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