109. We now proceed to treat the subject of
Annuities Algebraically.
I. ON ANNUITIES CERTAIN.
Let
denote the simple interest of L.1 for one year.
, any sum put out at interest.
, the number of years for which it is lent.
, its amount in that time.
, an annuity for the same time (3 and 4.)
, 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.
, the present value of the same annuity (6.)
110. Then, reasoning as in the first number of
this article, it will be found that . And
by No. 2. it appears, that the present value of
pounds to be received certainly at the expiration of
years, is , or .
111. The amount of L. 1 in years being ,
its increase in that time will be , and when
it is considered that this increase arises entirely from
the simple interest () 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 is the amount of an annuity
of pounds in that time, but ,
which, therefore, is equal to , the
amount of an annuity of pounds in years.
112. Reasoning as in No. 8. it will be found, that
since , the present value of a perpetual
annuity of pounds is .
113. If two persons, A and B, purchase a perpetuity
of pounds between them, which A and his
heirs or assigns are to enjoy during the first 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 , 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 years, will be ,
(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 , the present
value of an annuity of pounds for the term of
years certain.
114. If the annuity is not to be entered upon until
the expiration of years, but is then to continue
years, its value at the time of entering upon it will
be , as has just been shown; therefore
its present value will be
, (110.)
115. In the same manner, it appears that, when
the entrance on a perpetuity of pounds is deferred
years, its present value will be (110 and 112.)
116. being any number whatever, whole, fractional,
or mixed, let denote its logarithm, and
the arithmetical complement of that logarithm;
so that these equations may obtain, .
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.
, (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.
, (111.)
3. Value of a lease or an annuity.
(113.)
4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease.
, (114.)
5. Value of a deferred perpetuity, or the reversion of an estate in fee simple, after an assigned term.
, (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
times in the year, at equal intervals, since the interest
of L. 1 for one of these intervals will be , (109),
and the number of conversions of interest into principal
in years ; to adapt the formula in No. 110.
to this case, we have only to substitute for , and
for , in the equation there given,
whereby it will be transformed to this, .
118. According as 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 be equal to , , , or ; so shall
,
,
,
and
being the number whereof is the hyperbolic
logarithm, and its logarithm in
Briggs’ System, and the common tables.
119. From No. 117 and 110, it follows, that the
present value of pounds to be received at the end of
years, when the interest is convertible into principal
at equal intervals in each year, is .
120. When the present values and the amounts of
annuities are desired, let the interest be convertible
into principal at equal intervals in the year,
while the annuity is payable at intervals therein,
the amount of each payment being .
121. Case I. being any whole number not greater
than , let , so that the interest may be
convertible into principal 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 be (117), and the interest of
L. 1 for the same time will be ;
whence the present value of the perpetuity will be
(8), and the value of the same deferred
years, will be (119), therefore
the present value of the annuity to be entered
upon immediately, and to continue years, will be
.
122. Case 2. being any whole number greater
than , let , so that the annuity may be payable
times in each of the intervals between the
payments of interest, or the conversion thereof into
principal.
Then, at the expiration of the th of a year, when
the interest on the purchase-money is first payable
or convertible, the interest on all the payments
of the annuity previously made, will be
; to which, adding the payments of each (including the one only then due), the
sum, , is the simple interest which
the value of the perpetuity should yield at the expiration
of each 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 ,
this last expression
will be the value of such perpetuity with
immediate possession (8); the value of the same
deferred years, will therefore be
(119). Whence it appears, that the
present value of the annuity to be entered upon immediately,
and to continue years, will be
.
123. Case 3. When, in consequence of the annuity
being always payable at the same time that the interest
is convertible, .
Since the interest of L. 1 at the expiration of the
period will be , the value of the perpetuity will
be (8), whence, proceeding as before, we
obtain the present value of the annuity,
. When , and consequently
, the values of , given in the two preceding
cases, will be found to coincide with this.
124. According as and 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 be equal to , , ,
or , then will
,
,
,
and
, 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 years, the
number of periods of conversion will be , in the
1st Case, Where the interest is convertible times
in each of the intervals between the payments of the
annuity, we have ,
(117, 121, and 125). In the 2d Case, when
the annuity is payable times, in each interval between
the conversions of interest,
,
(117, 122, and 125).
And, in the 3d Case, when the annuity is always
payable at the same time that the interest is convertible,
, (117,
123, and 125).
127. According as and 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 be denoted by , ,
, or ;
then will
,
,
,
and
; 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:
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:
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:
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, will be equal to 1,
, and ; that formula will therefore become
; and the operation will be thus:
No. 130.
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:
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 , and the
operation will be as follows:
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 , , , &c.; while lives
years older than those respectively, are denoted thus:
, , , &c. and the numbers that attain to their
ages, by the symbols , , , &c.; also let lives
years younger than A, B, C, &c. he denoted thus:
, , , &c., while the numbers which the tables
show to attain to those younger ages, are designated
by the symbols , , , &c.
Then, if A be 21 years of age, and we use the
Carlisle Table, we shall have , and ,
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 years from the age of A, will be
, that is in 14 years, ; 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 years.
Solution. 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 (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 ; and the whole number
for its either happening or failing is ; 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 .
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 , or 0·95454.
136. Since the number that die in years from
the age of A is (134), it appears, in the same
manner, that the probability of that life failing in
years will be 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 years will be , and
that of B surviving the same term will be ; it appears
from the doctrine of probabilities that or
will be the measure of the probability that these
lives will both survive years.
In the same manner it may be shown, that the probability
of the three lives A, B, and C all surviving
years, will be measured by or
. And, universally, that any number of lives A,
B, C, &c. will jointly survive years, the probability
is
138. Let , , ; also
let ; so that the probabilities
of A, B, C, &c. surviving years may be denoted
by , , , &c. respectively; and that of all
those lives jointly surviving that term, by
Then will the probability that none of those lives
will survive years, be
139. But the probability that some one or more of
these lives will survive 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
140. Corol. 1. When there is only one life A, this
will be .
141. Corol. 2. When there are two lives A and B,
it becomes .
142. Corol. 3. When there are three lives A, B,
and C, it becomes .
143. When three lives A, B, and C are proposed, that at the expiration of years there will be
living
dead
the probability is
ABC
none
AB
C
AC
B
BC
A
And the sum of these four ,
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 ;
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 , &c.
Then will the value of an annuity on the joint continuance
of the three lives A, B, and C, be denoted
by .
And on the joint continuance of the two A and B,
by .
145. Also Iet and denote the value of annuities
on lives respectively older and younger than A,
by years: While designates the value of
an annuity on the joint continuance of lives years
older than A, B, C, &c. respectively; and
that of an annuity on the same number of joint lives,
as many years younger than these respectively.
146. Let , the present value of L. 1 to be received
certainly at the expiration of a year, be denoted by .
Then will be the present value of that sum certain
to be received at the expiration of 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 , and
will therefore be .
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
.
147. Let us denote the sum of any series,
as thus,
, by prefixing the italic capital to the
general term thereof. Then, from what has just
been advanced, it will be evident, that .
When there are but three lives A, B, and C; this becomes
When there are but two, A and B, it becomes
.
And in the same manner it appears, that for a
single life A, .
148. (138), where
the denominator ( &c.) is constant, while the numerator
varies with the variable exponent . 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
for each value of , and then to divide the sum
of all these values by &c.; for
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
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
for any single life of the same age.
But this labour may be greatly abridged as follows:
Prob. i.
149. Given , the value of an annuity on
any number of joint lives, to determine , &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 , &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 , &c.
therefore, in that case, the required present value of
the annuity would be , (146.)
But the probability of the lives A, B, C, &c. jointly
surviving one year, is less than certainty, in the
ratio of to unity; therefore .
150. Corol. 1. When there are but three lives, A,
B, and C, this becomes .
151. Corol. 2. When there are only two, A and B,
.
152. Corol. 3. And for a single life A, it appears,
in the same manner, that .
153. Hence, in logarithms, we have these equations,
&c. &c. &c.
Upon which it may be observed, that ,
the sum of the first two logarithms that are employed
in determining from , also enters the operation
whereby is determined from . And
that , the sum of the first three
logarithms that serve to determine from , is
also required to determine from ; 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
years be denoted by
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 ,
(145), its present value would be ,
(146).
But the measure of the probability that all the
lives will survive the term is , therefore
.
In the same manner, it appears, that for a single
life A, .
157. Let an annuity for the term of years only,
dependent upon the joint continuance of any
number of lives, A, B, C, &c. be denoted by
; 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 ;
whence, .
And for a single life 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, .
Solution.
The probability that the survivor of these two lives
will outlive the term of years, was shown in No.
141, to be ; therefore, reasoning
as in No. 146, it will be found, that the present
value of the th year’s rent of this annuity is
, and the value of all the rents
thereof will be or ; so that
(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, .
Solution.
The present value of the th year’s rent is
(142 and 146); whence, it appears, as in the preceding
number, that ,
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, :
Solution.
The present value of the th year's rent is
(143 and
146); whence, reasoning as in the two preceding
numbers, we infer, that ,
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 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 th year, provided that B be then
dead, and A living; but the probability of B being
then dead is , and that of A being then living
, and these two events are independent; therefore,
the probability of their both happening, or that of the
rent being received, is ; the
present value of that rent is, therefore, ;
whence, it follows, that the required value of the annuity
on the life of A after that of B, is ,
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 at
the end of each year, while both the lives survive;
since 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, , whence we have .
164. But if one premium is to be paid down
now, and an equal premium at the end of each year
while both the lives survive, we shall have ,
and .
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 th year there will be
the probability, multiplied by C’s proportion of the annuity in that circumstance, is
dead
living
A
BC
B
AC
and the sum of these being ,
the value of C’s interest is .
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 th year there will be
The probability, multiplied by B’s proportion of the annuity in that circumstance, is
dead
living
A
BC
AC
B
; and the
sum of these being , the
value of B’s interest is .
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 years there will be
The probability, multiplied by A’s proportion of the annuity in that circumstance, is
dead
living
none
ABC
C
AB
B
AC
BC
A
; and
the sum of these being ,
the required value of A’s interest is .
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 years there will be
The probability is
dead
living
AB
C
AC
B
BC
A
; and the
sum of all these being ,
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 denote the probability that the last survivors out of lives A, B, C, &c. will
jointly survive the term of years. And when ,
the expression will become the probability
that the lives will all survive the term (138).
When it will become , the
measure of the probability that the last survivor of
them will outlive the term; which it will be better to
write thus, , retaining the vinculum, but
omitting the unit over it, as in the notation of
powers.
Also let denote the value of an annuity
on the joint continuance of the same number of last
survivors out of the same lives. Then, if be equal
to 0, it will be , &c. the value of an annuity on
the joint continuance of all the lives; when , it
will be 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 survivors
out of these lives, according as it is limited to
the term of years, or deferred during that term, will
also he denoted by or (156
and 157.)
Prob. X.
172. An annuity certain for the term off years,
is to be enjoyed by P and his heirs during the joint
existence of the last survivors out of lives,
A, B, C, &c.; and if that joint existence fail before
the expiration of 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
years, is manifestly equal ta the whole present value
of the annuity certain for that term; that is, equal to
(113 and 146); and the value of P’s interest
for the term of years, is (171); therefore
the value of Q’s interest for the same term is
The present value of the annuity certain for years
after years is (114 and 146); and Q and
his heirs will receive this annuity, if the joint continuance
of the last survivors above mentioned fail
before the expiration of years; but the probability
of their joint continuance failing in the term, is
; therefore, the value of Q’s interest
in the annuity to be received after years, is
; and the whole value
of Q’s interest, is
173. Corol. 1. When the whole annuity certain is
a perpetuity, = 0, and the value of Q’s interest
is
174. Corol. 2. When the term is not less than
the greatest joint continuance of any of the proposed
lives, according to the tables of mortality adapted
to them, , and
therefore, in that case, the general formula
of No. 172 becomes ; that
is, the excess of the value of an annuity certain for
the whole term , above that of an annuity on
the whole duration of joint continuance of the last
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 the excess of
the value of the perpetuity above the value of an annuity
on the joint lives of the last 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
But
And ,
(68.)
Therefore
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 .
But
Subtract
(Tab. VI.)
remains
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,
(174 and 171).
But by the last example
and by No. 66.
So that the required value is
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 , and that of an assurance on
the joint continuance of any number of lives A, B, C,
&c. by Also, let the value of an assurance
on the joint continuance of any of them, out
of the whole number be denoted by
180. And, in every case, let us designate the annual
premium (83) for an assurance, by prefixing the
character to the symbol for the single premium;
so that may denote the annual premium for an
assurance on the life of A; the same
for an assurance on the joint continuance of all the
lives, A, B, C, &c.; and the annual
premium for an assurance on the joint continuance of
the last survivors out of the whole number
of those lives.
181. Then will and ,
and , and
designate the single and annual premiums for assurances
on the same life or lives for the term of years
only.
Prob. xi.
182. To determine the present
value of an assurance on the last survivors out of
lives A, B, C, &c. for the term of years only;
that is, the present value of L. 1, to be received upon
the joint continuance of these last 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 survivors
out of these lives may fail in the term, will be
of the same present value as 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
; whence it
is manifest, that
183. Since the annual premium for this assurance
must be paid at the commencement of every year in
the term, while the last 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 survivors
all outlive it; but the present value of L. 1 to
be received upon their surviving that last year is
, therefore all the future premiums are
now worth years’ purchase,
and the present value of all the premiums, or the
total present value of the assurance, is
, whence
we have
.
184. Corol. 1. When () the term of the assurance is not less than the greatest possible joint duration of any of the proposed lives, , and the general formulæ of the two preceding numbers become respectively
185. Corol. 2. In the same manner it appears, that,
for a single life,
,
and
186. Corol. 3. Also that
or .
And
that is,
.
187. Corol. 4. When the assurance is on the joint
continuance of all the lives, the formulæ of No. 184
become respectively
and .
And those of numbers 182 and 183,
,
and .
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
,
and ;
or ,
and .
And those of numbers 182 and 183 become
,
and ;
or
and respectively.
Where , (141).
and , (142).
For the values of , , , and , 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
Those of numbers 182 and 183,
and .
Where ,
(143).
For the values of and see numbers
157, 160, and 161.
190. the
value of an assurance on any life or lives for the term
of years, which was given in No. 182, may also be
expressed thus:
And this, in words at length, is the rule given in
No. 93.
191. When is not less than the greatest possible
joint duration of any of the proposed lives, the
last expression becomes
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 for (146) in the last expression, it becomes , or . And is the proposition enunciated
in No. 81; 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,
adding
, and subtracting unity,
we have
, 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)
, 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.
Subtract
from
remains
,
agreeably to No. 96.
What has been advanced from numbers 99 to 109,
needs no algebraical illustration.
(U.)