Guaranteed Life Annuity¶

Definition¶

An annual life annuity-due with an $n$ -year guarantee pays at times $0,\ldots,n-1$ irrespective of survival. Payments at $n,n+1,\ldots$ are made only while the annuitant survives. After an early death, guaranteed payments continue to the designated recipient.

Interpretation¶

The guarantee protects against losing the entire payment stream through early death. It converts the first part into an annuity certain and leaves a deferred life annuity afterwards.

Timeline of an annual annuity-due with a five-year guarantee: solid arrows at times 0 to 4 are paid regardless, dashed arrows from time 5 are paid only while the annuitant is alive.

In the figure the solid arrows at times $0,\ldots,4$ form the annuity certain; they are paid even when death $T_x$ falls inside the guarantee period, as marked. The dashed arrows from time 5 are paid only on survival and form the deferred life annuity. With the numbers from the Examples, the actuarial present value (APV) is $4.5+8=12.5$ $12.5$ .

Move $n$ : the first $n$ payments ignore survival and form $\ddot a_{\actuarialterm{n}}$ , the rest form ${}_{n|}\ddot a_x$ . The defaults give about $4.5+8=12.5$ $12.5$ , above $\ddot a_x$ without a guarantee.

Key Results¶

APV is $\ddot a_{\actuarialterm{n}}+{}_{n|}\ddot a_x$ $_{n ∣} \overset{a}{¨}_{x}$ .
With the same unit payment and timing, APV is at least that of a whole life annuity-due without a guarantee.
The guarantee is not the same as a temporary annuity: a temporary annuity stops on death, while guaranteed payments do not.
Beneficiary and payment-timing conventions must be stated in the contract.

Examples¶

If five certain payments have present value (PV) $4.5$ and the five-year deferred life annuity has APV $8$ , the guaranteed contract has APV $12.5$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 6

Aliases: life annuity with a guarantee period, certain-and-life annuity, guaranteed annuity, life annuity-due with a guarantee, life annuity with period certain Keywords: guarantee period, annuity certain plus deferred life annuity, payments continue to a beneficiary after early death, five-year certain and life annuity

Last updated: 17 October 2026 21:53