Joint-Life Survivor Annuity Valuation

Introduction

A pension payable while a retired member is alive may continue at a different amount to an eligible spouse after the member dies. Its value depends on more than the member’s lifetime: the spouse must also be alive when a survivor payment is due. The valuation must therefore distinguish periods when both are alive from periods when only the spouse is alive.

This tutorial values a hypothetical pension with a stated annual amount while the member survives and a stated survivor amount after the member’s death. The focus is the joint-life calculation and the timing of payments. Eligibility is assumed to have been established, and all amounts are hypothetical.

Definition: A joint-life survivor annuity in this setting pays amount $M$ while the member is alive and amount $S$ after the member has died, provided the spouse is alive. Once both are dead, no payment is due. Here $M$ and $S$ are fixed annual payment amounts; a payment is made only on a scheduled payment date when its qualifying life or lives are alive.

Model and Valuation Formula

First use a discrete annual model. Let the member’s future lifetime be $T_x$, and the spouse’s future lifetime be $T_y$, measured in years from the valuation date. Payments are annual in arrears, at times $k=1,2,\ldots$. Assume the two future lifetimes are independent, and use an effective annual interest rate $i$, with $v=1/(1+i)$. Define ${}_kp_x=P(T_x>k)$ and ${}_kp_y=P(T_y>k)$. The same formula applies to a fixed term by ending the sum at its final payment date.

At time $k$, the member’s payment is due with probability ${}_kp_x$. A survivor payment is due when the member has died by time $k$ and the spouse is alive at time $k$. Under independence, that probability is $(1-{}_kp_x){}_kp_y$. Thus the expected payment at time $k$ is $M{}_kp_x+S(1-{}_kp_x){}_kp_y$.

\operatorname{EPV} = \sum_{k=1}^{n}v^k\left[M\,{}_kp_x+S(1-{}_kp_x){}_kp_y\right].

For a whole-life annuity, replace the finite upper limit $n$ by an infinite sum, provided the series converges. Another useful form separates spouse-life payments from joint-life payments. Define the annuity-immediate values

a_x=\sum_{k\geq 1}v^k{}_kp_x,\qquad a_y=\sum_{k\geq 1}v^k{}_kp_y,\qquad a_{xy}=\sum_{k\geq 1}v^kP(T_x>k,T_y>k).

The value $a_{xy}$ is the present value of one unit at each payment date when both lives must be alive. With independent lifetimes, $P(T_x>k,T_y>k)={}_kp_x{}_kp_y$.

Theorem: For annual payments in arrears, amount $M$ while the member is alive and amount $S$ after the member’s death while the spouse is alive, the expected present value is $$ M a_x+S(a_y-a_{xy}). $$ For a finite term, use the corresponding sums only over the scheduled payment dates in that term. Independence is needed to replace joint survival by the product ${}_kp_x{}_kp_y$, but not to use the formula if $a_{xy}$ is calculated directly from the joint survival probabilities.

Why the Decomposition Works

At a payment date, the event “spouse alive but member dead” is the spouse-alive event with the both-alive event removed. This makes the survivor portion $S(a_y-a_{xy})$, rather than $Sa_y$. The same argument also provides a check on direct year-by-year calculations.

Write the payment indicator.
At time $k$, the payment is $M$ if the member is alive, plus $S$ if the spouse is alive and the member is dead.

Rewrite the survivor condition.
The indicator for “spouse alive and member dead” equals the indicator for “spouse alive” minus the indicator for “both alive.”

Take expectations and discount.
The expected present values of these indicators, summed over payment dates, are $a_x$, $a_y$, and $a_{xy}$, giving $Ma_x+S(a_y-a_{xy})$.

This derivation shows exactly what each term counts. If the lifetimes are independent, the joint-life survival probability at time $k$ is the product of the two individual survival probabilities. If independence is not assumed, use the actual joint probability instead; multiplying the marginal survival probabilities would not generally be justified.

Worked Valuations

Worked Example: Three Years of Payments in Arrears

A hypothetical pension pays $12,000 annually while a member is alive. After the member dies, it pays $6,000 annually while the spouse is alive. There are three possible payment dates, at the ends of years 1, 2, and 3. Assume an effective annual interest rate of 5%, independent annual mortality with constant $q_x=0.10$ for the member and $q_y=0.20$ for the spouse, and no payment after year 3. Find the expected present value. Use the direct payment-probability method.

The annual survival probabilities are $p_x=0.90$ and $p_y=0.80$. At time $k$, member survival is $p_x^k$, and the probability of member death by $k$ with spouse survival is $(1-p_x^k)p_y^k$. The discount factors are $v=1/1.05$, $v^2=1/1.1025$, and $v^3=1/1.157625$.

Year 1. The member-alive probability is $0.90$; the survivor-payment probability is $(1-0.90)(0.80)=0.0800$. The expected payment is $12{,}000(0.90)+6{,}000(0.0800)=10{,}800+480=\$11{,}280$. Its present value is $11{,}280/1.05=\$10{,}742.86$.

Year 2. The member-alive probability is $0.90^2=0.81$; the survivor-payment probability is $(1-0.81)(0.80^2)=0.19(0.64)=0.1216$. The expected payment is $12{,}000(0.81)+6{,}000(0.1216)=9{,}720+729.60=\$10{,}449.60$. Its present value is $10{,}449.60/1.1025=\$9{,}478.10$.

Year 3. The member-alive probability is $0.90^3=0.729$; the survivor-payment probability is $(1-0.729)(0.80^3)=0.271(0.512)=0.138752$. The expected payment is $12{,}000(0.729)+6{,}000(0.138752)=8{,}748+832.512=\$9{,}580.512$. Its present value is $9{,}580.512/1.157625=\$8{,}276.01$.

Adding the three discounted expected payments gives $10{,}742.86+9{,}478.10+8{,}276.01=\boxed{\$28{,}496.97}$ using the displayed rounded components. Using unrounded values throughout gives $\boxed{\$28{,}496.96}$, which is the result rounded to the nearest cent. The method avoids treating a spouse who is alive while the member is alive as already receiving the survivor amount. The calculation checks against the decomposition $12{,}000a_x+6{,}000(a_y-a_{xy})$ over these same three payment dates.

Worked Example: Payments in Advance

A hypothetical plan pays $18,000 immediately at retirement and at each of the next two annual payment dates, provided the member is alive. After the member’s death, it pays $9,000 at a scheduled payment date if the spouse is alive. Assume an effective annual interest rate of 4%, independent annual mortality with constant $q_x=0.20$ and $q_y=0.10$, and no payments after time 2. Find the expected present value. Use the scheduled-date probabilities; do not shift the payment dates.

At time 0, both are alive, so the immediate payment is $18,000. At time 1, the member-alive probability is $0.80$, and the survivor-payment probability is $(1-0.80)(0.90)=0.18$. The expected payment is $18{,}000(0.80)+9{,}000(0.18)=14{,}400+1{,}620=\$16{,}020$. Its present value is $16{,}020/1.04=\$15{,}403.85$.

At time 2, the member-alive probability is $0.80^2=0.64$, and the survivor-payment probability is $(1-0.64)(0.90^2)=0.36(0.81)=0.2916$. The expected payment is $18{,}000(0.64)+9{,}000(0.2916)=11{,}520+2{,}624.40=\$14{,}144.40$. Its present value is $14{,}144.40/(1.04)^2=14{,}144.40/1.0816=\$13{,}077.29$.

The expected present value is $18{,}000+15{,}403.85+13{,}077.29=\boxed{\$46{,}481.14}$, rounded to the nearest cent. This uses the annuity-due timing correctly: the time-0 payment is certain under the stated starting condition and is not discounted. The later survivor payments remain conditional on the spouse being alive at their scheduled dates.

Worked Example: Infinite-Term Valuation with Constant Mortality

A hypothetical pension pays $20,000 at each year-end while the member is alive and $10,000 at each year-end after the member’s death if the spouse is alive. Assume an effective annual interest rate of 5%, independent lives, and constant annual death probabilities $q_x=0.10$ and $q_y=0.20$ for every future year. Calculate the whole-life expected present value. Use geometric series for the annuity values.

Here $p_x=0.90$, $p_y=0.80$, and $v=1/1.05$. The member-life annuity has ratio $p_xv=0.90/1.05=6/7$, so $\displaystyle a_x=\sum_{k=1}^{\infty}(6/7)^k=(6/7)/(1-6/7)=6.$

The spouse-life annuity has ratio $p_yv=0.80/1.05=16/21$, so $\displaystyle a_y=(16/21)/(1-16/21)=16/5=3.2.$

The joint-life annuity has ratio $p_xp_yv=(0.90)(0.80)/1.05=24/35$. Therefore $\displaystyle a_{xy}=(24/35)/(1-24/35)=24/11.$ The survivor-only annuity factor is $a_y-a_{xy}=16/5-24/11=(176-120)/55=56/55$.

Thus the value is $20{,}000(6)+10{,}000(56/55)=120{,}000+10{,}181.818\ldots=\boxed{\$130{,}181.82}$, rounded to the nearest cent. As a check, the survivor factor $56/55$ is positive and smaller than the spouse-life factor $16/5$, as it must be: the survivor is paid only during spouse-alive years when the member is no longer alive.

Payment Timing and Common Traps

For payments in arrears, the first scheduled payment is at time 1, so each annuity sum begins at $k=1$. For payments in advance, include a time-0 payment if the contract provides one and the relevant life is alive then. At time 0, the member is alive under the usual valuation-at-retirement setup, while the survivor-only condition cannot yet hold. Later payment dates must still be discounted to the valuation date using their actual times.

A frequent error is to value the member pension as $Ma_x$ and then add $Sa_y$. That second term pays the survivor amount during every period the spouse is alive, including periods when the member is also alive and the full member amount is already being paid. For the first worked example, this incorrect calculation gives approximately $12{,}000a_x+6{,}000a_y=\$37{,}367.02$ over its three payment dates. It overstates the correct value because it fails to subtract the joint-life period. The correction is $Ma_x+S(a_y-a_{xy})$, or the direct probability of spouse alive and member dead at each date.

The formula also depends on the benefit design. It assumes the member amount is payable while the member survives and the stated survivor amount replaces it after the member’s death; it does not treat the survivor amount as an additional payment while both are alive. It assumes no payment after both lives have died and no adjustment for partial years, escalation, or a guaranteed payment period. If any of those conditions differ, the cash flows and event probabilities must be changed before valuation.

Key takeaway: At each scheduled payment date, separate “member alive” from “spouse alive and member dead.” With independent lifetimes, the survivor-payment probability is $(1-{}_kp_x){}_kp_y$; equivalently, value the survivor portion as spouse-life payments minus joint-life payments.

Check Your Understanding

For each problem, use the stated annual payment dates and assumptions. Amounts are hypothetical.

A pension pays $10,000 at each of the next two year-ends while the member is alive and $5,000 after the member’s death while the spouse is alive. Let $i=0$, with independent annual death probabilities $q_x=0.10$ and $q_y=0.20$. Find the expected present value. (Answer: $18,108.00)
A pension pays $8,000 at time 0 and at time 1 while the member is alive, and $4,000 at time 1 if the member has died and the spouse is alive. Let $i=5\%$, $q_x=0.25$, and $q_y=0.10$, with independent lifetimes. Find the value of these payments. (Answer: $14,571.43)
A hypothetical three-year pension pays $15,000 at each year-end while the member is alive, or $6,000 if the member has died and the spouse is alive. Let $i=0$, $q_x=0.20$, and $q_y=0.25$, independently. Calculate its expected present value. (Answer: $32,630.25)
Under independent lives with constant annual death probabilities $q_x=0.20$, $q_y=0.10$, and $i=5\%$, calculate the whole-life value of $12,000 annually while the member lives and $3,000 annually after the member’s death while the spouse lives. Payments are in arrears. (Answer: $49,854.55)
A pension pays $10,000 at each of the next two year-ends while the member is alive, and $5,000 at a payment date if the member has died and the spouse is alive. Let $i=10\%$, $q_x=0.20$, and $q_y=0.10$, with independent lifetimes. Find the expected present value to the nearest cent. (Answer: $14,585.12)

What You'll Learn