Withdrawal and Death Lump-Sum Values

Why Cause and Payment Timing Both Matter

A pension plan may pay a refund when an active member withdraws and a separate lump sum if the member dies in service. These benefits are not payable on the same event: a withdrawal refund is contingent on withdrawal, while a death-in-service lump sum is contingent on death before withdrawal or another exit. Their expected present values must therefore use the probability of the relevant cause, not simply the probability of leaving active service for any reason.

The calculation also depends on when each benefit is paid. In the discrete model used here, probabilities are specified for each year conditional on active status at its start. Unless a problem states a different date, a benefit arising from an exit during that year is paid at the end of the year. The payment is discounted to the valuation date using that actual payment time.

Definition: For year $k+1$, let $q_k^{(w)}$ be the probability of withdrawal during the year and $q_k^{(d)}$ the probability of death during the year, each conditional on being an active member at the start of year $k+1$. If withdrawal and death are the only possible exits, the probability of remaining active through that year is $p_k^{(a)}=1-q_k^{(w)}-q_k^{(d)}$. Define ${}_kp^{(a)}$ as the probability of being active at the start of year $k+1$, with ${}_0p^{(a)}=1$. Thus ${}_{k+1}p^{(a)}={}_kp^{(a)}p_k^{(a)}$.

These cause probabilities are conditional on being active at the beginning of the relevant year. They are not probabilities for the original group without regard to whether members have already left. To obtain an unconditional probability of a particular cause in a later year, multiply its conditional probability by the probability of still being active at the start of that year.

Discrete Lump-Sum Valuation Formulas

Suppose a withdrawal refund of $B_k^{(w)}$ is paid at the end of year $k+1$ if withdrawal occurs in that year. Suppose a death benefit of $B_k^{(d)}$ is paid at the end of that same year if death occurs. With an effective annual interest rate $i$, let $v=1/(1+i)$. The separate present values over $n$ years are:

\operatorname{EPV}_{w} = \sum_{k=0}^{n-1} {}_kp^{(a)}q_k^{(w)}B_k^{(w)}v^{k+1}, \qquad \operatorname{EPV}_{d} = \sum_{k=0}^{n-1} {}_kp^{(a)}q_k^{(d)}B_k^{(d)}v^{k+1}.

The combined value is the sum of the two cause-contingent values. This follows from the additivity of expected value: in each year, the withdrawal and death payments are triggered by mutually exclusive causes, and each payment contributes its discounted amount only when its own cause occurs. Consequently, the combined EPV is $\operatorname{EPV}_{w}+\operatorname{EPV}_{d}$; it is not necessary to treat either benefit as payable on every exit.

Timing condition: The factor $v^{k+1}$ applies when the payment is made at the end of year $k+1$. If the benefit is paid at another specified time, discount to that time instead. The cause probability and active-member probability are still determined by the year in which the exit occurs.

How to Organize the Calculation

A duration-by-duration table makes it easier to keep conditional probabilities, unconditional cause probabilities, benefit amounts, and payment dates distinct. The active probability at the start of a year is carried forward only after applying the probability of remaining active in the preceding year.

Set the valuation basis.
Record the annual effective interest rate, the modeled years, the exit causes, and the stated payment dates.

Find active-member probabilities.
For each year, calculate $p_k^{(a)}=1-q_k^{(w)}-q_k^{(d)}$, then multiply sequentially to obtain ${}_kp^{(a)}$.

Value each cause separately.
For each year and cause, multiply the start-of-year active probability, the cause-specific conditional probability, and the corresponding benefit.

Discount and reconcile.
Discount each expected payment from its actual payment date, sum by cause, and check that the combined EPV equals the sum of the separate EPVs.

Worked Valuations

Worked Example: Two-Year Refund and Death Benefit

A hypothetical plan has no interest, so $i=0$ and $v=1$. During year 1, the conditional withdrawal probability is $0.06$ and the conditional death probability is $0.02$. During year 2, the corresponding probabilities are $0.10$ and $0.03$. A withdrawal refund of \$1,500 and a death-in-service benefit of \$8,000 are each paid at the end of the year in which the exit occurs. Find both EPVs and their combined value.

Method. Calculate the probability of active membership at the start of each year, then multiply by the appropriate conditional cause probability and benefit. With $i=0$, discounting does not change the amounts.

In year 1, the probability of remaining active is $1-0.06-0.02=0.92$. Therefore, the probability of being active at the start of year 2 is $0.92$.

The withdrawal EPV is

(1)(0.06)(1500)+(0.92)(0.10)(1500) =90+138 =\$228.00.

The death-in-service EPV is

(1)(0.02)(8000)+(0.92)(0.03)(8000) =160+220.80 =\$380.80.

Thus the combined EPV is $228.00+380.80=\$608.80$. Check: the year 2 probabilities are applied to the 0.92 who are still active, not to the original membership. A common error is to use the year 2 conditional probabilities as unconditional probabilities.

Worked Example: Changing Benefits and Positive Interest

A hypothetical pension arrangement is valued for three years at an effective annual interest rate of $5\%$. Conditional withdrawal and death probabilities are, respectively, $0.08$ and $0.02$ in year 1, $0.10$ and $0.03$ in year 2, and $0.12$ and $0.04$ in year 3. A withdrawal refund of \$1,000, \$1,200, and \$1,400 is payable at the end of years 1, 2, and 3, respectively. The death benefits at those dates are \$10,000, \$9,000, and \$8,000. Find the separate and combined EPVs.

Method. Use the conditional cause probabilities to build active-member probabilities, then value each end-of-year payment with the matching power of $v=1/1.05$.

The annual probabilities of remaining active are $0.90$, $0.87$, and $0.84$. Therefore, the active probabilities at the starts of years 1, 2, and 3 are $1$, $0.90$, and $0.90(0.87)=0.783$. The discount factors are $v=0.952381$, $v^2=0.907029$, and $v^3=0.863838$, shown to six decimal places for the intermediate calculations.

For withdrawal, the expected payments before discounting are $1(0.08)(1000)=\$80$, $0.90(0.10)(1200)=\$108$, and $0.783(0.12)(1400)=\$131.544$. Hence

\operatorname{EPV}_{w} =80(0.952381)+108(0.907029)+131.544(0.863838) =\$287.78.

For death, the corresponding expected payments are $1(0.02)(10000)=\$200$, $0.90(0.03)(9000)=\$243$, and $0.783(0.04)(8000)=\$250.56$. Thus

\operatorname{EPV}_{d} =200(0.952381)+243(0.907029)+250.56(0.863838) =\$627.33.

The combined EPV is $\$287.78+\$627.33=\$915.11$. Amounts are rounded to the nearest cent only at the final step. Check: year 3 begins with $78.3\%$ of the original membership still active, not $90\%$; both prior years' active-retention probabilities must be applied.

Worked Example: A Death Benefit Paid One Year After Exit

A hypothetical plan is valued for two years at an effective annual interest rate of $4\%$. In year 1, withdrawal and death probabilities conditional on active status are $0.05$ and $0.01$. In year 2 they are $0.08$ and $0.02$. A withdrawal refund of \$2,500 is paid at the end of the year of withdrawal. A death benefit of \$12,000 is paid one year after death. Find the EPVs and combined value.

Method. First determine exit probabilities and active membership by year. Then use the payment date for each benefit when selecting its discount factor. Here $v=1/1.04=0.961538$, $v^2=0.924556$, and $v^3=0.888996$.

The year 1 active-retention probability is $1-0.05-0.01=0.94$, so $0.94$ is active at the start of year 2. The withdrawal refund for a year 1 exit is paid at time 1; a year 2 withdrawal refund is paid at time 2. Therefore,

\operatorname{EPV}_{w} =(1)(0.05)(2500)v+(0.94)(0.08)(2500)v^2 =125(0.961538)+188(0.924556) =\$294.01.

A death in year 1 occurs during $[0,1)$, and its payment is made at time 2. A death in year 2 occurs during $[1,2)$, and its payment is made at time 3. The corresponding expected payments are $1(0.01)(12000)=\$120$ and $0.94(0.02)(12000)=\$225.60$. Thus

\operatorname{EPV}_{d} =120v^2+225.60v^3 =120(0.924556)+225.60(0.888996) =\$311.50.

The combined EPV is $\$294.01+\$311.50=\$605.51$. Check: discounting the death benefits by $v$ and $v^2$ would incorrectly treat them as paid at the end of the year of death. The correct powers are $v^2$ and $v^3$, because payment is delayed by one year.

Common Errors and Valuation Checks

The most consequential errors usually come from mixing up conditional and unconditional probabilities or from assigning the wrong payment date. Before accepting a result, check the following:

Use cause-specific probabilities. A withdrawal refund uses $q_k^{(w)}$; a death benefit uses $q_k^{(d)}$. The total exit probability $q_k^{(w)}+q_k^{(d)}$ is not a substitute for either.
Carry active membership forward correctly. Each year's cause probability is conditional on active status at the start of that year. Multiply by ${}_kp^{(a)}$ to obtain the probability of that cause from the original membership.
Check the probability balance. If withdrawal and death are the only exits, $q_k^{(w)}+q_k^{(d)}\leq 1$, and $p_k^{(a)}=1-q_k^{(w)}-q_k^{(d)}$. If the model includes another exit cause, it must also be accounted for before calculating active retention.
Match discounting to payment dates. The discount exponent measures time from valuation to payment, not merely the year in which the exit occurs.
Keep benefits attached to their causes. A refund and a death benefit can have different amounts, escalation rules, or payment dates; value each on its own terms before combining the EPVs.

These calculations are useful whenever a benefit is triggered by one of several mutually exclusive active-member exits. Correctly separating the causes prevents one benefit from being assigned to exits that do not trigger it. The same structure also makes a valuation auditable: each term identifies the probability of reaching the year, the probability of the specified cause, the amount payable, and the date of payment.

Key takeaway: For each cause, multiply active-member probability at the start of the year by that cause's conditional exit probability and its benefit amount, then discount to the benefit's payment date. Add separate cause-specific EPVs only after each has been valued consistently.

Check Your Understanding

For each problem, use the stated payment timing and report money to the nearest cent. These exercises are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

A two-year model has $i=0\%$. In year 1, withdrawal and death probabilities are $0.06$ and $0.02$; in year 2 they are $0.10$ and $0.03$. A \$1,500 refund is paid on withdrawal and an \$8,000 benefit is paid on death, both at year-end. Find each cause-specific EPV and the combined EPV. (Answer: withdrawal \$228.00; death \$380.80; combined \$608.80)
At $i=0\%$, three years have withdrawal/death probabilities $(0.04,0.01)$, $(0.05,0.02)$, and $(0.06,0.03)$. A \$1,000 refund and a \$5,000 death benefit are paid at the end of the year of exit, at any duration. Find both EPVs and their combined value. (Answer: withdrawal \$140.51; death \$277.53; combined \$418.04)
A plan has $i=10\%$. In year 1, withdrawal and death probabilities are $0.10$ and $0.02$; in year 2 they are $0.05$ and $0.03$. A \$2,000 refund and a \$10,000 death benefit are each paid at the end of the year of exit. Find both EPVs and the combined EPV. (Answer: withdrawal \$254.55; death \$400.00; combined \$654.55)
At $i=0\%$, year 1 withdrawal and death probabilities are $0.10$ and $0.05$; in year 2 they are $0.20$ and $0.10$. A withdrawal refund $R$ is paid at year-end, and the death benefit is \$5,000 paid at year-end. If the combined two-year EPV is \$1,215, find $R$ and the separate EPVs. (Answer: $R=\$2,000$; withdrawal \$540.00; death \$675.00; combined \$1,215.00)
A plan has $i=4\%$. Year 1 withdrawal and death probabilities are $0.05$ and $0.01$; year 2 probabilities are $0.08$ and $0.02$. A \$2,500 refund is paid at the end of the year of withdrawal, while a \$12,000 death benefit is paid one year after death. Find each EPV and the combined value. (Answer: withdrawal \$294.01; death \$311.50; combined \$605.51)

What You'll Learn