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A rent of this annuity will only be payable at the end of the $$n$$th year, provided that B be then dead, and A living; but the probability of B being then dead is $$1 - {}_n\!b$$, and that of A being then living $${}_n\!a$$, and these two events are independent; therefore, the probability of their both happening, or that of the rent being received, is $$(1 - {}_n\!b) {}_n\!a = {}_n\!a - {}_n\!(ab)$$; the present value of that rent is, therefore, $$\left [ {}_n\!a - {}_n\!(ab) \right ] v^n$$; whence, it follows, that the required value of the annuity on the life of A after that of B, is $$\text{A} - \text{AB}$$, agreeably to No. 60.

163. If the payment for the annuity which was the subject of the last problem, is not to be made in present money, but by a constant annual premium $$p$$ at the end of each year, while both the lives survive; since $$\text{AB}$$ is the number of years purchase (6) that an annuity on the joint continuance of those lives is worth, the value of p will be determined by this equation, $$\text{p}\,.\,\text{AB} = \text{A} - \text{AB}$$, whence we have $$\text{p} = \frac{\text{A}}{\text{AB}} - 1$$.

164. But if one premium $$p$$ is to be paid down now, and an equal premium at the end of each year while both the lives survive, we shall have $$p\,.\,(1 + \text{AB}) = \text{A} - \text{AB}$$, and $$p = \frac{\text{A} - \text{AB}}{1 + \text{AB}} - 1$$.

165. For numerical examples illustrative of the formulæ given from No. 158 to the present; see Nos. 66—74.

166. A and B are in possession of an annuity on the life of the survivor of them, which, if either of them die before a third person C, is then to be divided equally between C and the survivor during their joint lives; to determine the value of C’s interest.

and the sum of these being $$\tfrac{1}{2} {}_n\!(ac) + \tfrac{1}{2} {}_n\!(bc) - {}_n\!(abc)$$, the value of C’s interest is $$\tfrac{1}{2} \text{AC} + \tfrac{1}{2} \text{BC} - \text{ABC}$$.

167. An annuity after the decease of A, is to be equally divided between B and C during their joint lives, and is then to go entirely to the last survivor for his life; it is proposed to find the value of B’s interest therein.

sum of these being $${}_n\!b - {}_n\!(ab) - \tfrac{1}{2} {}_n\!(bc) + \tfrac{1}{2} {}_n\!(abc)$$, the value of B’s interest is $$\text{B} - \text{AB} - \tfrac{1}{2}\text{BC} + \tfrac{1}{2}\text{ABC}$$.

168. A, B, and C purchase an annuity on the life of the last survivor of them, which is to be divided equally at the end of every year among such of them as may then be living; what should A contribute towards the purchase of this annuity?

the sum of these being $${}_n\!a - \tfrac{1}{2} {}_n\!(ab) - \tfrac{1}{2} {}_n\!(ac) + \tfrac{1}{3} {}_n\!(abc)$$, the required value of A’s interest is $$\text{A} - \tfrac{1}{2}\text{AB} - \tfrac{1}{2}\text{AC} + \tfrac{1}{3}\text{ABC}$$.

169. As soon as any two of the three lives, A, B, and C, are extinct, D or his heirs are to enter upon an annuity; which they are to enjoy during the remainder of the survivor’s life; to determine the value of D’s interest therein.

sum of all these being $${}_n\!a + {}_n\!b + {}_n\!c - 2 {}_n\!(ab) - 2 {}_n\!(ac) - 2 {}_n\!(bc) + 3 {}_n\!(abc)$$, the value of D’s interest is

170. The last four may be sufficient to show the method of proceeding with any similar problems.

171. Let $$\overset{m}{\overline{{}_t\!(abc,\ \&\text{c.})}}$$ denote the probability that the Rh