Vesting Probability Under Constant Competing Intensities

Why Vesting Probability Is a Competing-Risk Question

A pension benefit may become vested only if a member remains employed through a specified service date. Before that date, the member might withdraw or die. For a member who is active now, the probability of reaching the vesting date depends on avoiding both exits throughout the required period. It is not the probability of avoiding withdrawal alone, nor the probability of avoiding death alone.

This tutorial models withdrawal and death as competing exits from the active state. The target is the probability that the member is still active at a fixed future time \(T\). The result is a survival factor that may be used as an eligibility probability when valuing a benefit conditional on vesting. It is not an interest discount factor.

The model assumes that the two intensities remain constant over the period and that an exit removes the member from the active state. The rates and \(T\) must use the same time unit. For example, annual intensities require \(T\) in years. A member who withdraws cannot subsequently die as an active member in this model; the first exit determines the competing cause. The model does not include other exits, such as disability or retirement before vesting.

Deriving the Active-State Survival Factor

Write \(p(t)\) for the probability that a member is still active \(t\) time units after the starting date. Over a short interval of length \(h\), the conditional probability of withdrawal is \(\lambda h+o(h)\), and the conditional probability of death is \(\mu h+o(h)\). The probability of either exit in that interval is therefore \((\lambda+\mu)h+o(h)\). The probability of two exits in an interval of length \(h\) is of smaller order and does not affect the derivative.

Subtract \(p(t)\), divide by \(h\), and let \(h\) approach zero. Since the member starts active, \(p(0)=1\), so solving the resulting differential equation gives the vesting survival factor.

The sum \(\lambda+\mu\) is the total intensity of leaving the active state. The survival factor is also the product \(e^{-\lambda T}e^{-\mu T}\). That product can be motivated by independent exponential waiting times for withdrawal and death. More generally, the differential-equation derivation uses the cause-specific intensities conditional on remaining active; it does not require treating a withdrawn member as still exposed to active-member mortality.

Exit Probabilities by Cause

The active-state probability answers whether the member reaches vesting. It does not, by itself, identify which exit occurred among members who leave before \(T\). Under the same constant-intensity model, the probability of withdrawal in a short interval around time \(t\), with the member active until then, has density \(\lambda p(t)\). The corresponding death density is \(\mu p(t)\). Integrating these densities gives the cumulative probabilities of each cause before the vesting date.

The two cause probabilities add to \(1-e^{-(\lambda+\mu)T}\), the probability of any exit before \(T\). Along with the probability of reaching vesting, they sum to one. Conditional on an exit occurring before \(T\), the probability that the exit is withdrawal is \(\lambda/(\lambda+\mu)\), and the probability that it is death is \(\mu/(\lambda+\mu)\). These cause shares follow from the constant rates; they are not additional probabilities of reaching vesting.

Worked Applications

Interpretation, Limits, and Exam Traps

The factor \(e^{-(\lambda+\mu)T}\) is an eligibility or active-state probability. If a benefit amount is payable only to members who reach vesting, this probability can identify the fraction expected to qualify under the model. A complete valuation may also require benefit amounts, payment timing, interest, and post-vesting contingencies; none of those is included in the survival factor itself.

The constant-intensity assumption matters. If withdrawal or mortality intensity varies with time, the active-state probability generally depends on the accumulated total intensity, not simply one rate multiplied by the full period. In that setting, the constant-rate formula should not be used without modification. Similarly, if a member can leave the active state through another cause before vesting, that cause must be included in the total exit intensity when its intensity is specified by the model.

• Adding survival probabilities instead of intensities. The causes compete while the member is active. Add the intensities, then exponentiate: \(e^{-(\lambda+\mu)T}\).
• Using inconsistent time units. Monthly rates go with months; annual rates go with years. The product of intensity and duration must be dimensionless.
• Confusing an exit probability with a cause intensity. The cause share \(\lambda/(\lambda+\mu)\) divides the probability of any exit; it is not the probability of withdrawal by itself.
• Treating survival as discounting. Interest discounting and the probability of reaching vesting are conceptually different factors, even if a valuation later uses both.