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Exam ALTAM · Ancillary Benefits and Options · Tutorial 377 of 999

Pension Commencement Options Under Competing Risks

Learn to value pension commencement choices by separating the chance of reaching retirement from the value of each option at retirement.

Advanced 9 min read

What You'll Learn

  • Calculate the probability of remaining active until a specified retirement date under annual competing decrements.
  • Discount conditional pension and lump-sum values to an earlier valuation date.
  • Include benefits payable on withdrawal or death before retirement without double counting.
  • Compare retirement options on a consistent expected-present-value basis.
  • Recognize common errors involving conditional probabilities and the timing of cash flows.

Introduction

A pension election is usually described by the benefits available if the member reaches retirement. But a member who withdraws or dies before that date may never make the election. Comparing a lump sum and a pension therefore requires two distinct calculations: the probability of remaining active until retirement, and the value of each option conditional on reaching that date.

This tutorial develops that comparison using annual competing decrement probabilities. It also shows how to include benefits payable on earlier exits. The options must be valued at a common date and on a consistent interest and post-retirement assumption basis. A conditional value at retirement is not automatically the same as an expected present value before retirement.

Definition: Suppose a member is active at the start of each year before a specified retirement date. Let \(p_j\) be the conditional probability of remaining active through year \(j\), given active status at its start. Let \(q_{j,r}\) be the conditional probability of exit from cause \(r\) during year \(j\). Under an exhaustive set of mutually exclusive outcomes, \(p_j+\sum_r q_{j,r}=1\). The probability of reaching retirement after \(n\) years is \(\prod_{j=1}^{n}p_j\).

Valuing the Choices at a Common Date

Let \(i\) be the effective annual interest rate and \(v=1/(1+i)\). Suppose the retirement options are available at the end of year \(n\). Write \(V_R^{(o)}\) for the value at retirement of option \(o\), conditional on reaching retirement. For a lump sum, this may simply be the amount payable at retirement. For a pension, it is the value at retirement of the pension payments, using the applicable post-retirement assumptions.

If a benefit \(B_{j,r}^{(o)}\) is payable at the end of year \(j\) when the member exits through cause \(r\), the expected present value at the start of year 1 is

$$ \operatorname{EPV}^{(o)} = v^n\left(\prod_{j=1}^{n}p_j\right)V_R^{(o)} + \sum_{j=1}^{n}v^j \left(\prod_{k=1}^{j-1}p_k\right) \sum_r q_{j,r}B_{j,r}^{(o)}. $$

The empty product for \(j=1\) is 1. The first term values the retirement choice only on the event that the member reaches retirement. Each term in the sum values an earlier exit benefit using the probability of reaching that year and then exiting from the specified cause.

1
Specify the valuation date and timeline.
Identify when retirement occurs, when earlier exit benefits are paid, and how many years each payment is discounted.
2
Find the probability of reaching retirement.
Multiply the conditional active-continuation probabilities for all pre-retirement years.
3
Value each retirement option conditionally.
Use the lump-sum amount or the pension value at retirement, with post-retirement assumptions applied only within that conditional value.
4
Add any earlier exit benefits.
For each year and cause, multiply the conditional exit probability by the probability of remaining active to that year, then discount the payment to the valuation date.

Why the Valuation Formula Holds

The product of active-continuation probabilities follows from the multiplication rule for conditional probabilities. To reach retirement, the member must remain active in year 1, remain active in year 2 conditional on surviving year 1, and continue this way through year \(n\). Thus

$$ P(\text{reach retirement}) = p_1p_2\cdots p_n. $$

The formula for expected present value then follows by partitioning the possible outcomes into reaching retirement and exiting in each pre-retirement year through each cause. These outcomes are mutually exclusive. On each outcome, multiply its probability by the present value of the corresponding payment; summing those probability-weighted values gives the expectation. This reasoning assumes the stated probabilities are conditional on being active at the start of the relevant year, and that the cash-flow timing is as specified. If payments occur at another time, their discount factors must change accordingly.

Worked Example: Comparing Two Choices Three Years Before Retirement

A hypothetical member is three years from retirement. The conditional annual probabilities of remaining active are 0.92, 0.90, and 0.95. The effective annual interest rate is 4%. At retirement, the member may take a lump sum of $80,000 or a pension with a conditional value of $84,000. No benefit is payable on an earlier exit. Find the probability of reaching retirement and the expected present value of each choice at the start of the three-year period.

Method. First multiply the active-continuation probabilities. Then multiply each retirement value by that probability and discount for three years.

The probability of reaching retirement is

$$ 0.92(0.90)(0.95)=0.7866. $$

The common present-value factor is

$$ \frac{0.7866}{1.04^3} = \frac{0.7866}{1.124864} \approx 0.6992845358. $$

Therefore, the pension choice has expected present value

$$ \$84{,}000(0.6992845358) = \$58{,}739.900\ldots \approx \$58{,}739.90, $$

and the lump-sum choice has expected present value

$$ \$80{,}000(0.6992845358) = \$55{,}942.762\ldots \approx \$55{,}942.76. $$

The conditional pension value exceeds the lump sum by $4,000 at retirement, so its expected present value exceeds the lump-sum option by \(\$4{,}000(0.6992845358)=\$2{,}797.14\). The results are checked by using the same probability and discount factor for both options. A common mistake is to round the factor too early or to discount by the wrong number of years; keeping the full factor until the final step gives the stated amounts. Dollar values are rounded to the nearest cent.

Earlier Exits and Benefits Payable Before Retirement

An earlier exit benefit is not part of the retirement option value unless the plan explicitly makes it so. It is a separate cash flow, payable only if an exit occurs. The probability of exit in a later year is conditional: it must be multiplied by the probability of remaining active through all preceding years. This is particularly important when several exit causes compete.

If an exit benefit is identical under both retirement choices, its expected present value is added to each option. It affects each total value but does not change the difference between the two choices. If the exit benefit differs by option or cause, it can affect which option has the greater total expected present value.

Worked Example: Including Benefits on Withdrawal or Death

A hypothetical member is two years from retirement. In each year, conditional on being active at the year's start, the probability of withdrawal is 0.06 and the probability of death is 0.04. These causes are mutually exclusive, so the active-continuation probability is \(1-0.06-0.04=0.90\) in each year. Interest is 0%. A $2,000 benefit is payable at the end of the year of either type of exit, under both choices. At retirement, the pension has value $42,000 and the lump sum is $40,000. Find each total expected present value.

The probability of reaching retirement is

$$ 0.90^2=0.81. $$

The probability of exit in year 1 is \(0.06+0.04=0.10\). The probability of reaching year 2 active and then exiting is \(0.90(0.10)=0.09\). Thus the expected value of the earlier exit benefits is

$$ \$2{,}000(0.10)+\$2{,}000(0.09) = \$200+\$180 = \$380. $$

The pension option's total expected present value is

$$ \$42{,}000(0.81)+\$380 = \$34{,}020+\$380 = \$34{,}400. $$

The lump-sum option's total expected present value is

$$ \$40{,}000(0.81)+\$380 = \$32{,}400+\$380 = \$32{,}780. $$

The totals differ by $1,620, which is the $2,000 difference in retirement values multiplied by the 0.81 probability of reaching retirement. A common mistake is to count the second-year exit benefit as \(\$2{,}000(0.10)\), forgetting that the member must first remain active through year 1. The correct second-year probability is \(0.90(0.10)=0.09\). With zero interest, no further discounting is needed.

Valuing a Pension Conditional on Retirement

A pension's value at retirement may itself depend on post-retirement survival. That mortality is different from the pre-retirement probability of reaching retirement. To illustrate the distinction, suppose a hypothetical pension pays $5,000 annually in advance for life at retirement. Assume a constant annual post-retirement mortality probability of 0.05, independent from year to year, and an effective annual interest rate of 5%. The value at retirement is an annuity-due value: the first payment is immediate, and each later payment is made only if the member survives to that payment date.

If \(p=0.95\) denotes the annual post-retirement survival probability, the value at retirement per dollar of annual pension is

$$ \ddot{a} = \sum_{k=0}^{\infty}v^kp^k = \frac{1}{1-vp} = \frac{1}{1-(0.95/1.05)} = 10.5. $$

The first equality sums the expected discounted payments: the payment at time \(k\) is made with probability \(p^k\), and is discounted by \(v^k\). The final equality uses the sum of a convergent geometric series, since \(vp=0.95/1.05<1\).

Worked Example: Including Post-Retirement Mortality in the Pension Option

A hypothetical member is three years from retirement. The annual active-continuation probabilities are 0.95, 0.90, and 0.92. The effective annual interest rate is 5% both before and after retirement. The pension pays $5,000 annually in advance for life, with independent annual post-retirement mortality probability 0.05. A competing lump-sum option is $50,000, payable at retirement. Find the expected present value of each option now.

Method. Calculate the pension's value conditional on reaching retirement, then multiply by the pre-retirement survival probability and discount for three years. Do not use post-retirement mortality to alter the probability of reaching retirement.

The probability of reaching retirement is

$$ 0.95(0.90)(0.92)=0.7866. $$

As established above, the conditional annuity-due factor is 10.5. Therefore, the pension value at retirement is

$$ \$5{,}000(10.5)=\$52{,}500. $$

The expected present value of the pension now is

$$ \frac{0.7866}{1.05^3}(\$52{,}500) = 0.679494655\ldots(\$52{,}500) = \$35{,}673.47. $$

The expected present value of the lump sum is

$$ \frac{0.7866}{1.05^3}(\$50{,}000) = \$33{,}974.73. $$

On these assumptions, the pension's expected present value is $1,698.74 higher. The calculation is checked by noting that the options share the same probability and discount factor, so the difference must equal \(\$2{,}500(0.679494655\ldots)=\$1{,}698.74\), subject to rounding. Dollar values are rounded to the nearest cent.

Common Exam Traps

  • Using unconditional exit probabilities in every year. A year-2 exit probability applies only to members still active at the start of year 2. Multiply it by the probability of remaining active through year 1.
  • Forgetting the retirement contingency. A value payable only at retirement must be multiplied by the probability of reaching retirement before discounting to an earlier date.
  • Discounting the wrong cash flow period. A payment at the end of year \(j\) is discounted by \(v^j\); a payment at the end of year \(n\) is discounted by \(v^n\).
  • Mixing pre- and post-retirement survival. Pre-retirement active continuation determines whether the option is reached. Post-retirement mortality belongs in the conditional pension value.
  • Comparing values on inconsistent bases. The pension and lump sum should be valued at the same date and with compatible interest and benefit assumptions. A greater expected present value is a model-based comparison, not by itself a statement about an individual member's preference.
Key takeaway: Separate the probability of reaching retirement from the value of the pension or lump sum conditional on reaching it. Then add any earlier exit benefits using their own conditional probabilities, payment dates, and discount factors. This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated timing and assumptions. Round dollar answers to the nearest cent where needed.

  1. A member is two years from retirement, with annual active-continuation probabilities 0.88 and 0.93. The effective annual interest rate is 3%, and a pension has conditional value $70,000 at retirement. Find the probability of reaching retirement and the expected present value now. (Answer: probability 0.8184; expected present value $53,999.43)
  2. A member is three years from retirement, with active-continuation probabilities 0.96, 0.91, and 0.94. Interest is 0%. The conditional pension value is $67,000 and the lump sum is $63,000. Find the expected present value of each option. (Answer: pension $55,019.33; lump sum $51,734.59)
  3. A member is two years from retirement. In each year, conditional withdrawal probability is 0.05 and conditional death probability is 0.03; the causes are mutually exclusive. Interest is 0%. An exit benefit of $1,500 is paid at either cause of exit at the end of the exit year, under both choices. The pension value at retirement is $48,000 and the lump sum is $45,000. Find the total expected present value of each choice. (Answer: pension $40,857.60; lump sum $38,318.40)
  4. A member is two years from retirement, with active-continuation probabilities 0.90 and 0.80. Interest is 5%. The pension's conditional value at retirement is $60,000 and the lump sum is $58,000. No earlier exit benefit is payable. Find the expected present value of each choice and their difference. (Answer: pension $39,183.67; lump sum $37,877.55; difference $1,306.12)

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