How Decrement Assumptions Change Pension Values

Why a Decrement Change Affects More Than One Benefit

In a pension valuation with competing exits, a change to one cause of exit has two effects. It changes the probability of an exit from that cause, and it changes how long members remain exposed to the other causes. For example, a higher withdrawal assumption can increase withdrawal refunds while reducing the probability that an active member remains long enough to die in service. The value of the death benefit may therefore fall even though the death assumption itself is unchanged.

The direction of the change in the total value is not determined by probabilities alone. It also depends on benefit amounts, payment timing, discounting, and any benefits that depend on remaining active. A sound comparison changes one assumption at a time, recalculates the associated probabilities and exposure, and then revalues each cash flow under the stated payment rules.

Definition: Under competing decrements, an individual is exposed to several mutually exclusive causes of exit while active. A cause-specific force $\mu_j$ describes the instantaneous rate of exit from cause $j$, conditional on remaining active immediately before that time. The total force is $\lambda=\sum_j\mu_j$. A change to one cause-specific force changes the total force and can therefore change the probabilities of all causes.

Cause Probabilities Under Constant Forces

Suppose there are $r$ causes of exit during a one-year period, and each cause-specific force $\mu_j$ is constant during that period. Assume the member is active at the start of the year, the causes are mutually exclusive, and there are no other exits. The probability of remaining active for $t$ years is $e^{-\lambda t}$, where $\lambda=\sum_{j=1}^{r}\mu_j$. The probability of exit from cause $j$ during the year is

q_j=\int_0^1 e^{-\lambda t}\mu_j\,dt =\frac{\mu_j}{\lambda}\left(1-e^{-\lambda}\right).

The common factor $1-e^{-\lambda}$ is the probability of any exit during the year. The fraction $\mu_j/\lambda$ allocates that probability to cause $j$. The probability of remaining active through the year is $e^{-\lambda}$, and the cause probabilities sum to $1-e^{-\lambda}$.

Thus, increasing one force while holding the others fixed generally increases the probability of exit from that cause, but reduces the cause probabilities for the unchanged causes. This is not a contradiction: the increased force shortens the time members are exposed to every competing cause.

Worked Example: A Higher Withdrawal Force and the Death Probability

A hypothetical pension plan has constant annual death and withdrawal forces of $0.04$ and $0.06$. Find the one-year death and withdrawal probabilities. Then increase the withdrawal force to $0.12$, leave the death force unchanged, and recalculate both probabilities.

Method. Use the constant-force competing-risk formula, changing only the withdrawal force in the second calculation. The common mistake to avoid is treating the death probability as unchanged merely because the death force is unchanged.

Before the change: The total force is $\lambda=0.04+0.06=0.10$. The probability of any exit is $1-e^{-0.10}=0.09516258$. Therefore,

q_d=\frac{0.04}{0.10}(0.09516258)=0.03806503,\qquad q_w=\frac{0.06}{0.10}(0.09516258)=0.05709755.

The probability of remaining active is $e^{-0.10}=0.90483742$. The check is $0.03806503+0.05709755+0.90483742=1.00000000$, to the displayed precision.

After the change: The total force is now $\lambda=0.04+0.12=0.16$, and the probability of any exit is $1-e^{-0.16}=0.14785621$. Thus,

q_d=\frac{0.04}{0.16}(0.14785621)=0.03696405,\qquad q_w=\frac{0.12}{0.16}(0.14785621)=0.11089216.

The death probability decreases from $0.03806503$ to $0.03696405$, while the withdrawal probability increases from $0.05709755$ to $0.11089216$. The probabilities of death, withdrawal, and remaining active after the change sum to $0.03696405+0.11089216+e^{-0.16}=1.00000000$, to the displayed precision.

Why Competing-Cause Probabilities Move in Opposite Directions

The integral expression for $q_j$ makes the exposure effect explicit. For an unchanged cause $j$, its force $\mu_j$ stays fixed, but increasing a competing force increases $\lambda$. The survival factor $e^{-\lambda t}$, which measures the chance of still being active and exposed at time $t$, then decreases. Consequently, fewer members remain exposed to cause $j$ later in the year.

The direction can also be verified by differentiating the cause probability with respect to a competing force $\mu_k$, where $k\ne j$:

\frac{\partial q_j}{\partial \mu_k} =\int_0^1 -t\mu_j e^{-\lambda t}\,dt < 0.

The integrand is negative for $0<t\leq1$, so the derivative is negative. For the cause whose own force is increased, write $q_k=\int_0^1\mu_k e^{-\lambda t}\,dt$. Differentiating with respect to $\mu_k$ gives

\frac{\partial q_k}{\partial \mu_k} =\int_0^1 e^{-\lambda t}(1-\mu_k t)\,dt > 0,

because $\mu_k\leq\lambda$, and $1-\mu_k t\geq 1-\lambda t$. Moreover,

\int_0^1 e^{-\lambda t}(1-\lambda t)\,dt =\left[t e^{-\lambda t}\right]_0^1=e^{-\lambda}>0.

So the probability of the changed cause rises, while the probability of every unchanged competing cause falls. This conclusion concerns cause probabilities under the specified constant-force model; it does not, by itself, determine the direction of change in a benefit value.

From Cause Probabilities to Benefit Values

For a year-end benefit $B_j$ paid only if exit $j$ occurs during the year, the expected present value per active member at the start of the year is $vB_jq_j$, where $v=1/(1+i)$ and $i$ is the effective annual interest rate. The total for several exit benefits is $v\sum_j B_jq_j$. Recalculate each $q_j$ after changing an assumption; do not apply the change only to the benefit triggered by that cause.

Worked Example: Revaluing Death and Withdrawal Benefits

A hypothetical plan pays a death-in-service benefit of $\$15{,}000$ and a withdrawal benefit of $\$6{,}000$, each at the end of the year of exit. The constant annual forces are initially $\mu_d=0.02$ and $\mu_w=0.08$. The effective annual interest rate is $i=4\%$. Find the one-year expected present value per active member. Then recalculate it after increasing $\mu_w$ to $0.16$, with $\mu_d$ unchanged. Round final dollar values to the nearest cent.

Method. Calculate each cause probability from the total force, multiply each by its benefit, add the expected payments, and discount the year-end total by $v=1/1.04$. This avoids the trap of changing the withdrawal probability but incorrectly holding the death probability fixed.

Initial assumptions: The total force is $0.02+0.08=0.10$, so $1-e^{-0.10}=0.095162582$. The cause probabilities are

q_d=0.2(0.095162582)=0.019032516,\qquad q_w=0.8(0.095162582)=0.076130066.

The undiscounted expected payment is

15{,}000(0.019032516)+6{,}000(0.076130066) =285.487746+456.780394 =\$742.268140.

Since both benefits are paid at year-end, the expected present value is $\$742.268140/1.04=\$713.719365$, or $\$713.72$ to the nearest cent.

After increasing the withdrawal force: The new total force is $0.02+0.16=0.18$, and $1-e^{-0.18}=0.164729789$. Therefore,

q_d=\frac{0.02}{0.18}(0.164729789)=0.018303310,\qquad q_w=\frac{0.16}{0.18}(0.164729789)=0.146426479.

The new expected payment before discounting is

15{,}000(0.018303310)+6{,}000(0.146426479) =274.549650+878.558874 =\$1{,}153.108524.

Discounting gives $\$1{,}153.108524/1.04=\$1{,}108.758196$, or $\$1{,}108.76$. The total value increases even though the death probability and death-benefit value decrease: the increase in the withdrawal benefit value is larger. The result also checks against the probability of remaining active, $e^{-0.18}=0.835270211$, since the two exit probabilities and this survival probability sum to one.

Benefits Paid at the Time of Exit

A benefit paid immediately when an exit occurs is not discounted as if it were paid at the end of the year. Suppose the force of interest is constant at $\delta$, and a benefit $B_j$ is paid at the time of cause $j$. The probability density of exit $j$ at time $t$ is $\mu_j e^{-\lambda t}$. Multiplying by the discount factor $e^{-\delta t}$ and integrating over the year gives the expected present value:

B_j\int_0^1 \mu_j e^{-\lambda t}e^{-\delta t}\,dt = B_j\frac{\mu_j}{\lambda+\delta} \left(1-e^{-(\lambda+\delta)}\right).

This formula assumes constant cause-specific forces and a constant force of interest over the year, with the benefit paid at the exit time. It does not apply unchanged to a benefit paid at year-end; that benefit is valued using its year-end payment probability and the annual discount factor.

Worked Example: Exit-Time Benefits When Withdrawal Increases

A hypothetical plan pays $\$50{,}000$ at death in service and $\$20{,}000$ at withdrawal, both immediately at exit. Initially, the constant annual forces are $\mu_d=0.04$ and $\mu_w=0.06$. The force of interest is $\delta=0.05$. Revalue the one-year benefits after increasing the withdrawal force to $0.12$, leaving the death force unchanged. Round final amounts to the nearest cent.

Method. Integrate each benefit against its cause-specific exit density and continuous discount factor. The common mistake is to use $1/1.05$, which discounts a year-end payment rather than a payment made at the actual exit time.

Initially, $\lambda+\delta=0.04+0.06+0.05=0.15$, and $1-e^{-0.15}=0.139292024$. The common discounted exposure factor is $0.139292024/0.15=0.928613491$. Thus, the death component is $50{,}000(0.04)(0.928613491)=\$1{,}857.23$, and the withdrawal component is $20{,}000(0.06)(0.928613491)=\$1{,}114.34$. The total is $\$1{,}857.23+\$1{,}114.34=\mathbf{\$2{,}971.56}$, using unrounded components for the total.

After the change, $\lambda+\delta=0.04+0.12+0.05=0.21$, and $1-e^{-0.21}=0.189415755$. The common factor is $0.189415755/0.21=0.901979786$. The death component is $50{,}000(0.04)(0.901979786)=\$1{,}803.96$, while the withdrawal component is $20{,}000(0.12)(0.901979786)=\$2{,}164.75$. The total is $\$3{,}968.71$. As in the year-end example, the higher total does not mean every component increased: the death component falls because withdrawal reduces exposure to death.

Use the Model That Matches the Assumptions

A multiple-decrement table may instead state annual probabilities directly: $q_j$ is the probability of exit from cause $j$ during the year, conditional on being active at its start. Those probabilities are mutually exclusive, and the probability of remaining active is $1-\sum_j q_j$. If a problem changes one stated annual decrement probability while holding another annual probability fixed, use those stated probabilities as given. Do not automatically convert them to constant forces unless the problem supplies or requires a specific conversion assumption.

Under either approach, a multi-year valuation must account for the number still active at the start of each year. If the probability of remaining active in year $s$ is $p_s$, then the probability of being active at the start of year $k$ is the product of the prior annual persistence probabilities, $\prod_{s=1}^{k-1}p_s$. A changed decrement assumption can therefore affect benefits in later years even when those benefits are not triggered by the changed cause.

Worked Example: Two-Year Year-End Benefits Under Annual Decrement Probabilities

A hypothetical plan has, in each of two years, a death probability of $0.02$ and a withdrawal probability of $0.08$ for each member active at the start of that year. These are direct annual multiple-decrement probabilities, not forces. Death pays $\$20{,}000$ and withdrawal pays $\$5{,}000$, both at year-end. The effective annual interest rate is $5\%$, and assumptions are identical in each year. Find the two-year expected present value per initial active member. Then increase the annual withdrawal probability to $0.18$, retaining the death probability $0.02$. Round final values to the nearest cent.

Method. Find the expected year-end payment conditional on being active at the start of a year. Multiply the second-year amount by the probability of surviving active through year one, and discount each year-end amount. The trap is to value year two as though every initial member were still active.

Initial probabilities: The annual persistence probability is $1-0.02-0.08=0.90$. The expected payment conditional on being active at the start of a year is $0.02(20{,}000)+0.08(5{,}000)=400+400=\$800$. The year-one present value is $800/1.05=\$761.904762$. The probability of being active at the start of year two is $0.90$, so the year-two present value is $0.90(800)/(1.05)^2=\$653.061224$. The total is $\$761.904762+\$653.061224=\mathbf{\$1{,}414.97}$.

After the change: Annual persistence falls to $1-0.02-0.18=0.80$. The conditional expected payment per year is now $0.02(20{,}000)+0.18(5{,}000)=400+900=\$1{,}300$. Year one has present value $1{,}300/1.05=\$1{,}238.095238$. The year-two present value is $0.80(1{,}300)/(1.05)^2=\$943.310658$. The total is $\$1{,}238.095238+\$943.310658=\mathbf{\$2{,}181.41}$. Although fewer members reach year two, the increased withdrawal payments more than offset the reduced later exposure in this example.

Practical Checks Before Comparing Values

When one decrement assumption changes, keep the comparison controlled and explicit:

State which assumption changes and which assumptions remain fixed.
Recalculate the total force or the annual persistence probability under the correct model.
Recalculate every competing cause probability, not just the probability for the changed cause.
Track exposure into later years if the valuation extends beyond one year.
Apply the payment timing and interest convention specified for each benefit.
Compare benefit components as well as the total; a rising total can conceal a falling component.

These steps are particularly important for pension values because an exit can trigger one benefit while preventing another. A higher withdrawal assumption may raise refund values, lower death-in-service values, and reduce the number of members who remain active to earn further service. The net value depends on the benefit design and cannot be inferred from the direction of a single decrement probability.

Key takeaway: A competing decrement affects both the likelihood of its own benefit and the exposure available for other benefits. Revalue all causes and all relevant future periods under the changed assumptions before interpreting the change in total pension value. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated model and retain unrounded probabilities during calculations. Round final probabilities to four decimal places and final dollar values to the nearest cent.

Under constant forces, death is $0.03$ and withdrawal is $0.07$. Find the death probability before and after the withdrawal force increases to $0.10$, with death unchanged. State the direction of change. (Answer: $0.0285$ before; $0.0281$ after; it decreases.)
Under constant annual forces of $0.05$ for death and $0.05$ for withdrawal, death pays $\$10{,}000$ and withdrawal pays $\$2{,}000$ at year-end. If $i=4\%$, find the one-year expected present value per active member. (Answer: $\$549.02$.)
Benefits of $\$25{,}000$ at death and $\$5{,}000$ at withdrawal are paid immediately at exit. The constant forces are $0.02$ and $0.08$, respectively, and the force of interest is $0.04$. Find the one-year expected present value. (Answer: $\$839.84$.)
In each of two years, direct annual probabilities are $0.02$ for death and $0.08$ for withdrawal. Death pays $\$20{,}000$, withdrawal pays $\$5{,}000$, both at year-end, and $i=5\%$. Recalculate the two-year value if withdrawal probability rises to $0.18$ each year while death probability remains $0.02$. (Answer: $\$2{,}181.41$.)
Explain why an unchanged death force can produce a lower death probability when the withdrawal force rises, under a constant-force competing-risk model. (Answer: The higher total force reduces the probability of remaining active and exposed to death during the year, so the death cause probability falls even though its own force is unchanged.)

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