Decomposing Expected Benefits by Exit Cause

Why Decompose Benefits by Exit Cause?

A pension arrangement may pay different amounts when a member dies, withdraws, becomes disabled, or retires. The expected present value of the benefit package is therefore built from different possible outcomes. Separating those outcomes by cause makes the valuation easier to check: each cause has its own probability, timing, and benefit, and the cause-specific values should add to the total.

The partition check is essential. During a stated valuation period, a member might exit through one of several mutually exclusive causes, or remain active through the end of the period. If the model omits the no-exit outcome, or counts one outcome under two causes, the probabilities and the resulting expected value will be wrong.

Definition: Let $E_j$ be the event that exit occurs for cause $j$, and let $C_j$ be the present value of all benefits payable under that cause, measured at the valuation date. If the cause events are mutually exclusive, then the total present value $Z$ is $C_j$ on $E_j$, with any separately defined no-exit value applying outside those events. The cause-specific expected present value is $E[C_j\mathbf{1}_{E_j}]$, where $\mathbf{1}_{E_j}$ equals 1 when $E_j$ occurs and 0 otherwise.

When the cause events are mutually exclusive and cover every outcome that can produce a benefit, the expected present value is the sum of the cause-specific expected present values. If there is a no-exit outcome with a benefit, it must be included as another term. If no benefit is payable on that outcome, its present value is zero, but the outcome still belongs in the probability partition.

E[Z]=\sum_j E[C_j\mathbf{1}_{E_j}] =\sum_j P(E_j)E[C_j\mid E_j],

The second equality applies when $P(E_j)>0$ and the conditional expected value is defined. If the payment under cause $j$ is a fixed amount $B_j$ paid at time $T_j$, with effective annual interest rate $i$, then $C_j=B_jv^{T_j}$, where $v=1/(1+i)$. The corresponding term is $P(E_j)E[B_jv^{T_j}\mid E_j]$. This form keeps the probability of the cause and the discounted value conditional on that cause distinct.

Why the Decomposition Holds

The result follows from the indicator representation of the total present value. Suppose exactly one of $E_1,\ldots,E_m$ occurs whenever an exit benefit is payable, and let $C_j$ be the present value conditional on cause $j$. Then the total value is the sum of the cause-specific values multiplied by their event indicators. Taking expectations and using linearity gives the decomposition.

Z=\sum_{j=1}^{m} C_j\mathbf{1}_{E_j} \quad\Longrightarrow\quad E[Z]=\sum_{j=1}^{m}E[C_j\mathbf{1}_{E_j}].

Mutual exclusivity is what makes this representation match a single realized outcome: only one cause indicator can equal 1. If the events overlap, the sum may count more than one benefit when only one should be payable. Exhaustiveness is also important. If there are outcomes that can produce benefits but are not represented by any event, the sum omits those benefits.

For a horizon ending at time $T$, a complete set of outcomes might be “exit by cause 1 before $T$,” through “exit by cause $m$ before $T$,” and “no exit before $T$.” The last event has zero contribution only if the contract pays no benefit for remaining active to $T$. If a benefit is paid then, its present value belongs in the total.

Discrete Annual Decrements

In an annual model, specify the cause probabilities conditional on being active at the start of each year. For one year, the probabilities of all exits during the year and remaining active to year-end must sum to 1. In later years, a cause’s unconditional probability includes the probability of surviving in the active state to the start of that year.

Worked Example: Two Years of Competing Exit Benefits

A hypothetical pension fund values exit benefits over two years. At each year-end, conditional on being active at the start of that year, the member dies with probability 0.10, withdraws with probability 0.20, becomes disabled with probability 0.05, and remains active with probability 0.65. These probabilities are the same in both years, and exactly one outcome occurs in a year. A death benefit of $12,000, withdrawal benefit of $4,000, or disability benefit of $9,000 is paid immediately at the end of the year of exit. The member receives no benefit if still active at the end of year 2. The effective annual interest rate is 5%. Find and reconcile the cause-specific expected present values.

Method and partition check. First check one year’s conditional outcomes: $0.10+0.20+0.05+0.65=1.00$. An exit in year 1 cannot also be an exit in year 2. The probability of being active at the start of year 2 is 0.65, and the probability of remaining active through year 2 is $0.65^2=0.4225$. Thus the two-year outcomes partition the possibilities: each cause can occur in year 1 or year 2, or the member can remain active through both years.

Death. The probability of death in year 1 is 0.10. The probability of death in year 2 is $0.65(0.10)=0.065$, since the member must first remain active through year 1. Therefore,

PV_{\text{death}} =\frac{0.10(12{,}000)}{1.05} +\frac{0.065(12{,}000)}{1.05^2} =\frac{1{,}200}{1.05}+\frac{780}{1.1025} =1{,}850.34.

Withdrawal. The year 1 probability is 0.20; the year 2 probability is $0.65(0.20)=0.13$. Hence,

PV_{\text{withdrawal}} =\frac{0.20(4{,}000)}{1.05} +\frac{0.13(4{,}000)}{1.05^2} =\frac{800}{1.05}+\frac{520}{1.1025} =1{,}233.56.

Disability. The year 1 probability is 0.05; the year 2 probability is $0.65(0.05)=0.0325$. Thus,

PV_{\text{disability}} =\frac{0.05(9{,}000)}{1.05} +\frac{0.0325(9{,}000)}{1.05^2} =\frac{450}{1.05}+\frac{292.5}{1.1025} =693.88.

Adding the cause-specific values gives $1{,}850.34+1{,}233.56+693.88=\$3{,}777.78$. As a direct check, the expected undiscounted payment conditional on being active at the start of a year is $1{,}200+800+450=\$2{,}450$. The total present value is therefore $2{,}450/1.05+0.65(2{,}450)/1.05^2=\$3{,}777.78$. Both calculations agree to the nearest cent.

Common mistake avoided: Do not use the unconditional year 1 probabilities as the year 2 probabilities. Each year 2 probability must include the 0.65 probability of remaining active through year 1.

Continuous Competing Intensities

In a continuous-time model, suppose a member starts active, and the cause-specific exit intensities $\lambda_j$ are constant over the valuation period $0\leq t\leq T$. Let $\Lambda=\sum_j\lambda_j$. Under this model, the probability of remaining active through time $t$ is $e^{-\Lambda t}$, and the density of exit from cause $j$ at time $t$ is $e^{-\Lambda t}\lambda_j$. The cause-specific density includes both survival in the active state up to $t$ and an exit of type $j$ at $t$.

f_j(t)=e^{-\Lambda t}\lambda_j,\qquad P(\text{no exit by }T)=e^{-\Lambda T}.

If a fixed benefit $B_j$ is paid immediately on exit and the force of interest is constant at $\delta$, the discount factor at time $t$ is $e^{-\delta t}$. Multiplying the cause-specific exit density by the discounted benefit and integrating over the possible exit times gives its expected present value.

PV_j=\int_0^T e^{-\delta t}B_j f_j(t)\,dt =B_j\lambda_j\int_0^T e^{-(\delta+\Lambda)t}\,dt =\frac{B_j\lambda_j}{\delta+\Lambda} \left(1-e^{-(\delta+\Lambda)T}\right).

This formula assumes constant intensities, a constant force of interest, and payment at the instant of exit. It does not apply unchanged if the intensities vary with time, if payment is delayed after exit, or if the benefit amount depends on the time or history of exit. In those cases, the corresponding time-dependent density, discount factor, or benefit amount must be included in the integral.

Worked Example: Continuous Exit Benefits and the No-Exit Outcome

Consider a hypothetical two-year valuation period. The constant cause-specific intensities are 0.02 per year for death, 0.03 per year for withdrawal, and 0.01 per year for disability. A fixed benefit is paid immediately at exit: $10,000 for death, $4,000 for withdrawal, and $7,000 for disability. The force of interest is $\delta=0.04$ per year. Find the cause-specific present values and check the outcome partition.

Method and partition check. The total exit intensity is $\Lambda=0.02+0.03+0.01=0.06$. The probability of no exit by year 2 is $e^{-0.06(2)}=e^{-0.12}=0.8869$ to four decimal places. The probabilities of exit by each cause by year 2 are

P(E_j)=\int_0^2 e^{-0.06t}\lambda_j\,dt =\frac{\lambda_j}{0.06}(1-e^{-0.12}).

Using $1-e^{-0.12}=0.1130796$, the death, withdrawal, and disability probabilities are respectively $0.0376932$, $0.0565398$, and $0.0188466$. Their sum is $0.1130796$; adding the no-exit probability $0.8869204$ gives 1 before rounding.

Death benefit. Here $\delta+\Lambda=0.10$, so the present value is

PV_{\text{death}} =\frac{10{,}000(0.02)}{0.10}(1-e^{-0.20}) =2{,}000(0.18126925) =\$362.54.

Withdrawal benefit.

PV_{\text{withdrawal}} =\frac{4{,}000(0.03)}{0.10}(1-e^{-0.20}) =1{,}200(0.18126925) =\$217.52.

Disability benefit.

PV_{\text{disability}} =\frac{7{,}000(0.01)}{0.10}(1-e^{-0.20}) =700(0.18126925) =\$126.89.

The cause-specific values sum to $\$362.54+\$217.52+\$126.89=\$706.95$. Directly, the weighted intensity of benefit payments is $0.02(10{,}000)+0.03(4{,}000)+0.01(7{,}000)=\$390$ per year. Multiplying this by $(1-e^{-0.20})/0.10=1.8126925$ gives $\$706.95$, confirming the result.

Common mistake avoided: The three exit probabilities sum to the probability of any exit, not to 1. The remaining probability, 0.8869, is the no-exit outcome; it has zero benefit here, but it is necessary for a complete partition.

When One Cause Has Several Benefit Components

A cause-specific benefit can itself comprise several payments. For example, a death exit might trigger both a lump sum and a survivor pension. If both are payable on the same death event, their present values are added within the death outcome. They are not treated as separate causes. The total expected present value is then found by weighting the combined death benefit by the probability of death, alongside the other cause-specific values.

Worked Example: Combining Components Within Each Exit Cause

A hypothetical plan pays, at the end of the year of exit, a $14,000 lump sum plus a $8,000 survivor-benefit value on death, an $8,000 combined benefit on disability, and $3,000 on withdrawal. In each year, conditional on being active at the start of the year, death, disability, withdrawal, and remaining active have probabilities 0.04, 0.06, 0.10, and 0.80, respectively. The same probabilities apply in both years. The effective annual interest rate is 25%, and no benefit is paid if the member remains active through year 2. Find the total expected present value by cause.

Method and partition check. The annual probabilities sum to $0.04+0.06+0.10+0.80=1$. The probability of remaining active through year 2 is $0.80^2=0.64$. Thus the unconditional probabilities of each cause over the two years are: death $0.04+0.80(0.04)=0.072$, disability $0.06+0.80(0.06)=0.108$, and withdrawal $0.10+0.80(0.10)=0.18$. Together with 0.64 for no exit, these sum to 1.

The death benefit is the sum of its two components, $\$14{,}000+\$8{,}000=\$22{,}000$. The discount factors are $v=1/1.25=0.8$ and $v^2=0.64$. Year 2 exit probabilities must include the 0.80 probability of remaining active in year 1.

Death.

PV_{\text{death}} =0.04(22{,}000)(0.8)+0.80(0.04)(22{,}000)(0.64) =704+450.56 =\$1{,}154.56.

Disability.

PV_{\text{disability}} =0.06(8{,}000)(0.8)+0.80(0.06)(8{,}000)(0.64) =384+245.76 =\$629.76.

Withdrawal.

PV_{\text{withdrawal}} =0.10(3{,}000)(0.8)+0.80(0.10)(3{,}000)(0.64) =240+153.60 =\$393.60.

The cause-specific total is $\$1{,}154.56+\$629.76+\$393.60=\$2{,}177.92$. As a check, the expected benefit conditional on being active at the start of a year is $0.04(22{,}000)+0.06(8{,}000)+0.10(3{,}000)=\$1{,}660$. The direct total is $1{,}660(0.8)+0.80(1{,}660)(0.64)=\$2{,}177.92$, in agreement.

Common mistake avoided: Do not count the lump sum and survivor benefit as separate exit causes. Both are triggered by the same death event, so they are combined before applying the death probability.

Common Checks Before Finalizing a Valuation

A cause-specific decomposition is most useful when it is accompanied by explicit checks. In particular, confirm that the benefit, timing, probability, and discount factor in each term refer to the same outcome.

Check exclusivity. Determine whether two stated causes can occur together under the model. If they can, define disjoint outcomes or specify how overlapping benefits are coordinated before summing them.
Check completeness. Include the no-exit or continued-active outcome in the partition, even when it has no benefit.
Check conditioning. In a later year, multiply the conditional exit probability by the probability of being active at the start of that year.
Check timing. Discount to the valuation date using the actual payment time. A benefit paid at exit and one paid at the next year-end do not have the same present value.
Check benefit aggregation. Add components triggered by the same event within that event’s benefit value; do not label them as separate causes.

The central identity is simple, but it depends on correctly defining the outcomes. Once the causes are mutually exclusive and the set of modeled outcomes is complete, linearity of expectation makes the total expected present value the sum of its cause-specific values.

Check Your Understanding

For each problem, check both the cause-specific calculation and whether the stated outcomes cover the modeled possibilities.

A one-year hypothetical plan pays $5,000 on death, $2,000 on withdrawal, and $8,000 on disability at year-end. Conditional probabilities are 0.10, 0.15, and 0.05, respectively; the member remains active with probability 0.70. The effective annual interest rate is 5%. Find the expected present value of the exit benefits. (Answer: $1,142.86)
In each of two years, a member who is active at the start of the year dies with conditional probability 0.10, withdraws with conditional probability 0.20, and otherwise remains active. Find the unconditional probabilities of death, withdrawal, and no exit through the end of year 2, and verify the partition. (Answer: death 0.17, withdrawal 0.34, no exit 0.49; total 1.00)
Over one year, a continuous-time model has constant death and withdrawal intensities of 0.01 and 0.02 per year. Find the probabilities of death, withdrawal, and no exit by year 1. Give probabilities to four decimal places. (Answer: death 0.0099, withdrawal 0.0197, no exit 0.9704; the unrounded probabilities sum to 1)
A plan pays $10,000 on death and $4,000 on withdrawal at each year-end. In each of two years, conditional on being active at the start of the year, death and withdrawal probabilities are 0.05 and 0.10, and the remaining probability is continued activity. The effective annual interest rate is 5%, with no benefit if still active at the end of year 2. Find the total expected present value. (Answer: $1,551.02)

What You'll Learn