Annuities with Guaranteed Payment Periods

Why a Guarantee Changes an Annuity’s Value

A life annuity normally pays only while the annuitant is alive at the scheduled payment dates. A guaranteed payment period changes the contract: payments during that period are made even if the annuitant dies. If the annuitant survives beyond the guarantee, payments continue for life. The value therefore combines payments that are certain under the contract with later payments that depend on survival.

The key to valuing this benefit is to separate those two parts without counting any payment twice. Payment timing matters: an annuity-immediate pays at the end of each year, whereas an annuity-due pays at the beginning. A guarantee of $n$ annual payments means that exactly $n$ payments are certain, regardless of death.

Definition: A life annuity with an $n$-payment guarantee makes its first $n$ scheduled payments whether or not the annuitant survives to receive them. After those $n$ payments, further scheduled payments are made only if the annuitant is alive on each payment date. Unless a contract specifies otherwise, assume the guaranteed payments go to a beneficiary or the annuitant’s estate if the annuitant dies before receiving them.

This tutorial considers level annual payments and a single effective annual interest rate $i$, with $v=1/(1+i)$. For the formulas, assume payments occur on the standard annual dates and that the survival probabilities and any conditional annuity factors are based on the same mortality and interest assumptions.

Valuing the Guaranteed and Contingent Parts

Let $B$ be the annual payment, and let the annuitant be age $x$ at issue. For an annuity-immediate, payments are scheduled at times $1,2,3,\ldots$. With an $n$-payment guarantee, payments at times $1,\ldots,n$ are certain. Payments at times $n+1,n+2,\ldots$ require survival to their respective payment dates.

\text{EPV}=B\left(a_{\overline{n}|}+v^n\,{}_np_x\,a_{x+n}\right).

Here $a_{\overline{n}|}$ is the present value of $n$ certain payments of 1 at the ends of years $1$ through $n$. The term $v^n\,{}_np_x\,a_{x+n}$ values the payments after the guarantee: survival to time $n$ has probability ${}_np_x$, and, conditional on being alive then, the future annuity-immediate has value $a_{x+n}$ at time $n$. Discounting that conditional value to issue gives the factor $v^n$.

For an annuity-due, payments are scheduled at times $0,1,2,\ldots$. The first $n$ payments, at times $0$ through $n-1$, are certain. If the annuitant survives to time $n$, payments from time $n$ onward form a life annuity-due at age $x+n$.

\text{EPV}=B\left(\ddot{a}_{\overline{n}|}+v^n\,{}_np_x\,\ddot{a}_{x+n}\right).

The guaranteed-period factors $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$ value certain payments, not life-contingent payments. In contrast, $a_{x+n}$ and $\ddot{a}_{x+n}$ value payments conditional on being alive at the start of the post-guarantee period. The survival probability and discount factor bring that conditional value back to issue.

Why the Decomposition Works

The decomposition follows by valuing each scheduled payment separately and then grouping the terms. Let $K$ be the number of complete future years lived by an annuitant aged $x$, so ${}_kp_x=P(K\ge k)$. For an annuity-immediate, the payment at time $k$ is made only if $K\ge k$, except that it is guaranteed when $k\le n$. Thus the expected present value per dollar is

\sum_{k=1}^{n}v^k+\sum_{k=n+1}^{\infty}v^k\,{}_kp_x.

The first sum is $a_{\overline{n}|}$. For each term in the second sum, survival to time $n$ is necessary. Conditional on survival to time $n$, the probability of surviving a further $j$ years is ${}_jp_{x+n}$, where $j=k-n$. The multiplication rule for conditional probabilities gives ${}_kp_x={}_np_x\,{}_jp_{x+n}$. Therefore,

\sum_{k=n+1}^{\infty}v^k\,{}_kp_x = v^n\,{}_np_x\sum_{j=1}^{\infty}v^j\,{}_jp_{x+n} = v^n\,{}_np_x\,a_{x+n}.

Adding the certain and contingent parts proves the annuity-immediate formula. For an annuity-due, the guaranteed payments occur at times $0,\ldots,n-1$, and the post-guarantee annuity begins with a payment at time $n$. The corresponding conditional payments are at times $0,1,\ldots$ when viewed from time $n$, which is why the factor is $\ddot{a}_{x+n}$, not $a_{x+n}$.

Fix the payment dates.
For an annuity-immediate, the first payment is at time 1. For an annuity-due, it is at time 0.

Mark the guaranteed payments.
Count exactly $n$ payments. Do not apply a survival probability to these payments.

Value later payments conditionally.
Use the appropriate life annuity factor at age $x+n$, multiplied by ${}_np_x$.

Discount the later value to issue.
Multiply by $v^n$, then add the value of the guaranteed payments.

Worked Valuations

Worked Example: An Annuity-Immediate with Four Scheduled Payment Dates

A hypothetical contract pays \$1,000 at the end of each year. It guarantees the first two payments and then pays only if the annuitant is alive on the payment date. Assume $i=10\%$, so $v=1/1.10$. The probabilities of being alive at the third and fourth payment dates are ${}_3p_x=0.80$ and ${}_4p_x=0.60$, respectively. Assume there are no payments after time 4. Find the expected present value.

Method. Value the first two payments as certain, then weight the next two by their survival probabilities. The stated probabilities are unconditional probabilities of survival from issue to each payment date.

The discount factors are $v=0.9090909$, $v^2=0.8264463$, $v^3=0.7513148$, and $v^4=0.6830135$. The first two payments have expected present value

\$1{,}000(v+v^2) = \$1{,}000(0.9090909+0.8264463) = \$1{,}735.54.

The third payment contributes $\$1{,}000(0.80)v^3=\$1{,}000(0.80)(0.7513148)=\$601.05$. The fourth contributes $\$1{,}000(0.60)v^4=\$1{,}000(0.60)(0.6830135)=\$409.81$. Thus the total is $\$1{,}735.54+\$601.05+\$409.81=\$2{,}746.40$, rounded to the nearest cent.

Check. The guarantee covers the payments at times 1 and 2; only the payments at times 3 and 4 receive survival weights. A common mistake is to multiply the first two terms by survival probabilities as well, which would understate the guaranteed value.

Worked Example: An Annuity-Due with a Two-Payment Guarantee

A hypothetical annuity-due pays \$600 at the beginning of each year, beginning immediately. The first three payments are guaranteed; later payments require survival to the payment date. Assume $i=4\%$, ${}_3p_x=0.90$, and ${}_4p_x=0.75$, with no payments after time 4. Find the expected present value.

Method. The guaranteed payments are at times 0, 1, and 2. The later payments, at times 3 and 4, are survival-contingent. With $v=1/1.04=0.9615385$, the value per dollar is

1+v+v^2+{}_3p_xv^3+{}_4p_xv^4.

The needed powers are $v^2=0.9245562$, $v^3=0.8889964$, and $v^4=0.8548042$. The guaranteed part per dollar is $1+0.9615385+0.9245562=2.8860947$. The contingent part per dollar is $0.90(0.8889964)+0.75(0.8548042)=0.8000967+0.6411031=1.4411998$. Therefore the total factor is $2.8860947+1.4411998=4.3272945$, and the present value is

\$600(4.3272945)=\$2{,}596.38.

Check. The immediate payment at time 0 is part of the guarantee and has discount factor 1. Treating this as an annuity-immediate would incorrectly shift every payment one year later.

Worked Example: Using a Conditional Life Annuity Factor

A hypothetical annuity-immediate guarantees the first five annual payments of \$1,200. Assume $i=5\%$, ${}_5p_x=0.82$, and that the life-table value of the conditional annuity-immediate at age $x+5$, on this same interest basis, is $a_{x+5}=7.2$. Find the expected present value.

Method. Add the value of five certain end-of-year payments to the discounted, survival-weighted value of the annuity after time 5. First, $v^5=(1.05)^{-5}=0.7835262$. The certain factor is

a_{\overline{5}|} = \frac{1-v^5}{i} = \frac{1-0.7835262}{0.05} = 4.3294767.

The post-guarantee factor at issue is $v^5\,{}_5p_x\,a_{x+5}=0.7835262(0.82)(7.2)=4.6259385$. The total factor is $4.3294767+4.6259385=8.9554152$. Multiplying by the annual payment gives $\$1{,}200(8.9554152)=\$10{,}746.50$, rounded to the nearest cent.

Check. The factor $a_{x+5}$ is conditional on reaching age $x+5$, so it must be multiplied by ${}_5p_x$. The guarantee does not remove the need for that survival adjustment on payments after the fifth guaranteed payment.

Timing and Interpretation Traps

The formulas are specific to the payment convention. An $n$-payment guarantee for an annuity-immediate covers times 1 through $n$; for an annuity-due it covers times 0 through $n-1$. After the guarantee, the immediate annuity’s first conditional payment is one year after time $n$, while the due annuity’s first conditional payment is at time $n$. That distinction determines whether the conditional factor is $a_{x+n}$ or $\ddot{a}_{x+n}$.

Do not confuse guaranteed with life-contingent payments. A guaranteed payment is made even if death occurs before its scheduled date, under the contract assumption stated here.
Do not shift the guarantee boundary. For an annuity-due, the payment at time $n$ is not one of the first $n$ payments; it belongs to the post-guarantee annuity.
Do not apply survival twice. If a post-guarantee annuity factor is conditional on being alive at age $x+n$, multiply by ${}_np_x$ once, not once for every later payment.
Keep the assumptions aligned. The interest rate and mortality basis used for the conditional annuity factor must match those used in the valuation.

A guaranteed period increases the value relative to the otherwise identical life annuity without a guarantee, because payments that would have stopped at death may now continue to the beneficiary during the guarantee. It does not make payments after the guarantee certain. The same decomposition is useful when a stated annuity value or benefit is converted into a net single premium under the equivalence principle.

Key takeaway: Value the guaranteed payments as an annuity-certain factor. Then value payments beyond the guarantee as a life annuity at age $x+n$, multiplied by the probability of surviving $n$ years and discounted $n$ years to issue. Match the life-annuity factor to the payment timing.

Check Your Understanding

For each problem, identify the guaranteed dates before calculating. Round money to the nearest cent.

A hypothetical annuity-immediate pays \$500 annually, guarantees two payments, and has no payments after time 4. At $i=5\%$, ${}_3p_x=0.70$, and ${}_4p_x=0.40$, what is its expected present value? (Answer: \$1,396.59)
A hypothetical annuity-due pays \$900 annually. Its first two payments are guaranteed, and only payments at times 2 and 3 may be made afterward; ${}_2p_x=0.85$ and ${}_3p_x=0.65$. At $i=8\%$, find its expected present value. (Answer: \$2,853.59)
A hypothetical annuity-immediate pays \$750 annually with a three-payment guarantee. Assume $i=4\%$, ${}_3p_x=0.88$, and $a_{x+3}=6.1$ on the same interest and mortality basis. Find its expected present value. (Answer: \$5,660.42)
A hypothetical annuity-due has four guaranteed annual payments, followed by a conditional life annuity-due at age $x+4$. Write its expected present value per dollar using the appropriate survival probability and discount factor. (Answer: $\ddot{a}_{\overline{4}|}+v^4\,{}_4p_x\,\ddot{a}_{x+4}$)

What You'll Learn