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/655



Upon which it may be observed, that $$\lambda\;v + \lambda\;{}_\prime a$$, the sum of the first two logarithms that are employed in determining $$\text{A}$$ from $${}'\!\text{A}$$, also enters the operation whereby $$\text{AB}$$ is determined from $${}'\!(\text{AB})$$. And that $$ \lambda\;v + \lambda\;{}_\prime a + \lambda\;{}_\prime b$$, the sum of the first three logarithms that serve to determine $$\text{AB}$$ from $${}'\!(\text{AB})$$, is also required to determine $$\text{ABC}$$ from $${}'\!(\text{ABC})$$; which observation may be extended in a similar manner to any greater number of joint lives.

154. By these means it is easy to complete a table of the values of annuities on single lives of all ages; beginning with the oldest in the table, and proceeding regularly age by age to the youngest.

Also a table of the values of any number of joint lives, the lives in each succeeding combination, in any one series of operations, (according to the retrograde order of the ages in which they are computed), being one year younger respectively than those in the preceding combination.

And, if a table of single lives be computed first, then of two joint lives, next of three joint lives, and so on; the calculations made for the preceding tables will be of great use for the succeeding.

155. Having shown how to compute tables of the values of annuities on single and joint lives, we shall, in what follows, always suppose those values to be given.

156. Let the value of an annuity on the joint continuance of any number of lives, A, B, C, &c. that is not to be entered upon until the expiration of $$t$$ years be denoted by $${}_{\neg t}\!(\text{ABC }\&{c.})$$

Then, if it were certain that all the lives would survive the term, since the value of the annuity at the expiration of the term would be $${}^t\!(\text{ABC }\&{c.})$$, (145), its present value would be $$v^t \cdot {}^t\!(\text{ABC }\&{c.})$$, (146).

But the measure of the probability that all the lives will survive the term is $${}_t\!(abc\ \&\text{c.})$$, therefore $${}_{\neg t}\!(\text{ABC }\&{c.}) = {}_t\!(abc\ \&\text{c.}) v^t\,.\,{}^t\!\text{ABC }\&{c.})$$.

In the same manner, it appears, that for a single life A, $${}_{\neg t}\!\text{A} = {}_t\!a v^t\,.\,{}^t\!\text{A}$$.

157. Let an annuity for the term of $$t$$ years only, dependent upon the joint continuance of any number of lives, A, B, C, &c. be denoted by $${}_{t\urcorner}\!(\text{ABC }\&\text{c.})$$; and, since this temporary annuity, together with an annuity on the joint continuance of the same lives deferred for the same term, will evidently be of the same value as an annuity to be entered upon immediately, and enjoyed during their whole joint continuance, we have $${}_{t\urcorner}\!(\text{ABC }\&\text{c.}) + {}_{\neg t}\!(\text{ABC }\&{c.}) = \text{ABC }\&\text{c.}$$; whence, $${}_{t\urcorner}\!(\text{ABC }\&\text{c.}) = \text{ABC }\&\text{c.} - {}_{\neg t}\!(\text{ABC }\&{c.})$$.

And for a single life A, $${}_{t\urcorner}\!\text{A} = \text{A} - {}_{\neg t}\!\text{A}$$.

158. To determine the present value of an annuity on the survivor of the two lives A and B, (155), which we designate thus, $$\overline{\text{AB}}$$.

The probability that the survivor of these two lives will outlive the term of $$n$$ years, was shown in No. 141, to be $${}_n\!a + {}_n\!b - {}_n\!(ab)$$; therefore, reasoning as in No. 146, it will be found, that the present value of the $$n$$th year’s rent of this annuity is $$\left [ {}_n\!a + b - {}_n\!(ab) \right ] v^n$$, and the value of all the rents thereof will be $$S\left [ {}_n\!a + b - {}_n\!(ab) \right ] v^n$$ or $$S {}_n\!av^n + S {}_n\!bv^n - S {}_n\!(ab)v^n$$; so that $$\overline{\text{AB}} = \text{A} + \text{B} - \text{AB}$$ (147), agreeably to No. 48.

159. To determine the present value of an annuity on the last survivor of three lives, A, B, and C, (155); which we denote thus, $$\overline{\text{ABC}}$$.

The present value of the $$n$$th year’s rent is $$\left [ {}_n\!a + {}_n\!b + {}_n\!c - {}_n\!(ab) - {}_n\!(ac) - {}_n\!(bc) + {}_n\!(abc) \right ] v^n$$ (142 and 146); whence, it appears, as in the preceding number, that $$\overline{\text{ABC}} = \text{A} + \text{B} + \text{C} - \text{AB} - \text{AC} - \text{BC} + \text{ABC}$$, agreeably to No. 52.

160. To determine the present value of an annuity on the joint existence of the last two survivors out of three lives, A, B, C, (155); which we denote thus, $$\overset{2}{\overline{\text{ABC}}}$$:

The present value of the $$n$$th year's rent is $$\left [{}_n\!(ab) + {}_n\!(ac) + {}_n\!(bc) - 2 {}_n\!(abc) \right ] v^n$$ (143 and 146); whence, reasoning as in the two preceding numbers, we infer, that $$\overset{2}{\overline{\text{ABC}}} = \text{AB} + \text{AC} + \text{BC} - 2\text{ABC}$$, as was demonstrated otherwise in No. 51.

161. Since the solutions of the last three problems were all obtained by showing each year’s rent (as for instance the $$n$$th) of the annuity in question, to be of the same value with the aggregate of the rents for the same year, of all the annuities (taken with their proper signs) on the single and joint lives exhibited in the resulting formula: if any term of years be assigned, it is manifest that the value of such annuity for the term, must be the same as that of the aggregate of the annuities above mentioned, each for the same term.

162. A and B being any two proposed lives now both existing, to determine the present value of an annuity receivable only while A survives B.