CHAPTER IV

ANNUITIES

1. A certain man is aged exactly x to-day. It is his birthday. He is to receive one dollar on every future anniversary of his birthday to which he may survive. What is the value now of these future payments? We designate this value by ax, the symbol for an immediate annuity on (x).

If he is living n years hence he will be then aged x+n and will receive one dollar, the value of which to-day is $1 Xv". But he may not be living n years hence, in which case that dollar will not be paid. The probability that he will live to receive it is

Therefore the real value to-day of the dollar which he may possibly receive when he is aged x+n is S1. Xv" X,,Px•

co

   Therefore ax= E+   = V/2 +v2 2px+v3.3Px+ n=1

2. Again, suppose that we have lx persons living at exact age x, and that each one of them is to receive a payment of one at the end of each year of his life, the value of all these future payments will be

vlx+1+v2lx+2+v3lx+3+ . . . to end of table, and the share of any one of the lx persons will be

   ax =   [vlx+1 +v21x+2 +v31x+3 + . . .

=vpx+v` • 2px+V3.3px+ . . . , as above.

3. Or again, if the annuitant were to die during the first year he would receive nothing. If he were to die during the second year, the value now of the payments he would receive is cif . If he were to die during the third year, the value now of his receipts is ¢z , and so on. The probabilities of death during the first, second, third, etc., years are