this inference is obvious.

The present value of L. 1, to be received at the expiration of a given term, provided that any given number of lives all survive it, may be found by multiplying the present value of L. 1 to be received certainly at the end of the term, by the continual product of the numbers in the table of mortality against the ages greater respectively by the number of years in the term, than the ages of the lives proposed; and dividing the last result of these operations, by the continual product of the numbers in the table of mortality against the present ages of the proposed lives.

And by a series of similar operations, the present value of an annuity on the joint continuance of all these lives might be determined.

But it should be observed, that, in calculating the value of a life-annuity in this way, the denominator of the fractions expressing the values of the several years rents, that is, the divisor used in each of the operations, remains always the same; the division should, therefore, be left till the sum of the numerators is determined, and one operation of that kind will suffice.

40. Enough has been said to show that these methods of constructing tables of the values of annuities on lives are practicable, though excessively laborious, and, in fact, all the early tables of this kind were constructed in that manner. We proceed now