Pure Endowment¶

Definition¶

A unit pure endowment pays at time $n\ge0$ only if the life aged $x$ survives to that time. Its present value is $Z=v^n\indicator_{\{T_x>n\}}$ $v^{n} 𝟙 1_{{T_{x} > n}}$ , and its actuarial present value (APV) is denoted ${}_nE_x=v^n{}_np_x$ $v^{n}_{n} p_{x}$ .

Interpretation¶

The insurer pays for survival rather than death. The factor ${}_nE_x$ combines financial discounting with the probability that the later payment is due.

Two timelines for a pure endowment of 1000 at time 10: if the life survives past time 10 the insurer pays 1000 then, and if it dies earlier nothing is paid.

The figure uses the example below: $i=0.05$ $0.05$ , ${}_{10}p_x=0.8$ and a benefit of 1000 at $n=10$ $10$ . Only the upper scenario, $T_x>10$ $10$ , triggers a payment; the lower one ends with nothing paid. Discounting the payment by $v^{10}$ and weighting it by the survival probability $0.8$ gives the APV $1000\,v^{10}{}_{10}p_x\approx491.13$ $491.13$ . Because only these two outcomes exist, $Z$ is a scaled Bernoulli variable, so for benefit $B$ its variance is $\operatorname{Var}(Z)=B^2v^{2n}{}_np_x(1-{}_np_x)$ $_{n} p_{x})$ . Here $B=1000$ $1000$ , giving variance $1000^2(1.05)^{-20}(0.8)(0.2)\approx60302.32$ $60302.32$ ; the unit-benefit formula below has $B=1$ $1$ .

Change $n$ , ${}_np_x$ and $i$ : the APV is $Bv^n{}_np_x$ . For fixed positive $n$ , benefit $B$ and interest rate $i$ , varying ${}_np_x$ gives variance $B^2v^{2n}{}_np_x(1-{}_np_x)$ $_{n} p_{x})$ , largest at ${}_np_x=\tfrac12$ $\frac{1}{2}$ . At $n=0$ $0$ , the life is alive at issue, so ${}_0p_x=1$ and the variance is zero; the ${}_np_x$ slider is not linked to $n$ , so set it to $1$ to see this case.

The actuarial survival and death probabilities supply ${}_np_x$ , the probability that the survival benefit is payable. Multiplying that probability by the discounted amount $v^n$ gives the APV, while the two possible payment values give its variance.

Key Results¶

For benefit $B$ , APV is $Bv^n{}_np_x$ .
$\operatorname{Var}(Z)=v^{2n}{}_np_x(1-{}_np_x)$ for a unit benefit.
${}_{n+m}E_x={}_nE_x\,{}_mE_{x+n}$ when conditioning on survival is defined.
A pure endowment is the survival component of an endowment insurance.

Examples¶

With $i=0.05$ and ${}_{10}p_x=0.8$ $0.8$ , a benefit of 1000 at time 10 has APV $1000(1.05)^{-10}(0.8)\approx491.13$ $491.13$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 3

Aliases: survival benefit, actuarial discount factor, pure endowment factor, n-year pure endowment Keywords: n E x = v^n n p x, payment only on survival, Bernoulli present value, variance v^2n n p x n q x

Last updated: 27 October 2026 11:52