Spouse Age Difference and Survivor-Pension Cost

Why Spouse Age Matters

A continuing survivor pension is paid to a spouse after the pension member dies, for as long as the spouse remains alive. Its expected cost therefore depends on two survival patterns: how quickly the member is expected to die and how long the spouse is expected to survive. If the member’s assumptions are held fixed, changing the spouse’s assumed age can change the expected payment period and therefore the value of the benefit.

The effect is not captured by valuing the spouse’s lifetime pension alone. The benefit is payable only after the member has died. A valuation must account for both the spouse being alive and the member having died by each payment date. The calculations below use a clearly stated annual payment convention so that these conditions are explicit.

Definition: Suppose a member and spouse are both alive at retirement, time 0. A level annual survivor pension of amount $B$ is payable at times $t=1,2,\ldots$ if the member has died by time $t$ and the spouse is alive at time $t$. Payments stop at the spouse’s death. Let ${}_tp_x$ and ${}_tp_y$ be the probabilities that the member and spouse, respectively, survive from time 0 to time $t$. If their future lifetimes are independent, the probability of a payment at time $t$ is $(1-{}_tp_x){}_tp_y$.

This convention values payments at annual dates, not at the exact time of the member’s death. It assumes no payment on the member’s death itself and no partial-year payment. In a practical valuation, the payment frequency and timing must match the benefit terms. Independence is also a modeling assumption, not a necessary fact: if the lives are dependent, the joint probability must be modeled directly rather than replaced by a product.

Present Value Under Annual Payment Timing

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. For a benefit payable for at most $n$ years, the expected present value at time 0 is the discounted sum of the benefit times its payment probability at each date.

\operatorname{EPV}_n = B\sum_{t=1}^{n}v^t(1-{}_tp_x){}_tp_y.

For a benefit payable for the spouse’s remaining lifetime, use the same sum over all positive integers $t$, provided the resulting series converges. Notice that $1-{}_tp_x$ is the probability the member has died by time $t$; it is not the probability the member dies during year $t$. The difference matters because a payment at time $t$ is based on whether death has occurred by that date.

Why the Spouse-Only Annuity Is Not the Benefit

Under independence, the payment probability can be separated into two terms:

(1-{}_tp_x){}_tp_y = {}_tp_y-{}_tp_x{}_tp_y.

The first term represents the spouse being alive at time $t$. The second represents both lives being alive, which must be excluded because the survivor pension has not yet started. Thus, the survivor pension’s value is the value of a spouse-only annuity less the value of payments made while both are alive. This is a useful derivation, but each term must use the correct survival probabilities and payment timing.

Result: Under independent lifetimes and the annual-in-arrears convention, the value of a continuing survivor pension is the present value of payments while the spouse is alive minus the present value of payments while both member and spouse are alive. This subtraction is valid because the survivor payment event is “spouse alive” excluding “both alive.”

Constant Annual Mortality Probabilities

A compact comparison is possible if the annual survival probabilities are constant. Suppose the member survives each year with probability $p_m$, the spouse survives each year with probability $p_s$, and the lives are independent. Then ${}_tp_x=p_m^t$ and ${}_tp_y=p_s^t$. For an unlimited survivor pension,

\begin{aligned} \operatorname{EPV} &=B\sum_{t=1}^{\infty}v^t(p_s^t-(p_mp_s)^t)\\ &=B\left[ \frac{vp_s}{1-vp_s} - \frac{vp_mp_s}{1-vp_mp_s} \right], \end{aligned}

provided $vp_s<1$ and $vp_mp_s<1$. These are geometric series: for example, the spouse-only term has first term $vp_s$ and common ratio $vp_s$, so its sum is $vp_s/(1-vp_s)$. Since $v=1/(1+i)$, the same factors can be written $p_s/(1+i-p_s)$ and $p_mp_s/(1+i-p_mp_s)$.

This constant-probability model is a simplifying assumption, not a complete age-based mortality table. It is useful for understanding the direction and size of an age assumption’s effect. In a fuller valuation, use the survival probabilities appropriate to each age and duration.

Worked Example: Comparing Younger, Same-Age, and Older Spouse Assumptions

A hypothetical plan promises a level annual survivor pension of $6,000, payable at each year-end after the member’s death if the spouse is alive. Use an effective annual interest rate of 5%. Assume independent lifetimes and constant annual death probabilities: 10% for the member; for the spouse, 5% under a younger-spouse assumption, 10% under a same-age assumption, or 20% under an older-spouse assumption. Find the value under each assumption.

Method. Apply the constant-probability formula. The member’s annual survival probability is $p_m=1-0.10=0.90$, and $v=1/1.05$. For each spouse assumption, calculate the spouse-only factor and subtract the joint-life factor.

Younger spouse. Here $p_s=0.95$. The spouse-only factor is $0.95/(1.05-0.95)=9.5$. The joint-life survival probability is $p_mp_s=0.90(0.95)=0.855$, giving a joint-life factor of $0.855/(1.05-0.855)=0.855/0.195=4.384615$. The survivor factor is $9.5-4.384615=5.115385$. The value is $6{,}000(5.115385)=\$30{,}692.31$, rounded to cents.

Same-age spouse. Here $p_s=0.90$. The spouse-only factor is $0.90/(1.05-0.90)=6$. The joint-life probability is $0.90(0.90)=0.81$, and its factor is $0.81/(1.05-0.81)=0.81/0.24=3.375$. The survivor factor is $6-3.375=2.625$, so the value is $6{,}000(2.625)=\$15{,}750.00$.

Older spouse. Here $p_s=0.80$. The spouse-only factor is $0.80/(1.05-0.80)=3.2$. The joint-life probability is $0.90(0.80)=0.72$, and its factor is $0.72/(1.05-0.72)=0.72/0.33=2.181818$. The survivor factor is $3.2-2.181818=1.018182$, so the value is $6{,}000(1.018182)=\$6{,}109.09$.

The results decrease as the assumed spouse mortality increases: $30,692.31, $15,750.00, and $6,109.09. This direction follows from the stated survival assumptions; it is not a rule that an age difference alone determines cost without a mortality basis.

Using Age-Specific Survival Probabilities

When survival probabilities vary by age or duration, use the finite-sum formula or the corresponding sum over all payment dates. Calculate cumulative survival to each payment date first. Then calculate the chance the member has died by that date, multiply by the spouse’s probability of being alive, and discount the result. If only a finite number of payments is covered, do not extend the sum beyond the stated term.

Worked Example: Three Payment Dates With No Interest

A hypothetical survivor pension pays $2,000 at each of times 1, 2, and 3 if the member has died by that date and the spouse is alive. Assume independent lifetimes and no interest. The member’s cumulative survival probabilities are ${}_1p_x=0.90$, ${}_2p_x=0.72$, and ${}_3p_x=0.54$. The spouse’s corresponding cumulative survival probabilities under a specified age assumption are ${}_1p_y=0.95$, ${}_2p_y=0.855$, and ${}_3p_y=0.684$. Find the expected present value.

Method. For each date, multiply the probability the member has died by the spouse’s cumulative survival probability. With no interest, each payment has discount factor 1.

At time 1, the member’s probability of having died is $1-0.90=0.10$. The payment probability is $0.10(0.95)=0.095$, giving expected payment $2{,}000(0.095)=\$190.00$.

At time 2, the member’s probability of having died is $1-0.72=0.28$. The payment probability is $0.28(0.855)=0.2394$, giving expected payment $2{,}000(0.2394)=\$478.80$.

At time 3, the member’s probability of having died is $1-0.54=0.46$. The payment probability is $0.46(0.684)=0.31464$, giving expected payment $2{,}000(0.31464)=\$629.28$.

The total is $\$190.00+\$478.80+\$629.28=\$1{,}298.08$. Because the interest rate is zero, this total is also the expected present value. The cumulative survival values are used directly; they are not multiplied together across dates.

Worked Example: Measuring the Effect With Finite, Age-Specific Data

A hypothetical plan pays $5,000 at each of times 1 and 2 if the member has died by that date and the spouse is alive. The effective annual interest rate is 5%, and the lives are independent. The member’s cumulative survival probabilities are 0.92 at time 1 and 0.80 at time 2. Compare two spouse-age assumptions: under the younger-spouse assumption, spouse survival is 0.97 and 0.92; under the same-age assumption, it is 0.94 and 0.88; under the older-spouse assumption, it is 0.90 and 0.81.

Method. The member’s probabilities of having died by the two dates are $1-0.92=0.08$ and $1-0.80=0.20$. The discount factors are $v=1/1.05$ and $v^2=1/1.1025$.

For the younger-spouse assumption, the payment probabilities are $0.08(0.97)=0.0776$ and $0.20(0.92)=0.184$. Thus $5{,}000[0.0776/1.05+0.184/1.1025]=5{,}000(0.07390476+0.16689342)=\$1{,}203.99$, rounded to cents.

For the same-age assumption, the probabilities are $0.08(0.94)=0.0752$ and $0.20(0.88)=0.176$. Thus $5{,}000[0.0752/1.05+0.176/1.1025]=5{,}000(0.07161905+0.15963719)=\$1{,}156.28$.

For the older-spouse assumption, the probabilities are $0.08(0.90)=0.072$ and $0.20(0.81)=0.162$. Thus $5{,}000[0.072/1.05+0.162/1.1025]=5{,}000(0.06857143+0.14693878)=\$1{,}077.55$.

The younger-spouse value exceeds the older-spouse value by $\$1{,}203.99-\$1{,}077.55=\$126.44$. This comparison holds the member survival, benefit, interest, and payment dates fixed; only the spouse survival assumption changes.

Common Mistakes and Practical Interpretation

A frequent error is to treat the spouse-only annuity as the survivor pension. In the older-spouse case of the first example, the spouse-only factor is 3.2, so its value is $6{,}000(3.2)=\$19{,}200.00$. That is not the survivor-pension value: it includes payments while both lives are alive. The joint-life factor is $0.72/0.33=2.181818$, worth $6{,}000(2.181818)=\$13{,}090.91$. Subtracting gives $\$19{,}200.00-\$13{,}090.91=\$6{,}109.09$, the correct survivor-pension value.

Other checks help prevent errors:

Use cumulative survival at each payment date. The chance the member has died by $t$ is $1-{}_tp_x$, not the one-year death probability unless $t=1$.
Apply the stated payment event. A benefit payable at year-end after member death uses the probability the member has died by that date and the spouse is alive then. A benefit payable immediately on death requires a different timing model.
Keep age and mortality assumptions connected. An assumed age difference affects cost through the spouse survival probabilities selected. Do not infer numerical survival changes without a stated mortality basis.
Check dependence and term. The product of member-death and spouse-survival probabilities requires independence. A finite benefit term requires stopping the sum at the final eligible payment date.

The calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a valuation, state the spouse-age assumption alongside the mortality basis, benefit percentage, payment timing, and interest basis. That makes the source of a cost difference auditable and helps separate the effect of age from changes in other plan terms.

Check Your Understanding

For each problem, use the stated annual payment timing and assumptions. Round money to cents.

A benefit of $1,000 is paid at times 1 and 2 if the member has died by the payment date and the spouse is alive. There is no interest. The member’s cumulative survival probabilities are 0.80 and 0.64; the spouse’s are 0.90 and 0.81. Assume independence. Find the value. (Answer: $471.60)
A hypothetical plan pays $3,000 at times 1 and 2. The effective annual interest rate is 4%. Member survival probabilities to those dates are 0.95 and 0.85; spouse survival probabilities are 0.98 and 0.94. Assume independence. Find the expected present value. (Answer: $532.43)
Assume a perpetual annual survivor pension of $2,500, paid in arrears, with a 5% effective annual interest rate. Annual survival probabilities are constant and independent: member $p_m=0.88$, spouse $p_s=0.82$. Find the value. (Answer: $3,419.74)
For a perpetual annual survivor pension of $1,000 at 5% interest, the member’s annual survival probability is 0.90. Compare spouse survival probabilities 0.95 and 0.85 under independent, constant mortality. What is the value under each assumption, and how much greater is the first value? (Answer: $5,115.38 under 0.95; $1,565.79 under 0.85; difference $3,549.60)

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