Probability of Vesting Before Competing Exit

Why Vesting Before Exit Is a Competing-Risk Probability

A pension plan may require a member to complete a stated period of service before a benefit becomes vested. Until that eligibility date, withdrawal, death, or retirement may end active membership. The key probability question is therefore not just whether any one exit occurs, but whether the member remains eligible and active through the vesting date before any competing exit takes place.

The calculation uses cause-specific exit probabilities among members active at the start of each period. These are conditional probabilities: a member who has already exited is not exposed to later periods in the active-member calculation. The vesting probability is consequently built period by period, weighting each period's exit probabilities by the probability of having remained active up to that point.

Definition: Suppose vesting occurs at the end of $n$ discrete periods. Let $q_{j,t}$ be the probability that an active member exits during period $t$ for cause $j$, conditional on being active at its start. Causes might be withdrawal, death, and retirement. If the causes are mutually exclusive and exhaustive for exits during that period, the probability of remaining active throughout period $t$ is $$ p_t=1-\sum_j q_{j,t}. $$ The probability of vesting before any of these exits is $$ P(\text{vest before competing exit})=\prod_{t=1}^{n}p_t =\prod_{t=1}^{n}\left(1-\sum_j q_{j,t}\right). $$

This formula assumes that the member begins the first period active and unvested; reaching the end of period $n$ while still active satisfies all stated vesting conditions; and an exit during any of the $n$ periods prevents vesting. The probabilities may vary by period. No assumption that the annual exit events are independent is needed: the period probabilities are already conditional on active status at the start of each period.

The timing convention matters. If vesting occurs at the start of period $n$, then exits during period $n$ occur after vesting and should not be counted as preventing it. The product would then include only periods before that date. For the examples and main formula here, vesting takes place at the end of the stated eligibility period, and an exit before that point prevents vesting.

Deriving the Survival-to-Vesting Formula

Let $A_t$ denote the event that the member is active at the start of period $t$, with $A_1$ certain. To vest at the end of period $n$, the member must be active at the start of every period and avoid every exit in each period. By the definition of $p_t$, $P(A_{t+1}\mid A_t)=p_t$. Applying the multiplication rule for conditional probabilities gives the result:

\begin{aligned} P(\text{active at end of period }n) &=P(A_2\mid A_1)P(A_3\mid A_2)\cdots P(A_{n+1}\mid A_n)\\ &=p_1p_2\cdots p_n\\ &=\prod_{t=1}^{n}\left(1-\sum_j q_{j,t}\right). \end{aligned}

Equivalently, let $s_t$ be the probability of still being active at the start of period $t$. The useful recursion is $s_1=1$ and $s_{t+1}=s_t p_t$. In a period, the unconditional probability of an exit from the original starting group for cause $j$ is $s_tq_{j,t}$, not simply $q_{j,t}$. This distinction supports a complete check: the vesting probability plus the probabilities of all pre-vesting exit causes must equal one.

Worked Example: Three-Year Vesting With Constant Cause Probabilities

A hypothetical plan vests members after three complete years of active service. In each year, among members active at the start of that year, the probabilities of withdrawal, death, and retirement are respectively 0.08, 0.01, and 0.03. Find the probability of vesting. Also calculate the probability of exiting before vesting by each cause.

Method. First find the probability of remaining active in one year, then multiply the conditional active probabilities across all three years. For cause attribution, weight each year's cause-specific probability by the probability of being active at the start of that year.

The total exit probability in each year is $0.08+0.01+0.03=0.12$, so the annual probability of remaining active is $1-0.12=0.88$. Thus:

P(\text{vest})=0.88^3=0.681472.

For the exit-cause calculation, the probabilities of being active at the start of years 1, 2, and 3 are $1$, $0.88$, and $0.88^2=0.7744$. The unconditional probabilities of each cause before vesting are:

\begin{aligned} P(\text{withdrawal})&=1(0.08)+0.88(0.08)+0.7744(0.08)=0.212352,\\ P(\text{death})&=1(0.01)+0.88(0.01)+0.7744(0.01)=0.026544,\\ P(\text{retirement})&=1(0.03)+0.88(0.03)+0.7744(0.03)=0.079632. \end{aligned}

The check is $0.212352+0.026544+0.079632=0.318528$, and $0.318528+0.681472=1$. Therefore the probability of vesting is 0.6815, to four decimal places. The cause totals are probabilities from the original starting group, not cause probabilities conditional on an exit having occurred.

Common mistake avoided: adding the three annual exit probabilities to obtain 0.36 and calling 0.64 the vesting probability. An exit in year 2 or 3 can occur only among members who reached that year active. The correct pre-vesting exit probability is 0.318528, obtained by weighting each period by its active-at-start probability.

Varying Exit Probabilities by Period

Cause-specific probabilities often change with age, service, or plan assumptions. In that case, calculate the active probability separately for every period. The cause probabilities in one period still sum to the total probability of exit conditional on being active at its start, but probabilities from different years should not be added without weighting.

Worked Example: Age-Dependent Decrements Over Four Years

A hypothetical member must complete four years to vest. The annual probabilities among members active at the start of each year are shown below. Find the vesting probability and the probability of a pre-vesting exit from each cause.

Year	Withdrawal	Death	Retirement	Remain active
1	0.06	0.01	0.03	0.90
2	0.07	0.02	0.03	0.88
3	0.08	0.02	0.05	0.85
4	0.10	0.03	0.07	0.80

For example, in year 3 the total exit probability is $0.08+0.02+0.05=0.15$, leaving $0.85$ active. The probability of reaching the starts of years 1 through 4 active is $1$, $0.90$, $0.90(0.88)=0.792$, and $0.792(0.85)=0.6732$. Therefore:

P(\text{vest})=0.90(0.88)(0.85)(0.80)=0.53856.

Weighting each cause-specific probability by the active probability at the start of its year gives:

\begin{aligned} P(\text{withdrawal})&=0.06+0.90(0.07)+0.792(0.08)+0.6732(0.10)=0.25368,\\ P(\text{death})&=0.01+0.90(0.02)+0.792(0.02)+0.6732(0.03)=0.064036,\\ P(\text{retirement})&=0.03+0.90(0.03)+0.792(0.05)+0.6732(0.07)=0.143724. \end{aligned}

The three exit probabilities sum to $0.25368+0.064036+0.143724=0.46144$. Adding the vesting probability gives $0.46144+0.53856=1$, as required. The probability of vesting is 0.5386, to four decimal places. This calculation avoids treating the year-specific probabilities as though each applied to the original cohort.

Conditional Questions and Cause Attribution

A question may ask not only for the probability of vesting but also which cause is most likely to end active membership before vesting. First calculate the unconditional probabilities of the competing exits from the original starting group. If the question asks for a cause conditional on exiting before vesting, divide that cause's unconditional probability by the total probability of any pre-vesting exit.

Worked Example: Cause of Exit Conditional on Not Vesting

A hypothetical plan vests after three years. For active members, annual withdrawal, death, and retirement probabilities are $(0.05,0.01,0.02)$ in year 1, $(0.08,0.02,0.04)$ in year 2, and $(0.04,0.01,0.05)$ in year 3. Find (a) the probability of vesting and (b) the probability that withdrawal is the cause, conditional on an exit before vesting.

The probabilities of remaining active are $0.92$, $0.86$, and $0.90$. Thus the probability of reaching the starts of the three years active is $1$, $0.92$, and $0.92(0.86)=0.7912$. The vesting probability is:

P(\text{vest})=0.92(0.86)(0.90)=0.71208.

The unconditional probability of withdrawal before vesting is $0.05+0.92(0.08)+0.7912(0.04)=0.155248$. For a check, death has probability $0.01+0.92(0.02)+0.7912(0.01)=0.036312$, while retirement has probability $0.02+0.92(0.04)+0.7912(0.05)=0.09636$. All pre-vesting exits therefore have probability $0.155248+0.036312+0.09636=0.28792$, which equals $1-0.71208$.

The requested conditional probability is:

P(\text{withdrawal}\mid\text{exit before vesting}) =\frac{0.155248}{0.28792} =0.539205\ldots

The answers are 0.7121 for vesting and 0.5392 for withdrawal conditional on a pre-vesting exit, both to four decimal places. The denominator in the conditional calculation is the probability of any pre-vesting exit, not the probability of failing to withdraw.

Checks, Interpretation, and Exam Traps

The product of annual active probabilities gives the chance of reaching the vesting date, not the value of the benefit and not the probability of any particular exit. Those are separate questions. In particular, a member who exits before vesting may receive a different benefit under plan rules; the calculation here classifies the timing event only and does not value any post-exit pension or refund.

A reliable solution can be organized as a short recurrence:

Set the eligibility horizon.
Identify the exact number of periods that must be completed and the point at which vesting occurs.

Find each period's active probability.
Subtract the sum of mutually exclusive cause-specific exit probabilities from one.

Update the active cohort.
Multiply the probability active at the start of the period by that period's active probability.

Calculate vesting and, if needed, causes.
The final active probability is the vesting probability; period-start active probabilities weight cause-specific exits.

Check the partition.
Vesting plus all mutually exclusive pre-vesting exit causes must total one.

A frequent error is to sum cause-specific probabilities over time without recognizing their conditional basis. Another is to omit a competing cause from the total period exit probability, which overstates the chance of reaching vesting. Always include every cause that prevents active status through the eligibility date, and make sure the cause probabilities for each period use the same conditioning group and time unit.

Key takeaway: For a fixed vesting date, multiply the period-by-period probabilities of remaining active. To allocate failures to withdrawal, death, or retirement, weight each period's cause-specific probability by the chance of still being active when that period begins.

Check Your Understanding

For each problem, treat the stated cause probabilities as conditional on being active at the start of that year. Assume the member starts active and vesting occurs at the end of the required period.

A member vests after two years. Withdrawal, death, and retirement probabilities are $(0.04,0.01,0.05)$ in year 1 and $(0.06,0.02,0.02)$ in year 2. What is the probability of vesting? (Answer: 0.8100)
A three-year vesting period has annual total exit probabilities of 0.05, 0.10, and 0.20. What is the probability of reaching vesting? (Answer: 0.6840)
Vesting occurs after two years. Withdrawal, death, and retirement probabilities are $(0.10,0.02,0.03)$ in year 1 and $(0.05,0.01,0.04)$ in year 2. What is the unconditional probability of withdrawal before vesting? (Answer: 0.1425)
For a two-year vesting period, cause probabilities are $(0.04,0.01,0.05)$ in year 1 and $(0.08,0.02,0.05)$ in year 2, in withdrawal, death, retirement order. Given an exit before vesting, what is the probability its cause is death? (Answer: 0.1191)
For a three-year vesting period, annual cause probabilities are $(0.05,0.01,0.04)$, $(0.06,0.02,0.02)$, and $(0.08,0.01,0.06)$. What is the unconditional probability of retirement before vesting? (Answer: 0.1066)

What You'll Learn