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/654



And the sum of these four $${}_n\!(ab) + {}_n\!(ac) + {}_n\!(bc) - 2 {}_n\!(abc)$$, is the measure of the probability that some two at the least, out of these three lives, will survive the term.

144. Let the number of years purchase that an annuity on the life of A is worth, that is, the present value of L. 1, to be received at the end of every year during the continuance of that life, be denoted by $$\text{A}$$; while the present value of an annuity on any number of joint lives A, B, C, &c. that is, of an annuity which is to continue during the joint existence of all the lives, but to cease with the first that fails, is denoted by $$\text{ABC}$$, &c.

Then will the value of an annuity on the joint continuance of the three lives A, B, and C, be denoted by $$\text{ABC}$$.

And on the joint continuance of the two A and B, by $$\text{AB}$$.

145. Also Iet $${}^t\!\text{A}$$ and $$\text{A}_t$$ denote the value of annuities on lives respectively older and younger than A, by $$t$$ years: While $${}^t\!(\text{ABC}\ \&\text{c.})$$ designates the value of an annuity on the joint continuance of lives $$t$$ years older than A, B, C, &c. respectively; and $$(\text{ABC}\ \&\text{c.})_t$$ that of an annuity on the same number of joint lives, as many years younger than these respectively.

146. Let $$\frac{1}{1+r}$$, the present value of L. 1 to be received certainly at the expiration of a year, be denoted by $$v$$.

Then will $$v^n$$ be the present value of that sum certain to be received at the expiration of $$n$$ years.

But if its receipt at the end of that time, be dependent upon an assigned life A, surviving the term, its present value will, by that condition, be reduced in the ratio of certainty to the probability of A surviving the term, that is, in the ratio of unity to $${}_n\!a$$, and will therefore be $${}_n\!an^n$$.

In the same manner it appears, that if the receipt of the money at the expiration of the term be dependent upon any assigned lives, as A, B, C, &c. jointly surviving that period, its present value will be $${}_n\!(abc\ \&\text{c.})v^n$$.

147. Let us denote the sum of any series, as $${}_1\!(abc)v + {}_2\!(abc)v^2 + {}_3\!(abc)v^3 + \&\text{c.}$$ thus, $$S {}_n\!(abc)v^n$$, by prefixing the italic capital $$S$$ to the general term thereof. Then, from what has just been advanced, it will be evident, that $$\text{ABC }\&\text{c.} = S {}_n\!(abc\ \&\text{c.})v^n$$.

When there are but three lives A, B, and C; this becomes $$\text{ABC} = S {}_n\!(abc)v^n.$$

When there are but two, A and B, it becomes $$\text{AB} = S {}_n\!(ab)v^n$$.

And in the same manner it appears, that for a single life A, $$\text{A} = S {}_n\!av^n$$.

148. $${}_n\!(abc\ \&\text{c.})v^n = \frac{{}^n\!(abc\ \&\text{c.})v^n}{abc\ \&\text{c.}}$$ (138), where the denominator ($$abc$$ &c.) is constant, while the numerator varies with the variable exponent $$n$$. And the most obvious method of finding the value of an annuity on any assigned single or joint lives, is to calculate the numerical value of the term $${}^n\!(abc\ \&\text{c.})v^n$$ for each value of $$n$$, and then to divide the sum of all these values by $$abc$$ &c.; for $$\frac{S {}^n\!(abc\ \&\text{c.})v^n}{abc\ \&\text{c.}} = S {}_n\!(abc\ \&\text{c.})v^n = \text{ABC}\ \&\text{c.}$$

In calculating a table of the values of annuities on lives in that manner, for every combination of joint lives, it would be necessary to calculate the term $${}^n\!(abc\ \&\text{c.})v^n$$ for as many years as there might be between the age of the oldest life involved and the oldest in the table; and the same number of the terms $${}^n\!av^n$$ for any single life of the same age.

But this labour may be greatly abridged as follows:

149. Given $${}'\!(\text{ABC},\ \&\text{c.})$$, the value of an annuity on any number of joint lives, to determine $$\text{ABC}$$, &c. that of an annuity on the same number of joint lives respectively one year younger than them.

If it were certain that the lives $$\text{ABC}$$, &c. would all survive one year, the proprietor of an annuity of L. 1, dependent upon their joint continuance, would, at the expiration of a year, be in possession of L. 1, (the first year’s rent,) and an annuity on the same number of lives, one year older respectively than $$\text{ABC}$$, &c. therefore, in that case, the required present value of the annuity would be $$v \left [ 1 + {}'\!(\text{ABC},\ \&\text{c.}) \right ]$$, (146.)

But the probability of the lives A, B, C, &c. jointly surviving one year, is less than certainty, in the ratio of $${}_\prime\!(abc\ \&\text{c.})$$ to unity; therefore $$\text{ABC }\&\text{c.} = {}_\prime\!(abc\ \&\text{c.})v[1 + {}'\!(\text{ABC}\ \&\text{c.})]$$.

150. ''Corol. 1. When there are but three lives, A'', B, and C, this becomes $$\text{ABC} = {}_\prime\!(abc)v[1 + {}'\!(\text{ABC})]$$.

151. ''Corol. 2. When there are only two, A and B'', $$\text{AB} = {}_\prime\!(ab)v[1 + {}'\!(\text{AB})]$$.

152. ''Corol. 3. And for a single life A'', it appears, in the same manner, that $$\text{A} = {}_\prime av(1 + {}'\!\text{A})$$.

153. Hence, in logarithms, we have these equations, Rh