Fractional-Year Benefit Timing Under Decrement Assumptions

Why Within-Year Timing Matters

An annual decrement probability tells us the chance of an exit during a year, but it does not by itself tell us when during that year the exit occurs. That distinction matters when a benefit is payable at exit: an exit early in the year produces a payment that is discounted less than an exit near year-end. To value such benefits, the annual decrement probabilities must be supplemented by an assumption about fractional-year timing.

This tutorial considers two standard timing assumptions: the uniform distribution of decrements (UDD) and constant forces of decrement. Each supplies a different within-year pattern while preserving the specified annual cause-specific probabilities. The benefits and decrement experience below are hypothetical.

Definition: Consider a single year and a group exposed to $r$ competing decrement causes. Let $q_j$ be the probability of decrement from cause $j$ during the year, and let $q=\sum_{j=1}^{r}q_j$ be the total probability of decrement. The probability of remaining in the initial state for the full year is $1-q$. A benefit payable at the time of decrement from cause $j$ is valued using both $q_j$ and an assumption about the fractional time $T$ of exit.

Uniform Distribution of Decrements

Under UDD, the number of decrements from each cause accumulated by time $t$ is proportional to $t$, for $0\leq t\leq1$. In probability terms, the probability of cause $j$ decrement by time $t$ is $tq_j$, and the probability of still being in the initial state at time $t$ is $1-tq$. Thus the unconditional density of a cause $j$ exit at time $t$ is $q_j$.

Suppose the effective annual interest rate is $i$, with $v=1/(1+i)$. A payment at fractional time $t$ has discount factor $v^t$. For a fixed benefit $B_j$ payable on cause $j$, the expected present value per person exposed at the start of the year is therefore

B_j\int_0^1 v^t q_j\,dt = B_jq_j\frac{1-v}{\ln(1+i)}.

The factor $(1-v)/\ln(1+i)$ is the UDD timing factor for a fixed exit benefit. If there is no interest, its limiting value is 1, so the value reduces to $B_jq_j$. For a benefit amount $B_j(t)$ that varies with the exit time, the corresponding value is $q_j\int_0^1 v^t B_j(t)\,dt$; the fixed-benefit factor cannot simply be applied to the whole amount unless the benefit is in fact constant.

Worked Example: Fixed Benefits Under UDD

A hypothetical pension plan has annual withdrawal probability $q_w=0.06$ and annual death probability $q_d=0.04$, so $q=0.10$. A withdrawal benefit of $4,000 or a death benefit of $12,000 is payable immediately at the relevant exit. Assume UDD and an effective annual interest rate of $i=0.06$. Find the expected present value per person exposed at the start of the year.

Method. Under UDD, use the cause-specific annual probability multiplied by the integral discount factor. Here $v=1/1.06=0.9433962264$, and $\ln(1.06)=0.0582689081$. Hence the timing factor is

\frac{1-v}{\ln(1.06)} = \frac{1-1/1.06}{\ln(1.06)} = \frac{0.0566037736}{0.0582689081} \approx 0.9714233.

The withdrawal contribution is $4{,}000(0.06)(0.9714233)=240(0.9714233)=\$233.1416$. The death contribution is $12{,}000(0.04)(0.9714233)=480(0.9714233)=\$466.2832$. Adding the mutually exclusive cause-specific contributions gives $233.1416+466.2832=\$699.4248$. Rounded to the nearest cent, the expected present value is $699.42.

Check. With no discounting, the corresponding expected benefit would be $240+480=\$720$. The discounted value of $699.42 is lower, as expected. The method avoids treating an exit-time payment as though every exit occurred at the end of the year.

Constant Forces of Decrement

Under the constant-force assumption, each cause has a constant force $\mu_j$ throughout the year. The total force is $\mu=\sum_j\mu_j$, and the probability of remaining in the initial state to time $t$ is $e^{-\mu t}$. The cause $j$ exit density at time $t$ is consequently $e^{-\mu t}\mu_j$.

When $0<q<1$, the finite total force that reproduces the annual total decrement probability is $\mu=-\ln(1-q)$. To reproduce the annual cause probabilities, set $\mu_j=(q_j/q)\mu$. Indeed, integrating the cause $j$ density over the year gives $\mu_j(1-e^{-\mu})/\mu=\mu_j q/\mu=q_j$. These conversion formulas require $0<q<1$. If $q=0$, there are no cause-specific exits and no cause-specific forces are needed. If $q=1$, the total force is infinite in the limiting representation; do not substitute $q=1$ into the finite-force formulas.

For fixed benefit $B_j$, and force of interest $\delta=\ln(1+i)$, the present value under constant forces follows by discounting the cause-specific density:

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

When $\delta+\mu=0$, the integral is interpreted by its limit. In the usual setting of nonnegative interest and decrement forces, the expression is well-defined, with the zero-force limit covering the no-discount, no-decrement case. For a time-varying benefit, retain the benefit inside the integral: $ \int_0^1 e^{-\delta t}e^{-\mu t}\mu_j B_j(t)\,dt$.

Derivation: The probability of being in the initial state at time $t$ under constant forces is $e^{-\mu t}$. Conditional on being there at time $t$, the probability of a cause $j$ exit in a short interval of length $dt$ is approximately $\mu_jdt$. Thus the unconditional probability of a cause $j$ exit in that interval is $e^{-\mu t}\mu_jdt$. Multiplying by the payment’s discount factor $e^{-\delta t}$ and integrating over $0\leq t\leq1$ gives the displayed present-value formula.

Worked Example: Converting Annual Probabilities to Constant Forces

Use the same hypothetical annual probabilities and exit benefits as before: $q_w=0.06$, $q_d=0.04$, a $4,000 withdrawal benefit, and a $12,000 death benefit. Now assume constant forces. Choose an effective annual interest rate of $i=1/9$, so $v=0.9$ and $\delta=\ln(10/9)$. Calculate the expected present value.

Method. The total decrement probability is $q=0.10$, which satisfies $0<q<1$. The total force is $\mu=-\ln(0.9)=\ln(10/9)$, so $\delta=\mu$. The cause-specific forces are

\mu_w=0.6\mu,\qquad \mu_d=0.4\mu.

The integral common to both causes is

\int_0^1 e^{-(\delta+\mu)t}\,dt = \frac{1-e^{-2\mu}}{2\mu} = \frac{1-0.9^2}{2\mu} = \frac{0.19}{2\mu} = \frac{0.095}{\mu}.

The withdrawal contribution is $4{,}000(0.6\mu)(0.095/\mu)=4{,}000(0.057)=\$228$. The death contribution is $12{,}000(0.4\mu)(0.095/\mu)=12{,}000(0.038)=\$456$. The total is $\$228+\$456=\mathbf{\$684.00}$.

Check. The un-discounted expected amount is $720. The constant-force value is below that amount. It is also slightly higher than the UDD value for these same data: constant forces place relatively more exits earlier in the year, where discounting is lighter. The method avoids applying UDD’s linear timing pattern to a constant-force model.

Time-Varying Benefits Under UDD

A benefit may depend on service completed, salary, or another quantity that changes during the year. The decrement assumption supplies the exit-time density; the benefit rule supplies the amount payable at that time. Both must appear in the valuation integral. The next example uses a simple linearly increasing benefit to show why replacing it by its year-end amount would be inappropriate.

Worked Example: A Benefit That Increases During the Year

A hypothetical plan pays a cause A exit benefit of $B_A(t)=\$8{,}000+\$2{,}000t$ at exit time $t$, measured in years from the start of the year. It pays a fixed $5,000 benefit on cause B exit. The annual probabilities are $q_A=0.12$ and $q_B=0.08$. Assume UDD and an effective annual interest rate $i=1$, so $v=1/2$ and $\delta=\ln 2$. Find the expected present value per person exposed.

Method. Under UDD, the cause-specific densities are $q_A$ and $q_B$. Define the two needed integrals

I_0=\int_0^1 e^{-(\ln 2)t}\,dt =\frac{1-1/2}{\ln 2} \approx0.7213475,

I_1=\int_0^1 t e^{-(\ln 2)t}\,dt = \frac{1-(1+\ln 2)/2}{(\ln 2)^2} \approx0.319336.

For cause A, the expected present value is

0.12\left[8{,}000I_0+2{,}000I_1\right] \approx 0.12\left[8{,}000(0.7213475)+2{,}000(0.319336)\right] = 0.12(5{,}770.780+638.672) = \$769.1342.

For cause B, it is $0.08(5{,}000I_0)=400(0.7213475)=\$288.5390$. The total is $769.1342+288.5390=\$1{,}057.6732$, or $1,057.67 to the nearest cent.

Check. The $2{,}000t$ increment is discounted and weighted by the exit time through $I_1$; it is not valued as if the maximum $2,000 increment applied to every cause A exit. The method avoids substituting a year-end benefit for the stated benefit at the actual exit time.

Comparing the Timing Information

The annual cause probabilities are the same under either assumption, but the assumptions distribute those exits differently within the year. Under UDD, the cause-specific density is constant. Under constant forces, the total population at risk declines exponentially, so the cause-specific density $e^{-\mu t}\mu_j$ declines with time. Conditional on an exit during the year, the corresponding time densities are $1$ under UDD and $\mu e^{-\mu t}/(1-e^{-\mu})$ under constant forces.

Feature	UDD	Constant forces
Probability of cause $j$ exit by time $t$	$tq_j$	$(\mu_j/\mu)(1-e^{-\mu t})$
Cause $j$ exit density	$q_j$	$e^{-\mu t}\mu_j$
Annual probabilities reproduced?	Yes, by construction	Yes, with $\mu=-\ln(1-q)$ and $\mu_j=(q_j/q)\mu$, for $0<q<1$
Fixed-benefit valuation	$B_jq_j(1-v)/\ln(1+i)$	$B_j\mu_j[1-e^{-(\delta+\mu)}]/(\delta+\mu)$

Neither assumption is universally preferable. The appropriate one is determined by the valuation basis or problem statement. In either case, the timing assumption describes events within the year; it does not change the annual probabilities that the model is required to reproduce.

Common Exam Trap: Year-End Discounting

A frequent error is to multiply each annual cause probability by the benefit and by $v$, as if every exit benefit were paid at the end of the year. For example, in the first worked example this incorrect calculation would be

(0.06)(4{,}000)v+(0.04)(12{,}000)v =720(0.9433962264) =\$679.25.

That calculation values a year-end payment, not a payment at exit. Under the stated UDD assumption, the correct timing factor is approximately $0.9714233$, giving $720(0.9714233)=\$699.42$. The difference arises because exits occur throughout the year, including before its end. Always identify whether the contract pays at exit or at a specified later time before selecting a discount factor.

Key takeaway: Annual decrement probabilities specify how often each cause occurs over the year; UDD or constant forces specify the timing pattern within the year. For an exit-time benefit, combine that pattern with the benefit amount and the discount factor at the actual fractional exit time.

Check Your Understanding

For each problem, state which timing assumption is being used and keep the payment time consistent with the wording.

Under UDD, a hypothetical cause has annual probability $0.02$, the benefit is $5,000 payable at exit, and $i=0.04$. Find the expected present value, to the nearest cent. (Answer: $100(1-1/1.04)/\ln(1.04)\approx$$98.06.)
Under UDD with $i=0$, annual probabilities for causes A and B are 0.06 and 0.04. The respective exit benefits are $3,000 and $10,000. Find the expected present value. (Answer: $580.00.)
Under constant forces, $q_A=0.05$, $q_B=0.15$, and $i=0.25$. Benefits of $8,000 and $2,000 are payable at exits A and B, respectively. Find the value. (Answer: $630.00.)
Under UDD with $i=0$, cause C has annual probability 0.10 and pays $B_C(t)=\$1{,}000(1+t)$ at exit. Find its expected present value. (Answer: $150.00.)
Annual probabilities are $q_d=0.04$ and $q_w=0.06$, with death and withdrawal benefits of $12,000 and $4,000. Let $i=1/9$. Compare UDD with constant forces, using the corresponding assumption separately for each valuation. (Answer: UDD $=720[0.1/\ln(10/9)]\approx\$683.37$; constant forces $=\$684.00$.)

Feature	UDD	Constant forces
Probability of cause \(j\) exit by time \(t\)	\(tq_j\)	\((\mu_j/\mu)(1-e^{-\mu t})\)
Cause \(j\) exit density	\(q_j\)	\(e^{-\mu t}\mu_j\)
Annual probabilities reproduced?	Yes, by construction	Yes, with \(\mu=-\ln(1-q)\) and \(\mu_j=(q_j/q)\mu\), for \(0<q<1\)
Fixed-benefit valuation	\(B_jq_j(1-v)/\ln(1+i)\)	\(B_j\mu_j[1-e^{-(\delta+\mu)}]/(\delta+\mu)\)

What You'll Learn