Policy Values   69 = Px (Nx —Nx+n) — CThis is called the "re-

Dx+n   Dx+n

serve" at the end of n years and is written n T z. But Px . Nx =

Dx+n

=A,+,= —Px • ax+n, which is the value of the benefit ultimately payable less the value of the premiums still to be paid.

The value or reserve at the end of n years on a whole life policy with annual premiums, can be obtained from two different points of view.

From the retrospective point of view, the reserve is the difference between the premiums paid in and the claims paid out, each accumulated at interest, with due allowance made for mortality.

From the prospective point of view, the reserve is the difference between the value of the future benefit and the value of the future premiums.

These two methods will produce the same result provided the same bases of interest and mortality are used in the valuation factors as were used in the deduction of the annual premium.

3. Again, consider a whole life policy taken out on (x) with an annual premium of P,.

The present value at age x of the n premiums which may be paid between the ages x and x+n is Px.ax

The present value at age x of the benefit obtainable during these n years following age x is A'x;-1.

Then the difference Ps . ax —Al represents the value at age x of the excess amount that may be paid in over and above the cost of the assurance for that n year period. But the value at age x of the reserve that may be held at age x+n is unnpx • n Vx

therefore ,jjx=(1+i)n(Px.axn —Ax,Z)„px 1 D   ((Px.axn —AI -)

Dx+n

Therefore „Vx = ~1x+n —Px Nx+n