Benefits Paid at Exit Versus Year-End

Why Payment Timing Changes the Value

A decrement benefit can be triggered by an exit during a year but paid on different schedules. A plan might pay a lump sum as soon as the member exits, or defer payment until the end of the year in which the exit occurs. The benefit amount and the probability of each exit can be identical in both designs, while the present values differ because the payment dates differ.

The comparison requires a clear within-year timing assumption. In this tutorial, forces of decrement are constant during each year. A member who exits partway through a year receives an immediate benefit at the exit time in the first design; in the second, payment is at the end of that year. Interest is consistent with the effective annual rate: if that rate is $i$, the force of interest is $\delta=\ln(1+i)$, so discounting for $t$ years uses $e^{-\delta t}$. This is the constant-force timing model; without a specified assumption about when decrements occur within the year, an immediate-payment value cannot in general be obtained from annual decrement probabilities alone.

Definition: A benefit paid immediately at exit is paid at the time the covered decrement occurs. A benefit paid at year-end is paid at the end of the calendar or policy year in which that decrement occurs, regardless of the exact exit time within that year. Both are contingent on the same decrement; only the payment date changes.

Annual Probabilities and Payment Factors

Consider one year during which the member is active at the start. Suppose there are $r$ mutually exclusive decrements with constant forces $\mu_1,\ldots,\mu_r$. The total force is $\mu=\sum_{j=1}^r\mu_j$. Under these assumptions, active survival to time $t$ within the year is $e^{-\mu t}$, and the probability density of decrement $j$ at time $t$ is $e^{-\mu t}\mu_j$. Integrating that density over the year gives the cause-specific annual decrement probability:

q_j=\int_0^1 e^{-\mu t}\mu_j\,dt =\frac{\mu_j}{\mu}(1-e^{-\mu}), \qquad \mu>0.

If a benefit of amount $B$ is paid at the end of the year, its expected present value at the start of the year is $Bq_jv$, where $v=1/(1+i)$. If it is paid immediately, an exit at time $t$ is discounted for $t$ years, so its expected present value is the integral of the benefit times the exit density times the discount factor.

\operatorname{EPV}_{\text{exit},j} =B\int_0^1 e^{-\mu t}\mu_j e^{-\delta t}\,dt =B\frac{\mu_j}{\mu+\delta}\left(1-e^{-(\mu+\delta)}\right).

These expressions apply to a year with constant cause forces and a constant benefit amount. For multiple years, apply the appropriate one-year factor in each year and discount from the start of the contract to the start of that year. If the forces or benefit amounts vary by year, use that year’s values; do not assume a single unchanged force or amount without support from the model.

Why the Immediate-Payment Formula Holds

The immediate-payment formula follows by combining three ingredients at each possible exit time. First, the member must remain active to time $t$, which has probability $e^{-\mu t}$. Second, the cause-$j$ exit density at that time is the active-survival probability multiplied by its force, $e^{-\mu t}\mu_j$. Third, a payment at time $t$ has present-value factor $e^{-\delta t}$. Multiplying the density by the discount factor and integrating over all possible exit times gives:

B\int_0^1 \underbrace{e^{-\mu t}\mu_j}_{\text{cause-}j\text{ exit density}} \underbrace{e^{-\delta t}}_{\text{discount factor}}\,dt =B\mu_j\int_0^1 e^{-(\mu+\delta)t}\,dt =B\frac{\mu_j}{\mu+\delta}\left(1-e^{-(\mu+\delta)}\right).

The year-end expression uses the same probability of cause $j$, but every such payment is discounted for a full year. This is why the cause probability is calculated first and then multiplied by $v$; it is not the immediate-payment integral.

Worked Comparisons

Worked Example: Three Years of Death Benefits

A hypothetical plan pays $10,000 on death during each of the next three years, with no payment on withdrawal. In every year, the constant death force is $0.40$ and the constant withdrawal force is $0.2931471806$, so the total force is $\ln 2=0.6931471806$. The annual effective interest rate is $5\%$. Find the EPV if the benefit is paid immediately on death and if it is paid at the end of the death year.

Method and assumptions. The forces are constant within each year and identical across years. The death benefit is paid only if death occurs before the end of the three-year term. Set $v=1/1.05=0.9523809524$ and $\delta=\ln(1.05)=0.0487901642$.

Step 1: Find the annual death probability. Since the total force is $\ln 2$, one-year active survival is $e^{-\ln 2}=0.5$. Thus:

q_{\text{death}} =\frac{0.40}{0.6931471806}(1-0.5) =0.2885390082.

Step 2: Value the year-end benefits. The three payments, if triggered by death, are made at times 1, 2, or 3. The EPV is:

10{,}000(0.2885390082)(v+v^2+v^3) =10{,}000(0.2885390082)(2.723248034) =\$7{,}857.63.

Step 3: Value the immediate benefits. The one-year immediate-payment factor per dollar is:

\frac{0.40}{0.6931471806+0.0487901642} \left(1-e^{-(0.6931471806+0.0487901642)}\right) =0.282400948.

The three years start at times 0, 1, and 2, so their discount factors are $1,v,v^2$. Therefore:

10{,}000(0.282400948)(1+v+v^2) =10{,}000(0.282400948)(2.859410431) =\$8{,}075.00.

Check and common mistake. The immediate value exceeds the year-end value by $217.37, as expected with positive interest and the same death events and benefit amounts. The death force $0.40$ is not itself the annual death probability: the competing withdrawal force reduces the chance of death before year-end. Using $0.40$ as the annual probability would overstate both values.

Worked Example: One-Year Immediate Benefit

A hypothetical contract pays $5,000 immediately on a specified cause of exit. The constant force for that cause is $0.40$, the total decrement force is $1.00$, and the annual effective interest rate is $5\%$. Find the one-year EPV.

Method. The other decrements are competing causes, so the specified-cause probability density at time $t$ is $0.40e^{-t}$. With $\delta=\ln(1.05)=0.0487901642$, discount an exit at time $t$ by $e^{-\delta t}$:

\begin{aligned} \operatorname{EPV} &=5{,}000\int_0^1 0.40e^{-t}e^{-\delta t}\,dt\\ &=5{,}000\frac{0.40}{1+\delta} \left(1-e^{-(1+\delta)}\right)\\ &=5{,}000(0.24776686) =\$1{,}238.83. \end{aligned}

Check and common mistake. The annual probability of the specified cause is $0.40(1-e^{-1})=0.25284822$. Discounting that probability for a full year would give approximately $5{,}000(0.25284822)/1.05=\$1{,}204.04$, which is a year-end value, not the requested immediate-payment value. The earlier payment has the higher EPV.

Worked Example: Different Causes, Different Payment Dates

A hypothetical one-year plan pays $10,000 immediately on cause A and $4,000 at year-end on cause B. The constant forces are $0.10$ for A and $0.20$ for B, and $i=4\%$. Find the combined one-year EPV.

Method. The total force is $0.30$, and $\delta=\ln(1.04)=0.0392207132$. Value each cause using its own payment timing.

For cause A, use the immediate-payment factor:

\begin{aligned} \operatorname{EPV}_A &=10{,}000\frac{0.10}{0.30+\ln(1.04)} \left(1-e^{-(0.30+\ln(1.04))}\right)\\ &=10{,}000(0.08480461) =\$848.05. \end{aligned}

For cause B, first find its annual decrement probability and discount it one year:

\begin{aligned} q_B&=\frac{0.20}{0.30}(1-e^{-0.30})=0.17278785,\\ \operatorname{EPV}_B&=4{,}000q_Bv =4{,}000(0.17278785)\frac{1}{1.04} =\$664.57. \end{aligned}

The combined value is $\$848.05+\$664.57=\$1,512.62$ when adding the displayed rounded amounts; using unrounded component values before final rounding gives $\$1,512.61$. The reported EPV is therefore $1,512.61. A common mistake is to apply one timing factor to both benefits. Cause A is discounted to the exit time; cause B is discounted to year-end.

Interpreting the Comparison

For the same decrement event and benefit amount, the payment dates determine the direction of the comparison. If $i>0$, then $e^{-\delta t}\geq e^{-\delta}$ for $0\leq t\leq1$, so a payment made at time $t$ is worth at least as much as that payment deferred to year-end. With positive probability of an exit before year-end, the immediate-payment EPV is strictly greater. If $i=0$, the two values are equal because discounting does not depend on the payment date.

This conclusion assumes the benefit amount is unchanged by the timing choice and there are no additional conditions tied to payment. It does not mean that every immediate-payment value can be inferred from an annual decrement table alone: the within-year distribution of exits is needed. Under the constant-force model used here, that distribution is specified by the forces. The earlier material on discrete cause-contingent EPV and continuous-time exit benefits supplies the underlying valuation ideas; the key task in this comparison is to align each cash flow with its actual payment date.

Key takeaway: For a year-end cause-contingent benefit, multiply the cause-specific annual probability by $v$. For an immediate benefit under constant forces, integrate the cause-specific exit density against continuous discounting. Keep the cause probability, within-year timing assumption, and payment date distinct.

Check Your Understanding

Use the stated constant-force assumptions for immediate payments. Round dollar answers to the nearest cent and probabilities to four decimals.

During one year, two competing causes have constant forces $0.12$ and $0.18$. What is the annual probability of the first cause? (Answer: $0.12/0.30(1-e^{-0.30})=0.1037$.)
A benefit of $8,000 is paid at year-end if a specified decrement occurs, and its annual decrement probability is $0.08$. If $i=4\%$, find its EPV. (Answer: $615.38.)
A $7,000 benefit is paid immediately on a cause with force $0.25$; the total decrement force is $0.50$, and $i=10\%$. Find the one-year EPV. (Answer: $7000\frac{0.25}{0.50+\ln(1.10)}[1-e^{-(0.50+\ln(1.10))}]=\$1,318.75$.)
In a one-year model, death and withdrawal forces are $0.20$ and $0.30$. A $12,000 death benefit is paid immediately, while a $2,000 withdrawal benefit is paid at year-end. If $i=4\%$, find the combined EPV. (Answer: $12000\frac{0.20}{0.50+\ln(1.04)}[1-e^{-(0.50+\ln(1.04))}]+2000\frac{0.30}{0.50}(1-e^{-0.50})/1.04=\$2,309.11$.)

What You'll Learn