Salary-Linked Pension Benefits at Exit

Why Salary Projection Matters

A pension or exit benefit may be defined as a fraction or multiple of salary at the date a member retires, withdraws, becomes disabled, or dies. That date is uncertain when several exits compete. Valuation must therefore combine two steps: project salary to each possible exit date, then weight the benefit by the probability of that exit occurring while the member is still active.

The salary used is determined by the benefit rule. A formula based on salary at exit uses projected salary, not current salary. If the benefit is a pension, first determine the pension amount at retirement and then value its post-retirement payments on the specified annuity basis. These steps keep salary growth, competing exits, and pension conversion separate.

Definition: Let $S_0$ be current salary. Under deterministic continuous salary growth at force $g$, salary at time $t$ is $S(t)=S_0e^{gt}$. Under deterministic annual salary growth at rate $g$, salary at the end of year $k$ is $S_k=S_0(1+g)^k$. A salary-linked exit benefit is a specified function of the salary at the date and cause of exit, such as $B_j(t)=c_jS(t)$, where $j$ identifies the exit cause and $c_j$ is a benefit factor.

The continuous and annual formulas describe different salary assumptions. Use the formula matching the problem’s stated growth convention; a rate described as an annual increase is not automatically a continuous force. Also apply any cap, floor, or eligibility condition to salary exactly where the benefit definition requires it.

Continuous Competing Exits and Salary-Linked Benefits

Suppose an active member can leave through causes $j=1,\ldots,m$, with cause-specific exit force $\lambda_j(t)$. Assume these forces describe exits from the active state. The probability of remaining active through time $t$ is ${}_tp^{(0)}=\exp\left(-\int_0^t\sum_j\lambda_j(s)\,ds\right)$. The probability of a cause-$j$ exit in a short interval around $t$, conditional on being active just before that interval, is approximately $\lambda_j(t)\,dt$. Thus the unconditional cause-$j$ exit density is ${}_tp^{(0)}\lambda_j(t)$.

Theorem: If the cause-$j$ benefit $B_j(t)$ is paid immediately upon exit, the force of interest is constant at $\delta$, and the valuation period is $0\leq t\leq T$, then the expected present value of all continuous exit benefits is

\operatorname{EPV} = \sum_{j=1}^{m}\int_0^T e^{-\delta t}\,{}_tp^{(0)}\,\lambda_j(t)\,B_j(t)\,dt.

The formula assumes that benefits are paid at exit and that the stated exit forces apply to active members over the valuation period. A benefit payable later, or conditional on a separate event after exit, must be valued at its actual payment time and under its applicable assumptions.

The structure follows directly from expected value. In a small interval of length $dt$, the probability of cause $j$ exit is approximately ${}_tp^{(0)}\lambda_j(t)dt$. The benefit is $B_j(t)$, and its present-value factor is $e^{-\delta t}$. Multiplying these terms gives the interval’s expected present value to first order. Summing over intervals and taking the limit gives the integral; summing across causes gives the displayed expression.

Project salary.
Use the stated salary-growth convention to obtain salary at each potential exit time.

Apply the benefit rule.
Calculate the amount for each cause, including any cap or pension conversion factor.

Weight by cause-specific exits.
Multiply by active-state survival and the relevant cause-specific force or probability.

Discount at the payment time.
Integrate for continuous exits or sum over the stated payment dates.

Worked Example: Continuous Salary-Linked Benefits by Exit Cause

Method. Apply the cause-specific continuous valuation integral, using projected salary for each exit benefit. In a hypothetical plan, current salary is $50,000 and grows continuously at force $g=0.03$. The effective annual interest rate is not used directly: the valuation force is $\delta=0.05$. Until time 5, active-member exit forces are constant: withdrawal $0.04$, death $0.01$, disability $0.02$, and retirement $0.03$. At exit, withdrawal pays $0.5S(t)$, death pays $2S(t)$, and disability pays $S(t)$ as a lump sum. Retirement provides a continuous pension of $0.25S(t)$ per year for life. On the retirement basis, the force of interest is $0.05$, retired mortality is constant at $0.05$, and the pension is level with no guarantee period. Find the EPV of exit benefits through time 5.

1. Convert the retirement pension to a value at retirement. Under the stated constant force of interest and retired mortality, the probability of surviving $u$ years after retirement is $e^{-0.05u}$, and the discount factor is also $e^{-0.05u}$. The value of a continuous level pension of one dollar per year is therefore

\bar{a}_{\text{ret}} = \int_0^\infty e^{-0.05u}e^{-0.05u}\,du = \int_0^\infty e^{-0.10u}\,du = 10.

Consequently, the retirement benefit’s capital value at retirement is $0.25S(t)\times10=2.5S(t)$. This conversion uses post-retirement mortality; it does not change the active-member exit forces.

2. Find active-state survival. The total active exit force is $0.04+0.01+0.02+0.03=0.10$, so ${}_tp^{(0)}=e^{-0.10t}$. Salary is $50{,}000e^{0.03t}$. Combining active survival, discounting, and salary projection gives $e^{-0.10t}e^{-0.05t}e^{0.03t}=e^{-0.12t}$.

3. Add the cause-specific benefit factors. Relative to salary at exit, the weighted factors are $0.04(0.5)=0.020$ for withdrawal, $0.01(2)=0.020$ for death, $0.02(1)=0.020$ for disability, and $0.03(2.5)=0.075$ for retirement. Their sum is $0.135$.

4. Integrate and check. The EPV is

50{,}000(0.135)\int_0^5 e^{-0.12t}\,dt = 6{,}750\left(\frac{1-e^{-0.60}}{0.12}\right) = 6{,}750(3.759903\ldots) = \$25{,}379.35.

The check is that each cause’s force is multiplied by its own benefit factor, while the common survival, discount, and salary-growth terms are included once. The result is positive and is less than the total salary-weighted benefit one would obtain by assuming an immediate exit for every member.

Year-End Exit Benefits

For benefits paid at year-end, use a discrete sum rather than the continuous cause-specific integral. Let $q_{j,k}$ be the conditional probability of cause $j$ exit in year $k$, given active status at the start of that year. If $p_{k-1}^{(0)}$ is the probability of remaining active to the start of year $k$, the year-$k$ contribution is the starting active probability multiplied by each conditional exit probability and its benefit at year-end, then discounted by $v^k$. The probabilities $q_{j,k}$ are conditional on being active at the start of year $k$; they are not unconditional probabilities from time 0.

Worked Example: Two Year-End Benefits with Salary Growth

Method. Sum the expected discounted benefits at the two year-ends, keeping conditional exit probabilities separate from the probability of reaching each year active. In a hypothetical plan, current salary is $60,000 and increases by 10% at each year-end. The effective annual interest rate is $i=0.10$, so $v=1/1.10$. In each of the two years, conditional on being active at the start of that year, withdrawal probability is $0.05$, death probability is $0.01$, and the probability of remaining active is $0.94$. Withdrawal pays $0.4$ times salary at exit; death pays $2$ times salary at exit. Members still active after year 2 receive no benefit. Find the EPV.

Year 1. Year-end salary is $60{,}000(1.10)=\$66{,}000$. The conditional expected benefit is $0.05(0.4)(66{,}000)+0.01(2)(66{,}000)=1{,}320+1{,}320=\$2{,}640$. Discounting gives $2{,}640/1.10=\$2{,}400$.

Year 2. Salary is $66{,}000(1.10)=\$72{,}600$. The probability of being active at the start of year 2 is $0.94$. Conditional on that status, the expected year-2 benefit is $0.05(0.4)(72{,}600)+0.01(2)(72{,}600)=1{,}452+1{,}452=\$2{,}904$. The unconditional expected payment at year 2 is $0.94(2{,}904)=\$2{,}729.76$, giving present value $2{,}729.76/(1.10)^2=\$2{,}256$.

Total and check. The EPV is $2{,}400+2{,}256=\boxed{\$4{,}656.00}$. The year-2 contribution correctly includes the 0.94 probability of reaching year 2 active; omitting that factor would overstate the value.

Retirement Pensions Based on Salary at Retirement

A retirement pension often has two valuation dates. First calculate the pension amount using salary at the retirement date. Then value that pension at retirement under the retired-life annuity assumptions. Finally, if retirement occurs only at a specified future date provided the member remains active, weight the retirement value by active-state survival and discount it to the valuation date.

This differs from treating retirement as a cause that can occur continuously throughout the period. In a fixed-date arrangement, the retirement payment is made at the specified date only if the member is then eligible and active; competing exits before that date remove the member from the active group.

Worked Example: Pension at a Fixed Retirement Date

Method. Project salary to the scheduled date, convert the pension to capital at that date, and multiply by the probability of surviving all active-state competing exits and by the discount factor. A hypothetical member currently earns $50,000. Salary grows continuously at force $0.02$. Retirement is scheduled at time 3 if the member remains active. Before then, constant active exit forces are withdrawal $0.04$, death $0.01$, and disability $0.02$. The valuation force of interest is $0.05$. At retirement the annual pension is $0.30$ times salary at retirement, paid continuously for life. Retired mortality has constant force $0.05$, and the retirement valuation force is $0.05$. Find the EPV of this retirement pension.

1. Project salary. At time 3, salary is $50{,}000e^{0.02(3)}=50{,}000e^{0.06}=\$53{,}091.83$ (rounded to cents). The annual pension is $0.30(50{,}000e^{0.06})=\$15{,}927.55$ (rounded to cents).

2. Value the pension at retirement. As in the stated retirement basis, discounting and retired survival each contribute a factor $e^{-0.05u}$. The continuous annuity factor is $1/(0.05+0.05)=10$. Using unrounded salary, the capital value at time 3 is

0.30(50{,}000e^{0.06})(10) = 150{,}000e^{0.06} = \$159{,}275.48.

3. Weight for active survival and discount. The total active exit force is $0.04+0.01+0.02=0.07$, so the probability of remaining active to time 3 is $e^{-0.07(3)}=e^{-0.21}$. Discounting to time 0 contributes $e^{-0.05(3)}=e^{-0.15}$. Thus

\operatorname{EPV} = 150{,}000e^{0.06}e^{-0.21}e^{-0.15} = 150{,}000e^{-0.30} = \$111{,}122.73.

The result can also be checked by multiplying the time-3 capital value by $e^{-0.36}$, the combined active-survival and discount factor. The unrounded intermediate values are used before rounding the final EPV to the nearest cent.

Common Mistakes and Exam Traps

The central exam trap is to use the right benefit formula with salary from the wrong date. Other frequent errors are to confuse conditional exit probabilities with unconditional probabilities, discount a benefit from the wrong payment date, or use a retired annuity factor as if it were an active-state survival probability. Keep the benefit amount, probability of reaching the exit date, and payment timing as distinct components until the final calculation.

A cap must also be applied before converting a pension to capital. For example, suppose the year-2 retirement pension is $0.4\min(S_2,\$60{,}000)$ per year, the annuity factor is 10, and retirement at year 2 is reached with probability $0.81$. With salary of $72,000, the capped annual pension is $0.4(60{,}000)=\$24{,}000$, so its capital value at retirement is $240,000. The year-2 present value contribution at $i=0.05$ is $0.81(240{,}000)/1.05^2$.

Wrong calculation: Ignoring the cap gives an annual pension of $0.4(72{,}000)=\$28{,}800$, or a capital value of $288,000. It overstates the contribution by

\frac{0.81(288{,}000-240{,}000)}{1.05^2} = \frac{38{,}880}{1.1025} = \$35{,}265.31.

Correction: Apply the stated cap to salary before calculating the pension and its capital value. The incorrect method does not merely misstate salary; it changes the benefit amount used in the valuation.

Key takeaway: For a salary-linked exit benefit, identify the salary basis and project it to the benefit-determination date. Then apply the cause-specific exit probability or force, account for active survival, convert any pension using its retirement basis, and discount from the actual payment date.

Check Your Understanding

For each problem, use the payment timing and salary-growth convention stated.

A hypothetical member earns $50,000 and salary increases by 4% at year-end. At the end of year 1, conditional withdrawal and death probabilities are 0.10 and 0.02. Withdrawal pays $0.5$ times salary and death pays $2$ times salary. The effective annual interest rate is 4%. Find the EPV. (Answer: $4,500.00)
A hypothetical member earns $30,000, with continuous salary growth force $0.02$. Constant active withdrawal and death forces are $0.03$ and $0.01$, the valuation force is $0.04$, and the horizon is 2 years. Immediate benefits are $0.5S(t)$ for withdrawal and $2S(t)$ for death. Find the EPV. (Answer: $1,978.89)
A member earns $60,000 and salary grows continuously at force $0.03$. Retirement occurs at time 4 only if active; withdrawal and death forces before then are $0.02$ and $0.01$. The valuation force is $0.04$, and the retirement pension is $0.25$ times salary at retirement with an annuity factor of 8. Find the EPV. (Answer: $102,257.25)
A member earns $80,000 and salary increases by 5% at each year-end. The effective annual interest rate is 5%. Conditional year-1 withdrawal and death probabilities are 0.04 and 0.01; conditional year-2 probabilities are 0.03 and 0.02. The year-1 probability of remaining active is 0.95. Withdrawal pays $0.25$ times salary and death pays $1.5$ times salary, each at year-end. Find the two-year EPV. (Answer: $4,850.00)
A hypothetical member earns $50,000, with 10% annual salary growth. At year 2, a pension is payable only if the member is active; the probability of reaching year 2 active is 0.81. The annual pension is $0.4\min(S_2,\$60{,}000)$, paid at retirement with a capital factor of 10. The effective annual interest rate is 5%. Find the present value of the year-2 retirement benefit. (Answer: $176,326.53)

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