Survivor Benefits After Retirement

Why Payment Timing Matters

After retirement, a pension plan may pay an amount to the retired member while that member is alive and a different benefit to an eligible survivor after the member dies. These payments have different contingencies. A member’s pension is contingent on the member being alive at the payment date; a survivor benefit is contingent on the member having died and the survivor being alive and eligible when payment is due.

The distinction affects both the probabilities and the timing used in a present-value calculation. In particular, a survivor pension that starts at the first scheduled payment date after death should not be valued as though it were simply another installment of the member’s pension. The examples below use hypothetical plan provisions and assumptions.

Definition: A retired-member pension is a payment made while the retired member is alive, under the stated payment timing. A survivor benefit after retirement is a payment triggered by the retired member’s death, made to a survivor who satisfies the plan’s stated conditions. The benefit amount, payment frequency, starting date, and conditions must be specified before its value can be calculated.

A Discrete-Time Valuation Model

For an annual model, suppose payments are made at the end of each year. Let $q_k$ be the probability that a retired member who is alive at the start of year $k$ dies during that year, and let $p_k=1-q_k$. Let $v=1/(1+i)$, where $i$ is the effective annual interest rate. Assume that the survivor is alive and eligible at the member’s death unless a separate survivor survival assumption is included.

If the member dies at the end of year $k$, and the first survivor installment is paid at the end of year $k+1$, then the first installment is one year after death. If the survivor’s probability of surviving one year is $r$, the probability of surviving $j$ years after the member’s death is $r^j$, when this probability is constant and the successive survival probabilities follow the stated model.

P(\text{member dies in year }k) = \left(\prod_{h=1}^{k-1}p_h\right)q_k.

For a survivor pension of $B$ per year, payable at the end of each year after the member’s death while the survivor remains alive, the present value is the sum over possible death years and subsequent survivor payment dates. This representation explicitly separates the member’s death timing from the survivor’s subsequent survival.

\operatorname{EPV}(\text{survivor pension}) = B\sum_{k=1}^{\infty} v^k \left(\prod_{h=1}^{k-1}p_h\right)q_k \sum_{j=1}^{\infty}v^j\,{}_jp_{\text{survivor}}.

Here ${}_jp_{\text{survivor}}$ is the probability that the survivor remains alive for $j$ years after the member’s death, conditional on being alive at that death. The formula assumes the stated payment pattern continues indefinitely; for a benefit with a term limit, replace the inner infinite sum by the appropriate finite sum. If survivor eligibility is not assured, its probability must also be included at the relevant payment dates. The formula does not apply unchanged when the first installment is paid at death, when payments are monthly, or when the benefit is a fixed-term guarantee rather than a life-contingent pension.

Why the Valuation Separates into Two Parts

Conditional on the member dying at the end of year $k$, the present value of survivor payments is first discounted to the date of death and then valued as a stream beginning one year later. The probability of that death event weights this conditional value. Summing over the mutually exclusive possible death years gives the total expected present value.

\begin{aligned} \operatorname{EPV} &=B\sum_{k=1}^{\infty} P(\text{death in year }k) \sum_{j=1}^{\infty}v^{k+j}\,{}_jp_{\text{survivor}}\\ &=B\sum_{k=1}^{\infty} v^kP(\text{death in year }k) \sum_{j=1}^{\infty}v^j\,{}_jp_{\text{survivor}}. \end{aligned}

The second line follows because $v^{k+j}=v^kv^j$, so the discount to the member’s death date can be separated from discounting the survivor’s later payments. This is a factorization of the cash-flow timing, not an assumption that the member and survivor have the same mortality. Independence between the member’s and survivor’s future lifetimes is not needed if the survivor survival probabilities are specified conditionally on being alive at the member’s death.

Worked Example: Separating the Member’s Pension and Survivor Pension

A hypothetical plan pays a retired member $ \$1{,}000 $ at each year-end if the member is alive then. On the member’s death, it pays the survivor $ \$600 $ at each subsequent year-end while the survivor is alive. Assume a constant annual member death probability of $0.10$, a constant annual survivor survival probability of $0.80$ after the member’s death, and an effective annual interest rate of $5\%$. Treat the two mortality models as stated and assume the survivor is alive and eligible at the member’s death. Find the expected present value of each benefit.

Method. Value the member’s pension using the probability the member is alive at each payment date. Value the survivor pension by first finding the discounted probability of each member death year, then multiplying by the present value of survivor payments after that death.

For the member’s pension, $v=1/1.05=20/21$, and the probability of being alive at the end of year $t$ is $0.9^t$. Thus:

\begin{aligned} \operatorname{EPV}(\text{member pension}) &=1{,}000\sum_{t=1}^{\infty}(0.9v)^t\\ &=1{,}000\frac{0.9v}{1-0.9v}\\ &=1{,}000\frac{6/7}{1/7} =\$6{,}000. \end{aligned}

For the survivor pension, the probability of member death in year $k$ is $0.9^{k-1}(0.10)$. The discounted death-year probabilities sum to:

\sum_{k=1}^{\infty}v^k0.9^{k-1}(0.10) = \frac{0.10v}{1-0.9v} = \frac{2/21}{1/7} = \frac{2}{3}.

Conditional on death, the survivor’s first payment is one year later. The present value factor for the survivor’s payments is:

\sum_{j=1}^{\infty}(0.8v)^j = \frac{0.8v}{1-0.8v} = \frac{16/21}{5/21} = \frac{16}{5}.

Therefore:

\operatorname{EPV}(\text{survivor pension}) = 600\left(\frac{2}{3}\right)\left(\frac{16}{5}\right) = \$1{,}280.

The combined expected present value is $ \$6{,}000+\$1{,}280=\$7{,}280 $. The calculation checks because the member’s pension uses member survival probabilities, whereas the survivor pension uses member death probabilities followed by survivor survival probabilities. The common mistake avoided here is applying the survivor’s survival probability to the member’s own pension.

Finite Payment Terms and Changing Death Probabilities

A survivor benefit may be limited to a specified number of installments, and member death probabilities need not be constant. For a finite horizon, a reliable approach is to list each possible member death year, calculate its probability, and then list the survivor payments arising from that death. The table makes it harder to accidentally count a payment both as a member pension and as a survivor pension.

Specify the event.
State which death year triggers the survivor benefit and the probability of that death year.

Specify payment dates.
Identify whether the first survivor installment is paid at death, one year later, or at another stated time.

Apply survivor conditions.
For each scheduled installment, include the probability the survivor is alive and eligible on its payment date.

Discount and sum.
Discount every contingent payment from its own payment date, then sum the expected present values.

Worked Example: A Three-Year Member Mortality Table

A hypothetical plan pays a retired member $ \$1{,}200 $ at the end of each of the next three years if the member is alive. The member’s death probabilities during years 1, 2, and 3, conditional on being alive at each year’s start, are $0.10$, $0.20$, and $0.25$. If the member dies at a year-end, a survivor pension of $ \$500 $ is paid at each of the next two year-ends, provided the survivor is alive. The survivor’s probability of surviving each year after the member’s death is $0.80$. The effective annual interest rate is $5\%$. Find the member pension value and the survivor pension value.

Method. Calculate member survival and death probabilities year by year. Value the member payments at years 1–3; value survivor installments at the two dates following each possible death.

The member’s survival probabilities to the ends of years 1, 2, and 3 are $0.90$, $0.90(0.80)=0.72$, and $0.72(0.75)=0.54$. With $v=20/21$:

\begin{aligned} \operatorname{EPV}(\text{member pension}) &=1{,}200\left(0.90v+0.72v^2+0.54v^3\right)\\ &=1{,}200\left(0.857142857+0.653061224+0.4664723\right)\\ &=\$2{,}372.01. \end{aligned}

The probabilities of death in years 1, 2, and 3 are respectively $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. Their discounted sum is:

0.10v+0.18v^2+0.18v^3 = \frac{142}{343}.

Following any of these deaths, the survivor’s two possible installments are one and two years after death. Their conditional present value factor is:

0.80v+(0.80v)^2 = \frac{16}{21}+\frac{256}{441} = \frac{592}{441}.

Thus:

\operatorname{EPV}(\text{survivor pension}) = 500\left(\frac{142}{343}\right)\left(\frac{592}{441}\right) = \$277.87.

The combined value is $ \$2{,}372.01+\$277.87=\$2{,}649.88 $. This check confirms that the survivor stream begins after, not at, the member’s death under the stated timing. The common mistake avoided is treating the year-2 death probability as $0.20$; it is $0.90(0.20)=0.18$, because the member must first survive year 1.

Repeated Payments and a Geometric Series

When survivor installments recur at regular intervals and the probability of reaching each next installment follows a constant ratio, their present value is a geometric series. The starting power of $v$ must match the first payment date. For example, payments beginning two years after the relevant event involve $v^2$, not $v$.

Worked Example: Biennial Survivor Installments

A hypothetical plan assigns $95/9$ benefit units, each paying $ \$600 $ per survivor installment. Installments are paid every two years, beginning two years after the triggering event. The probability of reaching the first installment is $0.95$. After each installment, the probability of reaching the next one two years later is multiplied by $0.90$ for continued eligibility and $0.95$ for survivor survival. Assume these two conditions are independent for each interval. The effective annual interest rate is $4\%$. Find the value at the triggering date.

Method. Write the first expected discounted installment and the common ratio between successive expected discounted installments, then sum the geometric series.

Here $v=1/1.04=25/26$. The factor for the first installment and its successors is:

\begin{aligned} F &=0.95v^2\left[1+(0.90)(0.95)v^2+ \big((0.90)(0.95)v^2\big)^2+\cdots\right]\\ &=\frac{0.95v^2}{1-(0.90)(0.95)v^2}. \end{aligned}

Substituting $v=25/26$ gives:

F = \frac{0.95(25/26)^2} {1-0.90(0.95)(25/26)^2} = \frac{4750}{1133}.

The benefit value is therefore:

600\left(\frac{95}{9}\right)\left(\frac{4750}{1133}\right) = \$26{,}551.93.

The result is rounded to the nearest cent. This calculation checks by using a two-year discount interval in both the first term and the common ratio. A tempting but incorrect substitution would produce a factor such as $38/9$; the correct substitution is $4750/1133$. The trap is simplifying the denominator before substituting the full two-year discount and continuation factors.

Common Traps in Survivor-Benefit Valuation

Using the wrong life contingency. Member pension payments require the member to be alive; survivor payments require the member’s death to have triggered the benefit and the survivor to satisfy the payment conditions.
Starting the survivor stream too early. If the first installment is one year after death, its discount exponent is $k+1$ for a death at time $k$, not $k$.
Ignoring survival to the death year. A conditional death probability in year $k$ must be multiplied by the probability the member survived to the start of that year.
Reusing the member’s pension after death. Once death occurs, payments to the member stop unless the plan explicitly provides a guarantee or another benefit.
Mixing payment frequencies. A biennial installment uses two-year discount intervals; an annual formula cannot be used unchanged.

A further arithmetic check is useful when several mutually exclusive death paths produce installments on different dates. With $z=6/7$, suppose the survivor cash-flow projection assigns probability $0.18$ to a $ \$8{,}925.19 $ installment at year 1, and probability $0.18$ to each of a $ \$10{,}500 $ installment at year 3 and a $ \$10{,}710 $ installment at year 4. These paths are mutually exclusive, and the listed survivor is alive for each installment. The year-1 contribution is $0.18(8{,}925.19)z=\$1{,}377.03$, rounded to cents. The later contributions are:

\begin{aligned} 0.18\left[10{,}500z^3+10{,}710z^4\right] &=0.18\left[6{,}612.244898+5{,}780.990004\right]\\ &=\$2{,}230.78. \end{aligned}

Adding the year-1 contribution gives a total survivor-benefit value of $ \$1{,}377.03+\$2{,}230.78=\$3{,}607.81 $. In particular, $10{,}710(6/7)^4$ is approximately $5{,}780.99$, not $5{,}775.59$. Keep the unrounded amounts through the sum and round only the final value to cents.

Key takeaway: Value the retired member’s pension and the survivor benefit as distinct contingent cash-flow streams. For each survivor installment, establish the death trigger, the payment date, and the survivor’s probability of being alive and eligible; then apply the correct discount factor.

Check Your Understanding

For each problem, identify the payment timing before calculating. Amounts are hypothetical.

A plan pays a retired member $ \$900 $ at each year-end while alive. The member survives each year with probability $0.8$, independently from year to year, and $i=5\%$. What is the expected present value of the member’s pension? (Answer: $ \$3{,}600.00 $)
A survivor receives $ \$400 $ at the end of the year after the member’s death and one more year later, if alive. The survivor survival probability is $0.9$ per year and $i=0\%$. What is the value at the member’s death, conditional on the survivor being alive then? (Answer: $ \$684.00 $)
A member’s conditional death probabilities in years 1 and 2 are $0.10$ and $0.20$. At $i=0\%$, what is the probability of death in year 2? (Answer: $0.1800$)
A survivor benefit is $ \$500 $ annually, first paid one year after member death, and continues while the survivor lives. Member death probability is $0.10$ per year, survivor survival probability is $0.80$ per year, and $i=5\%$, all constant. What is the expected present value of the survivor benefit at retirement? (Answer: $ \$1{,}066.67 $)
A survivor installment of $ \$2{,}000 $ is payable at the end of year 2 with probability $0.25$, and $i=4\%$. What is its expected present value? (Answer: $ \$462.28 $)

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