■:■ n:4l:l! : S-M ni ; ! 
I 
! I ' 
i ' ii ■*. • i 
ii i 1 1 
158 
LIFE ANNUITIES. 
If we wish to obtain the single and annual premium at the age of 101, 
we have 
M m _ M 10 ; _ .121941 
D m D 101 “ .133270 
M m „M 101 __ .121941 
.294532 
; N m _. N t 
= .914998=single premium, 
= . 4140l7=annual premium. 
193. The present value of an assurance of ¿Pi on two joint lives is, 
{.P(rn, mj), nttf, n } — 
^*C Af All*** Ai+1* Azt+O'D' (4,1+1 • A^+l“ Ai+2■ Irru+a) + ? ( Ai+2 - Aij+2 - ‘Ai-f-S'Aii-J-3)^'’'"^ C ', 
^m • Aij 
7' (Ai-Ai, A t +l-A 1 +l)+? (A+l • At t +1 ~ Ai+g ■ 4,, +g) + r (Ai+!*Ah4-S“ Ai+3 ^mj+a)'!'* 6 
L.L ,r™ ' 
m being supposed the older age. 
Assume m — 85, and 7^=80, the expression by the Northampton 
Table will become 
^(As-Ao 4)6'4») + “ Ar-Aa) “l-T^fAy-Ag—As .es) 4“ •• 4" T^fAi-Aa— 0) 
~ r m .l as .l B0 
The expression points out a mode of constructing a table for two 
joint lives similar to the column M for single lives, since, by finding the 
value of for every successive combination, and taking the suc 
cessive differences between each of these products, we have a table of 
mortality for joint lives similar to that for single lives; then multiplying 
the decrements at each combination by the present value of <£l due at 
the end of as many years as the age of the older, we obtain the ele 
ments for forming the table the same as in single lives. The following 
is an illustration, the rate of interest being 3 per cent:— 
An • As— 
469X186 
406X145 
346x111 
289X 83 
234X 62 
186X 46 
145x 34 
114 X 24 
83 X 16 
62 X 9 
46 X 4 
34 X 1 
Combinations 
of Living. 
= 81234 
= 58870 
= 38406 
23987 
14508 
= 8556 
= 4930 
= 2664 
= 1328 
= 558 
= 184 
= 34 
0 
Decrements. 
28364 
20464 
14419 
9479 
5952 
3626 
2266 
1336 
770 
374 
150 
34 
tl