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 last $$m$$ survivors out of $$m+\mu$$ lives A, B, C, &c. will jointly survive the term of $$t$$ years. And when $$\mu=0$$, the expression will become $${}_t\!(abc,\ \&\text{c.})$$ the probability that the lives will all survive the term (138).

When $$m=1$$ it will become $${}_t\!(\overset{1}{\overline{abc,\ \&\text{c.}}})$$, the measure of the probability that the last survivor of them will outlive the term; which it will be better to write thus, $${}_t\!(\overline{abc,\ \&\text{c.}})$$, retaining the vinculum, but omitting the unit over it, as in the notation of powers.

Also let $$\overset{m}{\overline{\text{ABC},\ \&\text{c.}}}$$ denote the value of an annuity on the joint continuance of the same number of last survivors out of the same lives. Then, if $$\mu$$ be equal to 0, it will be $$\text{ABC}$$, &c. the value of an annuity on the joint continuance of all the lives; when $$m=1$$, it will be $$\overline{\text{ABC},\ \&\text{c.}}$$ the value of an annuity on the last survivor of them. The values of annuities on the last survivor of two and of three hives, will be denoted as in Nos. 158 and 159 respectively; and that of an annuity on the joint continuance of the last two survivors out of three lives, as in No. 160.

The value of an annuity on the last $$m$$ survivors out of these $$m + \mu$$ lives, according as it is limited to the term of $$t$$ years, or deferred during that term, will also he denoted by $${}_{t\urcorner}\!\overset{m}{\overline{\text{ABC},\ \&\text{c.}}}$$ or $${}_{\neg t}\!\overset{m}{\overline{\text{ABC},\ \&\text{c.}}}$$ (156 and 157.)

172. An annuity certain for the term off $$t + \nu$$ years, is to be enjoyed by P and his heirs during the joint existence of the last $$m$$ survivors out of $$m + \mu$$ lives, A, B, C, &c.; and if that joint existence fail before the expiration of $$t$$ years, the annuity is to go to Q and his heirs, for the remainder of the term; to determine the value of Q’s interest in that annuity.

Q’s expectation may be distinguished into two parts:

2em

The sum of the present values of the interests of P and Q, together in the annuity for the term of $$t$$ years, is manifestly equal ta the whole present value of the annuity certain for that term; that is, equal to $$\frac{1 - v^t}{r}$$ (113 and 146); and the value of P’s interest for the term of $$t$$ years, is $${}_{t\urcorner}\!\overset{m}{\overline{\text{ABC},\ \&\text{c.}}}$$ (171); therefore the value of Q’s interest for the same term is $$\frac{1 - v^t}{r} - {}_{t\urcorner}\!\overset{m}{\overline{\text{ABC},\ \&\text{c.}}}$$

The present value of the annuity certain for $$\nu$$ years after $$t$$ years is $$\frac{v^t(1-v^\nu)}{r}$$ (114 and 146); and Q and his heirs will receive this annuity, if the joint continuance of the last $$m$$ survivors above mentioned fail before the expiration of $$t$$ years; but the probability of their joint continuance failing in the term, is $$1 - {}_t\!(\overset{m}{\overline{abc,\ \&\text{c.}}})$$; therefore, the value of Q’s interest in the annuity to be received after $$t$$ years, is $$\left [ 1 - {}_t\! ( \overset{m}{\overline{abc,\ \&\text{c.}}} ) \right ] \frac{v^t (1-v^\nu)}{r}$$; and the whole value of Q’s interest, is $$\frac{1}{r} \left [ 1 - v^{\nu+t} - v^t(1-v^\nu) \cdot {}_t\!(\overset{m}{\overline{abc,\ \&\text{c.}}}) \right ] - {}_{t\urcorner}\!\overset{m}{\overline{\text{ABC, }\&\text{c.}}}$$

173. ''Corol. 1. When the whole annuity certain is a perpetuity, $$v^{t+\nu}$$ = 0, and the value of Q''’s interest is $$\frac{1}{r} \left [ 1 - {}_t\!(\overset{m}{\overline{abc,\ \&\text{c.}}}) v^t \right ] - {}_{t\urcorner}\!\overset{m}{\overline{\text{ABC, }\&\text{c.}}}$$

174. ''Corol. 2''. When the term $$t$$ is not less than the greatest joint continuance of any $$m$$ of the proposed lives, according to the tables of mortality adapted to them, $${}_t\!(\overset{m}{\overline{abc,\ \&\text{c.}}}) = 0$$, and $${}_{t\urcorner}\!\overset{m}{\overline{\text{ABC, }\&\text{c.}}} = \overset{m}{\overline{\text{ABC, }\&\text{c.;}}}$$ therefore, in that case, the general formula of No. 172 becomes $$\frac{1-v^{\nu+t}}{r} - \overset{m}{\overline{\text{ABC, }\&\text{c.}}}$$; that is, the excess of the value of an annuity certain for the whole term $$\nu+t$$, above that of an annuity on the whole duration of joint continuance of the last $$m$$ surviving lives.

175. And if, in the case proposed in the last No. the annuity certain be a perpetuity, as in No. 173, the formula will become $$\frac{1}{r} - \overset{m}{\overline{\text{ABC, }\&\text{c.}}}$$ the excess of the value of the perpetuity above the value of an annuity on the joint lives of the last $$m$$ survivors; agreeably to No. 63.

176. Example 1. Required the present value of the absolute reversion of an estate in fee simple, after the extinction of the last survivor of three lives, A, B, C, now aged 50, 55, and 60 years respectively: reckoning interest at 5 per cent.

The general Algebraical expression of this value has just been shown to be $$\frac{1}{r} - \overline{\text{ABC.}}$$

the value required. And if the annual produce of the estate, clear of all deductions, were L. 100, the title to the reversion would now be worth L. 599, 18s.—, agreeably to No. 76.

177. Ex. 2. An annuity for the term of 70 years certain (from this time), is to revert to Q and his heirs at the failure of a life A, now 45 years of age; what is the present value of Q’s interest therein; reckoning the interest of money at 5 per cent.?