Survivor Benefits With Age-Dependent Annuity Values

Why the Spouse’s Age Matters

A survivor pension is generally valued using the expected duration of payments to the surviving spouse. That duration depends on the spouse’s age when the pension begins, not simply on the member’s age at the valuation date. When a member’s death can trigger benefits in different future years, the spouse will have a different attained age at each possible start date. The annuity factor must be selected separately for each date.

This tutorial develops a discrete-time valuation for a hypothetical pension arrangement. It focuses on applying a specified spouse-age relationship to select the correct survivor annuity value, then combining that value with the probability and timing of the trigger. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $x$ be the member’s age and $y$ the spouse’s age at the valuation date. Define the spouse-age difference as $r=y-x$, so $r>0$ means the spouse is older. If a survivor pension starts at the end of year $k$, the member’s attained age is $x+k$ and the spouse’s attained age is $y+k=x+k+r$. Use the annuity value for the spouse’s attained age at that pension start date.

Assume annual effective interest $i$, with $v=1/(1+i)$. Let $\pi_k$ denote the probability that the member is eligible at the start of year $k$ and experiences the event that triggers the survivor pension during that year. Let ${}_kp_y$ be the probability that the spouse, age $y$ at the valuation date, survives to the end of year $k$. If the two lives are independent, their joint probability of a trigger in year $k$ and a living spouse at that date is $\pi_k\,{}_kp_y$.

Let $B_k$ be the annual survivor pension and $\ddot a_{y+k}$ the annuity-due value per dollar of pension for a life aged $y+k$, on the stipulated mortality and interest basis. An annuity-due begins with a payment at the pension start date. Thus, if the pension starts immediately when the trigger occurs at the end of year $k$, its present value at that date is $B_k\ddot a_{y+k}$.

Model assumptions: The trigger occurs at the end of an integer policy year; the pension starts immediately then and is paid annually in advance while the spouse is alive; the factor $\ddot a_{y+k}$ is conditional on the spouse being alive at the start date; and the stated factors use the same interest and mortality basis as the valuation. Independence is required only when the joint trigger-and-spouse-survival probability is calculated as a product. If the lives are dependent, use the appropriate joint probability instead.

Valuation Formula and Its Justification

When the survivor pension amount is constant, the expected present value at the valuation date is the sum of the contributions from all possible trigger years:

\operatorname{EPV} = \sum_{k\geq 1} v^k\,\pi_k\,{}_kp_y\,B_k\,\ddot a_{y+k}.

The factor $v^k$ discounts from the end of year $k$ to the valuation date. The factor ${}_kp_y$ is the spouse’s probability of being alive when the pension begins; it does not represent survival during the subsequent annuity. That later survival is already reflected in $\ddot a_{y+k}$. Multiplying by a second spouse-survival factor for the years after commencement would count mortality twice.

Identify the trigger-year event.
For each year $k$, determine the probability $\pi_k$ that the member is eligible and experiences the specified trigger during that year.

Determine the spouse’s status and age at commencement.
Calculate the probability that the spouse is alive at the end of year $k$, and set the attained spouse age to $y+k$.

Select the matching annuity value.
Use the survivor annuity factor for age $y+k$, the stated payment timing, and the given basis.

Value and discount the contribution.
Multiply the joint trigger-and-eligibility probability by the pension and annuity factor, then by $v^k$. Sum across trigger years.

The formula follows by partitioning the possible outcomes according to the year in which the trigger first occurs. Let $I_k$ equal 1 if the trigger occurs in year $k$ and the spouse is alive at that date, and 0 otherwise. Conditional on $I_k=1$, the pension’s value at the trigger date is $B_k\ddot a_{y+k}$; its value at the valuation date is $v^kB_k\ddot a_{y+k}$. The present value is the sum of these discounted amounts multiplied by their event indicators. Taking expectations and using $E[I_k]=\pi_k{}_kp_y$ under independence gives the displayed sum. If the trigger-year events are mutually exclusive, this sum directly adds the value from each possible trigger date rather than valuing multiple pensions for the same member.

Worked Example: A Younger Spouse and Two Possible Trigger Years

A hypothetical member is age 60 and the spouse is age 56. A survivor pension of $6,000 per year starts immediately at the end of the member’s death year. The annual effective interest rate is 4%. Assume the member has no other exit before death, and use the following probabilities and annuity-due factors: death probability in year 1 is 0.030; conditional death probability in year 2, given survival through year 1, is 0.040; spouse survival to the end of year 1 is 0.990 and to the end of year 2 is 0.975. The lives are independent. The applicable factors are $\ddot a_{57}=12.4$ and $\ddot a_{58}=12.0$. Find the expected present value.

Method. Value each trigger year separately. The spouse is four years younger, so at the end of years 1 and 2 the spouse is age 57 and 58, respectively. The age-specific factors supplied for those ages are the ones to use.

Year 1. The joint probability of member death and spouse survival to the end of year 1 is $0.030(0.990)=0.0297$. The pension value at that date is $6{,}000(12.4)=74{,}400$. Discounting one year gives

v(0.0297)(74{,}400) = \frac{2{,}209.68}{1.04} = 2{,}124.69.

Year 2. The member must survive year 1 and die in year 2, so the trigger probability is $(1-0.030)(0.040)=0.97(0.040)=0.0388$. Including spouse survival to the end of year 2 gives $0.0388(0.975)=0.03783$. The pension value at that date is $6{,}000(12.0)=72{,}000$. Therefore,

v^2(0.03783)(72{,}000) = \frac{2{,}723.76}{1.04^2} = \frac{2{,}723.76}{1.0816} = 2{,}518.27.

The total expected present value is $2{,}124.69+2{,}518.27=\mathbf{\$4{,}642.96}$, rounded to the nearest cent. The year-2 member survival probability is included before the year-2 death probability; omitting it would incorrectly treat a member who died in year 1 as exposed to a year-2 trigger.

Applying the Age Relationship Correctly

The age difference remains constant under the usual assumption that both ages advance by one year per year. It is the attained age at the trigger date—not the age difference itself—that determines which factor to select. If a table gives factors only at integer ages but the problem specifies fractional ages, follow the stated table convention. For example, use linear interpolation only when that convention is specified or appropriate to the supplied basis; do not silently substitute the nearest age.

Worked Example: Spouse Older Than the Member

A member is age 64 and the spouse is age 70. The member survives year 1 and then dies during year 2 with conditional probability 0.050. The spouse’s probability of surviving to the end of year 2 is 0.960. Assume independence. A pension of $5,000 per year starts immediately at the trigger date. Use $i=4\%$ and $\ddot a_{72}=10.0$. Find the expected present value.

Method. The spouse is six years older than the member. At the end of year 2, the member is age 66 and the spouse is age $70+2=72$; therefore, the factor is $\ddot a_{72}$, not a factor for age 66.

The member’s probability of surviving year 1 and dying in year 2 is $0.950(0.050)=0.0475$. The joint probability that this trigger occurs and the spouse is alive is $0.0475(0.960)=0.0456$. The annuity value at the trigger date is $5{,}000(10.0)=50{,}000$. Discounting two years,

\operatorname{EPV} = \frac{0.0456(50{,}000)}{1.04^2} = \frac{2{,}280}{1.0816} = \mathbf{\$2{,}107.99}.

The result is rounded to the nearest cent. Using the member’s attained age to choose the annuity factor would apply the wrong mortality value, even though the member’s age is relevant to the probability of the trigger.

Worked Example: Interpolating at the Spouse’s Attained Age

A member is age 60 and the spouse is 1.5 years older. A trigger occurs at the end of year 2 with probability $0.98(0.020)$, where 0.98 is the member’s probability of remaining eligible through year 1 and 0.020 is the conditional trigger probability in year 2. The spouse survives to the end of year 2 with probability 0.990, independently. The pension is $4,000 per year, payable immediately and then annually in advance. Use $i=5\%$. The supplied annuity table gives $\ddot a_{63}=13.0$ and $\ddot a_{64}=12.6$, with linear interpolation specified. Find the expected present value.

At the trigger date the spouse is age $60+1.5+2=63.5$. Linear interpolation halfway between the two supplied ages gives

\ddot a_{63.5} = 13.0+\frac{63.5-63}{64-63}(12.6-13.0) = 13.0+0.5(-0.4) = 12.8.

The trigger probability is $0.98(0.020)=0.0196$. Including spouse survival gives $0.0196(0.990)=0.019404$. The value at the trigger date is $4{,}000(12.8)=51{,}200$, and discounting two years gives

\operatorname{EPV} = \frac{0.019404(51{,}200)}{1.05^2} = \frac{993.4848}{1.1025} = \mathbf{\$901.12}.

The result is rounded to the nearest cent. Using the member’s age, or rounding the spouse’s age to 63 before selecting a factor, would contradict the specified interpolation rule.

Timing, Assumptions, and Common Traps

The annuity factor and discount exponent answer different questions. The annuity factor values payments from the survivor pension’s start date onward. The discount factor moves that value back from the start date to the original valuation date. If the pension begins one year after the trigger rather than immediately, the payment timing changes: the supplied annuity factor must match the delayed start, and the trigger-date value must reflect the additional waiting period. Do not use an annuity-due factor merely because payments are annual; it is appropriate here because the problem specifies a payment at the start date.

Likewise, $\pi_k$ must match the event being valued. If other exits can prevent a member from reaching year $k$, the trigger probability must include survival in the eligible state through those earlier years. In a simple model with no other exits, conditional annual death probabilities $q_1,q_2,\ldots$ give a year-$k$ death probability ${}_{k-1}p_xq_k$. In a model with competing exits, use the probability of being eligible at the start of the year and then experiencing the specified trigger; do not replace it with a mortality-only survival probability unless that is the model’s assumption.

Common exam trap: Applying the member’s attained age to the survivor annuity factor. For a member aged 60 with a spouse aged 56, a trigger at the end of year 3 starts the pension when the spouse is age 59. The factor must be selected for age 59, not age 63. The member’s attained age may matter for the trigger probability, but it does not determine the spouse’s annuity factor.

A second frequent error is to include spouse survival twice: once in the probability that the spouse is alive at pension commencement and again as a separate multiplier on the life annuity factor. The annuity factor already allows for the spouse’s future survival after commencement. Conversely, if a problem supplies a factor that is unconditional on the spouse being alive at commencement, check its definition before multiplying by a separate survival probability.

Key takeaway: For each possible trigger date, first find the spouse’s attained age, then select the corresponding annuity factor. Multiply that factor by the pension and the probability of a valid trigger with an eligible spouse, and discount the resulting trigger-date value back to the valuation date.

Check Your Understanding

For each problem, use the stated age relationship and payment timing. Round monetary answers to the nearest cent.

A trigger occurs at the end of year 1 with probability 0.025. The spouse survives to that date with probability 0.980, independently. The pension is $4,000 per year, payable immediately, and the factor at the spouse’s attained age is 11.0. If $i=4\%$, find the expected present value. (Answer: $1,036.54)
A member has a 0.940 probability of being eligible at the start of year 3 and a 0.050 conditional trigger probability in that year. The spouse survives to the trigger date with probability 0.970, independently. The pension is $8,000 per year, the correct attained-age factor is 9.2, and $i=5\%$. Find the year-3 contribution. (Answer: $2,898.54)
A member is age 65 and the spouse is age 61. A trigger occurs at the end of year 2; the member survives year 1 with probability 0.970 and then has conditional trigger probability 0.030. The spouse survives to that date with probability 0.990. The applicable factor is $\ddot a_{63}=12.1$, the annual pension is $6,000, and $i=4\%$. Find the expected present value. (Answer: $1,933.74)
A member is age 70 and the spouse is three years older. A year-1 trigger has probability 0.020, and the spouse’s probability of being alive at the end of that year is 0.950, independently. The pension is $5,000 per year and starts immediately. The factor at the spouse’s attained age is 9.8, and $i=5\%$. Find the expected present value. (Answer: $886.67)
For a year-2 trigger, the joint probability of a valid member trigger and an eligible spouse alive at that date is given directly as 0.030; no independence assumption is needed. The annual pension is $10,000, the applicable annuity-due factor is 7.5, and $i=4\%$. Find the expected present value contribution. (Answer: $2,080.25)

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