Survivor Valuation With Conditional Mortality

Why Conditional Mortality Matters

A survivor pension is contingent on a sequence of events: the member dies while the benefit is eligible to be paid, the spouse is alive when that death occurs, and the spouse survives to each payment date. These probabilities describe different parts of the contingency. Combining them without keeping their conditions clear can overstate the value or discount a payment to the wrong time.

The key is to condition on the member’s death time. For each possible death year, first identify the probability of that death from the valuation date. Then apply the probability that the spouse is alive at that death, followed by the spouse’s conditional probability of surviving to each later payment. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let time 0 be the valuation date, and suppose a survivor pension is triggered by the member’s death. Let $D_k$ be the probability, measured from time 0, that the member dies in year $k$ while eligible for the benefit. Let $r_k$ be the probability that the spouse is alive at the member’s death, conditional on that death occurring in year $k$. Let $s_{k,j}$ be the probability that the spouse survives for $j$ years after the member’s death, conditional on the member’s death in year $k$ and the spouse being alive at that death. For a benefit $b_{k,j}$ payable $j$ years after that death, the payment occurs at time $k+j$.

The probabilities $D_k$, $r_k$, and $s_{k,j}$ answer different questions. In particular, $D_k$ is an unconditional probability from time 0; it is not necessarily the member’s death probability conditional on being alive at the start of year $k$. Also, $r_k$ accounts for the spouse being alive at the member’s death, while $s_{k,j}$ describes subsequent survival.

If the member’s year-$k$ death probability is supplied conditional on surviving to the start of that year, it must be multiplied by the probability the member survives to that point. For example, if annual conditional death probabilities are $q_1^M,q_2^M,\ldots$, and death is the only way to leave the model, then $D_k={}_{k-1}p^M q_k^M$, where ${}_{k-1}p^M$ is the member’s probability of surviving to the start of year $k$. If other exits affect eligibility, the probability of reaching year $k$ still eligible must also be reflected.

Expected Present Value Formula

Assume an effective annual interest rate $i$, with $v=1/(1+i)$, and benefits paid at the ends of the specified years. For a payment $j$ years after a member death in year $k$, its probability of being paid is $D_k r_k s_{k,j}$. The payment is discounted from time $k+j$, not merely from the member’s death date.

\operatorname{EPV} = \sum_k D_k r_k \sum_j b_{k,j}s_{k,j}v^{k+j}.

This expression can allow different spouse mortality assumptions for different member death years. It also allows a benefit amount or payment schedule to vary with the death year. A payment made immediately at the member’s death has $j=0$, so its discount factor is $v^k$ and its post-death survival factor is $s_{k,0}=1$, provided the spouse is alive at that instant.

Important condition: The formula uses the stated conditional probabilities directly and does not require the member’s and spouse’s lifetimes to be independent. If a problem supplies marginal life-table probabilities instead, converting them into the conditional quantities needed here requires an explicit joint-life assumption or other dependence information. Independence must not be silently imposed.

Worked Valuations

Worked Example: One Payment After a Year-2 Death

A hypothetical pension pays a spouse $2,000 exactly one year after the member dies. The probability, from time 0, that the member dies in year 2 while eligible is 0.12. Conditional on that death, the spouse is alive at the death date with probability 0.75 and survives the following year with probability 0.80. The effective annual interest rate is 5%. Find the expected present value.

Method. Multiply the unconditional member death probability by the spouse’s probability of being alive at death and then by the spouse’s conditional one-year survival probability. Discount from the payment date.

The payment is made one year after a death in year 2, so it occurs at time $2+1=3$. Its probability of being paid is

$$ 0.12(0.75)(0.80)=0.072. $$

The expected payment at time 3 is $2{,}000(0.072)=144$ dollars. Since $v=1/1.05$, the expected present value is

144v^3 = \frac{144}{(1.05)^3} = \frac{144}{1.157625} = 124.3926\ldots.

Thus the value is $124.39 to the nearest cent. As a check, the probability of payment is less than 0.12, and discounting reduces the resulting expected payment of $144, so a present value below $144 is consistent.

Common mistake avoided: Discounting for only one year because the payment is one year after death. The valuation starts at time 0, so this payment at time 3 is discounted for three years.

Worked Example: Death in Either of Two Years

A hypothetical plan pays $1,000 one year after the member’s death. From time 0, the probabilities of the member dying while eligible are 0.10 in year 1 and 0.08 in year 2. Conditional on death in year 1, the spouse is alive at death with probability 0.80; conditional on death in year 2, that probability is 0.75. In either case, the spouse’s conditional probability of surviving one year after the death is 0.90. The effective annual interest rate is 10%. Calculate the expected present value.

Method. Split the value by member death year. The year-1 death produces a payment at time 2; the year-2 death produces a payment at time 3.

For a year-1 death, the probability of a payment is

$$ 0.10(0.80)(0.90)=0.072. $$

The expected payment is $1{,}000(0.072)=72$ dollars at time 2. Its present value is

72v^2=\frac{72}{(1.10)^2} =\frac{72}{1.21} =59.504132\ldots.

For a year-2 death, the probability of a payment is

$$ 0.08(0.75)(0.90)=0.054. $$

The expected payment is $1{,}000(0.054)=54$ dollars at time 3. Its present value is

54v^3=\frac{54}{(1.10)^3} =\frac{54}{1.331} =40.570999\ldots.

Adding the mutually exclusive death-year cases gives

59.504132\ldots+40.570999\ldots =100.075131\ldots.

The expected present value is $100.08 to the nearest cent. The calculation uses the given year-2 death probability as an unconditional probability from time 0; it does not multiply it by year-1 survival a second time.

Common mistake avoided: Treating the year-2 death probability as conditional on survival to year 2 when it is explicitly given from time 0. Doing so would add an unsupported survival factor and understate the value.

Worked Example: Several Payments and Cumulative Survival

A hypothetical pension pays $4,000 at each of the first three anniversaries after a member’s death in year 1. The probability of that member death, measured from time 0, is 0.15. Conditional on the death, the spouse is alive at the death date with probability 0.80. Conditional also on being alive then, the spouse’s probabilities of surviving one, two, and three years after the death are 0.80, 0.60, and 0.30, respectively. The effective annual interest rate is 25%. Find the expected present value.

Method. Use the supplied cumulative survival probabilities for each payment duration. Do not multiply them together: the two- and three-year probabilities already describe survival from the death date to those durations.

The factor for the member’s death and the spouse being alive at that death is $0.15(0.80)=0.12$. Multiplying by the $4,000 benefit gives $4{,}000(0.12)=480$. The three payment dates are times 2, 3, and 4, respectively. Here $v=1/1.25=0.80$, so $v^2=0.64$, $v^3=0.512$, and $v^4=0.4096$. The value is

\begin{aligned} \operatorname{EPV} &=480\left[0.80v^2+0.60v^3+0.30v^4\right]\\ &=480\left[0.80(0.64)+0.60(0.512)+0.30(0.4096)\right]\\ &=480\left[0.512+0.3072+0.12288\right]\\ &=480(0.94208)\\ &=452.1984. \end{aligned}

The expected present value is $452.20 to the nearest cent. The intermediate terms are $245.76, $147.456, and $58.9824, which sum to $452.1984. This agrees with the total above.

Common mistake avoided: Multiplying the cumulative two-year survival probability 0.60 by the one-year survival probability 0.80. The 0.60 is already the probability of surviving two full years, conditional on being alive at the member’s death.

Why the Formula Holds

The formula follows by valuing each possible payment separately and taking its expectation. Let $I_{k,j}$ equal 1 if the member dies in year $k$, the spouse is alive at death, and the spouse survives to the payment $j$ years later; otherwise let it equal 0. The present value of that scheduled payment is $b_{k,j}v^{k+j}I_{k,j}$.

Find the payment probability.
By the definition of conditional probability, $P(I_{k,j}=1)=D_k r_k s_{k,j}$. Each factor conditions on the preceding event in the sequence.

Take the expected value of this payment.
Because $I_{k,j}$ is an indicator, $E[I_{k,j}]=P(I_{k,j}=1)$. Therefore its expected present value is $b_{k,j}v^{k+j}D_k r_k s_{k,j}$.

Add the scheduled payments.
Expectation is additive, so summing the expected present values over all death years and payment durations gives the stated formula. The events for different death years are mutually exclusive, and additivity also applies across the payments following any one death.

This reasoning also explains why independence is unnecessary when the conditional probabilities are given: the product is a chain of conditional probabilities, not an assertion that the member and spouse have independent lifetimes. What matters is that each conditional probability matches the event immediately before it.

Common Exam Traps

Using the wrong member-death basis. Check whether a death probability is unconditional from time 0 or conditional on being alive at the start of a year. Convert it only when required by the stated basis.
Omitting the spouse-at-death condition. Post-death survival is not a substitute for the probability the spouse is alive when the member dies. Both factors are needed when the pension requires the spouse to be alive at death.
Discounting from the death date instead of the valuation date. A payment $j$ years after a death in year $k$ occurs at time $k+j$.
Multiplying cumulative survival probabilities. If survival to duration $j$ is already supplied, use that probability directly. Multiply successive annual probabilities only when the data are conditional one-year probabilities.
Assuming independence without support. Conditional spouse mortality after the member’s death may differ from a marginal survival probability. Use only the assumptions provided or explicitly justified.

Key takeaway: For each survivor-pension payment, identify the member’s death probability from the valuation date, the spouse’s conditional probability of being alive at that death, the spouse’s subsequent conditional survival probability, and the payment time. Multiply the probability factors, then discount from that payment time.

Check Your Understanding

For each problem, treat the supplied probabilities as applying on the stated conditional or unconditional basis. Round dollar values to the nearest cent.

A hypothetical plan pays $1,000 one year after a member dies in year 1. The unconditional probability of that death is 0.10, the spouse’s probability of being alive at death is 0.70, and the spouse’s conditional one-year survival probability is 0.90. At an effective annual interest rate of 5%, find the expected present value. (Answer: $57.14)
A member’s unconditional probability of dying in year 2 while eligible is 0.06. Conditional on that death, the spouse is alive with probability 0.80 and survives the following two years with probability 0.50. A $3,000 payment is made two years after death. At 4% effective annual interest, find the expected present value. (Answer: $61.55)
A member’s unconditional probability of death in year 1 is 0.12. The spouse is alive at death with probability 0.75 and, conditional on that, survives one and two years after death with probabilities 0.90 and 0.72. A $2,000 pension is paid at each of those two dates. At 10% effective annual interest, calculate the value. (Answer: $231.25)
The member’s conditional death probabilities are 0.10 in year 1 and 0.20 in year 2, given survival to the start of each respective year. Death is the only exit. If death occurs in year 2, the spouse is alive then with probability 0.50 and survives one further year with probability 0.80. A $1,500 payment is made one year after that death. At 5% effective annual interest, find the value attributable to year-2 deaths. (Answer: $93.29)
The unconditional probabilities of member death while eligible are 0.10 in year 1 and 0.05 in year 2. The spouse’s probability of being alive at death is 0.80 for a year-1 death and 0.60 for a year-2 death; the corresponding conditional one-year survival probabilities are 0.75 and 0.50. A $2,500 payment is made one year after death. At 10% effective annual interest, calculate the combined value. (Answer: $152.14)

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