ON THE VALUE OF ANNUITIES. 
— log |l “ -f C 1 + ^ —log .616843 : 
log (1 + 0 
.169512 
log 1.05 
8 years. 
.021189 
62. To find (d) the number of years deferred 
Art. 61, (1 + i)~ d - (1 + 0 ~ 
i. e. (1+0 _d {l - (1 + 0"”} 
ip 
a 
by logarithms, — d xlog(l+ i) + log {1 — (1 + 0 ”1 — l°g ~~i 
by transposition, d log (I + %) — log {1 — (1 + 0 ”} ~ log 
ip 
dividing by log (1 + 0 
log{l - (1 +0"”} ->g 
d = 
log (1 + 0 
Example. A deferred annuity of £30 to continue 8 years is pur 
chased for £119 0 8^ when the interest of money is 5 per cent; it is 
required to determine how many years the annuity is deferred. 
p — 119.035 n — 8 ¿=.05 a = 30 
Table 4, 1.05' 
1.00000000 
.67683936 
119.035 
.05 
1 - 1.05- 8 = .32316064 30)5.95175 
.198392 = 
ip 
log {1 - (1 + 0 “ ”} — log 
<z__log .32316064 - log .198392 
log (1 + i) 
.50941 - .29752 
log 1.05 
.21189 
.021189 
.021189 
10 years. 
63. To find (¿) the annual rate of interest, 
(1 + i)~ d - (1 + ¿)- (d + n:) 
(Art. 59.) p—a 
multiply each side by -, 
(l + ¿)~ J — (l + 0“ ( +B)