Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica/Annuities/Part 2

PART II.

109. We now proceed to treat the subject of Annuities Algebraically.

I. ON ANNUITIES CERTAIN.

Let	$r$ denote the simple interest of L.1 for one year.
	$s$, any sum put out at interest.
	$n$, the number of years for which it is lent.
	$m$, its amount in that time.
	$a$, an annuity for the same time (3 and 4.)
	${\text{M}}$, the amount to which that annuity will increase, when each payment is laid up as it becomes due, and improved at compound interest until the end of the term.
	${\text{V}}$, the present value of the same annuity (6.)

110. Then, reasoning as in the first number of this article, it will be found that $m=s(1+r)^{n}$. And by No. 2. it appears, that the present value of $s$ pounds to be received certainly at the expiration of $n$ years, is $s{\frac {1}{(1+r)^{n}}}$, or $s(1+r)^{-n}$.

111. The amount of L. 1 in $n$ years being $(1+r)^{n}$, its increase in that time will be $(1+r)^{n}-1$, and when it is considered that this increase arises entirely from the simple interest ($r$) of L. 1 being laid up at the end of each year, and improved at compound interest during the remainder of the term; it must be obvious that $(1+r)^{n}-1$ is the amount of an annuity of $r$ pounds in that time, but $r:a::(1+r)^{n}-1:{\frac {a}{r}}\left[(1+r)^{n}-1\right]$, which, therefore, is equal to ${\text{M}}$, the amount of an annuity of $a$ pounds in $n$ years.

112. Reasoning as in No. 8. it will be found, that since $r:1::a:{\frac {a}{r}}$, the present value of a perpetual annuity of $a$ pounds is ${\frac {a}{r}}$.

113. If two persons, A and B, purchase a perpetuity of $a$ pounds between them, which A and his heirs or assigns are to enjoy during the first $n$ years, and B, and his heirs and assigns, for ever after. Since the value of the perpetuity to be entered upon immediately, has just been shown to be ${\frac {a}{r}}$, the present value of B’s share, that is, the present value of the same perpetuity when the entrance thereon is deferred until the expiration of $n$ years, will be ${\frac {a}{r}}(r+1)^{-n}$, (110); and the value of the share of A will be thus much less than that of the whole perpetuity (21), and therefore equal to ${\frac {a}{r}}\left[1-(1+r)^{-n}\right]={\text{V}}$, the present value of an annuity of $a$ pounds for the term of $n$ years certain.

114. If the annuity is not to be entered upon until the expiration of $d$ years, but is then to continue $n$ years, its value at the time of entering upon it will be ${\frac {a}{r}}\left[1-(1+r)^{-n}\right]$, as has just been shown; therefore its present value will be ${\frac {a}{r}}\left[(1+r)^{-d}-(1+r)^{-(d+n)}\right]={\text{V}}$, (110.)

115. In the same manner, it appears that, when the entrance on a perpetuity of $a$ pounds is deferred $d$ years, its present value will be ${\frac {a}{r}}(1+r)^{-d}$ (110 and 112.)

116. $q$ being any number whatever, whole, fractional, or mixed, let $\lambda q$ denote its logarithm, and $\varkappa q$ the arithmetical complement of that logarithm; so that these equations may obtain, $\lambda {\frac {1}{q}}=-\lambda q=\varkappa q$.

Then, for the resolution of the principal questions of this kind by logarithms, we shall have these formulæ.

1. Amount of a sum improved at interest.

$\lambda m=n\lambda (1+r)+\lambda s$, (110.)

2. Amount of an annuity when each payment is laid up as it becomes due, and improved at interest until the expiration of the term.

$\lambda {\text{M}}=\lambda \left[(1+r)^{n}-1\right]+\lambda a+\varkappa r$, (111.)

3. Value of a lease or an annuity.

$\lambda {\text{V}}=\lambda \left[1-(1+r)^{-n}\right]+\lambda a+\varkappa r$ (113.)

4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease.

$\lambda {\text{V}}=\lambda \left[(1+r)^{-d}-(1+r)^{-(d+n)}\right]+\lambda a+\varkappa r$, (114.)

5. Value of a deferred perpetuity, or the reversion of an estate in fee simple, after an assigned term.

$\lambda {\text{V}}=\lambda a+\varkappa r+d\varkappa (1+r)$, (115).

By means of each of these equations, it is manifest that any one of the quantities involved in it may be found, when the rest are given.

117. If the interest be convertible into principal $\nu$ times in the year, at $\nu$ equal intervals, since the interest of L. 1 for one of these intervals will be ${\frac {r}{\nu }}$, (109), and the number of conversions of interest into principal in $n$ years $\nu n$; to adapt the formula in No. 110. to this case, we have only to substitute ${\frac {r}{\nu }}$ for $r$, and $\nu n$ for $n$, in the equation $m=s(1+r)^{n}$ there given, whereby it will be transformed to this, $m=s\left(1+{\frac {r}{\nu }}\right)^{\nu n}$.

118. According as $\nu$ is equal to 1, 2, 4, or is infinite; that is, according as the interest is convertible into principal yearly, half-yearly, quarterly, or continually, let $m$ be equal to ${\text{Y}}$, ${\text{H}}$, ${\text{Q}}$, or ${\text{C}}$; so shall

	${\text{Y}}=s(1+r)^{n}$,
	${\text{H}}=s\left(1+{\frac {r}{2}}\right)^{2n}$,
	${\text{Q}}=s\left(1+{\frac {r}{4}}\right)^{4n}$,
and	${\text{C}}=s\,.\,{\text{N}}$

${\text{N}}$ being the number whereof $nr$ is the hyperbolic logarithm, and $nr\times 0{\cdot }43429448$ its logarithm in Briggs’ System, and the common tables.

119. From No. 117 and 110, it follows, that the present value of $s$ pounds to be received at the end of $n$ years, when the interest is convertible into principal at $\nu$ equal intervals in each year, is $s\left(1+{\frac {r}{\nu }}\right)^{-\nu n}$.

120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at $\nu$ equal intervals in the year, while the annuity is payable at $\pi$ intervals therein, the amount of each payment being ${\frac {a}{\pi }}$.

121. Case I. $\mu$ being any whole number not greater than $\nu$, let ${\frac {1}{\pi }}={\frac {\mu }{\nu }}$, so that the interest may be convertible into principal $\mu$ times in each of the intervals between the payments of the annuity.

Then will the amount of L. 1, at the expiration of the period ${\frac {1}{\pi }}$ be $\left(1+{\frac {r}{\nu }}\right)^{\mu }$ (117), and the interest of L. 1 for the same time will be $\left(1+{\frac {r}{\nu }}\right)^{\mu }-1$; whence the present value of the perpetuity will be ${\frac {{\frac {1}{\pi }}a}{\left(1+{\frac {r}{\nu }}\right)^{\mu }-1}}$ (8), and the value of the same deferred $n$ years, will be ${\frac {a}{\pi }}\cdot {\frac {\left(1+{\frac {r}{\nu }}\right)^{-\nu n}}{\left(1+{\frac {r}{\nu }}\right)^{\mu }-1}}$ (119), therefore the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be ${\frac {a}{\pi }}\cdot {\frac {1-\left(1+{\frac {r}{\nu }}\right)^{-\nu n}}{\left(1+{\frac {r}{\nu }}\right)^{\mu }-1}}={\text{V}}$.

122. Case 2. $\mu$ being any whole number greater than $\pi$, let ${\frac {1}{\nu }}={\frac {\mu }{\pi }}$, so that the annuity may be payable $\mu$ times in each of the intervals between the payments of interest, or the conversion thereof into principal.

Then, at the expiration of the ${\frac {1}{\nu }}$th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the $\mu -1$ payments of the annuity previously made, will be ${\frac {ar}{\pi \pi }}\left[(\mu -1)+(\mu -2)+(\mu -3)+\cdots \cdots \cdot \cdot +3+2+1\right]={\frac {a}{\pi }}\cdot {\frac {r\mu (\mu -1)}{2\pi }}$; to which, adding the $\mu$ payments of ${\frac {a}{\pi }}$ each (including the one only then due), the sum, ${\frac {a}{\pi }}\left[\mu +{\frac {r\mu (\mu -1)}{2\pi }}\right]$, is the simple interest which the value of the perpetuity should yield at the expiration of each $\nu$th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since ${\frac {r}{\nu }}:{\frac {a}{\pi }}\left[\mu +{\frac {r\mu (\mu -1)}{2\pi }}\right]::1:{\frac {a\nu }{r\pi }}\left[\mu +{\frac {r\mu (\mu -1)}{2\pi }}\right]=a\left({\frac {1}{r}}+{\frac {\mu -1}{2\pi }}\right)$, this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred $n$ years, will therefore be $a\left({\frac {1}{r}}+{\frac {\mu -1}{2\pi }}\right)\times \left(1+{\frac {r}{\nu }}\right)^{-\nu n}$ (119). Whence it appears, that the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be $a\left({\frac {1}{r}}+{\frac {\mu -1}{2\pi }}\right)\cdot \left[1-(1+{\frac {r}{\nu }})^{-\nu n}\right]={\text{V}}$.

123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, $\nu =\pi$.

Since the interest of L. 1 at the expiration of the period ${\frac {1}{\pi }}$ will be ${\frac {r}{\pi }}$, the value of the perpetuity will be ${\frac {{\frac {1}{\pi }}a}{{\frac {1}{\pi }}r}}={\frac {a}{r}}$ (8), whence, proceeding as before, we obtain the present value of the annuity, ${\frac {a}{r}}\left[1-(1+{\frac {r}{\nu }})^{\nu n}\right]={\text{V}}$. When $\nu =\pi$, and consequently $\mu =1$, the values of ${\text{V}}$, given in the two preceding cases, will be found to coincide with this.

124. According as $\nu$ and $\pi$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let ${\text{V}}$ be equal to $y$, $h$, $q$, or $c$, then will

	$y={\frac {a}{r}}\left[1-(1+r)^{-n}\right]$,
	$h={\frac {a}{r}}\left[1-(1+{\frac {r}{2}})^{-2n}\right]$,
	$q={\frac {a}{r}}\left[1-(1+{\frac {r}{4}})^{-4n}\right]$,
and	$c={\frac {a}{r}}\left[1-{\frac {1}{\text{N}}}\right]$, ${\text{N}}$ being as in No. 118.

125. The amount of an annuity is equal to the sum to which the purchase money would amount, if it were put out and improved at interest during the whole term.

For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the grantor, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase money must evidently receive the same improvement during the term, as if it had been laid up at interest at its commencement.

126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in $n$ years, the number of periods of conversion will be $\nu n$, in the

1st Case, Where the interest is convertible $\mu$ times in each of the intervals between the payments of the annuity, we have $\left(1+{\frac {r}{\nu }}\right)^{\nu n}{\text{V}}={\frac {a}{\pi }}\cdot {\frac {\left(1+{\frac {r}{\nu }}\right)^{\nu n}-1}{\left(1+{\frac {r}{\nu }}\right)^{\mu }-1}}={\text{M}}$, (117, 121, and 125). In the 2d Case, when the annuity is payable $\mu$ times, in each interval between the conversions of interest, $\left(1+{\frac {r}{\nu }}\right)^{\nu n}{\text{V}}=a\left[{\frac {1}{r}}+{\frac {\mu -1}{2\pi }}\right]\cdot \left[(1+{\frac {r}{\nu }})^{\nu n}-1\right]={\text{M}}$, (117, 122, and 125).

And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible, $\left(1+{\frac {r}{\nu }}\right)^{\nu n}{\text{V}}={\frac {a}{r}}\left[(1+{\frac {r}{\nu }})^{\nu n}-1\right]={\text{M}}$, (117, 123, and 125).

127. According as $\nu$ and $\pi$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let ${\text{M}}$ be denoted by $y'$, $h'$, $q'$, or $c'$;

then will		$y'={\frac {a}{r}}\left[(1+r)^{n}-1\right]$,
		$h'={\frac {a}{r}}\left[(1+{\frac {r}{2}})^{2n}-1\right]$,
		$q'={\frac {a}{r}}\left[(1+{\frac {r}{4}})^{4n}-1\right]$,
	and	$c'={\frac {a}{r}}({\text{N}}-1)$; ${\text{N}}$ being as in No. 118.

128. Example 1. What will L. 320 amount to, when improved at compound interest during 40 years; the rate of interest being 4 per cent. per annum?

By the first formula in No. 116, the operation will be as follows:

$1+r=1{\cdot }04\;\lambda \;0{\cdot }01703334$

$\times n=40$

---

$(1+r)^{n}\;\lambda \;0{\cdot }6813336$

$s=320\;\lambda \;2{\cdot }5051500$

---

$m=1536{\cdot }327\;\lambda \;3{\cdot }1864836$

And the answer is L. 1536, 6s. 6½d.

129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:

$1+{\frac {r}{2}}=1{\cdot }02\;\lambda \;0{\cdot }00860017$

$\times \nu n=80$

---

$(1+r)^{n}\;\lambda \;0{\cdot }6880136$

$s=320\;\lambda \;2{\cdot }5051500$

---

$m=1560{\cdot }14\;\lambda \;3{\cdot }1931636$

So that in this case the amount would be L. 1560, 2s. 9½d.

130. Ex. 3. Required the present value of an annuity of L. 250 for 30 years, reckoning interest at 5 per cent.

By the third formula in No. 116, the operation will be thus:

$\lambda \;(1+r)^{-1}=\varkappa \;1{\cdot }05={\overline {1}}{\cdot }9788107$

$\times n=30$

---

$(1+r)^{-n}={\cdot }23137704\;\lambda \;{\overline {1}}{\cdot }3643210$

---

$1-(1+r)^{-n}={\cdot }76862296\;\lambda \;{\overline {1}}{\cdot }8857133$

$a=250\;\lambda \;2{\cdot }3979400$

$r={\cdot }05\;\varkappa \;1{\cdot }3010300$

---

${\text{V}}=3843{\cdot }114\;\lambda \;3{\cdot }5846833$

And the required value is L. 3843, 2s, 3¼d.

131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122, $\nu$ will be equal to 1, $\pi =2$, and $\mu =2$; that formula will therefore become $a\left({\frac {1}{r}}+{\frac {1}{4}}\right)\cdot \left[1-(1+r)^{-n}\right]={\text{V}}$; and the operation will be thus:

$1-(1+r)^{-n}\;\lambda \;{\overline {1}}{\cdot }8857133$ No. 130. $a=250\;\lambda \;2{\cdot }3979400$ ${\frac {1}{r}}+{\frac {1}{4}}=20{\cdot }25\;\lambda \;1{\cdot }3064250$

---

${\text{V}}=3891{\cdot }15\;\lambda \;3{\cdot }5900783$

The value of the annuity will, therefore, in this case, be L. 3891, 3s.

132. Ex. 5. To what sum will an annuity of L. 120 for 20 years amount, when each payment is improved at compound interest, from the time of its becoming due until the expiration of the term; the rate of interest being 6 per cent.?

 

The operation by the second formula in No. 116 is thus:

$1+r=1{\cdot }06\;\lambda \;0{\cdot }025305865$

$\times n=20$

---

$(1+r)^{n}=3{\cdot }207135\;\lambda \;0{\cdot }5061173$

---

$(1+r)^{n}-1=2{\cdot }207135\;\lambda \;0{\cdot }3438289$

$a=120\;\lambda \;2{\cdot }0791812$

$r={\cdot }06\;\varkappa \;1{\cdot }2218487$

---

${\text{M}}=4414{\cdot }27\;\lambda \;3{\cdot }6448588$

And the amount required is L. 4414, 5s, 5d.

133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulæ given in No. 127; which, in this case, will become ${\frac {120}{0{\cdot }6}}\left[(1{\cdot }03)^{40}-1\right]$, and the operation will be as follows:

$1{\cdot }03\;\lambda \;0{\cdot }01283723$

$\times 40$

---

$(1{\cdot }03)^{40}=3{\cdot }26204\;\lambda \;0{\cdot }5134892$

---

$(1{\cdot }03)^{40}-1=2{\cdot }26204\;\lambda \;0{\cdot }3545003$

$120\;\lambda \;2{\cdot }0791812$

${\cdot }06\;\varkappa \;1{\cdot }2218487$

---

${\text{M}}=4524{\cdot }08\;\lambda \;3{\cdot }6555302$

So that the amount in this case, would be L. 4524, 1s, 7¼d.

II. ON THE PROBABILITIES OF LIFE.

134. Any persons A, B, C, &c. being proposed, let the numbers which tables of mortality (32) adapted to them, represent to attain to their respective ages, be denoted by the symbols $a$, $b$, $c$, &c.; while lives $n$ years older than those respectively, are denoted thus: ${}^{n}\!A$, ${}^{n}\!B$, ${}^{n}\!C$, &c. and the numbers that attain to their ages, by the symbols ${}^{n}\!a$, ${}^{n}\!b$, ${}^{n}\!c$, &c.; also let lives $n$ years younger than A, B, C, &c. he denoted thus: $A_{n}$, $B_{n}$, $C_{n}$, &c., while the numbers which the tables show to attain to those younger ages, are designated by the symbols $a_{n}$, $b_{n}$, $c_{n}$, &c.

Then, if A be 21 years of age, and we use the Carlisle Table, we shall have $a=6047$, and ${}^{14}\!a=5362$, the number that attain to the age of thirty-five, or that live to be fourteen years older than A.

Hence the number that are represented by the table to die in $n$ years from the age of A, will be $a-{}^{n}\!a$, that is in 14 years, $a-{}^{14}\!a$; and by the Carlisle Table, in 14 years from the age of 21, that is, between 21 and 35, it will be 6047 − 5362 = 685.

135. Problem. To determine the probability that a proposed life A, will survive $n$ years.

Solution. $a$ being the number of lives in the table of mortality, that attain to the age of that which is proposed, conceive that number of lives to be so selected, (of which A must be one,) that they may each have the same prospect with regard to longevity, as the proposed life and those in the table, or the average of those from which it was constructed; then will the number of them that survive the term be ${}^{n}\!a$ (134).

So that the number of ways all equally probable, or the number of equal chances for the happening of the event in question is ${}^{n}\!a$; and the whole number for its either happening or failing is $a$; therefore, according to the first principles of the doctrine of probabilities, the probability of the event happening, that is, of A surviving the term, is ${\frac {{}^{n}\!a}{a}}$.

If the age of A be 14, the probability of that life surviving 7 years, or the age of 21, will, according to the Carlisle Table, be ${\frac {{}^{7}\!a}{a}}={\frac {6047}{6335}}$, or 0·95454.

136. Since the number that die in $n$ years from the age of A is $a-{}^{n}\!a$ (134), it appears, in the same manner, that the probability of that life failing in $n$ years will be ${\frac {a-{}^{n}\!a}{a}}=1-{\frac {{}^{n}\!a}{a}}$ which probability, when the life, term, and table of mortality, are the same as in the last No. will be 0·04546.

137. If two lives A and B be proposed, since the probability of A surviving $n$ years will be ${\frac {{}^{n}\!a}{a}}$, and that of B surviving the same term will be ${\frac {{}^{n}\!b}{b}}$; it appears from the doctrine of probabilities that ${\frac {{}^{n}\!a}{a}}\cdot {\frac {{}^{n}\!b}{b}}$ or ${\frac {{}^{n}\!(ab)}{ab}}$ will be the measure of the probability that these lives will both survive $n$ years.

In the same manner it may be shown, that the probability of the three lives A, B, and C all surviving $n$ years, will be measured by ${\frac {{}^{n}\!a}{a}}\cdot {\frac {{}^{n}\!b}{b}}\cdot {\frac {{}^{n}\!c}{c}}$ or ${\frac {{}^{n}\!(abc)}{abc}}$. And, universally, that any number of lives A, B, C, &c. will jointly survive $n$ years, the probability is ${\frac {{}^{n}\!(abc,\&{\text{c.}})}{abc,\&{\text{c.}}}}$

138. Let ${\frac {{}^{n}\!a}{a}}={}_{n}\!a$, ${\frac {{}^{n}\!b}{b}}={}_{n}\!b$, ${\frac {{}^{n}\!c}{c}}={}_{n}\!c$; also let ${\frac {{}^{n}\!(abc,\&{\text{c.}})}{abc,\&{\text{c.}}}}={}_{n}\!(abc,\&{\text{c.}})$; so that the probabilities of A, B, C, &c. surviving $n$ years may be denoted by ${}_{n}\!a$, ${}_{n}\!b$, ${}_{n}\!c$, &c. respectively; and that of all those lives jointly surviving that term, by ${}_{n}\!(abc,\&{\text{c.}})$

Then will the probability that none of those lives will survive $n$ years, be $(1-{}_{n}\!a)\,.\,(1-{}_{n}\!b)\,.\,(1-{}_{n}\!c)\,.\,\&{\text{c.}}$

139. But the probability that some one or more of these lives will survive $n$ years, will be just what the probability last mentioned is deficient of certainty; its measure therefore, being just what the measure of that probability is deficient of unity, will be

$1-(1-{}_{n}\!a)\,.\,(1-{}_{n}\!b)\,.\,(1-{}_{n}\!c)\,.\,\&{\text{c.}}$

140. Corol. 1. When there is only one life A, this will be ${}_{n}\!{a}$.

141. Corol. 2. When there are two lives A and B, it becomes ${}_{n}\!{a}+{}_{n}\!{b}-{}_{n}\!(ab)$.

142. Corol. 3. When there are three lives A, B, and C, it becomes ${}_{n}\!a+{}_{n}\!b+{}_{n}\!c-{}_{n}\!(ab)-{}_{n}\!(ac)-{}_{n}\!(bc)+{}_{n}\!(abc)$.

143. When three lives A, B, and C are proposed, that at the expiration of $n$ years there will be

living	dead	the probability is
ABC	none	$.\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!.+{}_{n}\!(abc)$
AB	C	${}_{n}\!(ab)\,.\,(1-{}_{n}\!c)={}_{n}\!(ab)-{}_{n}\!(abc)$
AC	B	${}_{n}\!(ac)\,.\,(1-{}_{n}\!b)={}_{n}\!(ac)-{}_{n}\!(abc)$
BC	A	${}_{n}\!(bc)\,.\,(1-{}_{n}\!a)={}_{n}\!(bc)-{}_{n}\!(abc)$

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.

III. OF ANNUITIES ON LIVES.

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:

Prob. i.

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.

Solution.

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,

${\begin{aligned}\lambda \;{\text{A}}&=\lambda \;v+\lambda \;{}_{\prime }a+\lambda \;(1+{}'\!{\text{A}}),\\\lambda \;{\text{AB}}&=\lambda \;v+\lambda \;{}_{\prime }a+\lambda \;{}_{\prime }b+\lambda \;[1+{}'\!({\text{AB}})],\\\lambda \;{\text{ABC}}&=\lambda \;v+\lambda \;{}_{\prime }a+\lambda \;{}_{\prime }b+\lambda \;{}_{\prime }c+\lambda \;[1+{}'\!({\text{ABC}})],\end{aligned}}$
&c. &c. &c.

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, ${}_{\neg t}\!{\text{A}}={}_{t}\!av^{t}\,.\,{}^{t}\!{\text{A}}$.

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, ${}_{t\urcorner }\!{\text{A}}={\text{A}}-{}_{\neg t}\!{\text{A}}$.

Prob. ii.

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

Solution.

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.

Prob. iii.

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, ${\overline {\text{ABC}}}$.

Solution.

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.

Prob. iv.

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, ${\overset {2}{\overline {\text{ABC}}}}$:

Solution.

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.

Prob. v.

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.

Solution.

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.

Prob. vi.

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.

Solution.

That at the end of the $n$th year there will be		the probability, multiplied by C’s proportion of the annuity in that circumstance, is
dead	living
A	BC	${\tfrac {1}{2}}(1-{}_{n}\!a)\,.\,{}_{n}\!(bc)={\tfrac {1}{2}}[{}_{n}\!(bc)-{}_{n}\!(abc)]$
B	AC	${\tfrac {1}{2}}(1-{}_{n}\!b)\,.\,{}_{n}\!(ac)={\tfrac {1}{2}}[{}_{n}\!(ac)-{}_{n}\!(abc)]$

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 ${\tfrac {1}{2}}{\text{AC}}+{\tfrac {1}{2}}{\text{BC}}-{\text{ABC}}$.

Prob. vii.

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.

Solution.

That at the end of the $n$th year there will be		The probability, multiplied by B’s proportion of the annuity in that circumstance, is
dead	living
A	BC	$..\!.\!..\!.\!..\!.\!.+{\tfrac {1}{2}}{}_{n}\!(bc)-{\tfrac {1}{2}}{}_{n}\!(abc)$
AC	B	${}_{n}\!b-{}_{n}\!(ab)-{}_{n}\!(bc)+{}_{n}\!(abc)$; and the

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 ${\text{B}}-{\text{AB}}-{\tfrac {1}{2}}{\text{BC}}+{\tfrac {1}{2}}{\text{ABC}}$.

Prob. viii.

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?

Solution.

That at the end of $n$ years there will be		The probability, multiplied by A’s proportion of the annuity in that circumstance, is
dead	living
none	ABC	$.\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!..\!.\!.+{\tfrac {1}{3}}{}_{n}\!(abc)$
C	AB	$.\!.\!.+{\tfrac {1}{2}}{}_{n}\!(ab).\!.\!..\!.\!...-{\tfrac {1}{2}}{}_{n}\!(abc)$
B	AC	$.\!.\!..\!.\!..\!.\!..\!.\!.+{\tfrac {1}{2}}{}_{n}\!(ac)-{\tfrac {1}{2}}{}_{n}\!(abc)$
BC	A	${}_{n}\!a-{}_{n}\!(ab)-{}_{n}\!(ac)+{}_{n}\!(abc)$; and

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 ${\text{A}}-{\tfrac {1}{2}}{\text{AB}}-{\tfrac {1}{2}}{\text{AC}}+{\tfrac {1}{3}}{\text{ABC}}$.

Prob. ix.

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.

Solution.

That at the end of $n$ years there will be		The probability is
dead	living
AB	C	${}_{n}\!c-{}_{n}\!(ac)-{}_{n}\!(bc)+{}_{n}\!(abc)$
AC	B	${}_{n}\!b-{}_{n}\!(ab)-{}_{n}\!(bc)+{}_{n}\!(abc)$
BC	A	${}_{n}\!a-{}_{n}\!(ab)-{}_{n}\!(ac)+{}_{n}\!(abc)$; and the

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

${\text{A}}+{\text{B}}+{\text{C}}-2{\text{AB}}-2{\text{AC}}-2{\text{BC}}+3{\text{ABC}}$.

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 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.)

Prob. X.

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.

Solution.

Q’s expectation may be distinguished into two parts:

1st, That of enjoying the annuity during the term of 7 years.

2d, That of enjoying it after the expiration of that term.

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 ${\frac {1}{r}}-{\overline {\text{ABC.}}}$

But ${\frac {1}{r}}={\frac {1}{0{\cdot }5}}$	$=20{\cdot }000$
And ${\overline {\text{ABC}}}$,	$=14{\cdot }001$	(68.)
Therefore	$5{\cdot }999$	years’ purchase is

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.?

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

But $\lambda \,v=\varkappa \,1{\cdot }05$	$=$	${\overline {1}}{\cdot }9788107$
		$\times 70$
$v^{70}={\cdot }032866$	$\lambda$	${\overline {2}}{\cdot }5167490$
$1-v^{70}={\cdot }967134$
${\frac {1-v^{70}}{r}}={\frac {{\cdot }967134}{{\cdot }05}}$	$=$	$19{\cdot }34268$
Subtract ${\text{A}}$	$=$	$12{\cdot }64800$	(Tab. VI.)
remains		$6{\cdot }69468$	years’ purchase;

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).

But by the last example ${\frac {1-v^{70}}{r}}=$	$19{\cdot }34268$
and by No. 66. ${\overline {\text{AB}}}=$	$15{\cdot }06600$
So that the required value is	$4{\cdot }27668$	years’

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

IV. OF ASSURANCES ON LIVES.

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.

Prob. xi.

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.

Solution.

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 two preceding numbers become respectively

${\begin{aligned}{\overset {m}{\overline {{\mathfrak {ABC}},\ \&{\text{c.}}}}}&=v-(1-v){\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}\\{\text{and }}\odot {\overset {m}{\overline {{\mathfrak {ABC}},\ \&{\text{c.}}}}}&={\frac {1}{1+{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}}}+v-1\end{aligned}}$

185. Corol. 2. In the same manner it appears, that,

for a single life, ${\mathfrak {A}}$	$=v-(1-v){\text{A}}$,
and $\odot {\mathfrak {A}}$	$={\frac {1}{1+{\text{A}}}}+v-1$

186. Corol. 3. Also that ${}_{t\urcorner }\!{\mathfrak {A}}=v\left(1-{}_{t}\!av^{t}\right)-(1-v){}_{t\urcorner }\!{\text{A}}$ or ${}_{t\urcorner }\!{\mathfrak {A}}=v\left(1-{\frac {{}^{t}\!a}{a}}v^{t}\right)-(1-v)\,.\,\left({\text{A}}-{\frac {{}^{t}\!a}{a}}v^{t}\cdot {}^{t}\!{\text{A}}\right)$.

And $\odot {}_{t\urcorner }\!{\mathfrak {A}}$	$={\frac {1-{}_{t}\!av^{t}}{1-{}_{t}\!av^{t}+{}_{t\urcorner }\!{\text{A}}}}+v-1$
that is, $\odot {}_{t\urcorner }\!{\mathfrak {A}}$	$={\frac {1-{\frac {{}^{t}\!a}{a}}v^{t}}{1+{\text{A}}-{\frac {{}^{t}\!a}{a}}v^{t}\cdot (1+{}^{t}\!{\text{A}})}}+v-1$.

187. Corol. 4. When the assurance is on the joint continuance of all the lives, the formulæ of No. 184 become respectively

${\mathfrak {ABC}},\ \&{\text{c.}}=v-(1-v){\text{ABC}},\ \&{\text{c.}}$

and $\odot {\mathfrak {ABC}},\ \&{\text{c.}}={\frac {1}{1+{\text{ABC}},\ \&{\text{c.}}}}+v-1$.

And those of numbers 182 and 183, ${}_{t\urcorner }\!{\mathfrak {ABC}},\ \&{\text{c.}}=v\left(1-{\frac {{}^{t}\!(abc,\ \&{\text{c.}})}{abc,\ \&{\text{c.}}}}v^{t}\right)-(1-v)\times$ $\left[{\text{ABC}},\ \&{\text{c.}}-{\frac {t(abc,\ \&{\text{c.}})}{abc,\ \&{\text{c.}}}}v^{t}\cdot {}^{t}\!({\text{ABC}},\ \&{\text{c.}})\right]$, and $\odot {}_{t\urcorner }\!{\mathfrak {ABC}},\ \&{\text{c.}}=$ ${\frac {1-{\frac {{}^{t}\!(abc,\ \&{\text{c.}})}{abc,\ \&{\text{c.}}}}v^{t}}{1+{\text{ABC}},\ \&{\text{c.}}-{\frac {{}^{t}\!(abc,\ \&{\text{c.}})}{abc,\ \&{\text{c.}}}}v^{t}\left[1+{}^{t}\!({\text{ABC}},\ \&{\text{c.}})\right]}}+v-1$.

188. Corol. 5. According as the assurance is in the last survivor of two, or of three lives, the formulæ of No. 184 become respectively

${\overline {\mathfrak {AB}}}=v-(1-v){\overline {\text{AB}}}$,

and $\odot {\overline {\mathfrak {AB}}}={\frac {1}{1+{\overline {\text{AB}}}}}+v-1$;

or ${\overline {\mathfrak {ABC}}}=v-(1-v){\overline {\text{ABC}}}$,

and $\odot {\mathfrak {ABC}}={\frac {1}{1+{\overline {\text{ABC}}}}}+v-1$.

And those of numbers 182 and 183 become

${}_{t\urcorner }{\overline {\mathfrak {AB}}}=v\left[1-{}_{t}\!({\overline {ab}})v^{t}\right]-(1-v){}_{t\urcorner }{\overline {\text{AB}}}$,

and $\odot {}_{t\urcorner }{\overline {\mathfrak {AB}}}={\frac {1-{}_{t}({\overline {ab}})v^{t}}{1-{}_{t}\!({\overline {ab}})v^{t}+{}_{t\urcorner }\!{\overline {\text{AB}}}}}+v-1$;

or ${}_{t\urcorner }\!{\overline {\mathfrak {ABC}}}=v\left[1-{}_{t}\!({\overline {abc}})v^{t}\right]-(1-v){}_{t\urcorner }\!{\overline {\text{ABC,}}}$

and $\odot {}_{t\urcorner }\!{\overline {\mathfrak {ABC}}}={\frac {1-{}_{t}\!({\overline {abc}})v^{t}}{1-{}_{t}\!(abc)v^{t}+{}_{t\urcorner }\!{\overline {\text{ABC}}}}}+v-1$ respectively.

Where ${}_{t}\!({\overline {ab}})={}_{t}\!a+{}_{t}\!b-{}_{t}\!(ab)$, (141). and ${}_{t}\!({\overline {abc}})={}_{t}\!a+{}_{t}\!b+{}_{t}\!c-\left[{}_{t}\!(ab)+{}_{t}\!(ac)+(bc)\right]+{}_{t}\!(abc)$, (142).

For the values of ${\overline {\text{AB}}}$, ${\overline {\text{ABC}}}$, ${}_{t\urcorner }\!{\overline {\text{AB}}}$, and ${}_{t\urcorner }\!{\overline {\text{ABC}}}$, see numbers 157—159, and 161.

189. Corol. 6. When the assurance is on the joint continuance of the two last survivors out of the three lives A, B, C; the formulæ of No. 184 become respectively

${\begin{aligned}{\overset {2}{\overline {\mathfrak {ABC}}}}&=v-(1-v){\overset {2}{\overline {\text{ABC,}}}}\\{\text{and }}\odot {\overset {2}{\overline {\mathfrak {ABC}}}}&={\frac {1}{1+{\overset {2}{\overline {\text{ABC}}}}}}+v-1{\text{.}}\end{aligned}}$

Those of numbers 182 and 183,

${}_{t\urcorner }\!{\overset {2}{\overline {\mathfrak {ABC}}}}=v\left[1-{}_{t}\!{\overset {2}{\overline {(abc)}}}v^{t}\right]-(1-v){}_{t\urcorner }\!{\overset {2}{\overline {\text{ABC,}}}}$ and $\odot {}_{t\urcorner }\!{\overset {2}{\overline {\mathfrak {ABC}}}}={\frac {1-{}_{t}{\overset {2}{\overline {(abc)}}}v^{t}}{1-{}_{t}({\overset {2}{\overline {abc}}})v^{t}+{}_{t\urcorner }\!{\overset {2}{\overline {\text{ABC}}}}}}+v-1$.

Where ${}_{t}\!{\overset {2}{\left({\overline {abc}}\right)}}={}_{t}\!(ab)+{}_{t}\!(ac)+{}_{t}\!(bc)-2{}_{t}\!(abc)$, (143).

For the values of ${\overset {2}{\overline {\text{ABC}}}}$ and ${}_{t\urcorner }\!{\overset {2}{\overline {\text{ABC}}}}$ see numbers 157, 160, and 161.

190. $v\left[1-{}_{t}\!{\overset {m}{\overline {(abc,\ \&{\text{c.}})}}}v^{t}\right]-(1-v){}_{t\urcorner }\!{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}$ the value of an assurance on any life or lives for the term of $t$ years, which was given in No. 182, may also be expressed thus:

$\left[1+{}_{t\urcorner }\!{\overset {m}{\overline {({\text{ABC}},\ \&{\text{c.}})}}}-{}_{t}\!{\overset {m}{\overline {(abc,\ \&{\text{c.}})}}}v^{t}\right]v-{}_{t\urcorner }\!{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}$

And this, in words at length, is the rule given in No. 93.

191. When $t$ is not less than the greatest possible joint duration of any $m$ of the proposed lives, the last expression becomes $\left(1+{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}\right)v-{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}$ which is also equivalent to the first in No. 184; and, in words at length, is the rule given in No. 97, for determining the value of an assurance on any life or lives for their whole duration.

192. By substituting ${\frac {1}{1+r}}$ for $v$ (146) in the last expression, it becomes ${\frac {1+{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}}{1+r}}-{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}={\frac {1-r.{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}}{1+r}}$, or ${\frac {{\frac {1}{r}}-{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c}}}}}}{1+{\frac {1}{r}}}}$. And ${\frac {{\frac {1}{r}}-{\overset {m}{\overline {{\text{ABC}},\ \&{\text{c.}}}}}}{1+{\frac {1}{r}}}}={\overset {m}{\overline {{\mathfrak {ABC}},\ \&{\text{c.}}}}}$ is the proposition enunciated in No. 81; ${\frac {1}{r}}$ being the value of the perpetuity (112).

193. Examples of the determination of the single premiums for assurances, and of the derivation of the annual premiums from them, have been given in numbers 82—88, also in 95 and 96; but by the algebraical formulæ given here, the annual premiums may be determined directly, without first finding the total present values of the assurances.

194. Example 1. Required the annual premium for an assurance on the life A now 50 years of age, interest 5 per cent.

According to No. 185, the operation is thus,

$1+A=$ $12{\cdot }660$ $\;\lambda$ ${\overline {2}}{\cdot }8975663$ ${\frac {1}{1+A}}=$ $\;{\cdot }0789890$ $\;\varkappa$ adding $v=$ $\;{\cdot }9523809$ , and subtracting unity, we have $\odot {\mathfrak {A}}=$ $\;{\cdot }0313699$ , agreeably to No. 85.

195. Ex. 2. What should the annual premium be for an assurance on the last survivor of three lives A, B, C, now aged 50, 55, and 60 years respectively, rate of interest 5 per cent.?

Operation by No. 188.

(68) $1+{\overline {\text{ABC}}}=$ $\;15{\cdot }001$ $\;\lambda$ ${\overline {2}}{\cdot }8238798$ ${\frac {1}{1+{\overline {\text{ABC}}}}}=$ $\;{\cdot }0666622$ $\;\varkappa$ $v=$ $\;{\cdot }9523809$ $\odot {\overline {\mathfrak {ABC}}}=$ $\;{\cdot }0190431$ , agreeably to No. 88.

196. Ex. 3. Required the annual premium for an assurance for 10 years only, on a life now 45 years of age, interest 5 per cent.

Operation according to No. 186.

$v^{10}$	$=$	${\cdot }613913$	$\lambda \;{\overline {1}}{\cdot }7881068$
${}^{10}\!a=4073$			$\lambda \;3{\cdot }6099144$
$a=4727$			$\varkappa \;{\overline {4}}{\cdot }3254144$
${}_{10}\!av^{10}$	$=$	${\cdot }528976$	$\lambda \;{\overline {1}}{\cdot }7234356$
$1+{}^{10}\!A$	$=$	$11{\cdot }347$	$\lambda \;1{\cdot }0548811$
Subtract		$6{\cdot }002$	$\lambda \;0{\cdot }7783167$
from $1+A$	$=$	$13{\cdot }648$
remains		$7{\cdot }646$	$\varkappa \;{\overline {1}}{\cdot }1165657$
$1-{}_{10}\!av^{10}$	$=$	${\cdot }471024$	$\lambda \;{\overline {1}}{\cdot }6730430$
		${\cdot }061604$	$\lambda \;{\overline {2}}{\cdot }7896087$
$v$	$=$	${\cdot }952381$
$\odot {}_{10\urcorner }\!{\mathfrak {A}}$	$=$	${\cdot }013985$,	agreeably to No. 96.

What has been advanced from numbers 99 to 109, needs no algebraical illustration. (U.)