ON THE VALUE OF ANNUITIES. 
What sum would be required to purchase an annuity of £20 to con 
tinue 15 years in the Government Office, when the price of the 3 per 
cent consols is such as to yield an interest of £3 5 per cent ? 
Here the annuity and interest are both convertible half-yearly. 
m = 2 a ~ 20 i = .0325 n = 15 
2).0325 
.01625 
1. 
log 1.01625 
1.99299944 
30 
mn 
1.7899832 
.616571 
1. 
(1 + 0" 
.383429 = 1 - (1 + i)~ mn 
20 
£235 19 1 
,0325)7.66858(235.956 
650 
116S 
975 
~1935 
1625 
3108 
2925 
1830 
1625 
205 
which agrees with the rate in 
serted in the Government scale 
for granting life annuities. 
PERPETUITIES. 
56. When the annuity is to continue for ever, it is called a perpetuity, 
1 - (1 + i)~ n 
in which case n is infinite, and in the formula p = a 
given in Art. 50, the value of the quantity (1 + ?') _n is less than any 
that can be assigned ; that part of the formula therefore vanishes, and 
the expression becomes 
la, 
a.—r — —r — , the present value. 
Rule. Divide the annuity whose value is to be found by the interest 
of £l per annum. 
What is the present value of an estate in fee simple of ,£434 per 
annum, when interest of money is 4 per cent ? 
434 
wn 
int 
“ £10850 
.04