LIFE ANNUITIES. 
Ill 
105. The present value of a sum (?) to be received at the end of any 
number of years (?t), in the event of an individual aged m surviving that 
term, is found by multiplying the present value of that sum receivable 
at the end of the given term by the probability of the individual surviving 
that term. 
Example. A father wishes to provide for his daughter, aged 14, the 
sum of <£850 on her attaining the age of 21 : what sum should he pay 
to secure it, supposing the interest of money 3 per cent, and the rate of 
mortality the same as at Carlisle? (Table 1.) 
r” = 1.03' 7 l m = l H = 6335 1. 21 = 6047 s = 850 
Table 4, Part I.; 
1.03- 7 = .81309151 
058 = s inverted 
650473208 
40654576 
691.127784 by logarithms, 
7406 =r/^inverted logl.03~ 7 = 1.9101395 
= 2.9294189 
— 3.7815400 
= 4.1982534 
4146766704 
log s 
27645111 
log l n 
4837894 
log l u 
/ u =i 6335)4179249.709(659.708 
38010 
37824 
31675 
”61499 
57015 
44847 
44345| 
50209 
2.8193518 £659.708 
= £659 14 2 
106. If the money be receivable in the event of two persons both 
surviving the term, the present value of the sum due at the expiration of 
the term must be multiplied by the two fractions which express the 
probability of each surviving the term separately. (Probability, Art. 15.) 
In the preceding example, if the receipt of the money at the end o 
the seven years depended not only on a life aged 14 surviving that term, 
but also on another aged 16 surviving the same period, the value would 
evidently be diminished; and the result obtained on the supposition of 
the receipt of the money depending on the happening of the first event 
only, must be multiplied by the fraction which expresses the chance of 
the happening of the other event.