PURCHASE OF ANNUITIES. 
217 
or when ?»! — 1 is greater than m, 
1 f r(N OT _i J tn, —1 ^m + n—1, m,+n —1 mj—1 “f" ^jn+n, mj+ n—1^ 
S'- Dm, m. 
+N, 
m I-n, ntj+n 
.Hr—i >m . C-^tn, Mi “h 1, 
D„ 
t»i+n) ^ 
j; 
the annual premium for the same will be 
A, 
1 (m, *n,) 
(l) l n 
L 
m+n v mi+n 
I’m . 
1 + &'m t mi 
1 
m+n ^rw-i+n- 
L 
a, 
m+n, m^+n 
or by substituting in the denominator in the expressions for the D and 
N columns, N ffl _i, mrI - ■Nm+tt—1, for D m>mi . 
PURCHASE OF ANNUITIES, SECURING THE CAPITAL BY AN 
ASSURANCE. 
Let s — the sum, 
i “ annual interest of £l, 
p =r annual premium for assurance of £l, 
a = the annuity. 
If the annuity be supposed payable at the end of each year, 
1 -;;(!+?) 
s d • ■■. . _ j 
l+p(l+l) 
i + p(l + i) 
a - s ’ l-/»(l+0* 
• — a ~p( s + a ) 
' “ s+p(s+ct) * 
a—si 
V ~~ (s+a)(l+0 ’ 
If tlie annuity be supposed payable until the day of death. 
Cl" s. 
1 -V 
i+p ’ 
i+p 
1 -P 
a-p 0 + u) 
a — is 
s + a* 
VALUATION OF LIFE POLICIES. 
Let s ~ the sum assured, 
p m — tlie annual premium charged on a life aged in.