Term Life Insurance¶

Definition¶

An $n$ -year term insurance pays a death benefit only if death occurs within the coverage period $[0,n)$ . A unit benefit paid at death has present value $Z=v^{T_x}\indicator_{\{T_x<n\}}$ $v^{T_{x}} 𝟙 1_{{T_{x} < n}}$ . For integer $n$ and annual settlement, $Z=v^{K_x+1}\indicator_{\{K_x<n\}}$ $v^{K_{x} + 1} 𝟙 1_{{K_{x} < n}}$ .

Interpretation¶

The contract provides protection during a fixed term and pays nothing on survival to its end. The death condition and settlement time must both be specified.

Plot of the discounted death density under constant force, with the area up to the term n = 10 shaded as the term insurance APV and the area after n shaded grey as deaths that pay nothing.

With constant force $\lambda=0.02$ and $\delta=0.03$ $0.03$ , the death density is $f_x(t)=0.02\mathrm{e}^{-0.02t}$ and $v^t=\mathrm{e}^{-0.03t}$ $e^{- 0.03 t}$ , so the integrand is $0.02\mathrm{e}^{-0.05t}$ . The whole area under this curve is the whole life actuarial present value (APV) $\lambda/(\lambda+\delta)=0.4$ $0.4$ ; the 10-year term contract keeps only the red part left of the cutoff $n=10$ $10$ , which is $0.4(1-\mathrm{e}^{-0.5})\approx0.15739$ $0.15739$ , as in the example. The grey remainder, $0.4\mathrm{e}^{-0.5}\approx0.24261$ $0.24261$ , belongs to deaths after the term. Moving the cutoff to the right adds area, which is why the APV increases with the term.

Move the term $n$ : the APV keeps only the red area of $v^tf_x(t)$ before $n$ , $0.4(1-e^{-0.5})\approx0.15739$ at the defaults, and it grows with the term.

Whole life insurance covers deaths at every future duration. Restricting its death-payment integral or annual payment sum to the first $n$ years gives the term contract; deaths after the cutoff contribute zero.

Key Results¶

Immediate-payment APV is $\int_0^n v^tf_x(t)\,\mathrm{d}t$ .
Annual-settlement APV is $\displaystyle \sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}$ .
At constant force of mortality $\lambda>0$ and force of interest $\delta\ge0$ $0$ , the immediate APV is $\lambda[1-\mathrm{e}^{-(\lambda+\delta)n}]/(\lambda+\delta)$ $δ)$ .
Increasing the term increases APV for a nonnegative fixed benefit.

Examples¶

With $\lambda=0.02$ $0.02$ , $\delta=0.03$ and $n=10$ $10$ , the unit APV is $0.4(1-\mathrm{e}^{-0.5})\approx0.15739$ $0.15739$ .

Prerequisites¶

Whole Life Insurance

Sources¶

Life Insurance · Lecture 3

Aliases: temporary life insurance, n-year term assurance, term insurance, term assurance, temporary assurance Keywords: death within n years, A 1 x:n angle, pays nothing on survival, n-year term cover Symbols: delta, integral, lambda, sum

Last updated: 17 October 2026 21:53