Payments at Exit Versus Payments at Year-End

Why Payment Timing Matters

A cause-contingent benefit can be defined by the event that triggers it and by the time it is paid. A benefit triggered by withdrawal, for example, might be paid immediately when withdrawal occurs or at the end of the year in which it occurs. The trigger is the same, but the payment dates differ, so their actuarial values need not be equal.

For a payment at the end of the exit year, an annual decrement table supplies the probability of exit from each cause during that year. To value a payment made at the exit itself, the annual probabilities are not enough on their own: we also need an assumption about when within the year each exit occurs. This tutorial compares the two timings under the uniform distribution of decrements (UDD) and constant-force assumptions. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Suppose a life or member is in the decrement group at age $x$ at the start of a year. Let ${}_1q_x^{(j)}$ be the probability of exit from cause $j$ during that year, and let ${}_1p_x^\tau$ be the probability of remaining in the group through the year. A cause-contingent benefit paid at exit is paid at the actual exit time if the exit is from the specified cause. A cause-contingent benefit paid at year-end is paid at time 1 if that cause occurred during the year. The cause probabilities satisfy $\sum_j {}_1q_x^{(j)}+{}_1p_x^\tau=1$.

Let the benefit for cause $j$ be $b_j$, payable once, and let $i$ be the effective annual interest rate, with $v=1/(1+i)$. We assume the benefits are nonnegative and the exit occurs no later than the end of the year. A year-end payment has expected present value $b_jv\,{}_1q_x^{(j)}$. An immediate payment requires the assumed within-year exit-time distribution as well as the annual cause probability.

UDD: Integrating Over the Exit Time

Under UDD for multiple decrements, the probability of a cause-$j$ exit by fractional time $t$, for $0\leq t\leq1$, is $t\,{}_1q_x^{(j)}$. Thus the unconditional density of a cause-$j$ exit time within the year is constant at ${}_1q_x^{(j)}$. The value per starting member of a benefit $b_j$ paid at that exit is therefore

b_j\,{}_1q_x^{(j)}\int_0^1 v^t\,dt = b_j\,{}_1q_x^{(j)}\frac{1-v}{\ln(1+i)}, \qquad i>0.

The integral discounts each possible payment time separately. When $i=0$, its value is 1, so the immediate-payment value is $b_j{}_1q_x^{(j)}$. For comparison, the year-end value discounts every payment by $v$, giving $b_jv{}_1q_x^{(j)}$.

Result under UDD: For nonnegative benefits and $i>0$, an immediate payment has greater expected present value than the corresponding year-end payment, provided the cause has positive probability. The difference for cause $j$ is $b_j{}_1q_x^{(j)}\left((1-v)/\ln(1+i)-v\right)$. At zero interest, the two values are equal.

Worked Example: UDD With a Certain Exit During the Year

In a hypothetical closed group, every member exits from a specified cause during the year. Under UDD, each exit time is uniform on $[0,1]$. A benefit of $1,800 is paid either immediately at exit or at year-end. The effective annual interest rate is 5%. Find both present values and their difference.

Method. Use the UDD integral for the immediate value and discount the year-end payment by $v$. Here ${}_1q_x^{(j)}=1$, $v=1/1.05=0.9523809524$, and $\ln(1.05)=0.0487901642$.

The immediate-payment factor is $\left(1-v\right)/\ln(1.05)=0.0476190476/0.0487901642=0.9759969$. Thus the immediate value is $1800(0.9759969)=\$1,756.79$, rounded to the nearest cent.

The year-end value is $1800v=1800/1.05=\$1,714.29$, rounded to the nearest cent. Subtracting the unrounded values gives $1800(0.9759969-0.9523809524)=\$42.5087$, or $42.51 to the nearest cent.

The check is that $v^t\geq v$ for $0\leq t\leq1$ when $i>0$, so the immediate value must exceed the year-end value. The common mistake is to subtract the separately rounded displayed values: $\$1,756.79-\$1,714.29=\$42.50$. The correct difference is obtained from unrounded values and rounds to $42.51.

Several Causes in One Year

If different causes have different benefits, calculate the value for each cause and add the results. Under UDD, the same timing factor applies to each cause because the cause-specific exit probability is spread uniformly across the year. This does not mean the causes have equal probabilities; each cause retains its own annual probability and benefit amount.

Worked Example: Two Cause-Contingent Benefits Under UDD

A hypothetical contract covers a member who is in the decrement group at the start of a year. The annual probabilities are ${}_1q_x^{(A)}=0.12$, ${}_1q_x^{(B)}=0.08$, and ${}_1p_x^\tau=0.80$. A cause-A exit triggers $10,000 and a cause-B exit triggers $5,000. Assume UDD and an effective annual interest rate of 4%. Compare the expected present values when benefits are paid at exit and at year-end.

Check the decrement probabilities. They sum to $0.12+0.08+0.80=1.00$, as required. The total cause-contingent benefit amount weighted by annual exit probabilities is $10{,}000(0.12)+5{,}000(0.08)=1{,}200+400=\$1{,}600$.

For immediate payments, the UDD timing factor is $(1-1/1.04)/\ln(1.04)=0.0384615385/0.0392207132=0.980645$, to six decimal places. The expected present value is therefore $1{,}600(0.980645)=\$1,569.03$, to the nearest cent.

For year-end payments, $v=1/1.04=0.96153846$, so the expected present value is $1{,}600(0.96153846)=\$1,538.46$. Using unrounded factors, the immediate value exceeds the year-end value by $\$30.57$, to the nearest cent.

The method is to aggregate the cause-weighted benefits first because the same UDD timing factor applies to both causes. A common mistake is to use the total decrement probability, $0.20$, with one benefit amount; that loses the distinction between the $10,000 and $5,000 benefits.

Constant Force Within the Year

Under the constant-force assumption, each cause has a constant force $\mu_j$ throughout the year, and the total force is $\mu=\sum_j\mu_j$. The probability of remaining in the decrement group to time $t$ is $e^{-\mu t}$. The density of an exit from cause $j$ at time $t$ is $\mu_j e^{-\mu t}$: the member must remain in the group until $t$, then exit from cause $j$.

The annual cause probability is ${}_1q_x^{(j)}=(\mu_j/\mu)(1-e^{-\mu})$ when $\mu>0$. For an immediate benefit, multiply the cause-specific exit density by the discount factor $e^{-\delta t}$, where $\delta=\ln(1+i)$, and integrate over the year:

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

This formula applies when the cause-specific forces are constant over the year and $\mu+\delta\ne0$; if $\mu+\delta=0$, the integral is evaluated directly as $b_j\mu_j$. For year-end payment, the value remains $b_jv\,{}_1q_x^{(j)}$. Do not substitute the UDD timing factor into a constant-force calculation: the distribution of exit times is different.

Worked Example: Two Constant Cause Forces

A hypothetical member has constant annual cause forces $\mu_A=\ln(1.25)/3$ and $\mu_B=2\ln(1.25)/3$ during the year. A cause-A exit triggers $9,000 and a cause-B exit triggers $3,000. The effective annual interest rate is 5%. Compare immediate-payment and year-end values per member at the start of the year.

Method. The total force is $\mu=\ln(1.25)=0.22314355$. Thus the probability of remaining in the group for the year is $e^{-\mu}=0.8$, and the total probability of exit is $0.2$. The cause shares are one-third and two-thirds, so ${}_1q_x^{(A)}=0.2/3=1/15$ and ${}_1q_x^{(B)}=0.4/3=2/15$. These probabilities plus survival sum to $1/15+2/15+0.8=1$.

For year-end benefits, the probability-weighted amount is $9{,}000(1/15)+3{,}000(2/15)=600+400=\$1{,}000$. With $v=1/1.05=20/21$, the year-end value is $1{,}000(20/21)=\$952.38$.

For immediate benefits, $\delta=\ln(1.05)=0.04879016$, so $\mu+\delta=\ln(1.3125)=0.27193372$. The combined weighted force in the integral is $9{,}000\mu_A+3{,}000\mu_B=5{,}000\ln(1.25)$. Also, $1-e^{-(\mu+\delta)}=1-1/1.3125=5/21$. Therefore the immediate value is $5{,}000\ln(1.25)(5/21)/\ln(1.3125)=\$976.88$, to the nearest cent.

Using the unrounded values, immediate payment exceeds year-end payment by $\$24.50$, to the nearest cent. This is consistent with positive interest: each exit occurs no later than time 1, so discounting to the actual exit time cannot produce a smaller value than discounting the same benefit to year-end.

Derivation and Practical Checks

The key formula follows from the expected present value of a payment at a random time. If a cause-$j$ exit has density $g_j(t)$ over $0\leq t\leq1$, then the value per starting member is $b_j\int_0^1 v^t g_j(t)\,dt$. Under UDD, $g_j(t)={}_1q_x^{(j)}$. Under constant force, $g_j(t)=\mu_j e^{-\mu t}$. In each case the density integrates to the annual probability of the cause:

\int_0^1 g_j(t)\,dt={}_1q_x^{(j)}.

This provides a useful check: if the density is integrated without discounting, it must recover the annual cause probability. A second check is to set $i=0$. Then every discount factor is 1, so the immediate and year-end values must agree, regardless of the assumed timing distribution.

For multiple years, apply the one-year calculation within each year using that year's conditional cause probabilities or forces, and multiply by the probability of remaining in the decrement group to the start of that year. The payment time in year $k$ is $k+t$ for an immediate benefit and $k+1$ for a year-end benefit, so both values must use the same survival-to-year-$k$ and interest bases. This comparison concerns payment timing only; changing the decrement probabilities or the benefit amounts would change the comparison as well.

Identify the payment date.
Decide whether the benefit is paid at the exit time or at the end of the exit year.

State the timing assumption.
Use UDD or constant force only when that is the specified within-year model.

Value each cause separately.
Pair each cause probability or force with its own benefit, then add the cause-specific values.

Check and round last.
Verify cause probabilities and use unrounded values when calculating the difference between the two payment timings.

A frequent exam trap is to take an annual cause probability and simply multiply it by $v$ when the benefit is paid at exit. That calculation values a year-end payment, not an immediate one. Another trap is to assume that “immediate” means no discounting; exits occur at random times within the year, so each payment still has a positive time to discount unless it occurs at time zero. The correct method discounts over the exit-time distribution specified by the model.

Key takeaway: Annual cause probabilities determine how often benefits are triggered; the within-year decrement assumption determines when an immediate benefit is paid. Keep these roles separate, calculate cause-specific values on the same interest basis, and round only after comparing unrounded amounts.

Check Your Understanding

For each problem, calculate the requested values per member at the start of the year. Round dollar amounts to the nearest cent.

Under UDD, a cause has annual probability $0.25$ and triggers a $4,000 benefit. The effective annual interest rate is 4%. Find the immediate-payment value and the year-end value. (Answer: Immediate $980.65; year-end $961.54.)
Under UDD, two causes have annual probabilities $0.10$ and $0.05$, with benefits of $12,000 and $4,000, respectively. The effective annual interest rate is zero. Find the value under either payment timing. (Answer: $1,400.00 under either timing.)
Under constant forces, $\mu_1=0.04$ and $\mu_2=0.06$ per year. The benefits are $5,000 and $2,000, respectively, and the effective annual interest rate is zero. Find the immediate-payment and year-end values. (Answer: $304.52 under either timing.)
Under UDD, cause A has annual probability $0.08$ and a $10,000 benefit; cause B has annual probability $0.12$ and a $4,000 benefit. At an effective annual interest rate of 5%, find both values and their difference. (Answer: Immediate $1,249.28; year-end $1,219.05; immediate exceeds year-end by $30.23.)

What You'll Learn