72   Life Contingencies

7. By a similar line of reasoning, analogous values and relations can be deduced for any type of policy.

For an endowment assurance of 1 with annual premiums, we have

£Vxn =Ax+t:n—t—t; —Pxnl ax+t:n=~

= 1   [Px-,-T (Nx —Nx+t) — (Mx —Mx+t)] Dx+t

and   (t—lVxnl+Pxni)(1+i)—Qx+t-1(1—tVx1)=tT'xni.

For an n year term assurance of 1 with annual premiums, we have

tV ; =Ax+t:n—t~ —Pzn~'ax+t:n—t~

= 1    [F;(Nx—Nx+t) —(Mx—Mx+t)]

Dx+t

and   (1—1V,71 +P,t,) (1+i) —qx+t—1(1—iV;) =tVzn;.

For an n-year limited payment whole life assurance of 1 we have, when t <n

=   1   [nPx (Nx —Nx+t) — (Mx — Mx+t)]
Dx+t

and   11x+nPx) (1+i)—Qx+t—1(1—1:nitx)=t:nl'x•

Also, when t

= 1 [nPx (Nx —Nx+n) — (Mx — Mx+t)]

Dx+t

Also if t> n, Ax+t—1(1 +i) -4x+t—1(1—Ax+t) =Ax+t• But when t=n, we have

(n—1:n j'x+nPx) (1+i) —Qx+n—1 (1 —Ax+n) =Ax+n•

Expressions for the reserve under endowment, term and limited payment policies in terms of the A, a and P symbols analogous to those in section 4 should be obtained by the student.