46 
ON THE VALUE OF ANNUITIES. 
REVERSIONS. 
59. When an annuity is not to be entered upon until after the 
expiration of a certain number of years, it is called a Reversionary or 
Deferred Annuity, the present value of which may be obtained by 
finding the present value of an annuity to he entered upon immediately 
and continue until the expiration of the reversion, and subtracting 
therefrom the present value of an annuity to be continued only until 
the time of entering on possession of the reversion; for it is evident 
that if an annuity be deferred d years, and then continue n years, its 
present value will be less than that of an annuity to be received during 
both the d years and the n years by the present value of an annuity for 
d years. 
Let p = the present value, 
a — the annuity, 
n = number of years the annuity continues, 
d ~ number of years deferred, 
i = annual interest of £l; 
then Art. 50. p ~ a. 
1 ~ C 1 + i)- {i+n) 
i 
1 - (1 + i)~ d 
i 
(1 + i)- d — (1 + i)- {d + n) 
Rule, From the present value of £l, due the number of years de 
ferred, subtract the present value of £l, due at the same time as the last 
payment of the reversionary annuity, multiply the difference by the 
annuity, and divide by the annual interest of £l. 
Example. What is the present value of the reversion of £30 per 
annum for 8 years, to be entered upon after the expiration of the next 
10 years; interest 5 per cent ? 
(1 + i)~ d == 1.05- 10 = .613913 
(1 + ¿)-(<*+«) = 1.05 ~ 18 = .415521 
.198392 
30 
.05)5.95176 
119.035 = ¿£119 0 8^ 
60. To find (a) the annuity. 
(Art. 59.) p —a. 
(1 + i)~ d 
(1 + ¿)- (<i+,1) 
i 
multiply by 
i 
(1 +i)~ d - (1 +i)~ (d+n) 
ip 
" (1 + t)~ d - (1 +i)^+^ 
Rule. Divide the product of the present value of the annuity and 
the annual interest of £l by the difference between the present value