68   Life Contingencies

2. Consider 1x persons each aged exactly x and assume that each person takes out a whole life assurance for 1 at an annual premium of P.

The fund starts with I,Px which initial sum accumulated at interest rate i amounts to IxPx(1+i) at the end of the first year. The claims for the first year amount to 1 Xdx =dx, so that the net amount available for Ix+1 survivors at the end of the first year is 1xPx(1+i) —dx.

At the beginning of the second year, premiums aggregating lx+1Px are paid into the fund. The initial amount of the fund for the second year then stands at

lxPx(l +i) —dx+l.x+1Px

which at interest amounts at the end of the year to

[1xPx(l +i) —dx+1x+1Px](1 +i).

Deducting claims for the dx+1 deaths in that year, we have the net amount available for the 1x+2 survivors equal to

[lP(1 +i) —dx+1x+1Px] (1+i) -dx+1.

If this process be continued for n years, there will be available at the end of the n,years for the lx+n survivors a total of

lxPx(l+i)n—dx(1+i)s-1+1x+1Px(l+i)n-1—dx+1(1+i)" 2+ ... +lx+n-1Px(l+i) —dx+n-1

=Px [l(1 +i)"+l+1(1 +i)'1 + .... Flx+n_1(1+i)] —[dx(l+i)n-1+dx+1(1+i)"-2+ ... +dx+n-1]

=Px(l+2)x+"[Dx+Dx+1+ ... +Dx+n-ll

—(l+i)x+"[C77x+Cx+l+ . • • +Cx+n—11

=Px(1+i)x+n(A7x—x~~71'Ix+n) —(1+i)x+n(l1.—:11x+n) or an average amount per person of

1

[Px(I +i)x+n(~T

L~x —Nx+n) — (1 +i)x+n(A1x —Mx+n) lx+n