Cause-Contingent Present Values for Ancillary Benefits

Why the Exit Cause Matters

An ancillary benefit may be payable when a member leaves active service for one particular reason, such as death, withdrawal, or disability. Its expected present value depends on more than the benefit amount: the member must first leave for the qualifying cause, and the payment must be discounted from its actual payment date. A benefit payable only on death, for example, is not payable when withdrawal occurs first.

The valuation has a simple structure: for each possible exit time, multiply the probability that the specified cause occurs then by the benefit amount and the discount factor for its payment time. Sum over annual periods, or integrate over time in a continuous model.

Definition: A cause-contingent benefit is a payment made only if an exit occurs for a specified cause $j$. In an annual model, let $q_t^{(j)}$ be the probability of exit for cause $j$ during year $t$, conditional on being active at the start of that year. Let $L_{t-1}$ be the probability of still being active at the start of year $t$, with $L_0=1$. If the benefit $B_j(t)$ is paid at time $t$, its expected present value is the sum of the cause-specific exit probability, benefit amount, and time-$t$ discount factor over all covered years.

Annual Competing-Decrement Valuation

Assume exits are classified into mutually exclusive causes and that a member who exits cannot experience another active-member exit later. If $q_t^{(k)}$ denotes the conditional probability of exit for cause $k$ in year $t$, then the probability of remaining active through that year is $1-\sum_k q_t^{(k)}$. Consequently,

L_t=L_{t-1}\left(1-\sum_k q_t^{(k)}\right), \qquad P(\text{cause }j\text{ exit in year }t)=L_{t-1}q_t^{(j)}.

With an effective annual interest rate $i$, a payment at the end of year $t$ has discount factor $v^t=(1+i)^{-t}$. If $B_j(t)$ is the amount payable at that time, the expected present value (EPV) is

\operatorname{EPV}_j =\sum_{t=1}^{n} L_{t-1}q_t^{(j)} B_j(t)v^t.

This formula assumes the annual decrement probabilities are conditional on being active at the start of the relevant year, and that the stated benefit is paid at year-end. If payment is instead made at another specified time, the discount exponent must match that time. The cause-specific probability is not $q_t^{(j)}$ alone: that is conditional on reaching the start of year $t$ still active.

Worked Example: A Death Refund Over Two Years

A hypothetical pension plan pays a refund of $10,000 if an active member dies in year 1, or $12,000 if the member dies in year 2. Withdrawal is the only other exit. In each year, conditional on being active at the start of the year, the death probability is 0.02 and the withdrawal probability is 0.08. Benefits are paid at the end of the year of exit. The effective annual interest rate is 5%. Find the EPV of the death refund.

Method. Calculate the probability of still being active at each year’s start, multiply it by the conditional death probability to get the probability of a death exit in that year, then discount the corresponding refund from year-end.

In year 1, the member is active with probability $L_0=1$. The death-exit probability is $1(0.02)=0.02$, and the probability of remaining active is $L_1=1(1-0.02-0.08)=0.90$.

In year 2, the death-exit probability is $L_1(0.02)=0.90(0.02)=0.018$. The year-1 refund’s present-value contribution is $0.02(10{,}000)/1.05=190.4762$. The year-2 contribution is $0.018(12{,}000)/(1.05)^2=216/1.1025=195.9184$.

\operatorname{EPV}_{\text{death}} =190.4762+195.9184 = \$386.39.

The answer is rounded to the nearest cent. As a check, the probabilities of death in years 1 and 2 sum to 0.038, which is less than the total probability of leaving by either cause over the two years. A common mistake is to use 0.02 for the year-2 death probability without allowing for year-1 exits; the correct unconditional probability is $0.90(0.02)=0.018$.

Deriving the Continuous-Time Formula

In a continuous-time competing-risks model, let $\lambda_k(t)$ be the force of exit for cause $k$ at time $t$, conditional on being active immediately before $t$. Assume these intensities describe mutually exclusive causes, and that they are integrable over the valuation period. Define the total exit intensity as $\lambda(t)=\sum_k\lambda_k(t)$. The probability of remaining active to time $t$ is

s(t)=\exp\left(-\int_0^t \lambda(u)\,du\right).

Over a short interval of length $dt$, the probability of a cause-$j$ exit is approximately the probability of being active at $t$, multiplied by $\lambda_j(t)dt$. Thus the cause-$j$ exit-time density is $s(t)\lambda_j(t)$. If the benefit is paid immediately on exit and has amount $B_j(t)$, its payment-time discount factor under force of interest $\delta$ is $e^{-\delta t}$. Integrating the discounted payment against the cause-specific exit density gives

\operatorname{EPV}_j =\int_0^n e^{-\delta t}B_j(t)s(t)\lambda_j(t)\,dt.

This is a direct expectation calculation: at each time $t$, the discounted amount is multiplied by the probability density of the qualifying exit, and all possible exit times are accumulated. If the benefit is paid after a delay $c$, rather than immediately, its discount factor becomes $e^{-\delta(t+c)}$, provided the delay is fixed and the payment remains due after the qualifying exit.

Worked Example: Immediate Death Benefit Under Constant Intensities

A hypothetical plan pays $15,000 immediately if an active member dies within two years. Assume a constant death intensity of 0.02 per year and a constant withdrawal intensity of 0.08 per year. The force of interest is $\delta=0.10$ per year. Find the EPV.

Method. Apply the continuous competing-risks integral. The total exit intensity is $0.02+0.08=0.10$, so the probability of remaining active to time $t$ is $s(t)=e^{-0.10t}$. The death-exit density is $0.02e^{-0.10t}$. Because payment is immediate, the discount factor is $e^{-0.10t}$.

The discounted expected payment at time $t$, per unit of time, is therefore $15{,}000(0.02)e^{-0.10t}e^{-0.10t}=300e^{-0.20t}$. Integrating over the two-year term gives

\begin{aligned} \operatorname{EPV}_{\text{death}} &=\int_0^2 300e^{-0.20t}\,dt\\ &=300\left[\frac{1-e^{-0.20(2)}}{0.20}\right]\\ &=1{,}500(1-e^{-0.4})\\ &=1{,}500(1-0.670320)\\ &=\$494.52. \end{aligned}

The answer is rounded to the nearest cent. The total exit intensity determines the chance of remaining active; the death intensity alone determines the density of the qualifying exit. A common mistake is to use $e^{-0.02t}$ for survival, as if withdrawal could not occur. That would overstate the probability of reaching later times still active and overvalue the death benefit.

Payment Date and Benefit Amount Must Match

The exit date and payment date are not necessarily the same. In an annual model, an exit during year $t$ might generate a payment at year-end, at the beginning of the following year, or after a stated administrative delay. The probability of exit is still attached to the year in which the exit occurs; only the discount factor changes to reflect when the cash is paid.

Likewise, the benefit amount must be the amount applicable to the qualifying exit at that time. It may vary by service, salary, or exit year. If an amount is conditional on a qualifying cause, use that amount only in the corresponding cause-specific term. Do not apply it to members who leave through a nonqualifying cause.

Worked Example: A Delayed Withdrawal Benefit

A hypothetical plan pays $4,000 for a withdrawal in year 1 and $6,000 for a withdrawal in year 2. Death is the only other exit. In each year, the conditional withdrawal probability is 0.05 and the conditional death probability is 0.05. Each withdrawal payment is made three months after the end of its exit year. The effective annual interest rate is 8%. Find the EPV of these withdrawal payments.

Method. First calculate the unconditional probability of withdrawal in each year. Then discount each payment from its actual date: 1.25 years for a year-1 exit and 2.25 years for a year-2 exit.

The year-1 withdrawal probability is $1(0.05)=0.05$. The probability of remaining active after year 1 is $1-0.05-0.05=0.90$, so the year-2 withdrawal probability is $0.90(0.05)=0.045$. The present-value contributions are

\begin{aligned} \text{Year 1 contribution} &=\frac{0.05(4{,}000)}{(1.08)^{1.25}} =\frac{200}{1.1009806} =\$181.66,\\ \text{Year 2 contribution} &=\frac{0.045(6{,}000)}{(1.08)^{2.25}} =\frac{270}{1.1890551} =\$227.07. \end{aligned}

Thus the EPV is $181.66+227.07=\$408.73$, rounded to the nearest cent. The calculation is checked by confirming that each discount exponent is the payment time, not merely the exit year. A common mistake is to discount the year-1 payment for one year and the year-2 payment for two years, ignoring the additional three-month delay.

Practical Checks and Common Traps

A reliable valuation separates three questions: which members remain exposed to the decrement, which exit cause triggers the benefit, and when the payment is made. Keeping these distinct prevents most errors in cause-contingent calculations.

Establish the exposure.
Find the probability of being active at the start of each period, allowing for all competing exits.

Select the qualifying cause.
Multiply active exposure by the conditional probability or intensity of the specified cause only.

Apply the right benefit and discount.
Use the amount due for that cause and the discount factor to its actual payment date.

Aggregate the possible payments.
Sum over years in a discrete model or integrate over time in a continuous model.

Two checks are especially useful. First, cause-specific exit probabilities over a period cannot exceed the probability of leaving active service over that period. Second, if interest is positive, delaying an otherwise identical payment lowers its present value. A result that violates either check signals a likely error in exposure, cause selection, or timing.

Key takeaway: The EPV of a benefit payable only on cause $j$ uses the probability of being active just before the exit, the probability or intensity of cause $j$, the benefit amount triggered by that cause, and discounting to the payment date. Competing exits affect exposure even when they do not trigger the benefit.

Check Your Understanding

For each problem, value only the benefit payable on the specified cause.

A hypothetical plan pays $1,000 at the end of the year of death, for deaths in either of two years. Each year the conditional death probability is 0.03 and the other-exit probability is 0.07. The effective annual interest rate is 5%. Find the EPV. (Answer: $53.06)
A hypothetical plan pays $5,000 at the end of the year of withdrawal, for withdrawals within three years. Each year the conditional withdrawal probability is 0.08 and the other-exit probability is 0.02. The effective annual interest rate is 4%. Find the EPV. (Answer: $1,005.49)
A hypothetical benefit of $20,000 is paid immediately on death within four years. The constant death intensity is 0.03, the constant withdrawal intensity is 0.07, and the force of interest is 0.05. Find the EPV to the nearest cent. (Answer: $1,804.75)
A hypothetical plan pays $10,000 on a year-1 death or $12,000 on a year-2 death, at the end of the exit year. The conditional death probabilities are 0.04 and 0.06 in years 1 and 2; the other-exit probabilities are 0.06 and 0.04. The effective annual interest rate is 5%. Find the EPV. (Answer: $968.71)
A hypothetical plan pays immediately on withdrawal within two years. The constant withdrawal intensity is 0.02, the competing death intensity is 0.08, and the force of interest is 0.05. The benefit at time $t$ is $10{,}000(1+0.1t)$ dollars. Find the EPV to the nearest cent. (Answer: $378.41)

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