Whole Life Annuity Due¶

Definition¶

A unit annual whole life annuity-due pays at the beginning of each year while the life aged $x$ is alive, including a certain payment at issue. Its present value is $\displaystyle Y=\sum_{k=0}^{K_x}v^k$ and actuarial present value (APV) is $\ddot a_x=\mathbb{E}[Y]$ $E [Y]$ .

Interpretation¶

Each annual payment is a pure endowment for its payment date. A longer lifetime increases the number of payments, whereas a death benefit is generally discounted more heavily when death occurs later.

Timeline of one lifetime that ends between the integer times K and K + 1, with a unit payment at each of the times 0, 1, up to K and no payment at time K + 1.

For a fixed lifetime the annuity-due is no longer random: payments are made at $0,1,\ldots,K_x$ while the life is alive, and the payment that would be due at $K_x+1$ falls after death at $T_x$ . These $K_x+1$ payments form an annuity certain, so $\displaystyle Y=\sum_{k=0}^{K_x}v^k$ $k = 0 \sum K_{x} v^{k}$ . For $d\ne0$ this is $(1-v^{K_x+1})/d$ $v^{K_{x} + 1}) / d$ , and $\operatorname{Var}(Y)=\operatorname{Var}(v^{K_x+1})/d^2$ when finite. At zero interest, $Y=K_x+1$ and $\operatorname{Var}(Y)=\operatorname{Var}(K_x)$ when finite.

Change $v$ and $q$ : each arrow is the expected present value $v^k{}_kp_x$ of the payment at $k$ , and $A_x+d\,\ddot a_x=1$ throughout. At $v=0.95$ and $q=0.0351$ $0.0351$ , $A_x\approx0.4$ and $\ddot a_x\approx12$ $12$ .

For a realized curtate future lifetime $K_x$ , the payments at $0,\ldots,K_x$ have the same discounted sum as an annuity certain with $K_x+1$ payments; its geometric-sum formula gives $Y=(1-v^{K_x+1})/d$ when $d=1-v\ne0$ $0$ ; at $d=0$ the sum is $K_x+1$ $1$ . To average over possible lifetimes, write each payment using an indicator random variable: the payment at time $k$ is made only on survival to $k$ . Its expected discounted value is $v^k{}_kp_x$ , so adding these expectations gives the APV sum below.

Key Results¶

$\displaystyle \ddot a_x=\sum_{k=0}^{\infty}v^k{}_kp_x$ $k = 0 \sum \infty v^{k}_{k} p_{x}$ .
For $d=1-v\ne0$ $0$ , $Y=(1-v^{K_x+1})/d$ $v^{K_{x} + 1}) / d$ , so $A_x+d\ddot a_x=1$ $1$ , where $A_x$ is the APV of whole life insurance with annual settlement.
For $d\ne0$ $0$ , $\operatorname{Var}(Y)=\operatorname{Var}(v^{K_x+1})/d^2$ when finite; at $d=0$ $0$ , $Y=K_x+1$ and $\operatorname{Var}(Y)=\operatorname{Var}(K_x)$ when finite.
One-year recursion: $\ddot a_x=1+vp_x\ddot a_{x+1}$ when $p_x>0$ $0$ ; if $p_x=0$ $0$ , then $\ddot a_x=1$ $1$ .

Examples¶

If $A_x=0.4$ and $d=0.05$ $0.05$ , the unit annuity APV is $(1-0.4)/0.05=12$ $12$ . This uses annual death settlement for $A_x$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 5

Aliases: annual life annuity in advance, whole-life annuity-due, life annuity-due, life annuity in advance Keywords: ä x, A x + d ä x = 1, payments in advance while alive, ä x = 1 + v p x ä x+1 Symbols: infinity, sum

Last updated: 27 October 2026 11:52