Refunds on Death Before Vesting

Introduction

A pension arrangement may return a member’s contributions if the member dies before becoming vested. The key valuation question is not simply how likely the member is to die. It is whether death occurs while the member is still eligible for that particular refund, what amount the plan pays at that time, and how other exits affect the chance of reaching the death claim.

A contribution refund on death is a death-triggered cash flow. A withdrawal refund is triggered by withdrawal; a retirement benefit is triggered by retirement or pension commencement under the plan’s terms. These cash flows should not be combined merely because they relate to the same member contributions. This tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A refund on death before vesting is a payment of specified member contributions, possibly with credited interest, made if the member dies before satisfying the plan’s vesting condition. The amount and eligibility are determined by the plan’s terms. Death after vesting does not qualify as a pre-vesting death refund unless the plan separately provides for such a payment.

Model the Event That Triggers the Payment

Suppose a member is active at the start of each year and can leave through death or withdrawal. Let $q_k^{(D)}$ be the probability of death during year $k$, conditional on being active at its start, and let $q_k^{(W)}$ be the corresponding probability of withdrawal. In a multiple-decrement model, these are probabilities under the same competing-exit basis, so the probability of remaining active for that year is

p_k^{(\tau)}=1-q_k^{(D)}-q_k^{(W)}.

Here the superscript $(\tau)$ denotes remaining in the active state, not a single-decrement probability. The probability of being active at the start of year $k$, with year 1 indexed by $k=1$, is

{}_ {k-1}p^{(\tau)} =\prod_{j=1}^{k-1}p_j^{(\tau)},

where the empty product for $k=1$ is 1. If a death refund $B_k^{(D)}$ is paid at the end of year $k$, and that death occurs before vesting, its expected present value at entry is the probability of reaching the year, multiplied by the conditional death probability and the discounted refund. For a constant effective annual interest rate $i$, $v=1/(1+i)$.

\operatorname{EPV}_{D} =\sum_{k\ \text{eligible}} v^k\,{}_{k-1}p^{(\tau)}q_k^{(D)}B_k^{(D)}.

The summation includes only years in which death is still a qualifying pre-vesting event. The refund amount $B_k^{(D)}$ must follow the plan formula: it might be the contributions accumulated to the date of death, a fixed amount, or another specified amount. It is not automatically equal to the amount payable on withdrawal.

Why the Competing-Decrement Formula Works

For a death claim in year $k$, two conditions must both hold: the member must still be active at the start of that year, and death must then occur during that year. Under the stated multiple-decrement probabilities, the probability of this joint event is ${}_{k-1}p^{(\tau)}q_k^{(D)}$. The claim is paid at time $k$, so its present value if payable is $v^kB_k^{(D)}$. Multiplying the claim present value by the event probability gives the expected present value for that year. Adding across the eligible years gives the formula above.

The withdrawal probability affects the result through the chance of remaining active. It does not create a death-refund payment. This is why the death probability cannot be used without first accounting for the competing exits.

Worked Example: A Two-Year Refund Before Vesting

Consider a hypothetical plan with a two-year cliff-vesting period. At the start of each year, an active member pays a $1,000 contribution. The plan returns the total nominal contributions, with no credited interest, if the member dies before vesting. Withdrawal produces a different benefit and does not qualify for this death refund. In each year, the conditional death probability is 0.05 and the withdrawal probability is 0.10. The refund, if triggered, is paid at the end of the year. Use an effective annual valuation rate of 5%. Find the entry-date EPV of the death refund.

Method. Use annual multiple-decrement probabilities, count the contribution balance at the death date, and include only death during the two pre-vesting years. The active probability for each year is $p^{(\tau)}=1-0.05-0.10=0.85$.

At the end of year 1, the refund is $1,000. The probability of death in year 1 is $0.05$, so its EPV contribution is

v(0.05)(\$1{,}000) =\frac{\$50}{1.05} =\$47.6190.

At the end of year 2, the member must first remain active through year 1. The probability of death in year 2 is therefore $0.85(0.05)=0.0425$. Two contributions have been made, so the refund is $2,000. Its EPV contribution is

v^2(0.85)(0.05)(\$2{,}000) =\frac{\$85}{1.05^2} =\$77.0975.

Adding the two eligible death claims gives

\operatorname{EPV}_{D} =\$47.6190+\$77.0975 =\$124.7165 \approx \boxed{\$124.72}.

Check. The year-2 probability uses survival from both competing exits in year 1, and the year-2 amount includes both contributions. The result is the EPV of the death refund only. The withdrawal benefit is not added. A common mistake is to use $1-0.05=0.95$ as the year-1 active probability; that incorrectly ignores withdrawals and overstates the year-2 death claim.

Continuous-Time Valuation

A continuous-time model is useful when exits can occur at any time and the refund balance changes continuously. Suppose, before vesting, death and withdrawal have constant forces $\mu_D$ and $\mu_W$, respectively, and these are the only exits. Then the probability of remaining active for $t$ years is $\exp[-(\mu_D+\mu_W)t]$. The density of a death at time $t$, while the member is still active immediately before death, is

f_D(t)=\exp[-(\mu_D+\mu_W)t]\mu_D.

If the death refund at time $t$ is $B_D(t)$, and the force of interest is constant at $\delta$, discounting by $e^{-\delta t}$ gives

\operatorname{EPV}_{D} =\int_0^T e^{-\delta t} \exp[-(\mu_D+\mu_W)t]\mu_D B_D(t)\,dt,

where $T$ is the vesting time, and death at or after $T$ is not eligible under this model. This formula assumes constant competing forces up to $T$, continuous-time exits, and payment at the instant of death. If the benefit is paid later, it must also be discounted for the payment delay.

Worked Example: A Continuously Growing Contribution Balance

A hypothetical plan vests a member after 3 years. Before vesting, death and withdrawal have constant forces $\mu_D=0.03$ and $\mu_W=0.07$ per year. Contributions enter the refund account continuously at $2,000 per year, with no interest credited to the account, so the death refund at time $t$ is $B_D(t)=\$2{,}000t$. The valuation force of interest is $\delta=0.04$. Find the EPV of the refund.

The total exit force is $0.03+0.07=0.10$. The death density at time $t$ is $0.03e^{-0.10t}$, and the discount factor is $e^{-0.04t}$. Therefore

\operatorname{EPV}_{D} =\int_0^3 e^{-0.04t}e^{-0.10t}(0.03)(\$2{,}000t)\,dt =\$60\int_0^3 t e^{-0.14t}\,dt.

For $a>0$, integration by parts gives $\int_0^T te^{-at}\,dt=[1-(1+aT)e^{-aT}]/a^2$. Applying this with $a=0.14$ and $T=3$,

\begin{aligned} \operatorname{EPV}_{D} &=\$60\frac{1-(1+0.14\cdot3)e^{-0.14\cdot3}}{0.14^2}\\ &=\$60\frac{1-1.42e^{-0.42}}{0.0196}\\ &=\$60\frac{1-1.42(0.6570468)}{0.0196}\\ &=\$60\frac{0.0669935}{0.0196}\\ &=\$205.0822 \approx \boxed{\$205.08}. \end{aligned}

Check. The exponent $0.14$ combines the 0.10 total exit force and the 0.04 valuation force. Omitting withdrawal would replace the active-survival force with 0.03 and overstate the expected payment. The amount is rounded to the nearest cent only at the final step.

Vesting and Benefit Classification in a Discrete Model

An annual model needs an explicit convention for exits around a vesting anniversary. In the next example, the plan defines death during either of the first two years as occurring before the two-year vesting test; a member who remains active to the anniversary vests immediately afterward. This convention makes deaths during year 2 eligible. A different plan or valuation convention may classify an anniversary event differently, so state the rule rather than infer it.

The refund balance is also kept separate from other plan benefits. The member’s contributions may be relevant to a withdrawal benefit, but that does not make the withdrawal amount payable on death. Similarly, once vesting occurs, a pension entitlement may replace the pre-vesting refund under the plan. The death-refund calculation stops at the eligibility boundary.

Worked Example: Varying Decrements and a Vesting Boundary

A hypothetical plan vests at the end of two years. Under its annual timing convention, death in years 1 and 2 occurs before the vesting test and triggers a refund. The death probabilities in years 1, 2, and 3 are 0.03, 0.04, and 0.05; the withdrawal probabilities are 0.10, 0.08, and 0.06. The plan’s specified death-refund amounts are $1,200 in year 1 and $2,600 in year 2. Use $i=4\%$. Find the EPV at entry.

Method. Calculate the active probability at the start of each eligible year, then multiply by the conditional death probability and discounted amount. Year 1 begins with the member active, so its death probability is 0.03. The probability of remaining active through year 1 is $1-0.03-0.10=0.87$. Thus the year-2 death probability from entry is $0.87(0.04)=0.0348$. Death in year 3 is after vesting and does not trigger the pre-vesting refund.

\begin{aligned} \operatorname{EPV}_{D} &=v(0.03)(\$1{,}200)+v^2(0.87)(0.04)(\$2{,}600)\\ &=\frac{\$36}{1.04}+\frac{\$90.48}{1.04^2}\\ &=\$34.6154+\$83.6538\\ &=\$118.2692 \approx \boxed{\$118.27}. \end{aligned}

Check. The second-year term includes the probability of avoiding both death and withdrawal in year 1. The year-3 decrement rates are irrelevant to this death-refund EPV because death in year 3 is not eligible. The result does not include a withdrawal benefit or any vested retirement benefit. A common mistake is to continue the refund sum through year 3 simply because a death probability is provided for that year.

Common Mistakes and Practical Checks

A reliable valuation keeps the trigger, eligibility period, amount, and payment time distinct. Before calculating, identify each explicitly. In particular, use the following checks:

Check the cause. Only death probabilities or death forces generate the death-refund claim. Withdrawal affects the active survival probability but is not itself a death claim.
Check eligibility. Stop at the plan’s vesting boundary. Do not include post-vesting deaths unless a separate post-vesting death benefit is specified.
Check the refund formula. Apply the specified contribution balance and credited-interest basis at the date of death. Do not assume the death and withdrawal formulas are identical.
Check the payment date. A benefit paid at the end of year $k$ is discounted by $v^k$. If it is paid immediately on death in a continuous model, discount continuously to the event time.
Check the decrement basis. Annual cause probabilities must be consistent competing-decrement probabilities. Do not combine probabilities from incompatible bases without an appropriate conversion.

Key takeaway: The EPV of a pre-vesting death refund is the sum or integral of the discounted refund amount weighted by the probability density of a qualifying death while the member remains active. Competing exits reduce the chance of reaching that death claim; they do not change its trigger.

Check Your Understanding

For each problem, use only the death-triggered refund described. Round monetary answers to the nearest cent.

A one-year plan has death probability 0.04 and withdrawal probability 0.06. A death before vesting triggers a $1,500 refund at year-end. At $i=5\%$, find the EPV at the start of the year. (Answer: $57.14)
A member is unvested for two years. In each year, conditional on being active at the start, death probability is 0.02 and withdrawal probability is 0.08. Death refunds are $1,000 in year 1 and $2,200 in year 2. They are paid at each year-end. At $i=4\%$, find the entry-date EPV. (Answer: $55.84)
Before vesting at time 2, a hypothetical continuous-time plan has constant death force 0.02 and withdrawal force 0.08. Contributions accumulate continuously at $500 per year with no account interest, so the refund at time $t$ is $500$t$. The valuation force is $\delta=0.05$. Find the EPV. (Answer: $16.42)
A two-year plan vests at the end of year 2, with death during year 1 counted as pre-vesting. Year-1 death probability is 0.03 and withdrawal probability is 0.07. A death refund of $800 is paid at year-end. At $i=4\%$, what is the EPV of the pre-vesting death refund? (Answer: $23.08)
A member may receive death refunds only in years 1 and 2, before vesting. In each year, death probability is 0.04 and withdrawal probability is 0.06. Refunds are $900 and $1,900 at the respective year-ends. At $i=5\%$, calculate the EPV at entry. (Answer: $96.33)

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