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/658

 In No. 174, the Algebraical expression of the required value is shown to be $$\frac{1-v^{70}}{r} - \text{A}$$.

so that if the annuity were L. 1000, the value of the reversion would be L. 6694, 13s. 7d.

178. Ex. 3. An annuity for the term of 70 years certain from this time, is to revert to Q and his heirs at the extinction of the survivor of two lives, A and B, now aged 40 and 50 years respectively; the interest of money being 5 per cent., it is required to determine the value of Q’s interest in this annuity.

The algebraical expression of the value is, $$\frac{1 - v^{70}}{r} - \overline\text{AB}$$ (174 and 171).

purchase; and if the annuity be L. 1000, the present value of the reversion will be L. 4276, 13s. 7d.

179. Let the present value of the assurance (77 and 78) of L. 1 on the life of A be denoted by the Old English capital $$\mathfrak{A}$$, and that of an assurance on the joint continuance of any number of lives A, B, C, &c. by $$\mathfrak{ABC},\ \&\text{c.}$$ Also, let the value of an assurance on the joint continuance of any $$m$$ of them, out of the whole number $$m + \mu$$ be denoted by $$\overset{m}\overline{\mathfrak{ABC},}\ \&\text{c.}$$

180. And, in every case, let us designate the annual premium (83) for an assurance, by prefixing the character $$\odot$$ to the symbol for the single premium; so that $$\odot\mathfrak{A}$$ may denote the annual premium for an assurance on the life of A; $$\odot\mathfrak{ABC},\ \&\text{c.}$$ the same for an assurance on the joint continuance of all the lives, A, B, C, &c.; and $$\odot\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}}$$ the annual premium for an assurance on the joint continuance of the last $$m$$ survivors out of the whole number $$m + \mu$$ of those lives.

181. Then will $${}_{t\urcorner}\!\mathfrak{A}$$ and $$\odot {}_{t\urcorner}\!\mathfrak{A}$$, $${}_{t\urcorner}\!\mathfrak{ABC},\ \&\text{c.}$$ and $$\odot {}_{t\urcorner}\!\mathfrak{ABC},\ \&\text{c.}$$, $${}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}}$$ and $$\odot {}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}}$$ designate the single and annual premiums for assurances on the same life or lives for the term of $$t$$ years only.

182. To determine $$\left ( {}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}} \right )$$ the present value of an assurance on the last $$m$$ survivors out of $$m+\mu$$ lives A, B, C, &c. for the term of $$t$$ years only; that is, the present value of L. 1, to be received upon the joint continuance of these last $$m$$ survivors failing in the term.

By reasoning as in No. 79, it will be found, that a perpetuity, the first payment of which is to be made at the end of the year in which the last $$m$$ survivors out of these $$m+\mu$$ lives may fail in the term, will be of the same present value as $$\left ( 1 + \frac{1}{r} = \right ) \frac{1}{1-v}$$ pounds to be received in the same event (112 and 146); but, in No. 173, the value of the reversion of such a perpetuity in that event, was shown to be $$\frac{v}{1-v} \left [ 1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t \right ] - {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}}$$; whence it is manifest, that $${}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}} = v \left [ 1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t \right ] - (1-v) {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}}$$

183. Since the annual premium for this assurance must be paid at the commencement of every year in the term, while the last $$m$$ surviving lives all subsist (83); besides the premium paid down now, one must be paid at the expiration of every year in the term except the last, provided that these last $$m$$ survivors all outlive it; but the present value of L. 1 to be received upon their surviving that last year is $${}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t$$, therefore all the future premiums are now worth $${}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}} - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t$$ years’ purchase, and the present value of all the premiums, or the total present value of the assurance, is $$\odot {}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}} \left [ 1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t + {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}} \right ] =$$ $$v \left [ 1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t \right ] - (1-v) {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}} = 1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t$$ $$- (1-v) \,.\, \left [ 1 - {}_t(\overset{m}\overline{abc,\ \&\text{c.}}) v^t + {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}} \right ]$$, whence we have $$\odot {}_{t\urcorner}\!\overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}} = \frac{1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}} v^t}{1 - {}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) v^t + {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}}} + v - 1$$.

184. ''Corol. 1''. When ($$t$$) the term of the assurance is not less than the greatest possible joint duration of any $$m$$ of the proposed lives, $${}_t\!(\overset{m}\overline{abc,\ \&\text{c.}}) = 0$$, $${}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}} = \overset{m}\overline{\text{ABC},\ \&\text{c.}}$$ and the general formulæ of the