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Exam ALTAM · Ancillary Benefits and Options · Tutorial 388 of 999

Retirement Age Effects on Survivor Options

Learn to compare survivor-option values at different retirement ages without overlooking the pension amount, survival to retirement, or the survivor’s conditional payment period.

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What You'll Learn

  • Separate the value conditional on retirement from the cost measured at the valuation date.
  • Include the probability that the member and spouse both survive to each proposed retirement age.
  • Calculate a survivor-pension factor from conditional post-retirement mortality and interest assumptions.
  • Compare retirement ages when pension amounts and survivor factors change.
  • Identify errors caused by omitting survival weighting or applying it twice.

Why Retirement Age Changes Survivor-Option Value

A survivor option may pay a spouse a stated fraction of the member’s pension after the member dies. Comparing the option at two retirement ages requires more than comparing the pension amounts. A later retirement may produce a larger pension, but the member and spouse must survive longer to reach it. The survivor’s expected payment period may also change because the spouse and member are older at the later retirement age.

To make a consistent comparison, distinguish the value conditional on reaching retirement from the expected value measured at an earlier valuation date. The first reflects payments after retirement; the second also accounts for the probability of reaching retirement and the discounting to that date.

Definition: For retirement at time \(r\), let \(P_r\) be the member’s annual pension, \(\alpha\) the survivor fraction, and \(b_r\) the value at retirement of one dollar per year of survivor pension, conditional on both lives being alive at retirement. Let \(J_r\) be the probability that both are alive at time \(r\), and let \(v^r\) be the discount factor from time \(r\) to the valuation date. Under the assumptions stated below, the expected present value of the survivor option is $$ C_r=v^r J_r \alpha P_r b_r. $$

This formula assumes the option is available only if both the member and spouse are alive at retirement, payments are annual in arrears, and the survivor pension is \(\alpha P_r\) while the spouse is alive after the member’s death. It assumes no guarantee period, refund, or other ancillary payment. The factor \(b_r\) is conditional on both being alive at retirement, so survival to retirement must not be included in it again. If the option has different eligibility conditions, payment timing, or guarantees, the expression must be adjusted.

Valuing the Survivor Payment Period

Suppose the survivor pension is paid at the end of each year after retirement if the member has died by that payment date and the spouse is still alive. Write \({}_kp_{m,r}\) and \({}_kp_{s,r}\) for the conditional probabilities that, respectively, the member and spouse survive \(k\) years after retirement at time \(r\). If future lifetimes are conditionally independent, the value per dollar of annual survivor pension is

$$ b_r=\sum_{k=1}^{\infty}v^k\left(1-{}_kp_{m,r}\right){}_kp_{s,r}. $$

The term \(1-{}_kp_{m,r}\) is the probability that the member has died by the \(k\)th payment date, while \({}_kp_{s,r}\) is the probability the spouse is alive then. Independence allows these probabilities to be multiplied. If the lives are dependent, use the joint probability that the member is dead and the spouse alive instead.

Derivation: At payment time \(k\), the survivor pension is due exactly when the member has died by then and the spouse is alive. Under conditional independence, that event has probability \((1-{}_kp_{m,r}){}_kp_{s,r}\). Discounting the payment by \(v^k\), multiplying by the annual amount \(\alpha P_r\), and summing over payment dates gives the conditional value \(\alpha P_r b_r\). Reaching retirement is a separate event with probability \(J_r\), and discounting to the valuation date contributes \(v^r\). Thus the expected present value is \(v^rJ_r\alpha P_rb_r\).

For a useful special case, suppose post-retirement annual survival probabilities are constant, \(p_m\) for the member and \(p_s\) for the spouse, with independent lifetimes. Then \({}_kp_m=p_m^k\) and \({}_kp_s=p_s^k\). Provided both geometric series converge,

$$ b=\sum_{k=1}^{\infty}(vp_s)^k-\sum_{k=1}^{\infty}(vp_mp_s)^k =\frac{vp_s}{1-vp_s}-\frac{vp_mp_s}{1-vp_mp_s}. $$

This expression makes the competing effects visible: the spouse’s survival supports payments, while the member’s survival delays their start. A single factor may be used at alternative retirement ages only if the relevant conditional mortality and interest basis makes it appropriate at both ages.

Worked Comparisons Across Retirement Ages

The following examples use hypothetical pension arrangements. In each comparison, the expected present value is measured at time zero, and all stated survival probabilities refer to the relevant period and conditioning event.

Worked Example: Correctly Weighting Two Retirement Ages

A hypothetical plan is valued at 0% interest. The probability that both member and spouse survive each year before retirement, conditional on both being alive at the start of that year, is 0.95. The survivor fraction is 50%. At retirement after two years, the annual member pension is $24,000; after four years, it is $32,000. The conditional survivor factor is 2.4 at either retirement age. Compare the expected costs at time zero.

Method. Since interest is 0%, \(v^r=1\). The joint probability of reaching retirement is \(J_r=0.95^r\). The conditional option value at retirement is \(\alpha P_r b_r\).

For retirement after two years,

$$ J_2=0.95^2=0.9025,\qquad \alpha P_2b_2=0.50(24{,}000)(2.4)=28{,}800. $$

Therefore,

$$ C_2=1(0.9025)(28{,}800)=\$25{,}992.00. $$

For retirement after four years,

$$ J_4=0.95^4=0.81450625,\qquad \alpha P_4b_4=0.50(32{,}000)(2.4)=38{,}400. $$

Therefore,

$$ C_4=1(0.81450625)(38{,}400)=\$31{,}277.04. $$

The later retirement has the larger expected cost: $31,277.04 compared with $25,992.00. The larger pension more than offsets the lower probability of reaching retirement. Common mistake avoided: using the conditional value at retirement as if it were already the time-zero value. In particular, the two-year calculation is \(28{,}800(0.95)^2=\$25{,}992.00\), not $25,920.00.

Worked Example: Calculating the Conditional Survivor Factor

A hypothetical option has a 60% survivor fraction. Compare retirement after two or four years. The annual effective interest rate is 5%, so \(v=1/1.05=20/21\). Before retirement, the conditional probability that both lives survive each year is 0.96. After retirement, assume independent lifetimes with constant annual survival probabilities \(p_m=0.90\) for the member and \(p_s=0.95\) for the spouse. The annual member pensions are $30,000 at year two and $36,000 at year four. Payments are annual in arrears, with no guarantee period.

First calculate the conditional survivor factor. The first geometric-series term is

$$ \frac{vp_s}{1-vp_s} =\frac{(20/21)(19/20)}{1-(20/21)(19/20)} =\frac{19/21}{2/21} =\frac{19}{2}=9.5. $$

The second term is

$$ \frac{vp_mp_s}{1-vp_mp_s} =\frac{(20/21)(0.90)(0.95)}{1-(20/21)(0.90)(0.95)} =\frac{57/70}{13/70} =\frac{57}{13}. $$

Thus

$$ b=9.5-\frac{57}{13} =\frac{247-114}{26} =\frac{133}{26} \approx 5.1153846. $$

The pre-retirement joint survival and discounting combine to \(v^rJ_r=(0.96/1.05)^r=(32/35)^r\). At year two, the conditional value at retirement is

$$ 0.60(30{,}000)\frac{133}{26} =\$92{,}076.9231. $$

Therefore,

$$ C_2=\left(\frac{32}{35}\right)^2(92{,}076.9231) =\$76{,}968.79. $$

At year four, the conditional value at retirement is

$$ 0.60(36{,}000)\frac{133}{26} =\$110{,}492.3077. $$

Therefore,

$$ C_4=\left(\frac{32}{35}\right)^4(110{,}492.3077) =\$77{,}207.55. $$

The four-year option is slightly more costly, despite the lower probability of reaching retirement, because its pension is larger. Money amounts are rounded to the nearest cent only at the final step. Common mistake avoided: treating the survivor factor as an annuity payable whenever the spouse is alive; each payment also requires that the member has died by that date.

Worked Example: A Shorter Survivor Period Can Reverse the Comparison

A hypothetical plan is valued at 0% interest. The probability that both lives survive each pre-retirement year is 0.90. The survivor fraction is 50%. Retirement after two years gives an annual pension of $20,000 and a conditional survivor factor of 4.0; retirement after five years gives an annual pension of $29,000 and a conditional survivor factor of 3.0. The factors are the supplied values from the plan’s conditional post-retirement mortality basis, measured per dollar of annual survivor pension. Compare costs at time zero.

For the two-year option, the conditional value at retirement is

$$ 0.50(20{,}000)(4.0)=\$40{,}000. $$

The probability of reaching retirement is \(0.90^2=0.81\), so

$$ C_2=1(0.81)(40{,}000)=\$32{,}400.00. $$

For the five-year option, the conditional value at retirement is

$$ 0.50(29{,}000)(3.0)=\$43{,}500. $$

The probability of reaching that retirement date is \(0.90^5=0.59049\), so

$$ C_5=1(0.59049)(43{,}500)=\$25{,}686.315 \approx \$25{,}686.32. $$

Here the later pension is larger, but the shorter conditional survivor payment period and lower chance of reaching retirement make its expected cost smaller. This comparison shows why no single component determines the ranking. Common mistake avoided: assuming that a higher annual pension necessarily means a higher expected survivor-option value.

A Reliable Comparison Procedure

Use the same valuation date, payment timing, and assumptions when comparing retirement ages. The following sequence keeps conditional and unconditional quantities separate.

1
Specify the option at each age.
Record the annual pension \(P_r\), survivor fraction \(\alpha\), payment timing, and any guarantee or eligibility condition.
2
Value payments after retirement.
Calculate \(b_r\) using mortality conditional on both lives being alive at retirement and the applicable interest basis.
3
Weight for reaching retirement.
Use the joint survival probability \(J_r\). If independence is justified, this may be calculated as the product of the two survival probabilities; otherwise use the joint probability directly.
4
Discount and compare.
Calculate \(C_r=v^rJ_r\alpha P_rb_r\) for each age, then compare values measured at the same date.

Interpretation and Exam Traps

The formula separates four influences: discounting to retirement, joint survival to retirement, the annual survivor pension, and the conditional survivor payment factor. A later retirement can increase \(P_r\), decrease \(J_r\), and change \(b_r\) in either direction. The direction of the overall change must be determined from the combined calculation rather than inferred from one factor.

  • Omitting survival to retirement: \(\alpha P_rb_r\) is conditional on reaching retirement. It is not yet the value at time zero.
  • Applying survival twice: if \(b_r\) is conditional on both lives being alive at retirement, do not include pre-retirement survival in that factor as well as in \(J_r\).
  • Using the wrong survivor event: a payment requires the member to have died and the spouse to be alive at its payment date. A spouse-survival annuity alone does not represent this benefit.
  • Reusing a factor without checking its basis: age-dependent mortality or a changed interest basis can make \(b_r\) differ between retirement ages.
  • Comparing values at different dates: discount every alternative to a common valuation date before deciding which has the greater expected cost.
Key takeaway: For each proposed retirement age, value the survivor payments conditional on retirement, multiply by the probability both lives reach retirement and by the pension and survivor fraction, and discount to a common date. A larger pension alone does not establish a larger expected option value.

Check Your Understanding

For each problem, compare values at the stated valuation date and use annual payments in arrears unless the problem says otherwise.

  1. A plan is valued at 0% interest. Both lives survive each pre-retirement year with conditional probability 0.92. The survivor fraction is 50%, and the survivor factor is 2.5 at both ages. The annual pension is $18,000 at retirement after one year and $25,000 after three years. Which option has the larger expected cost? (Answer: One year: $20,700.00; three years: $24,310.40; the three-year option is larger.)
  2. At 4% effective annual interest, \(v=1/1.04\). The joint probability of reaching retirement is 0.95 per year. A 40% survivor fraction applies to a $28,000 annual pension at year two; the conditional survivor factor is 3.2. Find the time-zero expected cost. (Answer: \(v^2(0.95)^2(0.40)(28{,}000)(3.2)=\$29{,}232.16\), rounded to cents.)
  3. Assume 0% interest and independent post-retirement lifetimes with constant annual survival probabilities \(p_m=0.80\) and \(p_s=0.90\). Calculate the conditional survivor factor per dollar of annual pension. (Answer: \(0.90/(1-0.90)-(0.80)(0.90)/(1-0.80(0.90))=9-3.6=5.4\).)
  4. At 0% interest, the joint probability of reaching retirement is 0.97 per year. A 60% survivor fraction applies. At year two, \(P_2=\$22{,}000\) and \(b_2=3.5\); at year four, \(P_4=\$27{,}000\) and \(b_4=2.8\). Find both expected costs and identify the larger. (Answer: \(C_2=(0.97)^2(0.60)(22{,}000)(3.5)=\$42{,}626.46\); \(C_4=(0.97)^4(0.60)(27{,}000)(2.8)=\$38{,}470.38\); the two-year option is larger.)
  5. A valuation gives \(P_r=\$31{,}000\), \(\alpha=0.50\), and \(b_r=3.0\) at a retirement date six years away. The probability both lives survive to that date is 0.80, and the annual effective interest rate is 5%. Calculate the time-zero expected cost. (Answer: \((1/1.05)^6(0.80)(0.50)(31{,}000)(3.0)=\$30{,}329.40\), rounded to cents.)

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