Valuation Setup for Competing Exit Causes

Why the Valuation Setup Matters

A pension plan may pay different benefits when an active member retires, withdraws, dies, or becomes disabled. These events compete: in a model where each member leaves active membership only once, a member who exits through one cause cannot also exit through another cause during the same period. Valuation therefore requires more than listing benefit amounts. The actuary must identify the population at risk, determine which probability applies to each cause, and discount each payment to the valuation date.

The purpose here is to organize those inputs and express the total expected present value (EPV) as a sum of cause-specific values. The setup is discrete by year. It treats cause probabilities and active-member probabilities as valuation inputs; deriving those probabilities is a separate step. All examples use hypothetical plans and data.

Definition: The active-member population is the group of members who are active at the valuation date and whose future benefits are being valued. Let $a_{k-1}$ be the probability that a member from this original group is active immediately before year $k$. Let $q_k^{(j)}$ be the probability that a member active at the start of year $k$ exits through cause $j$ during that year. When the causes are mutually exclusive, $\sum_j q_k^{(j)}\leq 1$; the remaining probability represents members who stay active through the year.

Specify the Causes, Amounts, and Dates

Begin by defining the valuation date and the population included. For example, the population might be all active members at a particular date, or a representative member with specified age and service. State whether the model ends at a fixed horizon, such as retirement eligibility, and whether an exit removes a member from active membership permanently. The simple setup below assumes no re-entry after an exit. If reinstatement or re-employment is possible, it must be represented in the model rather than silently treated as a permanent exit.

Next, name the competing causes. A pension valuation might distinguish retirement, withdrawal, death in service, and disability. The cause probabilities are conditional on being active at the start of the year. They are not, by themselves, probabilities for the original population. The corresponding probability of being active at the start of year $k$ and then exiting by cause $j$ is $a_{k-1}q_k^{(j)}$.

Finally, translate each plan rule into its benefit amount and payment date. A withdrawal benefit may depend on completed service; a death benefit may be a fixed amount; and a retirement benefit may be paid later than the retirement date. If a cause produces several payments, list the amount and date of each payment. Do not assume that the payment date is the date of exit unless the plan rules say so.

Valuation setup: Let $i$ be the effective annual interest rate and $v=1/(1+i)$. If an exit by cause $j$ in year $k$ produces a single benefit $B_{k,j}$ at time $t_{k,j}$ years after the valuation date, its EPV contribution is $a_{k-1}q_k^{(j)}B_{k,j}v^{t_{k,j}}$. Add these contributions over all years and causes. If one cause produces payments $C_{k,j,r}$ at times $t_{k,j,r}$, its contribution is $a_{k-1}q_k^{(j)}\sum_r C_{k,j,r}v^{t_{k,j,r}}$.

This expression assumes that the benefit amounts and payment dates are specified for each cause and year, and that $a_{k-1}$ and $q_k^{(j)}$ use the same population and time intervals as the benefit rules. It also assumes that the cause-specific probability describes an exit during the stated year. If the benefit depends on the exact exit date within that year, a year-level probability alone may not be enough to value it; the timing information must also be modeled.

Why Cause-Specific Values Add

For a fixed year, the events “active at the start of the year, then exit by retirement,” “active at the start, then exit by withdrawal,” and so on are disjoint. A member cannot realize two mutually exclusive exit benefits from that year in this model. The expected present value of the total benefit is therefore the sum of the expected present values for the separate cause events.

For one cause $j$, the probability of the event “active at the start of year $k$ and exits by $j$” is the probability of being active at the start multiplied by the conditional probability of that exit, namely $a_{k-1}q_k^{(j)}$. If the benefit is $B_{k,j}$ at time $t_{k,j}$, its discounted value conditional on that event is $B_{k,j}v^{t_{k,j}}$. Multiplying the event probability by this discounted amount gives its expected contribution. Adding over causes and years follows from the additivity of expectation:

\operatorname{EPV} = \sum_k\sum_j a_{k-1}q_k^{(j)}B_{k,j}v^{t_{k,j}}.

This argument does not require the exit causes to be independent. They are competing and mutually exclusive in the stated model. The multiplication uses a conditional probability given active status; it is not a product of independent cause events.

Worked Example: Matching Each Exit Cause to Its Benefit

A hypothetical plan values one active member at the start of year 1. The annual effective interest rate is $i=5\%$. During year 1, the conditional probabilities of retirement, withdrawal, and death are $0.06$, $0.04$, and $0.00$, respectively. Retirement pays $12,000 at the end of the exit year, withdrawal pays $8,000 at the end of the exit year, and death would pay $30,000 at the end of the exit year. At the start of year 2, the probability of still being active is $0.90$. In year 2, conditional probabilities of retirement, withdrawal, and death are $0.10$, $0.05$, and $0.03$, with the same benefits paid at the end of the exit year. Find the two-year EPV.

Method. For each year, multiply the active probability by the conditional exit probability and the benefit for that same cause. Discount the resulting expected payment to its stated year-end payment date. Here $v=1/1.05$. Year 1 has active probability $a_0=1$; year 2 has $a_1=0.90$.

Year 1. The expected payment is $1(0.06)(\$12{,}000)+1(0.04)(\$8{,}000)+1(0.00)(\$30{,}000)=\$720+\$320+\$0=\$1{,}040$. The payment is at time 1, so its present value is $\$1{,}040/1.05=\$990.476190\ldots$.

Year 2. The expected payment from the original population is $0.90(0.10)(\$12{,}000)+0.90(0.05)(\$8{,}000)+0.90(0.03)(\$30{,}000)=\$1{,}080+\$360+\$810=\$2{,}250$. The payment is at time 2, so its present value is $\$2{,}250/(1.05)^2=\$2{,}040.816326\ldots$.

Total and check. Adding the unrounded present values gives $\$990.476190\ldots+\$2{,}040.816326\ldots=\$3{,}031.292517\ldots$, or $3,031.29 to the nearest cent. The year 1 probabilities sum to $0.10$, consistent with the stated $0.90$ active probability at the start of year 2.

Common mistake avoided. The year 1 withdrawal probability is multiplied by the withdrawal benefit, $8,000—not the death benefit, $30,000. A benefit amount must remain paired with its own cause throughout the calculation.

Worked Example: An Exit Benefit Paid After the Exit Year

A hypothetical plan uses $i=4\%$. At the start of year 2, $0.80$ of the original active-member group remains active. During year 2, conditional retirement and death probabilities are $0.15$ and $0.05$. Retirement pays $8,000 at the end of year 3, while death pays $20,000 at the end of year 2. Find the EPV of these year 2 exit benefits at the valuation date.

Method. First find each cause’s probability in the original group, then discount each benefit to its own payment date. The year 2 retirement probability in the original group is $0.80(0.15)=0.12$, and the year 2 death probability is $0.80(0.05)=0.04$.

The retirement contribution is $0.12(\$8{,}000)/(1.04)^3=\$960/1.124864=\$853.436\ldots$, because the payment is at time 3. The death contribution is $0.04(\$20{,}000)/(1.04)^2=\$800/1.0816=\$739.645\ldots$, because the payment is at time 2.

Adding the unrounded values gives $\$853.436\ldots+\$739.645\ldots=\$1{,}593.081\ldots$, or $1,593.08 to the nearest cent. The cause probabilities sum to $0.20$ conditional on being active at the start of year 2, so the setup allows for the remaining $0.80$ to stay active through that year.

Common mistake avoided. Discounting both benefits for two years would value the retirement payment too early. Its exit occurs in year 2, but its payment date is the end of year 3.

Benefits That Depend on Service

The benefit amount should be evaluated using the plan rule applicable to the member at the relevant exit. If a withdrawal benefit is a fixed amount per completed service year, a year 2 withdrawal uses the amount for two completed years, not the amount for one year. If an exit benefit has a cap, escalation rule, or eligibility condition, apply that rule before inserting the resulting amount into the EPV expression.

A useful way to prevent mismatches is to make a cause-and-year schedule before calculating. It should show the active probability, conditional cause probability, resulting benefit amount, and payment date. The example below uses this schedule logic while checking that the active probabilities are consistent with the earlier year’s decrements.

Worked Example: Service-Related Benefits With Three Competing Causes

A hypothetical plan has $i=5\%$. At the start of year 1, the active probability is $a_0=1$. In year 1, conditional probabilities of withdrawal, death, and retirement are $0.03$, $0.01$, and $0.04$. In year 2, the corresponding probabilities are $0.04$, $0.01$, and $0.03$. In year 3, they are $0.02$, $0.015$, and $0.025$. All benefits are paid at the end of the year of exit. Withdrawal pays $1,500 per completed service year; death pays $25,000; and retirement pays $9,000. Find the EPV through year 3.

Method. The cause probabilities in years 1 and 2 each sum to $0.08$. Thus $a_1=1(1-0.08)=0.92$ and $a_2=0.92(1-0.08)=0.8464$. At each exit year, the withdrawal amount is based on that year’s completed service.

Year 1. The conditional expected benefit is $0.03(\$1{,}500)+0.01(\$25{,}000)+0.04(\$9{,}000)=\$45+\$250+\$360=\$655$. Since $a_0=1$, the expected payment from the original group is $655. Its present value is $\$655/1.05=\$623.809523\ldots$.

Year 2. The withdrawal amount is $2(\$1{,}500)=\$3{,}000$. The conditional expected benefit is $0.04(\$3{,}000)+0.01(\$25{,}000)+0.03(\$9{,}000)=\$120+\$250+\$270=\$640$. Weighting by $a_1=0.92$ gives an expected payment of $0.92(\$640)=\$588.80$. Its present value is $\$588.80/(1.05)^2=\$534.058956\ldots$.

Year 3. The withdrawal amount is $3(\$1{,}500)=\$4{,}500$. The conditional expected benefit is $0.02(\$4{,}500)+0.015(\$25{,}000)+0.025(\$9{,}000)=\$90+\$375+\$225=\$690$. Weighting by $a_2=0.8464$ gives an expected payment of $0.8464(\$690)=\$584.016$. Its present value is $\$584.016/(1.05)^3=\$504.494\ldots$.

Total and check. Sum the unrounded year-specific present values: $\$623.809523\ldots+\$534.058956\ldots+\$504.494\ldots=\$1{,}662.36$ to the nearest cent. In each year the cause probabilities sum to less than one, and the calculated active probabilities reflect the probability of remaining active through the preceding years.

Common mistake avoided. Applying the one-year withdrawal amount in every year understates later withdrawal benefits. The amount is determined by completed service at the relevant exit, while its probability is determined by the cause probability for that same year.

Checks Before Summing

Before finalizing a competing-exit valuation, check that every probability and benefit refers to the same event. In particular, a conditional probability given active status must be multiplied by the probability of being active; it cannot be used directly as a proportion of the original group. Also check that the causes are defined to be mutually exclusive within the modeled period and that their probabilities do not sum to more than one.

Check payment dates independently of exit dates. A benefit paid immediately on exit may have a different discount factor from one paid at the end of the year or deferred to a later date. Finally, calculate the total using unrounded cause-specific values and round only the final EPV to the nearest cent. Rounding each component early can make the displayed component sum differ by a cent from the correctly rounded total.

This setup is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. It provides a consistent framework for later valuations in which the probabilities, payment timing, or benefit structure require more detailed modeling.

Check Your Understanding

For each hypothetical plan, identify the relevant cause probabilities and payment dates before calculating. Round monetary answers to the nearest cent.

A plan has $i=0\%$. At the start of year 1, one member is active. Retirement and withdrawal probabilities are $0.05$ and $0.03$, and their benefits are $4,000 and $1,000, respectively, paid at year-end. Find the EPV. (Answer: $230.00)
A plan has $i=5\%$. At the start of year 2, $0.75$ of the original group is active. Conditional death and withdrawal probabilities in year 2 are $0.02$ and $0.08$. Death pays $15,000 at the end of year 2; withdrawal pays $2,000 at the end of year 3. Find the EPV of these benefits. (Answer: $307.74)
A plan has $i=4\%$. At the start of year 1, one member is active; during that year, total exit probability is $0.10$, so $0.90$ are active at the start of year 2. Year 2 retirement and death probabilities conditional on active status are $0.12$ and $0.03$. Retirement pays $10,000 at the end of year 3, and death pays $20,000 at the end of year 2. Find the EPV of the year 2 benefits. (Answer: $1,459.38)
A plan has $i=3\%$. At the start of year 1, one member is active. In year 1, withdrawal and death probabilities are $0.02$ and $0.01$; in year 2, they are $0.04$ and $0.01$. Withdrawal pays $500 per completed service year, and death pays $10,000; both benefits are paid at year-end. Find the two-year EPV. (Answer: $234.80)

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