Survivor Pension Values After Member Death

Introduction

A survivor pension has two distinct stages. First, an active member must die while eligible for the benefit. Then, if the benefit is triggered, payments depend on the survivor’s age, the pension’s commencement terms, and the survivor’s subsequent survival. Valuation requires both stages: the member’s death is not itself the full stream of pension payments.

This tutorial develops a discrete annual valuation for a hypothetical pension plan. It assumes that member deaths and other active-member exits are described by competing decrement probabilities, while survivor payments depend on the survivor’s own mortality. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A survivor pension triggered by active-member death is a contingent benefit payable to an eligible survivor if the active member dies before another exit that terminates eligibility. Its value is the expected present value of the pension payments, allowing for the probability and timing of the triggering death and for the survivor’s survival after that death.

Separate the Trigger From the Payments

Use an annual model in which active-member decrements occur at the end of each year. Let $D_k$ be the probability, measured at time 0, that the member dies as an active member at the end of year $k$. If annual probabilities are specified conditionally on being active at the start of each year, then $D_k$ includes the probability of remaining active through the earlier years.

For example, if $q_{x+k-1}^{(d)}$ is the conditional probability of death during year $k$, given active status at its start, and ${}_{k-1}p_x^{(\tau)}$ is the probability of avoiding all active-member exits for the first $k-1$ years, then

D_k={}_{k-1}p_x^{(\tau)}q_{x+k-1}^{(d)}.

Here the superscript $(d)$ denotes death, while $(\tau)$ denotes survival free of every active-member decrement under consideration. A withdrawal or retirement before death prevents a benefit that is payable only on active-member death. Therefore, the death probability is not generally the member’s death probability on a basis that ignores competing exits.

Suppose the survivor is age $y$ at time 0, so that, under the annual-age convention, the survivor is age $y+k$ when the member dies at the end of year $k$. Write ${}_jp_{y+k}$ for the probability that the survivor lives a further $j$ years from that death date. Assume survivor mortality is described by the applicable survivor life table and is conditionally independent of the member’s prior decrement history, given the survivor’s age at the member’s death.

Timing convention: A life annuity-immediate $a_z$ pays at the end of each year, beginning one year after valuation, provided the life aged $z$ is then alive. If the survivor pension begins after a waiting period of $w$ full years following the member’s death, its first payment is $w+1$ years after death. Payments continue annually while the survivor is alive.

Value of the Survivor Pension After a Death

For a member death at the end of year $k$, a pension of amount $B$ per year that begins after $w$ full waiting years has value, measured at the death date,

B\sum_{j=w+1}^{\infty}v^j\,{}_jp_{y+k}.

The index $j$ counts years after the member’s death. It accounts for both discounting from the death date and the probability that the survivor is alive at the payment date. If the pension is lifelong and $w$ is a nonnegative integer, the same value can be written

Bv^w\,{}_wp_{y+k}\,a_{y+k+w}.

To see why, payments begin $w+1$ years after death. The survivor must first live $w$ years to reach age $y+k+w$; the remaining payments then have the value of an annuity-immediate at that age. The factor $v^w{}_wp_{y+k}$ discounts and weights that survival to the start of the annuity. When $w=0$, the expression reduces to $Ba_{y+k}$.

Combining this conditional value with the probability and timing of the trigger gives the time-0 expected present value:

\operatorname{EPV} =\sum_{k}D_kv^kB_k \sum_{j=w_k+1}^{\infty}v^j\,{}_jp_{y+k}.

The factor $v^k$ discounts from time 0 to the end of the member’s death year. The inner sum is already measured from that death date, so its discounting begins at $j=1$, not at $j=0$. This formula assumes end-of-year member deaths, annual survivor payments, a fixed annual effective interest rate, and no escalation or guarantee unless these are explicitly incorporated into the payment amounts.

Find the trigger probability.
For each death year, include the probability of remaining eligible through earlier years and the conditional probability of death in that year.

Set the survivor’s age and commencement date.
Advance the survivor’s age to the member’s death date, then apply any waiting period to determine the first payment date.

Value the survivor payments at the death date.
Use the survivor’s survival probabilities and discount each payment from the member’s death date.

Discount and weight the trigger.
Multiply the death-date value by the death-year probability and by the discount factor from time 0 to the death date, then sum over eligible death years.

Worked Valuations

Worked Example: A Waiting Period After a Year-1 Death

A hypothetical plan pays a survivor $ \$1{,}000$ annually for life if its active member dies. The member’s year-1 probability of death is $0.04$, and the year-1 probability of another active-member exit is $0.06$. These are competing decrement probabilities from time 0. The survivor is age 60 at time 0 and, for this calculation, has a constant annual survival probability of $0.80$ at every age. The pension’s first payment is two years after the member’s death. Interest is $i=5\%$ effective annually. Find the time-0 value attributable to year-1 deaths only.

Method. The member dies at the end of year 1 with probability $D_1=0.04$. The other-exit probability does not need to be subtracted from $0.04$: the stated $0.04$ is already the death decrement probability. It does mean the member does not remain eligible after that other exit.

The first payment is two years after death. Thus the survivor must survive two years for the first payment, three years for the second, and so on. With $v=1/1.05=20/21$, the value at the death date per dollar of annual pension is

\sum_{j=2}^{\infty}(0.80v)^j =\frac{(0.80v)^2}{1-0.80v} =\frac{(16/21)^2}{1-16/21} =\frac{256}{105}.

Discount this death-date value from the end of year 1 to time 0, multiply by the annual pension, and weight by the year-1 death probability:

0.04\left(\frac{20}{21}\right)(1{,}000)\left(\frac{256}{105}\right) =\frac{204{,}800}{2{,}205} =92.8798\ldots

Rounded to the nearest cent, the year-1 death cohort contributes $\$92.88$.

Common mistake avoided. Using $(0.80v)/(1-0.80v)$ would start the annuity one year after death, ignoring the stated two-year delay. That incorrect death-date factor is $16/5=3.2$, giving $0.04v(1{,}000)(3.2)=\$121.90$. The first payment is at $j=2$, so the correct geometric series starts with $(0.80v)^2$.

Worked Example: Two Possible Death Years and Competing Exits

A hypothetical plan pays $ \$2{,}000$ per year for the survivor’s lifetime, with the first payment one year after the member’s death. The member’s year-1 death and other-exit probabilities are $0.02$ and $0.08$. Conditional on still being active at the start of year 2, the year-2 death and other-exit probabilities are $0.03$ and $0.07$. The survivor’s annual survival probability is $0.90$ at every age. Interest is $i=4\%$. Find the time-0 value of benefits triggered by deaths in years 1 and 2 only.

Method. The year-1 death probability is $D_1=0.02$. The probability of remaining active through year 1 is $1-0.02-0.08=0.90$, so

$$ D_2=0.90(0.03)=0.027. $$

The survivor annuity-immediate factor is the same in either death year because survival is assumed constant across ages. With $v=25/26$,

a_{\text{survivor}} =\sum_{j=1}^{\infty}(0.90v)^j =\frac{0.90v}{1-0.90v} =\frac{45/52}{7/52} =\frac{45}{7}.

For year-1 deaths, discount the trigger by $v$:

D_1v(2{,}000)a_{\text{survivor}} =0.02\left(\frac{25}{26}\right)(2{,}000)\left(\frac{45}{7}\right) =\frac{45{,}000}{182} =247.2527\ldots

For year-2 deaths, discount by $v^2$ and use the unconditional probability $D_2=0.027$:

D_2v^2(2{,}000)a_{\text{survivor}} =0.027\left(\frac{25}{26}\right)^2(2{,}000)\left(\frac{45}{7}\right) =\frac{1{,}518{,}750}{4{,}732} =320.9535\ldots

The total is $247.2527\ldots+320.9535\ldots=568.2062\ldots$, or $\$568.21$ to the nearest cent.

Common mistake avoided. The year-2 death probability $0.03$ is conditional on reaching the start of year 2 active. Using it directly as $D_2$ omits the $0.90$ probability of surviving year 1 without another exit.

Worked Example: Age-Specific Survival and a Deferred, Limited Pension

A hypothetical plan pays $ \$1{,}200$ annually for at most three installments following an active member’s death. The first installment is two years after death, and each installment is payable only if the survivor is alive on its payment date. The member’s year-1 death and other-exit probabilities are $0.02$ and $0.08$. Conditional on being active at the start of year 2, the year-2 death probability is $0.04$, and the other-exit probability is $0.06$. The survivor is age 58 at time 0. The survivor’s annual survival probabilities are $0.95$ from age 58 to 59, $0.90$ from 59 to 60, $0.85$ from 60 to 61, and $0.80$ from 61 to 62. Interest is $i=5\%$. Find the value at time 0 of the three installments triggered by year-2 deaths only.

Method. The probability of remaining active through year 1 is $1-0.02-0.08=0.90$, so the year-2 death probability from time 0 is $D_2=0.90(0.04)=0.036$. The survivor is age 60 at the end of year 2. The three payments occur at times 2, 3, and 4 after that death. The corresponding probabilities of surviving to those payments are $0.95(0.90)=0.855$, $0.855(0.85)=0.72675$, and $0.72675(0.80)=0.5814$.

The payments are therefore four, five, and six years after time 0. With $v=20/21$, the expected present value is

\begin{aligned} D_2(1{,}200)\left(0.855v^4+0.72675v^5+0.5814v^6\right) &=43.2\left(0.855(0.822702474)+0.72675(0.783526167)+0.5814(0.746215397)\right)\\ &=43.2(0.703410615+0.569427641+0.433850632)\\ &=43.2(1.706688888)\\ &=73.72896\ldots \end{aligned}

Rounded to the nearest cent, the value is $\$73.73$. The result is a limited three-payment value, not a life-annuity value: no payments are included after the third installment.

Common mistake avoided. The survivor’s age at the member’s death is 60, not 58. Also, the first installment is two years after death, so its survival probability is $0.95(0.90)$, and its discount exponent from time 0 is $2+2=4$.

Checks Before Finalizing a Valuation

The model is useful because it separates assumptions that are easy to conflate. Member decrement probabilities determine whether and when the pension is triggered. Survivor mortality determines whether each post-death payment is made. Interest discounts each payment from its actual payment date. If the benefit amount varies by death year, use the applicable $B_k$ in that year’s contribution.

Check the probability basis. Determine whether a death probability is conditional on active status at the start of the year or already unconditional from time 0.
Check the first payment date. A payment one year after death is at $j=1$; after one full waiting year, the first payment is at $j=2$.
Check the survivor’s attained age. Use the survivor’s age at the member’s death, then advance age for any waiting period.
Check survival through deferral. A survivor must be alive on the payment date; survival during a waiting period is not optional when the first payment is deferred.
Check that discounting starts at time 0. Discount once to the member’s death date and then from the death date to each payment, or discount each payment directly from time 0. Do not omit or duplicate a discount factor.

Key takeaway: A survivor pension value is a sum over possible active-member death years. Each term combines the probability of an eligible death, discounting to the trigger date, and the survivor’s age- and commencement-dependent value of the payments that follow.

Check Your Understanding

For each problem, state the trigger probability and payment timing before calculating. Round monetary answers to the nearest cent.

A member has a year-1 death probability of $0.03$; other exits do not trigger a benefit. A survivor pension of $ \$5{,}000$ per year starts one year after a year-1 death and is lifelong. The survivor’s annual survival probability is $0.80$, and interest is $0\%$. Find the value at time 0 for year-1 deaths only. (Answer: $\$600.00$)
A member’s year-1 death probability is $0.025$. A $ \$1{,}500$ annual lifelong survivor pension begins two years after death. Survivor annual survival probability is $0.90$, and interest is $4\%$. Find the time-0 value for year-1 deaths only. (Answer: $\$200.60$)
Year-1 member death and other-exit probabilities are $0.02$ and $0.04$. Conditional on active status at the start of year 2, the death probability is $0.03$. Interest is $0\%$. A survivor pension pays two installments of $ \$1{,}000$, one year after death and two years after death. The survivor’s probabilities of surviving one and two years after the death are $0.80$ and $0.60$. Find the time-0 value for year-2 deaths only. (Answer: $\$39.48$)
A member dies at the end of year 3. The survivor is age 55 at time 0. A two-year waiting period follows the death, after which the pension’s first payment is made. What is the survivor’s age at the first payment under the annual-age convention? (Answer: age 60)
Year-1 death and other-exit probabilities are $0.04$ and $0.16$. Conditional on active status at the start of year 2, death and other-exit probabilities are $0.05$ and $0.05$. A $ \$1{,}000$ lifelong survivor pension starts one year after death. Survivor annual survival probability is $0.90$, interest is $0\%$, and only deaths in years 1 and 2 are included. Find the total value. (Answer: $\$720.00$)

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