Payment Timing for Benefits at Member Exit

Why Payment Timing Matters

An exit benefit can be triggered by the same event and have the same amount at exit, yet have different present values depending on when it is paid. A benefit paid immediately is discounted from the actual exit time. A benefit paid at the end of the exit year is discounted from a later, discrete date. An administrative delay creates another payment date, which may fall between those two dates or after both.

The central discipline is to define the payment time before applying interest. Let $T$ be the time of member exit, measured in years from the valuation date, and let $B(T)$ be the benefit amount determined at exit. With effective annual interest rate $i$, use $v=1/(1+i)$ and discount a payment at time $s$ by $v^s$. The benefit amount and payment time are separate parts of the valuation.

Definition: For an exit at time $T$, an immediate payment is made at $T$. A year-end payment is made at $\lceil T\rceil$, the end of the policy year containing the exit, provided exits exactly at a year boundary are assigned consistently. A payment after an administrative delay of $h$ years is made at $T+h$. Unless specified otherwise, the benefit is fixed at exit and does not earn interest during the delay.

For an exit at an exact policy-year boundary, the year-end convention must be interpreted explicitly. In the examples below, exits are either strictly within a year or occur at a stated deterministic time, so the payment date is unambiguous. The probability of an exit exactly at a year boundary is zero under a continuous exit-time distribution.

Valuing the Three Payment Conventions

Assume the interest rate is effective annually and applies consistently to fractional years through the discount function $v^t=(1+i)^{-t}$. If exit occurs at time $T$ and the benefit is $B(T)$, the present values are:

\begin{aligned} \text{Immediate payment:} \quad &B(T)v^T,\\ \text{Year-end payment:} \quad &B(T)v^{\lceil T\rceil},\\ \text{Payment after delay }h: \quad &B(T)v^{T+h}. \end{aligned}

These expressions presume no interest credit or benefit increase between exit and payment. If the plan instead increases the benefit during an administrative delay, that increase must be specified separately and included in the payment amount. It is not correct to change the discount date while silently changing the benefit.

Key result: If the exit-time distribution and exit benefit are unchanged, moving a fixed benefit from immediate payment to a fixed delay $h$ multiplies its immediate-payment present value by $v^h$. This follows because $B(T)v^{T+h}=v^hB(T)v^T$, where $v^h$ is constant.

A useful special case occurs when exit, conditional on occurring in a particular year, is uniformly distributed over that year. For year $k$, this means the conditional density of $T$ is 1 on $(k-1,k)$. The expected immediate discount factor conditional on exit in year $k$ is:

\int_{k-1}^{k} v^t\,dt = \frac{v^{k-1}-v^k}{\ln(1+i)}.

Because the interval has length one, this integral is also the conditional average discount factor. For a uniform exit time in year 1 it reduces to $(1-v)/\ln(1+i)$. For year-end payment, by contrast, the discount factor is simply $v^k$. The uniform-timing formula is not appropriate if exit is concentrated at a particular date or has a nonuniform distribution within the year.

Worked Example: One Exit Year and Three Payment Dates

Method: value the benefit at each specified payment date, then multiply by the probability of exit. A hypothetical plan pays a fixed benefit of $12,000 if a member exits during year 1. The probability of exit in year 1 is 0.20; conditional on exit in that year, $T$ is uniform on $(0,1)$. Otherwise no benefit is payable within the valuation period. The effective annual interest rate is 4%. Compare immediate payment, payment at the end of year 1, and payment one quarter of a year after exit.

Immediate payment. The conditional average discount factor is $(1-v)/\ln(1.04)$, where $v=1/1.04=0.961538462$ and $\ln(1.04)=0.039220713$. Thus the factor is $0.038461538/0.039220713=0.980643526$. The expected present value is:

0.20(12{,}000)(0.980643526) =2{,}400(0.980643526) =\$2{,}353.54.

Payment at year-end. Every eligible benefit is paid at time 1, so its discount factor is $v=0.961538462$. The expected present value is $0.20(12{,}000)(0.961538462)=\$2{,}307.69$.

Payment after a 0.25-year delay. The payment time is $T+0.25$. The delay multiplies the immediate value by $v^{0.25}=(1.04)^{-0.25}=0.990242735$. Therefore the expected present value is $2{,}353.544462(0.990242735)=\$2{,}330.58$.

The check is that the values are ordered as expected: $2{,}353.54>2{,}330.58>2{,}307.69$. For a positive interest rate and a fixed benefit, later payment has a lower present value.

Combining Exit Years and Benefit Amounts

When the benefit amount or exit probability varies by year, calculate each year’s expected contribution using that year’s benefit and payment-time rule, then add the contributions. Under a uniform-within-year assumption, the immediate-payment contribution for year $k$ is the probability of exit in that year multiplied by the benefit for that year and the uniform average discount factor. Do not use the year-1 average discount factor for exits in later years: the discount interval shifts with the exit year.

Worked Example: Service-Related Benefits in Two Exit Years

Method: calculate the expected value separately for each exit year under each payment convention. In a hypothetical plan, the probability of exit in year 1 is 0.15 and the exit benefit is $8,000. The probability of exit in year 2 is 0.25 and the benefit is $10,000. The remaining probability, 0.60, has no exit in these two years and produces no benefit in this calculation. Conditional on exit in either year, exit is uniform within that year. The effective annual interest rate is 4%.

For immediate payment, the average discount factors are $0.980643526$ for year 1 and $v(0.980643526)=0.942926467$ for year 2. The year-1 contribution is $0.15(8{,}000)(0.980643526)=1{,}200(0.980643526)=\$1{,}176.77$. The year-2 contribution is $0.25(10{,}000)(0.942926467)=2{,}500(0.942926467)=\$2{,}357.32$. Their sum is $3,534.09.

For year-end payment, discount the year-1 benefit by $v=0.961538462$, and the year-2 benefit by $v^2=0.924556213$. The contributions are $1{,}200(0.961538462)=\$1{,}153.85$ and $2{,}500(0.924556213)=\$2{,}311.39$. The total is $3,465.24.

If both benefits are paid 0.25 years after exit, multiply the immediate-payment total by $v^{0.25}=0.990242735$. This gives $3{,}534.088399(0.990242735)=$$3,499.60.

The last result is larger than the year-end value because a quarter-year delay after exit is not the same as waiting until the end of the exit year. In year 1, the delayed payment can occur before time 1; in year 2, it can occur before time 2. Applying $v^{0.25}$ to the immediate value correctly reflects that fixed delay.

Why the Uniform-Timing Formula Holds

The formula for uniform exit timing is an average over all possible exit times in the year, not a substitute assumption that everyone exits at the midpoint. If $T$ is uniform on $(k-1,k)$, its conditional density is 1. The expected discount factor is therefore the integral of $v^t$ over that interval. Since $v^t=e^{-t\ln(1+i)}$, integrating gives:

Write the conditional expectation.
$E[v^T\mid k-1<T<k]=\int_{k-1}^{k}v^t\,dt$, because the uniform density is 1 on an interval of length one.

Express discounting exponentially.
$v^t=e^{-t\ln(1+i)}$, so an antiderivative is $-e^{-t\ln(1+i)}/\ln(1+i)$.

Evaluate at the interval endpoints.
The result is $(v^{k-1}-v^k)/\ln(1+i)$, as stated. The formula averages the continuously varying discount factor across the year.

This derivation assumes $i>0$. At $i=0$, every discount factor equals 1 and the expected factor is 1; that is also the limit of the formula as $i$ tends to zero. If the conditional density is not uniform, replace the integral with the expectation under the actual density.

Worked Example: A Known Exit Date With a Shorter Administrative Delay

Method: identify the actual payment time in each case and discount from that time. A hypothetical member exits at $T=1.5$ years, triggering a fixed benefit of $24,000. The effective annual interest rate is 6%. Compare immediate payment, payment at the end of the exit year, and payment 0.25 years after exit. There is no benefit increase during the delay.

Immediate. The payment time is 1.5, so the present value is $24{,}000(1.06)^{-1.5}$. The discount factor is approximately $0.9163074$, giving $21,991.38.

Year-end. The exit is in year 2, so payment is at time 2. The present value is $24{,}000(1.06)^{-2}=24{,}000/1.1236=$$21,359.91.

Administrative delay. Payment occurs at $1.5+0.25=1.75$. Thus the present value is $24{,}000(1.06)^{-1.75}$. The discount factor is approximately $0.9030561$, giving $21,673.35.

The check is $21{,}991.38>21{,}673.35>21{,}359.91$. The administrative payment is earlier than year-end in this scenario, so its value lies between the immediate and year-end values. All dollar amounts are rounded to the nearest cent; calculations use unrounded discount factors.

Common Mistakes and Practical Checks

A frequent error is to discount every benefit paid “during year 1” by $v$, even when the benefit is paid immediately at a uniformly distributed exit time. That calculation values all exits as if payment occurred at time 1. In Worked Example 1, it gives $0.20(12{,}000)v=\$2{,}307.69$, which is the year-end value, not the immediate-payment value of $2,353.54. The correction is to integrate $v^T$ over the conditional exit-time distribution.

Other checks help prevent timing errors:

Write the payment date explicitly. For an exit at $T=1.4$, the three dates are $1.4$, $2$, and $1.4+h$, respectively.
Keep the benefit basis fixed. If the amount is determined at exit, do not apply interest to the benefit during a delay unless the plan terms say it grows.
Use year-specific discount intervals. Uniform exits in year 2 have discount times between 1 and 2, not between 0 and 1.
Check ordering. For a fixed benefit and positive interest, a later payment date cannot have a greater present value than an earlier date.
Keep probabilities attached to the right benefit. Each year’s probability multiplies that year’s benefit and its applicable discount factor.

Payment-timing distinctions matter in pension and ancillary-benefit valuation because payment rules can change the value even when eligibility and benefit formulas are unchanged. The same discipline applies whether the exit is a withdrawal, retirement, or another covered event. The focus here is timing; a valuation that further separates exit causes requires the corresponding cause-specific probabilities.

Key takeaway: First identify the exit time, then identify the contractual payment time, and only then discount. Immediate payment uses $v^T$, year-end payment uses $v^{\lceil T\rceil}$, and a fixed administrative delay uses $v^{T+h}$ when the benefit does not grow during the delay. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use an effective annual rate and state the payment time before calculating. Round dollar answers to the nearest cent.

A hypothetical plan pays $5,000 exactly at time 1. The effective annual interest rate is 4%. What is the present value, whether the payment is described as immediate at that exact exit time or at year-end? (Answer: $4,807.69.)
A $8,000 benefit is paid immediately on exit. Exit is uniform within year 1, conditional on exit in that year, and the effective annual interest rate is 4%. Find the expected present value conditional on exit. (Answer: $8{,}000(1-1/1.04)/\ln(1.04)=\$7{,}845.15$.)
A hypothetical benefit of $5,000 is paid at the end of the exit year. Exit probabilities are 0.20 in year 1 and 0.30 in year 2; there is no benefit for other outcomes in the two-year period. At 4% effective annual interest, find the expected present value. (Answer: $1{,}000/1.04+1{,}500/(1.04)^2=\$2{,}348.37$.)
A fixed $9,000 benefit is paid immediately on exit. Conditional on exit in year 2, exit is uniform within that year. The effective annual interest rate is 5%. Find the expected present value conditional on exit. (Answer: $9{,}000(1/1.05)(1-1/1.05)/\ln(1.05)=\$8{,}365.69$.)
A $12,000 benefit is fixed at exit. Exit probabilities are 0.10 in year 1 and 0.20 in year 2, with uniform exit timing within each year. All benefits are paid 0.25 years after exit. At 4% effective annual interest, find the expected present value. (Answer: $[1{,}200(0.980643526)+2{,}400(0.942926467)](1.04)^{-0.25}=\$3{,}406.23$.)

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