Withdrawal Benefits With Service-Based Formulas

Why Service-Based Withdrawal Benefits Need Careful Valuation

A withdrawal benefit can depend on how long a member has worked when active membership ends. The plan might pay nothing before a service threshold, a fixed amount within each service band, or an amount that increases with completed service and salary. Valuing such a benefit therefore requires two linked calculations: determine the benefit at each possible withdrawal date, then weight it by the probability that withdrawal occurs at that date while the member is still active.

The second calculation matters because withdrawal is not necessarily the only way active membership ends. Death, retirement, or another specified exit can remove a member from the group eligible to withdraw later. The annual withdrawal probability in the model below is conditional on being active at the start of that year; it is not itself the probability that a member from the original population withdraws in that year.

Definition: A service-based withdrawal benefit is an amount $B_k$ payable if a member withdraws at the end of service year $k$, where the amount depends on service attained by that date. A service-band formula assigns amounts by ranges of service; an accrued-service formula calculates an amount using a specified accrual rule, such as a percentage of salary for each year of service.

Annual Competing-Exit Model

Use an annual model in which a member is active at the start of each year and can experience one of several mutually exclusive exits during that year. Assume that exit probabilities are conditional on being active at the start of the year. For year $k+1$, let $q_{w,k}$ be the withdrawal probability and let $q_{j,k}$ be the probability of each other exit cause $j$. The probability of remaining active throughout that year is $p_k=1-q_{w,k}-\sum_j q_{j,k}$, which must be between zero and one.

Let $A_k$ be the probability that the member is still active at the start of year $k+1$, for a member active at time zero. Then $A_0=1$ and $A_{k+1}=A_kp_k$. If withdrawal at the end of year $k+1$ pays $B_{k+1}$, its probability, measured over the original member, is $A_kq_{w,k}$.

\operatorname{EPV} = \sum_{k=0}^{n-1} v^{k+1} A_k q_{w,k} B_{k+1}, \qquad A_0=1,\quad A_{k+1}=A_k\left(1-q_{w,k}-\sum_j q_{j,k}\right).

Here $n$ is the number of years in the projection and $v=1/(1+i)$, where $i$ is the effective annual interest rate. The formula assumes the benefit is paid at the end of the year of withdrawal, service is measured at that date, and the interest rate and benefit amounts are deterministic. If the benefit is zero before an eligibility threshold, set $B_{k+1}=0$ for those withdrawal dates. If the contract instead pays at another time, the discount factor must reflect that payment time.

Why the Expected Present Value Formula Holds

The formula follows by separating the mutually exclusive possible withdrawal years. The probability of reaching the start of year $k+1$ still active is $A_k$. Conditional on being active then, the probability of withdrawal during that year is $q_{w,k}$. Thus, the probability of receiving that year’s withdrawal benefit is their product, $A_kq_{w,k}$. If withdrawal occurs, the benefit is $B_{k+1}$ and its end-of-year present value is $v^{k+1}B_{k+1}$. Multiplying present value by its probability gives the expected contribution from that year. Adding the contributions over all years is valid by linearity of expectation; no independence between different years’ exit events is required.

Determine service and the benefit.
For each possible withdrawal year, apply the plan’s service-band or accrued-service rule at the assumed exit date.

Track active membership.
Multiply the prior active probability by the probability of no exit in that year.

Find withdrawal probability over the original group.
Multiply the probability active at the start of the year by the conditional withdrawal probability.

Discount and add.
Discount each possible benefit to time zero and sum the expected contributions.

Worked Valuations

Worked Example: Service Bands With Three Competing Exits

A hypothetical plan projects an active member for three years. At the end of each year, conditional on being active at the start of that year, the probabilities of withdrawal, death, and retirement are respectively $0.10$, $0.02$, and $0.08$. The plan pays nothing for less than two completed years of service, $4,000 for at least two but less than three years, and $6,000 for at least three years. A benefit is paid at the end of the withdrawal year. The effective annual interest rate is 10%. Calculate the expected present value over the three-year projection.

Method. Calculate the active probability year by year, then apply the service bands to the withdrawal dates. This avoids treating each year’s conditional withdrawal probability as if it applied to the original member.

The probability of remaining active for a year is $1-0.10-0.02-0.08=0.80$. Therefore, the active probabilities at the starts of years 1, 2, and 3 are $1$, $0.80$, and $0.80^2=0.64$. The first-year withdrawal benefit is zero because only one year of service has been completed.

At the end of year 2, the probability of withdrawal is $0.80(0.10)=0.08$, and the benefit is $4,000. At the end of year 3, the probability is $0.64(0.10)=0.064$, and the benefit is $6,000.

\begin{aligned} \operatorname{EPV} &= \frac{0.08(4{,}000)}{1.10^2} + \frac{0.064(6{,}000)}{1.10^3}\\ &= \frac{320}{1.21}+\frac{384}{1.331}\\ &=264.4628+288.5049\\ &=552.9677. \end{aligned}

The expected present value is $552.97, rounded to the nearest cent. The probability of remaining active through two years is $0.64$, which also checks the year-3 withdrawal probability $0.64(0.10)=0.064$. A common mistake is to assign the year-3 withdrawal probability as $0.10$ over the original group, ignoring prior exits.

Worked Example: Salary-Related Accrued-Service Benefit

A hypothetical plan pays a withdrawing member $8\%$ of final salary for each completed year of service. Salary is $50,000 at time zero and increases by $4\%$ at the end of each year. Project three years. Conditional on being active at the start of each year, the withdrawal probabilities are $0.10$, $0.12$, and $0.15$; the probabilities of all other exits combined are $0.05$, $0.08$, and $0.10$. Benefits are paid at the end of the withdrawal year. The effective annual interest rate is $4\%$. Find the expected present value.

Method. Calculate each year’s salary and accrued benefit, then weight it by the probability of withdrawal in that year. In this example, salary growth and interest happen to offset when discounting the benefit; this is a property of the chosen assumptions, not a general rule.

The probabilities of remaining active in years 1, 2, and 3 are $0.85$, $0.80$, and $0.75$. Thus, the probabilities of withdrawal over the original group are $0.10$ in year 1, $0.85(0.12)=0.102$ in year 2, and $0.85(0.80)(0.15)=0.102$ in year 3.

The salaries at the three exit dates are $52,000, $54,080, and $56,243.20. Applying the formula $B_k=0.08kS_k$, where $S_k$ is salary at the end of year $k$, gives benefits of $0.08(1)(52{,}000)=\$4{,}160$, $0.08(2)(54{,}080)=\$8{,}652.80$, and $0.08(3)(56{,}243.20)=\$13{,}498.368$.

\begin{aligned} \operatorname{EPV} &= \frac{0.10(4{,}160)}{1.04} + \frac{0.102(8{,}652.80)}{1.04^2} + \frac{0.102(13{,}498.368)}{1.04^3}\\ &=400+816+1{,}224\\ &=2{,}440. \end{aligned}

The expected present value is $2,440.00. For a check, the individual benefits discounted to time zero are $4,000, $8,000, and $12,000, respectively; multiplying these by the withdrawal probabilities gives $400, $816, and $1,224. Do not apply the annual withdrawal probabilities directly to the original group in years 2 and 3.

Worked Example: An Eligibility Threshold and a Higher Service Band

A hypothetical plan projects four years. At the end of each year, an active member withdraws with probability $0.20$, and experiences another exit with probability $0.10$. The plan pays nothing before two completed years of service, $5,000 for two but fewer than four years, and $9,000 for at least four years. The benefit is paid at the end of the withdrawal year. The effective annual interest rate is $5\%$. Calculate the expected present value.

Method. Apply the service bands at each year-end and include only withdrawal outcomes that occur while the member remains active. This makes the threshold and the effect of earlier competing exits explicit.

The probability of remaining active each year is $1-0.20-0.10=0.70$. The active probabilities at the starts of years 1 through 4 are $1$, $0.70$, $0.49$, and $0.343$. There is no payment for a first-year withdrawal. The probabilities of withdrawal in years 2, 3, and 4 over the original member are $0.70(0.20)=0.14$, $0.49(0.20)=0.098$, and $0.343(0.20)=0.0686$.

The year-2 and year-3 benefits are each $5,000; the year-4 benefit is $9,000. The expected present value is therefore

\begin{aligned} \operatorname{EPV} &= \frac{0.14(5{,}000)}{1.05^2} + \frac{0.098(5{,}000)}{1.05^3} + \frac{0.0686(9{,}000)}{1.05^4}\\ &= \frac{700}{1.1025} + \frac{490}{1.157625} + \frac{617.4}{1.21550625}\\ &=634.9206+423.2804+507.9365\\ &=1{,}566.1375. \end{aligned}

The expected present value is $1,566.14, rounded to the nearest cent. A tempting but incorrect calculation uses $0.20$ as the year-2, year-3, and year-4 withdrawal probabilities over the original member. That produces approximately $3,251.73 because it ignores members who left before those years. The $0.20$ probability is conditional on being active; it must be multiplied by the probability of reaching the year active.

Checks That Prevent Valuation Errors

Before finalizing a calculation, check that each annual set of mutually exclusive exit probabilities sums to no more than one, that active probabilities decrease or remain unchanged over time, and that every benefit amount corresponds to the service attained at the assumed withdrawal date. Also verify the power of $v$: a payment at the end of year $k$ is discounted by $v^k$, not $v^{k-1}$.

Service-band boundaries require particular attention. If a benefit changes at two completed years, a withdrawal at the end of the second service year belongs in the two-year band, not in the band for less than two years. For an accrued-service rule, use the stated service unit and salary definition consistently; do not substitute projected final salary for current salary unless the plan formula calls for it.

Key takeaway: A service-based withdrawal benefit is valued by combining the benefit at each possible withdrawal date with the probability of reaching that date active and then withdrawing. The conditional withdrawal probability alone is not the probability of payment.

Check Your Understanding

For each problem, assume benefits are paid at the end of the withdrawal year and use the stated annual competing-exit model.

A two-year projection has annual conditional withdrawal probability $0.10$ and competing-exit probability $0.10$. The benefit is zero before two completed years of service and $2,000 at two years. At an effective annual interest rate of $10\%$, find the expected present value. (Answer: $132.23)
For three years, conditional withdrawal probability is $0.10$ and the probability of all other exits is $0.20$ each year. The benefit at the end of year $k$ is $1,000 times $k$ completed years of service. At $5\%$ interest, find the expected present value. (Answer: $349.21)
For three years, withdrawal probability is $0.20$ and another exit probability is $0.10$ each year, conditional on being active. The benefit is zero before two years, $3,000 for two but fewer than three years, and $7,000 at three years. At $0\%$ interest, find the expected present value. (Answer: $1,106.00)
A member’s salary is $40,000 at time zero and increases by $5\%$ at each year-end. The benefit is $10\%$ of salary at withdrawal for each completed service year. For two years, withdrawal probabilities are $0.15$ and $0.20$, and other-exit probabilities are $0.05$ and $0.10$, conditional on being active. At $5\%$ interest, find the expected present value. (Answer: $1,880.00)
For three years, conditional withdrawal and other-exit probabilities are $0.10$ and $0.10$ each year. Benefits are $1,000, $3,000, and $5,000 for withdrawals at the ends of years 1, 2, and 3, respectively. At $10\%$ interest, find the expected present value. (Answer: $529.68)

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