CHAPTER VIII POLICY VALUES

1. Consider lx persons each aged exactly x and assume that each person takes out a whole life assurance for 1 at a single premium of A. An initial fund of lxAx is set up. The interest earned by the end of the year is ilxAX and the total accumulated amount of the fund at the end of the year is (1+i)lxAx.

Out of this sum there are dx claims of 1 each to be paid. Therefore, the 1x+1 survivors will have to their credit

(1+i)lxAx—dx=(1+i) (vdx+v2dx+1+ . . . ) —dx = vdx+1 +tied x+2 + . . . lx+lA x+1

and the share of each will be Ax+1. This is exactly the premium which a new man of the same age x+1 must pay if he wishes to join the fund a year after it started.

Or again, consider (x) who pays a single premium Ax for a whole life assurance of 1.

If (x) should die during the year, the probability of which is qx, a sum of 1 will become payable. If (.v) should survive the year, the probability of which is px, there must be Ax+1 to his credit in the fund,

therefore   Ax =vqx X 1 +vpxA x+1

or   Ax(l+i) =Ax+1+qx(1—Ax+1).

This statement has already been proved. It means that at the end of the year the fund must have on hand in respect of this policy A,+1 and further, in the case of death, it will have to provide the difference between the claim of 1 and the amount held Ax+1 or a net sum of (1—Ax+l).

The amount, 1—Ax+1, which is the difference between the sum assured and the value of the policy at age x+1, is called the "net amount at risk" for the year x to x+l,

qx(l —Ax+1) is called the "cost of insurance" or "death strain" for the year x to x+1.