REVERSION. 
Problem 4. To find the value of an an- 
auity certain for a given term after the ex- 
tinction of any given life or lives. 
Subtract the value of the life or lives 
from the perpetuity, and reserve the re- 
mainder: then say, as the perpetuity is to 
the present value of the annuity certain, so 
as the said reserved remainder to a fourth 
proportional, which will he the number of 
years purchase required. 
Example. Suppose A and his heirs are 
entitled to an annuity certain for 14 years, 
to commence at the death of B, aged 25. — 
What is the present value of A’s interest in 
this annuity ? — The value of the life of B is 
13,567 which subtracted from 20 (the per- 
petuity) leaves 6,433 for the remainder: 
therefore, as 20 is to 9,198 the value of an 
annuity certain for 14 years, so is 6,433 to 
3,183 the number of years purchase re- 
quired. 
Problem 5. B, who is of a given age, 
will, if he lives till the decease of A, whose 
age is also given, become possessed of an 
estate of a given value ; what is the worth 
of his expectation in present money ? 
Find the value of an annuity on two equal 
joint lives whose common age is equal to 
the age of the oldest of the two proposed 
lives, which value subtract from the per- 
petuity, and take half the remainder ; then 
say, as the expectation of duration of the 
younger of the two lives is to that of the 
older, so is the said half remainder to a 
fourth proportional ; which will be the 
number of years purchase required when 
the life of B in expectation is the older of 
the two; but if B be the younger, then add 
the value so found to that of the joint lives 
A and B, and let the sum be subtracted 
from the perpetuity, which gives the answer 
in this case. 
Example 1. Suppose the age of A to 
be 20, and that of B 30 years; and the 
annual value of the estate 50/. Then the 
value of two equal joint lives aged 30 being 
10,255, and the perpetuity 20, the differ- 
ence will be 9,745, the half of which is 
4,872. Therefore, as 33,43, the expecta- 
tion of A, is to 28,27 the expectation of B, 
so is 4,872 to 4,119 years purchase, which 
being multiplied by 50, the given annual 
value, we have 205/. 19s. for the required 
value of B’s expectation. 
Example 2. Let the age of A be 30, that 
of B 20 years; and the rest as in the pre- 
ceding example. Then, the value of the 
joint lives is 10,707, which being added to 
4,119 found above, the sum is 14,826; and 
this subtracted from 20 the perpetuity, and 
multiplied by 50, gives 258/. 14s. for the 
value in this case. 
Problem 6. To find the value of a given 
estate at the death of B, provided that 
should happen after the death of A. 
Find the value of an annuity upon the 
longest of two equal lives whose common 
age is that of the older of the two lives, A 
and B, which value subtract from the per- 
petuity and take half the remainder. Then, 
as the expectation of duration of the younger 
of the lives is to that of the older, so is the 
said half remainder to tlie number of years 
purchase required, when B is the older of 
the two. But if B be the younger, then 
to the number of years purchase thus found 
add the value of an annuity on the longest 
Of the lives, A and B, and subtract the sum 
fiom the perpetuity for the answer in this 
case. 
Example 1. Let the age of A be 30, and 
that of B 60 years ; the given estate 120/. 
per annum. Then the value of an annuity 
on the longest of two lives aged 60 each, 
will be found to be 10,896, which taken 
from 20 the perpetuity, leaves 9,104 for 
the remainder. Therefore it will be as 
28,27, the expectation of A, is to 13,21 the 
expectation of B, so is 4,552 the half re- 
mainder, to 2,127 the number of years pur- 
chase required, which, being multiplied by 
120, gives 255/. 4s. 9d. for the present 
value. 
Example 2. Let the age of A be 60 and 
that of B 30 years ; then, to the number of 
years purchase found in the preceding ex- 
ample, add 14,172 the value of an annuity 
on the longest of the two lives, the sum is 
16,299, and this subtracted from 20 the 
perpetuity, and multiplied by 120, gives 
444/. 2s. 4d. for the value in this case. 
The solutions of the two last problems 
comprehend all the cases of survivorship 
between two lives for their whole duration • 
but an expectation dependent on survi' 
vorship is sometimes restricted to a term 
of years less than the whole duration of the 
lives. Those who have occasion for the 
rules for resolving questions of this descrip- 
tion, or of the various cases which may 
arise when three or more lives are concern- 
ed, are referred to Mr. T. Simpson’s Doctrine 
of Annuities, Dr. Price’s Treatise on Re. 
versionary Payments, or Mr. W. Morgan’s 
Treatise on Annuities and Assurances. 
Reversionary interests being a species of 
property of which purchasers are not always 
readily found, those who have occasion to 
Pp 2 
