Whole Life Insurance¶

Definition¶

A unit whole life insurance pays on death whenever it occurs. For immediate payment its present value is $Z=v^{T_x}$ and actuarial present value (APV) is $\bar A_x$ . For payment at the end of the year of death, $Z=v^{K_x+1}$ and APV is $A_x$ , where $v=(1+i)^{-1}$ $i)^{- 1}$ .

Interpretation¶

The contract covers the entire remaining lifetime. Its value depends on whether settlement occurs at death or later at an annual boundary.

Timeline of one lifetime ending at time 3.4, with the death benefit paid either immediately at time 3.4 or at the end of the year of death, time 4.

The figure follows one illustrative lifetime with death at $T_x=3.4$ $3.4$ , so $K_x=3$ complete years are lived. Immediate settlement pays at the moment of death, with present value $Z=v^{T_x}=v^{3.4}$ $v^{3.4}$ ; annual settlement waits until the end of the year of death, $K_x+1=4$ $4$ , with $Z=v^{K_x+1}=v^4$ $v^{4}$ . Since $T_x\le K_x+1$ for every lifetime, the immediate payment is never later, which is why $\bar A_x\ge A_x$ when interest is nonnegative.

Change $\lambda$ and $\delta$ : $\bar A_x=\lambda/(\lambda+\delta)$ is the red area under $v^tf_x(t)$ . Drag the probe along $T_x$ , and switch on end-of-year settlement ( $m =$ $m=1$ $1$ ) to see $A_x\le\bar A_x$ $\overset{ˉ}{A}_{x}$ . The displayed window is finite; compare its shown contribution with the full lifetime value and omitted tail in the readout.

The curtate future lifetime $K_x$ determines the annual settlement date $K_x+1$ $1$ . For immediate settlement, the force of mortality converts the survival probability into the death density ${}_tp_x\mu_{x+t}$ used in the APV integral.

Key Results¶

$\bar A_x=\int_0^\infty v^t{}_tp_x\mu_{x+t}\,\mathrm{d}t$ $\int_{0}^{\infty} v^{t}_{t} p_{x} μ_{x + t} d t$ .
$\displaystyle A_x=\sum_{k=0}^\infty v^{k+1}{}_kp_xq_{x+k}$ $k = 0 \sum \infty v^{k + 1}_{k} p_{x} q_{x + k}$ .
For nonnegative interest, $\bar A_x\ge A_x$ because immediate settlement is earlier.
At constant force of mortality $\lambda>0$ and force of interest $\delta\ge0$ $0$ , $\bar A_x=\lambda/(\lambda+\delta)$ $δ)$ .

Examples¶

With $\lambda=0.02$ and $\delta=0.03$ $0.03$ , a benefit of 100 paid at death has APV $100(0.02/0.05)=40$ $40$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 3

Aliases: whole-life assurance, insurance payable at death, whole of life insurance, whole of life assurance Keywords: A bar x, A x with end-of-year settlement, v^(K+1), lambda/(lambda + delta) at constant force, payable whenever death occurs Symbols: delta, infinity, integral, lambda, mu, sum

Last updated: 27 October 2026 11:52