Introduction
A member’s death after pension payments have begun does not, by itself, determine what a survivor receives. The outcome depends on the payment form elected at retirement, the plan’s eligibility conditions, the member’s date of death relative to scheduled payment dates, and whether the survivor is alive when a continuing payment is due. Translating those rules into a timeline is essential before calculating any benefit or value.
This tutorial represents hypothetical post-retirement survivor pensions as event-contingent cash flows. It focuses on the survivor benefit after retirement, rather than benefits triggered by a death before retirement. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Start with the Elected Payment Form
A life-only pension pays the member while the member is alive and provides no continuing survivor pension under that form. A joint-and-survivor pension generally pays the member during the member’s lifetime and, after the member’s death, pays an eligible surviving beneficiary a specified fraction of the member’s pension. A form with a guaranteed period may continue payments after the member’s death until the guarantee is satisfied; its rules may specify a different amount after that period.
The election is important because the member’s pension amount and the survivor’s amount need not be the same. If the member’s pension under the elected form is \(B\) per year and the survivor percentage is \(r\), then a continuing survivor pension is \(rB\) per year when the plan’s conditions are met. Apply \(r\) to the pension actually payable under the elected form, not automatically to an unreduced benefit or to a different pension option.
| Payment form | Effect of member’s death after retirement | Key rule to identify |
|---|---|---|
| Life-only | Member payments stop; no continuing survivor pension under this form | Whether a separate guarantee or other provision applies |
| Joint-and-survivor | An eligible survivor may receive the elected fraction of the member’s pension | Survivor eligibility, percentage, and payment dates |
| Guaranteed period with survivor continuation | Payments may continue at one amount through the guarantee and then at another amount | How many guaranteed payments remain and what follows the guarantee |
Use the plan’s precise terms. In particular, distinguish a survivor who is eligible under the election from one who merely has a personal relationship with the member. Eligibility might depend on a condition specified at retirement or at the member’s death. The condition must be checked at the time the plan says it applies; it should not be presumed.
Build the Payment Timeline
Before calculating amounts, set out the retirement date, payment dates, member’s death date, survivor’s eligibility, and survivor’s status at each due date. For an annual pension paid in arrears, payments are scheduled at anniversaries of retirement. If the plan provides no proration, a death between anniversaries does not create an extra partial-year installment. The next payment follows the stated post-death rule.
For a joint-and-survivor pension with an annual member amount \(B\), survivor fraction \(r\), and scheduled payment dates \(t_k\), the survivor payment at a due date is \(rB\) only if the survivor is eligible and alive at that date, and the member has died by then. This is a representation of the plan terms, not a claim that all plans use the same payment timing.
Record the form, member pension amount under that form, survivor fraction, and any guarantee.
Apply the stated survivor condition at the date specified by the plan. If the condition fails, do not assign a survivor pension under that provision.
Identify the scheduled payment dates that occur after death and apply any guarantee rules.
Use the correct amount at each date; for an expected present value, include the probability the survivor is alive to receive it.
Worked Example: Joint-and-Survivor Payments After a Midyear Death
A hypothetical plan pays an annual pension in arrears on each retirement anniversary. A member retires at time 0 with an elected joint-and-survivor pension of $30,000 per year and a 50% survivor continuation. The eligible survivor is alive at the member’s death. The member dies at time 2.4, and the plan does not prorate payments. Determine the scheduled payments at times 1 through 4.
Method. Set up the annual payment dates first, then apply the death trigger and the elected survivor percentage. The member’s $30,000 payments due at times 1 and 2 were due before death, so each is $30,000.
The next scheduled date is time 3. Since \(2 < 2.4 < 3\), the member died after the time-2 payment but before the time-3 payment. Under the stated joint-and-survivor terms, the survivor amount is \(0.50 \times \$30{,}000 = \$15{,}000\).
Thus the payment at time 3 is $15,000, and, if the survivor remains eligible and alive, the payment at time 4 is also $15,000. The schedule is $30,000 at time 1, $30,000 at time 2, $15,000 at time 3, and $15,000 at time 4.
Check and common mistake. The post-death payment is half of the elected $30,000 pension, not half of a prorated amount for the 0.6 years between death and the next anniversary. The stated plan has annual scheduled payments and no proration.
Guarantees and Changes in Payment Amount
A guaranteed-payment provision requires a separate timeline. If the member dies before the guarantee ends, payments may continue at the member amount for the remaining guaranteed period. After that period, a survivor continuation may begin at the elected fraction, if the survivor qualifies and the plan specifies that continuation. Do not count the guarantee as a survivor pension for the survivor’s lifetime: it is a promise about a specified number or period of payments.
Worked Example: A Guarantee Followed by a Survivor Pension
A hypothetical pension pays $24,000 at each year-end, beginning one year after retirement. The elected form guarantees the first 10 annual payments to the member or eligible survivor; if the member dies during that period, the eligible survivor receives the remaining guaranteed payments at $24,000 each. After the guarantee, the survivor receives 50% of the member pension while alive. The member dies at time 6.5, and the survivor satisfies the eligibility condition and remains alive through time 11. Identify the payments at times 7 through 11.
Method. Count the guaranteed payment dates, not the years elapsed since the death. The first 10 due dates are times 1 through 10. At death, payments at times 1 through 6 have already been made, so the remaining guaranteed dates are 7, 8, 9, and 10.
The survivor receives four guaranteed payments of $24,000. Their total is \(4 \times \$24{,}000 = \$96{,}000\). At time 11, the guarantee has ended. The survivor pension is \(0.50 \times \$24{,}000 = \$12{,}000\).
Therefore, the payments at times 7 through 11 are $24,000, $24,000, $24,000, $24,000, and $12,000. The time-11 payment is payable only if the survivor is still eligible and alive under the stated rules.
Check and common mistake. The four remaining guaranteed payments are counted from the original retirement payment schedule. Counting 10 years from the member’s death would incorrectly extend the guarantee.
Valuing Payments Conditional on Survival
A survivor payment is contingent on the survivor being alive when it is due. Suppose the member has already died, the survivor is eligible and alive at that death, and the remaining survivor payments are made at future times \(1,\ldots,n\) measured from the death date. Let \(C_k\) be the amount due at future time \(k\), and let \({}_kp_y\) be the probability that a survivor aged \(y\) survives \(k\) years from that date. With effective annual interest rate \(i\), set \(v=1/(1+i)\). The expected present value at the death date is
This expression assumes the amounts and dates are as specified, interest is effective annually, and the survival probabilities are appropriate for the survivor and period. It values only the listed finite payments. For a continuing pension beyond the listed dates, additional terms are needed; if payments are not annual, the discounting and survival periods must match the actual payment timing.
The reason for multiplying by survival probability can be seen directly. The payment at time \(k\) has present value \(C_kv^k\) if it is made and zero otherwise. Let \(I_k\) equal 1 if the survivor is alive at that date and 0 otherwise. The random present value of that payment is \(C_kv^kI_k\). Since \(E[I_k]={}_kp_y\), its expected present value is \(C_kv^k{}_kp_y\). Adding the expected values of the individual payments gives the displayed sum. This calculation is conditional on the survivor already being alive and eligible at the member’s death; it does not multiply again by a probability of the member surviving to that death date.
Worked Example: Expected Value of Three Continuing Payments
A hypothetical member has an elected annual pension of $28,800 after an early-retirement adjustment. The election provides an eligible survivor with two-thirds of that amount after the member’s death. The member dies at time 2 after retirement. Conditional on the survivor being alive and eligible at the death date, the probabilities of surviving to the next three annual payment dates are 0.97, 0.92, and 0.86. Payments are in arrears at times 1, 2, and 3 after death. The effective annual interest rate is 4%. Find the expected present value at the death date of these three payments.
Method. First calculate the survivor amount from the pension payable under the election: \(C = \frac{2}{3}\times \$28{,}800 = \$19{,}200\). The discount factor is \(v=1/1.04=0.96153846\), so \(v^2=0.92455621\) and \(v^3=0.88899636\).
The three expected present-value terms are:
Adding the unrounded terms gives an expected present value of $48,918.16, rounded to the nearest cent. As a check, the calculation is also \(\$19{,}200(0.93269231+0.85059172+0.76453687)=\$48{,}918.16\), where each quantity in parentheses is the survival probability multiplied by the relevant discount factor.
Check and common mistake. The survival probabilities are conditional on the survivor being alive at the member’s death, so no additional probability for survival to time 2 after retirement belongs in this conditional value. Also, the two-thirds fraction applies to $28,800, the pension after the stated adjustment.
Common Interpretation Errors
The main difficulty is often not arithmetic but translating the election and timing rules correctly. A complete solution should show the pension amount used, the eligibility test, the payment dates, and the treatment of survival. These checks prevent several errors:
- Applying the survivor fraction to the wrong base. Use the pension payable under the elected form and any stated adjustment, then apply the survivor percentage.
- Starting the survivor pension at the death instant without checking payment timing. If payments are annual in arrears and no proration is specified, use the next scheduled due date and the plan’s stated rule.
- Ignoring eligibility. A person’s relationship to the member is not sufficient unless it meets the plan’s stated condition.
- Confusing a guarantee with lifetime continuation. A guarantee covers remaining specified payments; a survivor pension after the guarantee is a separate provision.
- Using survival probabilities from the wrong starting date. In a value conditional on the member’s death and the survivor’s survival to that date, use probabilities measured forward from that date.
Check Your Understanding
For each problem, identify the applicable rule before calculating. Unless stated otherwise, annual payments are in arrears, there is no proration, and money amounts are rounded to the nearest cent where necessary.
- A hypothetical joint-and-survivor pension pays $18,000 per year, with a 50% continuation to an eligible survivor. The member dies at time 4.2, between the time-4 and time-5 payment dates. What amount is payable at time 5 if the survivor is alive and eligible? (Answer: $9,000.)
- A member’s pension under the elected form is $26,000 per year, and the survivor fraction is 75%. The member dies after retirement, and the eligible survivor is alive at the next due date. Find the annual survivor amount. (Answer: $19,500.)
- A plan guarantees eight annual payments of $20,000, with any remaining guaranteed payments going to an eligible survivor if the member dies during the guarantee. After the guarantee, the survivor receives 40% of the member pension. The member dies after the fifth payment but before the sixth; the survivor is alive through time 9. State the payments at times 6 through 9. (Answer: $20,000 at times 6, 7, and 8; $8,000 at time 9.)
- An eligible survivor receives $12,000 at each of the next two annual due dates after the member’s death. Conditional survival probabilities to those dates are 0.90 and 0.80. At an effective annual interest rate of 5%, calculate the expected present value at the death date. (Answer: $18,993.20.)
- A hypothetical election provides a survivor pension only if the survivor met a specified plan condition on the member’s retirement date. The person meets the plan’s relationship description at the later date of the member’s death but did not meet the stated condition at retirement. Under this rule, what survivor pension is payable? (Answer: $0 under this survivor provision, because the specified retirement-date eligibility condition is not met.)