Whole Life Annuity Immediate¶

Definition¶

A unit annual whole life annuity-immediate pays at times $1,2,\ldots$ only while the annuitant survives. Its present value is $\displaystyle Y^*=\sum_{k=1}^{K_x}v^k$ and actuarial present value (APV) is $\displaystyle a_x=\sum_{k=1}^{\infty}v^k{}_kp_x$ $k = 1 \sum \infty v^{k}_{k} p_{x}$ .

Interpretation¶

There is no guaranteed initial payment. Compared with the corresponding life annuity-due, the contract omits exactly the payment at time zero.

Three timelines for one lifetime: the annuity-due pays at times 0 to K, the annuity-immediate at times 1 to K, and the one-year shift of the due stream also pays at K + 1, after death.

In the figure the life dies between $K_x$ and $K_x+1$ $1$ . The annuity-immediate makes the same survival-contingent payments as the annuity-due except the certain one at time $0$ , so $Y^*=Y-1$ for every lifetime and the two have the same variance. Shifting the whole due stream by one year, $vY$ , instead pays at $1,\ldots,K_x+1$ $1$ : its last payment (orange) falls after death, so $v\ddot a_x$ is generally larger than $a_x$ .

Change $v$ and $q$ : the immediate row drops only the payment at 0, so $a_x=\ddot a_x-1$ $1$ . The shifted row $v\,\ddot a_x$ keeps each payment's earlier survival condition, and its orange parts would be paid after death.

Key Results¶

$Y^*=Y-1$ $1$ , so $a_x=\ddot a_x-1$ and both present values have the same variance.
$a_x=vp_x\ddot a_{x+1}$ by conditioning on survival to the first payment.
Multiplying $\ddot a_x$ by $v$ alone is generally wrong: moving a survival-contingent payment also changes its survival condition.

Examples¶

If the annuity-due APV is 12, the matching annuity-immediate APV is 11. Their variance is equal because the omitted initial payment is deterministic.

Prerequisites¶

Whole Life Annuity Due

Temporary Life Annuity

Sources¶

Life Insurance · Lecture 5

Aliases: annual life annuity in arrears, whole-life annuity-immediate, life annuity-immediate, life annuity in arrears Keywords: a x = ä x - 1, payments at the end of each year, a x = v p x ä x+1 Symbols: infinity, sum

Last updated: 17 October 2026 21:53