Why Vesting and Refunds Require Multiple-Decrement Probabilities
A pension or employment benefit may depend on both how long a member has completed service and why the member leaves active service. A withdrawal before vesting might trigger a refund of contributions, while a withdrawal after vesting might trigger a larger vested benefit. Death may instead trigger a separate benefit, and members who remain active receive no exit payment at that time.
These conditions make the payment depend on a joint event: the member must reach a particular service duration and then exit for a particular cause. The probability of an exit in a given year is therefore not generally just that year’s cause-specific decrement probability. It must also reflect the probability the member remained active up to that year. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
The cause-specific probabilities in a year are conditional probabilities from the same active starting population. Causes are mutually exclusive over that year, so their sum cannot exceed 1. The residual probability \(1-\sum_j q_k^{(j)}\) is the probability of staying active through the year. In the examples below, exits and payments occur at the end of each service year, and annual decrement probabilities are given directly. They are not derived from continuous transition intensities.
Set the Service and Payment Rules Before Calculating
Before valuing a benefit, establish what “two years of service,” for example, means for an exit at the end of year 2. If vesting occurs upon completing two years, a member who exits at that year-end qualifies. If a plan instead requires more than two years, the same exit may remain nonvested. Apply the stated rule to completed service at the payment date, not to a vague count of years begun.
Also identify the payment attached to each combination of duration and cause. A refund might be available only for nonvested withdrawals; a vested withdrawal benefit might replace, rather than supplement, that refund. A death benefit may apply regardless of vesting. Do not assume that a payment is made for every decrement unless the product rules say so.
| Exit event | Service condition | Possible payment to value |
|---|---|---|
| Withdrawal | Before vesting | Refund under the stated contribution and interest rules |
| Withdrawal | At or after vesting | Vested benefit if the rules provide one |
| Death | Any duration, if specified | Death or survivor benefit under the stated rule |
| No exit | Active at year-end | No exit-contingent payment at that date |
A refund based on contributions is not automatically equal to the sum of contributions paid. If the refund includes interest, accumulate each contribution to the exit date under the specified account-crediting basis. Then apply the probability of the qualifying exit and discount the payment using the valuation interest rate. The account’s crediting rate and the valuation rate are separate assumptions unless the problem explicitly equates them.
Deriving the Expected Present Value
For a payment \(C_{k,j}\) made at the end of year \(k\) if cause \(j\) occurs in that year, let \(i\) be the effective annual valuation interest rate. Under the annual model above, the expected present value is
This formula assumes payments are made at year-end, the stated payment amounts are known for each cause and duration, and the model covers all relevant exits through year \(n\). If payment timing differs, or if a payment is made while the member remains active, its timing and probability must be represented separately.
Multiply the previous years’ conditional stay-active probabilities. For year \(k\), this is \({}_{k-1}p^A\).
Multiply \({}_{k-1}p^A\) by \(q_k^{(j)}\) to obtain the unconditional probability of cause \(j\) in year \(k\).
Check completed service, vesting eligibility, and any separate refund or death-benefit rule.
Multiply the expected payment by \(v^k\) and sum across years and causes.
The probability step follows from conditional probability. An exit for cause \(j\) in year \(k\) can occur only if the member was active at the start of that year. Thus
The same logic explains why the active probabilities multiply across years: the probability of being active through another year is the probability of reaching that year multiplied by the conditional probability of staying active in it.
Worked Example: Valuing Refunds and Vested Benefits by Exit Cause
A hypothetical plan pays a $900 contribution refund for a withdrawal before two completed years of service. A withdrawal at or after two completed years instead pays a vested lump sum of $10,000. A death benefit of $4,000 is payable at any duration. All payments occur at year-end. The conditional withdrawal and death probabilities are, respectively, 0.12 and 0.08 in year 1, 0.10 and 0.10 in year 2, and 0.08 and 0.12 in year 3. The valuation rate is 4% effective annually. Calculate the expected present value of payments through year 3.
Method. For each year, calculate the unconditional probability of each exit by multiplying its conditional decrement probability by the probability of being active at the start of that year. The vesting rule means the year-2 withdrawal receives $10,000: it occurs after two completed years.
The probabilities of remaining active through years 1 and 2 are \(1-0.12-0.08=0.80\) and \(1-0.10-0.10=0.80\). Therefore the probability of being active at the start of year 2 is \(0.80\), and at the start of year 3 it is \(0.80(0.80)=0.64\).
In year 1, a withdrawal receives the $900 refund and death triggers the $4,000 death benefit. The expected payment at year-end is
Its present value is \(428/1.04=\$411.54\), rounded to cents.
In year 2, the unconditional probabilities are \(0.80(0.10)=0.08\) for withdrawal and \(0.80(0.10)=0.08\) for death. The expected payment at year-end is \(0.08(10{,}000)+0.08(4{,}000)=800+320=\$1{,}120\). Its present value is \(1{,}120/(1.04)^2=1{,}120/1.0816=\$1{,}035.50\), rounded to cents.
In year 3, the unconditional probabilities are \(0.64(0.08)=0.0512\) for withdrawal and \(0.64(0.12)=0.0768\) for death. The expected payment at year-end is \(0.0512(10{,}000)+0.0768(4{,}000)=512+307.20=\$819.20\). Its present value is \(819.20/(1.04)^3=819.20/1.124864=\$728.27\), rounded to cents.
Adding the unrounded present values and rounding only at the end gives an expected present value of $2,175.31. The rounded components also sum to \(411.54+1{,}035.50+728.27=\$2{,}175.31\). The method avoids treating each year’s conditional decrement probability as though it applied to the original population.
Worked Example: Accumulating Contributions Before a Refund
In a hypothetical plan, a member contributes $600 at the beginning of each year while active. The account earns 5% effective annual interest. Vesting requires three completed years, so a withdrawal at the end of either year 1 or year 2 receives the account balance as a refund. In year 1, the conditional withdrawal and death probabilities are 0.10 and 0.15. In year 2, they are 0.08 and 0.17. Death does not receive this withdrawal refund. The valuation rate is 4% effective annually. Find the expected present value of refunds through year 2.
Method. First accumulate the contributions to each possible withdrawal date. Then multiply each refund by the probability of reaching the year and withdrawing, and discount at the valuation rate. The account-crediting rate is 5%; the valuation rate is 4%.
At the end of year 1, the $600 contribution has earned one year of interest, so the refund is \(600(1.05)=\$630\). The probability of withdrawal in year 1 is 0.10. Thus the year-1 refund present value is \(0.10(630)/1.04=63/1.04=\$60.5769\ldots\).
The probability of remaining active through year 1 is \(1-0.10-0.15=0.75\). A member active at the start of year 2 makes another $600 contribution at the beginning of that year. At the end of year 2, the refund is
The unconditional probability of a year-2 withdrawal is \(0.75(0.08)=0.06\). Therefore the year-2 refund present value is \(0.06(1{,}291.50)/(1.04)^2=77.49/1.0816=\$71.6439\ldots\). The total is \(60.5769\ldots+71.6439\ldots=\$132.2208\ldots\), or $132.22 to cents.
This calculation avoids two errors: omitting year-1 active survival before the year-2 withdrawal, and using the account-crediting rate as the discount rate. Death is a competing decrement here; it reduces the probability of reaching year 2 but does not generate a refund under the stated rules.
Worked Example: Applying a Vesting Threshold at the Exit Date
A hypothetical plan pays a $1,200 refund for a withdrawal before two completed service years, a $9,000 vested benefit for a withdrawal at or after two completed years, and a $3,000 death benefit at any duration. Conditional withdrawal and death probabilities are 0.15 and 0.05 in year 1, 0.10 and 0.10 in year 2, and 0.08 and 0.12 in year 3. Payments occur at year-end. At a 3.5% effective annual valuation rate, calculate the expected present value through year 3.
Method. Apply the threshold to service completed at the exit date. The year-1 withdrawal is nonvested and receives $1,200; year-2 and year-3 withdrawals receive $9,000. Death receives $3,000 in every year. The conditional stay-active probabilities are \(0.80\), \(0.80\), and \(0.80\), so the probabilities of reaching the starts of years 1, 2, and 3 are \(1\), \(0.80\), and \(0.64\).
For year 1, the expected payment is \(0.15(1{,}200)+0.05(3{,}000)=180+150=\$330\). Its present value is \(330/1.035=\$318.84\).
For year 2, the unconditional withdrawal and death probabilities are \(0.80(0.10)=0.08\) and \(0.80(0.10)=0.08\). The expected payment is \(0.08(9{,}000)+0.08(3{,}000)=720+240=\$960\). Its present value is \(960/(1.035)^2=960/1.071225=\$896.17\).
For year 3, the unconditional probabilities are \(0.64(0.08)=0.0512\) for withdrawal and \(0.64(0.12)=0.0768\) for death. The expected payment is \(0.0512(9{,}000)+0.0768(3{,}000)=460.80+230.40=\$691.20\). Its present value is \(691.20/(1.035)^3=691.20/1.108717875=\$623.42\).
Adding the unrounded present values gives \(\$318.8406\ldots+\$896.1703\ldots+\$623.4228\ldots=\$1{,}838.43\), rounded to cents. The year-2 payment is vested because the member completes two years by that exit date; classifying it as a refund would apply the threshold incorrectly.
Common Exam Trap: Forgetting Prior Active Survival
A conditional decrement probability applies only to members active at the beginning of its year. For example, in the first worked example the year-2 withdrawal probability is 0.10 conditional on being active at the start of year 2. The unconditional probability of a year-2 withdrawal is \(0.80(0.10)=0.08\), not 0.10.
The wrong calculation for the year-2 vested benefit would be \(0.10(10{,}000)/(1.04)^2=\$924.56\), because it omits the 0.80 probability of reaching year 2 active. Including that survival probability gives \(0.80(0.10)(10{,}000)/(1.04)^2=\$739.65\) for the withdrawal benefit alone. In the full example, the year-2 death benefit is also included, producing a total year-2 present value of $1,035.50.
A second frequent error is to apply the vesting threshold to the start of a year rather than the exit date. A member who exits at the end of a year has completed that year of service under the assumptions here. A third is to discount a refund before accumulating its contributions to the exit date. Keep the sequence clear: determine eligibility, determine the amount at payment, calculate the exit probability, and then discount.
Check Your Understanding
For each problem, use the stated annual timing and apply the vesting or refund rule to the exit date.
- In year 1, conditional withdrawal and death probabilities are 0.10 and 0.20. In year 2, they are 0.15 and 0.10. Find the unconditional probability of withdrawal in year 2. (Answer: 0.1050)
- A member vests after completing three years. The probabilities of remaining active through years 1 and 2 are 0.80 and 0.75, respectively. The conditional year-3 withdrawal probability is 0.10. Find the probability of reaching the year-3 withdrawal as a vested exit. (Answer: 0.0600)
- A plan refunds contributions for withdrawals before three completed years. A member contributes $400 at the beginning of each active year; the account earns 4% annually. Year-1 withdrawal and death probabilities are 0.18 and 0.12. The conditional year-2 withdrawal probability is 0.12. Find the expected refund paid at the end of year 2, before discounting. (Answer: $71.29)
- A $15,000 vested benefit is payable for withdrawal at the end of year 3, after vesting at the end of year 2. The member remains active through years 1 and 2 with conditional probabilities 0.90 and 0.80. The conditional year-3 withdrawal probability is 0.10. At 5% effective annual interest, find the expected present value of this year-3 benefit. (Answer: $932.94)
- A plan pays a $500 nonvested withdrawal refund in year 1, a $2,000 death benefit in either year, and an $8,000 vested withdrawal benefit in year 2. Year-1 conditional withdrawal and death probabilities are 0.10 and 0.05; year-2 conditional probabilities are 0.12 and 0.08. Payments are at year-end and the valuation rate is 4%. Find the expected present value of all payments through year 2. (Answer: $1,024.41)