Vesting Thresholds and Expected Benefit Cost

Why a Vesting Threshold Changes Expected Cost

A service-based vesting rule can affect expected benefit cost in two distinct ways. Raising the threshold usually means fewer members qualify for a payment, but it may also increase the benefit amount for members who do qualify. The first effect tends to reduce cost; the second may offset that reduction. Valuing only the eligible group or only the benefit formula can therefore give the wrong comparison.

The model below treats the benefit as payable when a member withdraws after reaching the threshold. This makes the timing of payment explicit: reaching the threshold creates eligibility, but does not itself trigger payment. All scenarios and values are hypothetical.

Definition: Let $h$ be the service threshold, and let $B(h)$ be the lump sum fixed by the plan for a member who satisfies that threshold. The member must remain active through service duration $h$. If the member later withdraws, the plan pays $B(h)$ immediately at withdrawal. Death does not produce this withdrawal benefit.

Assume the member starts active, with constant withdrawal intensity $\lambda_w$ and constant death intensity $\lambda_d$. These are competing exits: the first exit ends active membership. Assume the intensities are independent of service duration and each other in the standard competing-intensity model. Let $\delta$ be a constant force of interest, so a payment at time $t$ has present-value factor $e^{-\delta t}$. Write $\alpha=\lambda_w+\lambda_d$. The formula requires constant intensities and force of interest, a payment at withdrawal, and no payment at death; it does not apply unchanged if any of these assumptions are altered.

Theorem: Under these assumptions, the expected present value of the withdrawal benefit with threshold $h$ is $$ C(h)=B(h)\int_h^\infty \lambda_w e^{-(\lambda_w+\lambda_d+\delta)t}\,dt =\frac{\lambda_w}{\lambda_w+\lambda_d+\delta}B(h)e^{-(\lambda_w+\lambda_d+\delta)h}. $$

Why the Formula Holds

For a payment at time $t$, the member must remain active until $t$, which has probability $e^{-\alpha t}$. Conditional on being active just before $t$, the probability of withdrawal in a short interval of length $dt$ is approximately $\lambda_w dt$. Multiplying by the discount factor $e^{-\delta t}$ gives the expected discounted payment contribution $B(h)\lambda_w e^{-(\alpha+\delta)t}dt$, for $t\ge h$. Integrating over every possible eligible withdrawal time gives the formula.

Equivalently, define the discounted withdrawal factor $q(h)=\frac{\lambda_w}{\lambda_w+\lambda_d+\delta}e^{-(\lambda_w+\lambda_d+\delta)h}$. Then $C(h)=B(h)q(h)$. The factor $q(h)$ is not just the probability of reaching the threshold: it also accounts for the possibility of a later withdrawal rather than death, and for discounting until payment.

P(\text{active at }h)=e^{-(\lambda_w+\lambda_d)h}, \qquad P(\text{withdrawal after }h)=\frac{\lambda_w}{\lambda_w+\lambda_d}e^{-(\lambda_w+\lambda_d)h}.

These are eligibility probabilities, not expected present values. The expected present value uses the additional discounting and the withdrawal timing shown in $q(h)$. Keeping these quantities separate is central to comparing thresholds.

Separating Eligibility and Amount Effects

Suppose a threshold changes from $h_1$ to $h_2$, and the corresponding benefit amounts are $B(h_1)$ and $B(h_2)$. The exact change in expected present value can be decomposed by first changing eligibility while holding the old amount fixed, then changing the amount at the new threshold:

C(h_2)-C(h_1) = \underbrace{B(h_1)\bigl(q(h_2)-q(h_1)\bigr)}_{\text{eligibility effect}} + \underbrace{\bigl(B(h_2)-B(h_1)\bigr)q(h_2)}_{\text{benefit-amount effect}}.

This identity follows by adding and subtracting $B(h_1)q(h_2)$. The decomposition is exact, but its allocation of the interaction depends on the order chosen: here eligibility changes first, and benefit amount second. State that convention when reporting the two components. The total change does not depend on the decomposition order.

Calculate the exit and discount rate.
Set $k=\lambda_w+\lambda_d+\delta$, using rates per year if service is measured in years.

Calculate the discounted withdrawal factor.
For each threshold, evaluate $q(h)=\frac{\lambda_w}{k}e^{-kh}$.

Value each threshold and decompose.
Calculate $C(h)=B(h)q(h)$, then apply the eligibility-first identity to identify the two effects.

Worked Example: Isolating the Eligibility Effect

A hypothetical plan pays a fixed \$10,000 on withdrawal, provided the member has completed the service threshold. The threshold increases from 1 year to 2 years. Withdrawal and death intensities are $\lambda_w=0.10$ and $\lambda_d=0.05$ per year, respectively, and the force of interest is $\delta=0.05$ per year. Calculate the change in expected present value. Identify the method and the main mistake it avoids.

Method. Since the benefit amount is unchanged, the entire change is an eligibility effect. First calculate $k=0.10+0.05+0.05=0.20$. The discounted withdrawal factors are $q(1)=\frac{0.10}{0.20}e^{-0.20}=0.5(0.81873075)=0.40936538$ and $q(2)=0.5e^{-0.40}=0.5(0.67032005)=0.33516002$.

The expected present values are $C(1)=\$10{,}000(0.40936538)=\$4{,}093.65$ and $C(2)=\$10{,}000(0.33516002)=\$3{,}351.60$, rounded to cents. Thus the change is $\$3{,}351.60-\$4{,}093.65=-\$742.05$. The threshold increase reduces expected present value by \$742.05.

Check. With a fixed benefit amount, the ratio of the factors is $q(2)/q(1)=e^{-0.20}<1$, so the higher threshold must lower the cost. The computed change is negative, as expected.

Common mistake avoided. Using $B e^{-(\lambda_w+\lambda_d)h}$ would value the benefit as though it were paid automatically at the threshold. For $h=1$, that incorrect calculation is $\$10{,}000e^{-0.15}=\$8{,}607.08$, not the correct withdrawal-benefit value of \$4,093.65. Reaching the threshold establishes eligibility; a later withdrawal is still required for payment.

When a Higher Benefit Can Offset Lower Eligibility

If $B(h)$ increases with service, a higher threshold has two opposing effects. The expected present value can rise, fall, or remain unchanged. The eligibility-first decomposition shows precisely how large the amount effect must be to offset the eligibility effect.

Worked Example: Separating Opposing Effects

A hypothetical plan sets the withdrawal benefit to $B(h)=\$2{,}000+\$1{,}000h$, where $h$ is the vesting threshold in years. The threshold rises from 2 to 3 years. Let $\lambda_w=0.08$, $\lambda_d=0.02$, and $\delta=0.05$ per year. Calculate the old and new expected present values, and separate the change into eligibility and amount effects.

Method. Calculate $k=0.08+0.02+0.05=0.15$, so $q(h)=\frac{0.08}{0.15}e^{-0.15h}=\frac{8}{15}e^{-0.15h}$. At $h=2$, $q(2)=\frac{8}{15}(0.74081822)=0.39510305$. At $h=3$, $q(3)=\frac{8}{15}(0.63762815)=0.34006835$.

The plan amounts are $B(2)=\$2{,}000+\$1{,}000(2)=\$4{,}000$ and $B(3)=\$2{,}000+\$1{,}000(3)=\$5{,}000$. Therefore $C(2)=\$4{,}000(0.39510305)=\$1{,}580.41$ and $C(3)=\$5{,}000(0.34006835)=\$1{,}700.34$, rounded to cents. The total increase is \$119.93.

Using the eligibility-first convention, the eligibility effect is $\$4{,}000(0.34006835-0.39510305)=-\$220.14$. The amount effect is $(\$5{,}000-\$4{,}000)(0.34006835)=\$340.07$. Adding the rounded components gives $-\$220.14+\$340.07=\$119.93$, matching the direct comparison.

Check. The benefit amount rises by 25%, while the discounted withdrawal factor falls by less than 25%; the total expected present value can therefore increase. The calculation avoids the trap of assuming that a stricter threshold must always reduce expected cost.

Worked Example: Comparing Thresholds With Different Intensities

A hypothetical plan pays $B(h)=\$2{,}500+\$750h$ on withdrawal after vesting. Compare thresholds of 2 and 4 years, with $\lambda_w=0.06$, $\lambda_d=0.04$, and $\delta=0.05$ per year. Give the total change and its eligibility-first components.

Method. Here $k=0.06+0.04+0.05=0.15$, and $q(h)=\frac{0.06}{0.15}e^{-0.15h}=0.4e^{-0.15h}$. Thus $q(2)=0.4(0.74081822)=0.29632729$ and $q(4)=0.4(0.54881164)=0.21952466$. The benefits are $B(2)=\$4{,}000$ and $B(4)=\$5{,}500$.

The expected present values are $C(2)=\$4{,}000(0.29632729)=\$1{,}185.31$ and $C(4)=\$5{,}500(0.21952466)=\$1{,}207.39$, rounded to cents. The total change is an increase of \$22.08.

The eligibility effect is $\$4{,}000(0.21952466-0.29632729)=-\$307.21$. The amount effect is $(\$5{,}500-\$4{,}000)(0.21952466)=\$329.29$. Their sum is $-\$307.21+\$329.29=\$22.08$, consistent with the direct values.

Check. The higher threshold reduces the discounted withdrawal factor, but the resulting eligibility cost reduction is slightly smaller than the benefit-amount increase. The decomposition accounts for both effects rather than inferring the direction of total cost from the threshold change alone.

Common Traps and Limits of the Model

Confusing eligibility with payment. A member who reaches $h$ is eligible, but the modeled benefit is paid only at a later withdrawal.
Using an eligibility probability as a present-value factor. The probability of surviving in service to $h$ omits both the withdrawal condition and discounting to the payment time.
Changing the threshold but not the benefit formula. If $B(h)$ depends on service, calculate its value separately at each threshold before comparing total costs.
Mixing time units. Service in years requires intensities and force of interest per year. Convert monthly or other-period rates consistently before using the continuous-time formula.
Ignoring the payment condition. If the plan pays at the vesting date, at retirement, or on death, the payment-time distribution changes and so does the valuation formula.

The competing-intensity result is useful because it separates a threshold's effect on eligibility from its effect on the amount for eligible members. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: in a valuation problem, first identify the trigger and payment time, then match the decrement assumptions and benefit formula to that event.

Check Your Understanding

For each problem, treat withdrawal and death as competing constant intensities. Benefits are paid immediately on withdrawal after vesting, and there is no benefit on death. Give money answers to the nearest cent.

A plan pays a fixed \$12,000 after a 1-year threshold. The threshold rises to 2 years; $\lambda_w=0.05$, $\lambda_d=0.05$, and $\delta=0.10$ per year. Find the change in expected present value. (Answer: $-\$445.23$)
A plan sets $B(h)=\$1{,}500+\$500h$. Compare thresholds of 2 and 3 years when $\lambda_w=0.12$, $\lambda_d=0.03$, and $\delta=0.05$ per year. Find the change in expected present value. (Answer: $-\$17.62$)
A plan pays a fixed \$8,000 after vesting. Find the eligibility-only change when the threshold increases from 0.5 to 1.5 years, with $\lambda_w=0.09$, $\lambda_d=0.01$, and $\delta=0.05$ per year. (Answer: $-\$620.29$)
With $\lambda_w=0.08$, $\lambda_d=0.02$, and $\delta=0.05$ per year, a 2-year threshold gives a \$4,000 benefit. What benefit amount at a 3-year threshold would give the same expected present value? (Answer: $\$4{,}647.34$)

What You'll Learn