Guarantee Periods in Pension Options

Why a Guarantee Period Needs Its Own Valuation

A pension option may promise that payments continue for a minimum period even if the retired member dies soon after retirement. This promise changes the value of the pension: some installments are payable regardless of the member’s survival, while later installments may depend on the member’s or a spouse’s survival. The key is to keep those promises separate.

This tutorial uses annual payments made at the ends of years. A guarantee of $n$ years means that the first $n$ installments are paid in full, even if the member dies before they are due. Any installments after that period are valued under the stated continuation terms. Unless specified otherwise, the member is alive at retirement, and a guaranteed installment unpaid at the member’s death goes to the designated beneficiary or estate.

Definition: A guarantee period of $n$ years ensures that the first $n$ scheduled pension installments are paid, whether or not the retired member survives to their payment dates. A continuing survivor pension is a separate benefit: after the guarantee period, it pays a stated fraction of the pension if the member has died and an eligible spouse is alive.

A guarantee does not, by itself, create a lifetime pension for a spouse. For example, the guarantee may ensure full installments for a fixed period, while a separate survivor provision pays a fraction only after that period. The contract terms determine whether these benefits apply; the valuation must reflect those terms without overlapping or omitting payments.

Valuation Model and Main Formula

Let the annual pension installment be $B$, the effective annual interest rate be $i$, and $v=1/(1+i)$. Payments are made at the ends of years $1,2,\ldots$. The first $n$ payments are guaranteed. After year $n$, the member receives $B$ if alive at the payment date. If the member is dead and an eligible spouse is alive, the spouse receives fraction $r$ of $B$, where $0\le r\le1$.

Write ${}_kp_x$ and ${}_kp_s$ for the probabilities that the member and spouse, respectively, survive from retirement to the end of year $k$. For the following formula, assume their survival is independent. The spouse must be alive at the payment date to receive the continuing survivor pension.

\operatorname{EPV} = B\sum_{k=1}^{n}v^k + B\sum_{k=n+1}^{\infty}v^k \left[ {}_kp_x+r(1-{}_kp_x){}_kp_s \right].

The first sum is certain: each of those installments is paid, regardless of either life’s survival. After the guarantee expires, the two payment events are mutually exclusive. The member receives the full installment if alive; otherwise, an eligible living spouse receives the fraction $r$. Independence gives the probability of the second event as $(1-{}_kp_x){}_kp_s$.

Condition: If the member’s and spouse’s survival are dependent, do not replace the probability that the member is dead and the spouse alive by $(1-{}_kp_x){}_kp_s$. Use the appropriate joint probability $P(\text{member dead by }k,\text{ spouse alive at }k)$ instead.

The formula also assumes that the pension installments continue indefinitely unless the contract specifies an end date. For a temporary pension, replace the infinite sum by a sum over the actual remaining payment dates. If payments are monthly or are made at the start of each period, adjust the payment dates and discount factors rather than applying this end-of-year formula unchanged.

Why the Post-Guarantee Formula Works

The expected present value of a payment at time $k$ is its amount multiplied by the probability it is paid and by its discount factor. After the guarantee period, let the payment amount be $B$ when the member is alive and $rB$ when the member is dead but the spouse is alive. These events cannot both occur at the same payment date.

\begin{aligned} E[\text{payment at }k] &=B\,P(\text{member alive at }k) +rB\,P(\text{member dead by }k,\text{ spouse alive at }k)\\ &=B\left[{}_kp_x+r(1-{}_kp_x){}_kp_s\right]. \end{aligned}

The second equality uses independence. Multiplying the expected payment by $v^k$ gives its present value contribution; summing over the eligible payment dates gives the post-guarantee term. During the guarantee period, the payment amount is $B$ with probability one, so its expected present value contribution is simply $Bv^k$.

When annual survival probabilities are constant, say $p_x$ for the member and $p_s$ for the spouse, then ${}_kp_x=p_x^k$ and ${}_kp_s=p_s^k$. For an indefinite pension, the post-guarantee series can then be evaluated with geometric sums, provided the relevant ratios are less than one:

\sum_{k=n+1}^{\infty}v^k p_x^k = \frac{(vp_x)^{n+1}}{1-vp_x}, \qquad \sum_{k=n+1}^{\infty}v^k p_s^k = \frac{(vp_s)^{n+1}}{1-vp_s}.

For the independent-life model, the survivor component is $r$ times the difference between the spouse-survival series and the series in which both lives survive. This follows because, at time $k$, the probability the spouse is alive but the member is not is $p_s^k-(p_xp_s)^k$.

Worked Valuations

Worked Example: Lifetime Pension With a Two-Year Guarantee and Survivor Pension

A hypothetical pension pays $B=\$1{,}200$ at each year-end. The first two payments are guaranteed. After year 2, the member receives the full pension while alive; if the member has died, an eligible spouse alive at the payment date receives 50% of the pension. The effective annual interest rate is 5%. Assume independent lifetimes and constant annual survival probabilities $p_x=0.80$ and $p_s=0.90$. Find the expected present value, rounded to the nearest cent.

Method. Value the two certain payments, then value the member and survivor streams separately from year 3 onward. Here $v=1/1.05=20/21$, $vp_x=16/21$, $vp_s=6/7$, and $vp_xp_s=24/35$.

The guaranteed-payment factor is $v+v^2=20/21+400/441=820/441=1.85941043$. The member’s post-guarantee factor is $(16/21)^3/[1-(16/21)]=4096/2205=1.85759637$.

For the spouse-survivor stream, the probability factor at year $k$ is $0.90^k-0.72^k$: the spouse is alive, but the member is not. Discounting and summing from year 3 gives $(6/7)^3/[1-6/7]-(24/35)^3/[1-24/35]=216/49-13824/13475=4.40816327-1.02589981=3.38226345$. The spouse receives half, so this contributes $0.5(3.38226345)=1.69113173$.

The total factor is $1.85941043+1.85759637+1.69113173=5.40813853$. Therefore the expected present value is $\$1{,}200(5.40813853)=\$6{,}489.77$, rounded to the nearest cent.

Check. The first two installments contribute exactly $\$1{,}200(v+v^2)$; mortality affects only payments after year 2. The survivor component is positive but smaller than the full spouse-survival series because it excludes dates when the member is also alive.

Worked Example: Finite Pension With a Guarantee and a Survivor Continuation

A hypothetical pension pays $\$1{,}800$ at the ends of years 1 through 4 only. The first two payments are guaranteed. For years 3 and 4, the member receives the full installment if alive; otherwise a living spouse receives 40%. There is no interest. Assume independent lives with constant annual survival probabilities $p_x=0.80$ and $p_s=0.90$. Find the expected present value to the nearest cent.

Method. Since $i=0$, each discount factor is 1. The guaranteed installments total $2(\$1{,}800)=\$3{,}600$. For each remaining date, calculate the member-alive probability and the probability that the member is dead while the spouse is alive.

At year 3, the member-alive probability is $0.80^3=0.512$. The probability the member is dead and the spouse alive is $(1-0.512)(0.90^3)=0.488(0.729)=0.355752$. Thus the expected year-3 payment is $\$1{,}800[0.512+0.40(0.355752)]=\$1{,}800(0.6543008)=\$1{,}177.74144$.

At year 4, the member-alive probability is $0.80^4=0.4096$. The probability the member is dead and the spouse alive is $(1-0.4096)(0.90^4)=0.5904(0.6561)=0.38736144$. The expected year-4 payment is $\$1{,}800[0.4096+0.40(0.38736144)]=\$1{,}800(0.564544576)=\$1{,}016.1802368$.

Adding the guaranteed installments and the two expected later payments gives $\$3{,}600+\$1{,}177.74144+\$1{,}016.1802368=\$5{,}793.9216768$, or \$5,793.92 to the nearest cent.

Check. The year-3 and year-4 expected payment factors are each below 1 because there is a chance neither the member nor the spouse is alive at the payment date. The first two payments are not reduced for mortality.

Worked Example: Measuring the Cost of a Two-Year Guarantee

A hypothetical life-only pension pays $\$1{,}600$ annually at year-end while the member is alive. Compare its value with the value when the first two installments are guaranteed, but there is no survivor pension. Use an effective annual interest rate of 4% and constant annual member survival probability $p_x=0.75$. Round final dollar amounts to the nearest cent.

Method. The life-only pension has expected payment factor $\sum_{k=1}^{\infty}(vp_x)^k$. Under the guarantee, replace the first two mortality-weighted installments by certain installments. Here $v=1/1.04=25/26$ and $vp_x=75/104$.

The life-only factor is $(75/104)/[1-(75/104)]=75/29=2.58620690$. Its value is $\$1{,}600(75/29)=\$4{,}137.93$.

For the guaranteed pension, the first two installments have factor $v+v^2=25/26+625/676=1.88609467$. From year 3 onward, the member-only factor is $(75/104)^3/[1-75/104]=421875/313664=1.34499018$. The total factor is $1.88609467+1.34499018=3.23108485$, giving a value of $\$1{,}600(3.23108485)=\$5{,}169.74$.

The guarantee increases value by $\$5{,}169.74-\$4{,}137.93=\$1{,}031.81$. Directly, the increase in factor is $v(1-0.75)+v^2(1-0.75^2)=25/104+4375/10816=0.64487796$, which gives $\$1{,}600(0.64487796)=\$1{,}031.80$ before rounding from unrounded values. Using the unrounded totals, the difference is $\$1{,}031.8047$, which rounds to $\$1{,}031.80$. Thus the values rounded separately differ by one cent from the correctly rounded difference; the guarantee’s incremental value is **\$1,031.80**.

Check. The incremental value is the present value of the extra probability of payment in years 1 and 2: $1-{}_kp_x$ for each guaranteed installment. The later installments have the same terms in both options and cancel in the comparison.

Common Exam Traps

A frequent error is to apply the member’s survival probability to guaranteed installments. That incorrectly reduces payments that are promised even after early death. Another is to value a continuing survivor pension as though the spouse is certain to survive whenever the member dies.

For instance, in the finite example, the incorrect year-3 factor $0.512+0.40(1-0.512)=0.7072$ assumes the spouse always survives after the member dies. It gives an expected year-3 payment of $\$1{,}272.96$, rather than the correct $\$1{,}177.74$. The omitted spouse-survival probability is $0.729$. Likewise, adding a survivor fraction to a guaranteed installment during the guarantee period can count the same payment twice: the guarantee already provides the full installment.

Mark the guarantee dates.
List the first $n$ scheduled payments; value each as certain unless the contract says otherwise.

Apply the post-guarantee terms.
At each later payment date, distinguish member alive from member dead with an eligible spouse alive.

Discount and check coverage.
Discount each expected payment to retirement and confirm that no date is omitted or counted twice.

Key takeaway: A guarantee period makes specified installments certain; it does not automatically promise a lifetime survivor pension. Value guaranteed payments separately, then apply the stated post-guarantee survival and survivor-benefit conditions. This framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated payment timing and assumptions. Round dollar results to the nearest cent where needed.

A pension pays $\$900$ at each year-end. The first three installments are guaranteed, and there are no other benefits. At an effective annual interest rate of 4%, find the present value of the guaranteed installments. (Answer: $\$2{,}497.58$)
A pension pays $\$1{,}000$ annually at year-end, with the first two installments guaranteed and no survivor pension thereafter. The member survives each year with probability 0.80, independently across years; $i=0$. Find the expected present value of the lifetime pension. (Answer: $\$4{,}560.00$)
A pension pays $\$1{,}500$ at the ends of years 1 through 3. The first two payments are guaranteed. At year 3, the member receives the full amount if alive; otherwise an eligible living spouse receives 50%. At $i=0$, assume independent constant annual survival probabilities $p_x=0.75$ and $p_s=0.80$. Find the expected present value. (Answer: $\$3{,}854.81$)
A lifetime pension pays $\$1{,}000$ at each year-end, with one guaranteed installment. Thereafter, a living member receives the full amount and a living spouse receives 50% if the member has died. At $i=0$, assume independent constant annual survival probabilities $p_x=0.50$ and $p_s=0.80$. Find the expected present value. (Answer: $\$2{,}966.67$)
A pension pays $\$1{,}000$ annually, with the first two installments guaranteed. At year 3, the member receives the full installment if alive; otherwise a living spouse receives 40%. There is no interest. Given $P(\text{member alive at year 3})=0.65$, $P(\text{spouse alive at year 3})=0.72$, and $P(\text{both alive at year 3})=0.50$, find the expected value of the first three installments. (Answer: $\$2{,}738.00$)

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