Deferred Life Insurance¶

Definition¶

A death insurance deferred $u$ years pays only for a death after duration $u$ . For a unit deferred whole life benefit paid at death, $Z=v^{T_x}\indicator_{\{T_x>u\}}$ $v^{T_{x}} 𝟙 1_{{T_{x} > u}}$ , with actuarial present value (APV) ${}_{u|}\bar A_x=\mathbb{E}[Z]$ $E [Z]$ . The discount is from the original issue date, not from the end of the deferment.

Interpretation¶

The insured must first survive the waiting period. Conditional on doing so, the remaining contract is a whole life insurance issued at the older age.

Timeline with a grey five-year waiting period and a covered period after it; a death after time 5 pays 1, and the cover worth 0.4 at age x+5 is worth 0.28 at issue.

The figure uses the Examples case with $u=5$ $5$ . A death in the grey waiting period pays nothing; a death after time 5, such as the one marked, pays 1 at $T_x$ . For a life alive at age $x+5$ the remaining cover is a whole life insurance worth $\bar A_{x+5}=0.4$ $0.4$ , and multiplying by ${}_5E_x=0.7$ brings it back to the issue date: ${}_{5|}\bar A_x=0.28$ $0.28$ .

Move the deferment $u$ : deaths before $u$ pay nothing, and ${}_{u|}\bar A_x={}_uE_x\bar A_{x+u}$ $_{u} E_{x} \overset{ˉ}{A}_{x + u}$ . At constant force $\bar A_{x+u}=0.4$ for every $u$ ; here ${}_5E_x=e^{-0.25}\approx0.779$ rather than the Examples' $0.7$ .

The pure endowment factor ${}_uE_x=v^u{}_up_x$ values survival to the end of the waiting period. Multiplying it by the death-insurance APV at age $x+u$ brings that conditional later contract back to the original issue date.

Key Results¶

${}_{u|}\bar A_x=v^u{}_up_x\bar A_{x+u}={}_uE_x\bar A_{x+u}$ $_{u} E_{x} \overset{ˉ}{A}_{x + u}$ .
Whole life APV equals term APV over $u$ years plus the $u$ -year deferred whole life APV.
The same factorization applies to annual settlement for integer $u$ .
A deferred term policy additionally ends coverage after a finite period.

Examples¶

If ${}_5E_x=0.7$ and the unit whole life APV at age $x+5$ is $0.4$ , the deferred contract has APV $0.7(0.4)=0.28$ $0.28$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 4

Aliases: deferred assurance, deferred whole-life insurance, deferred insurance, deferred death benefit Keywords: waiting period, u| A x = u E x A x+u, whole life = term + deferred insurance

Last updated: 14 October 2026 08:21