Temporary Life Annuity¶

Definition¶

A unit $n$ -year temporary life annuity pays while the life survives, for at most $n$ payments. For integer $n\ge1$ $1$ , the due version has payments at $0,\ldots,n-1$ and the immediate version at $1,\ldots,n$ .

Interpretation¶

The term limits payment duration even if the annuitant lives longer. The final immediate payment requires survival to time $n$ , whereas the due version starts with a certain payment at time zero.

Two timelines for two-year temporary life annuities: the due version pays at times 0 and 1, the immediate version at times 1 and 2, and later whole life payments are greyed out.

The figure uses the example values $n=2$ $2$ , $v=0.95$ $0.95$ , ${}_1p_x=0.9$ and ${}_2p_x=0.8$ $0.8$ . Each red arrow is a unit payment conditional on survival, labelled with its contribution $v^k{}_kp_x$ to the actuarial present value (APV); the grey arrows are whole life payments that the term cuts off. The due version keeps the certain payment at time $0$ and stops before $n$ , giving $1.855$ ; the immediate version starts at time $1$ and includes the payment at $n=2$ $2$ , which needs survival to $n$ , giving $1.577$ . Their difference, $1-0.95^2(0.8)=0.278$ $0.278$ , is $1-{}_2E_x$ $_{2} E_{x}$ .

Move $n$ , $v$ and the survival model: the due version pays at $0,\ldots,n-1$ and the immediate one at $1,\ldots,n$ , and they differ by $1-{}_nE_x$ $_{n} E_{x}$ . The defaults reproduce $1.855$ and $1.577$ .

Truncating the payment sum of a whole life annuity-due after times $0,\ldots,n-1$ gives the due version. Truncating a whole life annuity-immediate after times $1,\ldots,n$ gives the immediate version. The identity involving endowment insurance uses the same stopping time: an annual death-or-survival payment occurs at $\min(K_x+1,n)$ $1, n)$ , and the discounted annuity payments before it form a finite geometric sum. Taking expectations yields $d\ddot a_{x:\actuarialterm{n}}=1-A_{x:\actuarialterm{n}}$ $A_{x : n}$ , where $d=1-v$ and $A_{x:\actuarialterm{n}}$ is the annual-settlement endowment APV.

Key Results¶

$\displaystyle \ddot a_{x:\actuarialterm{n}}=\sum_{k=0}^{n-1}v^k{}_kp_x$ and $\displaystyle a_{x:\actuarialterm{n}}=\sum_{k=1}^nv^k{}_kp_x$ $k = 1 \sum n v^{k}_{k} p_{x}$ .
$\ddot a_{x:\actuarialterm{n}}-a_{x:\actuarialterm{n}}=1-{}_nE_x$ $_{n} E_{x}$ .
For $d=1-v\ne0$ $0$ , $d\ddot a_{x:\actuarialterm{n}}=1-A_{x:\actuarialterm{n}}$ $A_{x : n}$ , where $A_{x:\actuarialterm{n}}$ is annual-settlement endowment APV.
The whole life annuity is the limit as the term increases, when finite.

Examples¶

For $n=2$ $2$ , $v=0.95$ $0.95$ , ${}_1p_x=0.9$ and ${}_2p_x=0.8$ $0.8$ , the due APV is $1+0.95(0.9)=1.855$ and the immediate APV is $0.95(0.9)+0.95^2(0.8)=1.577$ $1.577$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 5

Aliases: n-year life annuity, temporary annuity due and immediate, temporary annuity, temporary life annuity-due, temporary life annuity-immediate Keywords: ä x:n angle, d ä x:n = 1 - A x:n, due minus immediate equals 1 - n E x, annuity for at most n years Symbols: sum

Last updated: 14 October 2026 08:21