Endowment Insurance¶

Definition¶

An $n$ -year endowment insurance pays a unit benefit at death within the term or at time $n$ on survival. With immediate death settlement, $Z=v^{\min(T_x,n)}$ $v^{m i n (T_{x}, n)}$ . With annual death settlement and integer $n$ , $Z=v^{\min(K_x+1,n)}$ $v^{m i n (K_{x} + 1, n)}$ . The actuarial present values (APVs) $\mathbb{E}[Z]$ are denoted $\bar A_{x:\actuarialterm{n}}$ and $A_{x:\actuarialterm{n}}$ , respectively.

Interpretation¶

Every lifetime leads to a payment. The uncertainty concerns when that payment occurs, up to the fixed maturity time.

Plot of the present value of a 10-year endowment against the time of death at force of interest 0.03: falling until time 10, then constant at exp(-0.3), with the mean 0.764 dashed.

The figure plots the present value $Z=v^{\min(T_x,n)}$ against $T_x$ for $n=10$ and $\delta=0.03$ $0.03$ , so $v=\mathrm{e}^{-0.03}$ $e^{- 0.03}$ . A death before time 10 gives $v^{T_x}$ , which falls from 1 to $\mathrm{e}^{-0.3}\approx0.741$ $0.741$ ; survival to time 10 gives the fixed maturity value $v^{10}=\mathrm{e}^{-0.3}$ $e^{- 0.3}$ . Every outcome lies between these two values, and with $\lambda=0.02$ the mean is the Examples value $\mathbb{E}[Z]\approx0.76392$ $0.76392$ .

Key Results¶

APV equals the APV of an $n$ -year term insurance plus ${}_nE_x$ .
Death and survival benefits occur on disjoint events; their product is zero. Their covariance is generally negative, not zero.
To calculate the second moment, square each discounted payment on its own event. Do not add component variances without a covariance term.

Examples¶

At $\lambda=0.02$ $0.02$ , $\delta=0.03$ and $n=10$ $10$ , immediate death-or-survival APV is $0.4(1-\mathrm{e}^{-0.5})+\mathrm{e}^{-0.5}\approx0.76392$ $0.76392$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 4

Aliases: n-year endowment assurance, death-or-survival benefit, endowment assurance, n-year endowment insurance, endowment policy Keywords: term insurance plus pure endowment, v^min(T x, n), A x:n angle, negative covariance of death and survival benefits Symbols: delta, lambda

Last updated: 14 October 2026 08:21