ENDOWMENTS. 
133 
30 
If we add these quantities perpendicularly, the sum of those in the 
first column (by Art. 112) will be the present value of an annuity on a 
life aged m; those in the second, of an annuity on a life aged ; in the 
third on a life aged &c.;—the total value of these expressions is 
therefore 
@m~J~ &C. @m, m 2 &C, “1“ Cf m> Wg &C, 
144. When there are three lives, it becomes 
a m +ffmj + @m 2 - a m> m[ —a m> m —a mi , mg + a„ t mi , mg . 
Rule. Find the value of the annuity on each of the single lives ; to 
their sum add the value of an annuity on the three joint lives, and sub 
tract the sum of the values on each pair of joint lives. 
Example. What is the present value of an annuity of £50 payable 
until the death of the last survivor of three lives respectively, aged 18, 
27, and 36 years? (Northampton, 3 per cent.) 
145. As there are no tables of annuities on three lives, we approxi 
mate by the following rule, which is given by Mr. Baily in his Trea 
tise on Life Annuities :—Take the value of an annuity on the joint 
lives of the two oldest, and find the age of a single life of the same 
value. Then find the value of an annuity on the joint lives of the one 
just found and the remaining life of the three, which diminished by .05 
will give 
very nearly the true value. 
a io 
=19.0131 
a a , v = 13.7363 Tables. 
a-27 
= 17.4674 
a 18 , 36 =: 12.7635 do. 
@36 
= 15.7288 
36 = 12.2295 = a 5l do. 
®i8.27.36— 10.3887 
= c 18 ,5i— .05 38.7293 
62.5980 
38.7293 
23.8687 
X 50 — 1193.435 = £l 193 8 9. 
146. When the annuity is on the longest of two lives, the formula 
becomes 
a m + a mi — a m , • 
Rule. From the sum of the values on each of the single lives, sub 
tract the value of the annuity on the joint lives. 
What is the present value of an annuity of £30 on the longest of 
two lives aged 39 and 43 ? (Northampton 3 per cent.) 
Table 7, 
@39 — 
15.0750 
do. 
@i'3 — 
14.1626 
29.2376 
Table 8, 
°'39.« ” 
10.5485 
18.6891