Maturity and Termination Profits in Signatures

Why Termination Profits Need Careful Placement

A policy can generate profit while it is in force and also generate a profit or loss when it ends. A surrender payment, death benefit, or maturity amount belongs to the duration in which the termination occurs. The key accounting question is whether that amount is an incremental adjustment to ordinary duration profit or whether it is the total profit for the terminating policy in that duration.

A signature entry should include each profit exactly once. If a termination amount is incremental, add it to the ordinary profit for the duration. If it is already the total terminal profit for that policy, do not add it on top of ordinary profit for the same policy. In either case, only policies in force at the start of the duration can contribute to that duration’s experience.

Definition: In this tutorial, the cohort profit signature entry for duration $t$, denoted $\Psi_t$, is the expected aggregate profit attributed to duration $t$ for the cohort being valued. A termination profit is attributed to the duration in which the policy terminates. The cohort entering the next duration consists only of policies that remain in force.

Ordinary Profit and Termination Adjustments

Suppose $N_t$ policies are in force at the start of duration $t$. Let $g_t$ be the ordinary profit per policy for that duration, calculated on the stated basis. Assume each terminating policy experiences one of several mutually exclusive termination events, indexed by $j$, with probability $p_{tj}$ conditional on being in force at the start. Let $c_{tj}$ be the incremental profit adjustment for event $j$, over and above the ordinary profit already counted for that policy.

\Psi_t=N_tg_t+N_t\sum_j p_{tj}c_{tj}.

This form applies when ordinary profit $g_t$ is counted for every policy in force at the start, including policies that terminate during the duration, and each $c_{tj}$ is explicitly an additional adjustment. If termination outcomes are given as total profits instead, use the total amount for terminating policies rather than adding it to their ordinary profit. Either accounting convention is valid when applied consistently.

Important distinction: If a terminal amount $T_{tj}$ is the total profit for a policy terminating through event $j$, and ordinary profit $g_t$ has already been counted for every policy entering the duration, the adjustment is $c_{tj}=T_{tj}-g_t$. Adding $T_{tj}$ to $g_t$ would count the ordinary component twice.

The next duration’s starting cohort is determined by the policies that continue. If the termination events listed are the only exits, the expected number continuing is $N_t(1-\sum_j p_{tj})$. For a cohort model with fractional expected policy counts, this expected number can be carried forward directly. If an actual observed cohort is being followed, use its actual count at the next duration instead.

Why the Cohort Formula Holds

The formula follows from adding the profit contributions across policies and taking expectations. Each of the $N_t$ policies contributes ordinary profit $g_t$, for an ordinary-profit total of $N_tg_t$. For a particular termination event $j$, each policy has probability $p_{tj}$ of experiencing that event and, if it does, contributes an additional amount $c_{tj}$. The expected incremental contribution per starting policy from that event is therefore $p_{tj}c_{tj}$.

E[\text{incremental profit from event }j] =N_t p_{tj}c_{tj}.

Summing over the mutually exclusive termination events and adding the ordinary component gives $\Psi_t=N_tg_t+N_t\sum_j p_{tj}c_{tj}$. This uses linearity of expectation; it does not require the profit amounts from different policies to be independent. The event probabilities must, however, be defined on the same basis and conditional on the same starting in-force group.

Worked Examples

Worked Example: Surrender and Death Adjustments in One Duration

A hypothetical cohort has 120 policies at the start of a one-year duration. Ordinary profit is $5 per policy, including policies that terminate during the year. During the year, 10% surrender and each surrender produces an incremental profit of $3. A further 5% die and each death produces an incremental loss of $20. These termination events are mutually exclusive. Find the signature entry and the number expected to continue.

Method. First count ordinary profit on the 120 policies entering the duration. Then add the expected incremental adjustment for each termination event.

Ordinary profit. The ordinary component is $120(5)=\$600$.

Surrender adjustment. Expected surrenders are $120(0.10)=12$. Their incremental profit is $12(3)=\$36$.

Death adjustment. Expected deaths are $120(0.05)=6$. Their incremental loss is $6(-20)=-\$120$.

Signature entry. Combining the components gives $\Psi_1=600+36-120=\$516$.

Continuing policies. The continuation proportion is $1-0.10-0.05=0.85$, so the expected number continuing is $120(0.85)=102$.

Check and common mistake. The 12 surrenders and 6 deaths total 18 exits, leaving $120-18=102$, consistent with the continuation calculation. A common error is to add the surrender profit but omit the death loss; that would give $636. The signed death adjustment is negative, so the correct entry is $516.

Worked Example: Surrender in Year One and Maturity in Year Two

A hypothetical cohort of 80 policies has two annual durations. In duration 1, ordinary profit is $4 per policy. At the end of that duration, 25% surrender, each with an incremental loss of $8; all other policies continue. In duration 2, ordinary profit is $6 per policy, and all policies in force at its start mature at the end, each producing an incremental profit of $5. Profits emerge at the ends of the respective durations. At an effective annual interest rate of 5%, find the two signature entries and their time-zero present value.

Duration 1 ordinary profit. All 80 policies entering duration 1 contribute ordinary profit, including those that surrender: $80(4)=\$320$.

Duration 1 surrender adjustment. Expected surrenders are $80(0.25)=20$. Their incremental loss is $20(-8)=-\$160$. Thus $\Psi_1=320-160=\$160$.

Duration 2 starting cohort. The number continuing is $80(1-0.25)=60$.

Duration 2 profit. Ordinary profit is $60(6)=\$360$. The maturity adjustment is $60(5)=\$300$. Therefore $\Psi_2=360+300=\$660$.

Present value. With end-of-year profit timing, the time-zero value is \[ 160v+660v^2,\qquad v=\frac{1}{1.05}. \] The two discounted terms are $160/1.05=\$152.380952\ldots$ and $660/(1.05)^2=660/1.1025=\$598.639455\ldots$. Their sum is $\$751.020408\ldots$, or $751.02 to the nearest cent.

Check and common mistake. Duration 1 leaves 60 policies, and the duration 2 ordinary and maturity components both use those 60 policies. Discounting duration 2 by $1.05$, rather than by $(1.05)^2$, would place its end-of-year-2 profit one year too early. The correctly timed present value is $751.02.

Worked Example: Multiple Terminations Before Maturity

A hypothetical cohort of 200 policies enters duration 1. Ordinary profit is $7 per policy. At the end of that duration, 15% surrender with an incremental profit of $12 each, and 5% die with an incremental loss of $40 each. All others continue. At the start of duration 2, ordinary profit is $9 per policy. At its end, 20% of the policies then in force surrender with an incremental loss of $5 each, 5% die with an incremental loss of $30 each, and all remaining policies mature with an incremental profit of $18 each. Events in each duration are mutually exclusive. Calculate both signature entries and the time-zero present value at an effective annual interest rate of 4%, assuming profits emerge at each year-end.

Duration 1 ordinary profit. The 200 starting policies produce $200(7)=\$1{,}400$.

Duration 1 termination adjustments. Expected surrenders are $200(0.15)=30$, giving $30(12)=\$360$. Expected deaths are $200(0.05)=10$, giving $10(-40)=-\$400$. Hence $\Psi_1=1{,}400+360-400=\$1{,}360$.

Policies entering duration 2. The continuation proportion is $1-0.15-0.05=0.80$. Thus $200(0.80)=160$ policies enter duration 2.

Duration 2 ordinary profit. The ordinary component is $160(9)=\$1{,}440$.

Duration 2 termination adjustments. There are $160(0.20)=32$ surrenders, contributing $32(-5)=-\$160$, and $160(0.05)=8$ deaths, contributing $8(-30)=-\$240$. The remaining $160(1-0.20-0.05)=120$ policies mature, contributing $120(18)=\$2{,}160$. Therefore $\Psi_2=1{,}440-160-240+2{,}160=\$3{,}200$.

Present value. The duration 1 and duration 2 entries are discounted one and two years, respectively: \[ \frac{1{,}360}{1.04}+\frac{3{,}200}{(1.04)^2} =1{,}307.692307\ldots+2{,}958.579881\ldots =\$4{,}266.272189\ldots. \] The time-zero present value is $4,266.27 to the nearest cent.

Check and common mistake. The duration 2 outcome counts reconcile: $32+8+120=160$. A frequent error is to apply the duration 2 percentages to the original 200 policies. They are conditional on being in force at the start of duration 2, so the correct base is 160.

When a Termination Amount Is a Total Profit

A problem may state that a maturity profit is a total amount for a policy at maturity, rather than an additional bonus. In that case, separate the terminating policies from the policies that do not terminate. For example, if ordinary profit is counted for all policies entering a duration and a terminating policy’s total profit is $T$, the termination adjustment to ordinary profit is $T-g_t$, not $T$.

This adjustment method gives the same total as directly adding ordinary profit for nonterminating policies and total terminal profit for terminating policies. The choice is an accounting convenience; the result must not change. Make the interpretation explicit before calculating, especially when a problem uses words such as “total profit,” “additional profit,” or “incremental loss.”

Common Errors and a Practical Sequence

Counting a terminal amount twice. If the stated maturity amount is already total profit for the policy, do not add it on top of ordinary profit for that same policy. Replace the ordinary component for that terminating policy, or use an adjustment equal to total terminal profit minus ordinary profit.
Using the wrong cohort size. Apply each duration’s termination probabilities to policies in force at that duration’s start, not automatically to the original cohort.
Dropping the sign on a loss. A termination loss is a negative adjustment. Keep its sign when multiplying by the expected number of terminations.
Mixing unconditional and conditional quantities. If an event probability is conditional on entering the duration, combine it with the in-force count at that duration. Do not apply it again to a count that already reflects the event probability.
Discounting to the wrong time. Under year-end profit emergence, duration $t$ is discounted by $v^t$. A different timing convention requires the discount exponent to match the stated emergence time.

Key takeaway: For each duration, identify the starting in-force cohort, count ordinary profit once, and then include each termination outcome once in that duration. Carry only continuing policies forward, and discount the completed signature entries according to their stated timing.

Check Your Understanding

For each problem, distinguish ordinary profit from any stated termination adjustment and use the correct cohort at each duration.

A hypothetical cohort of 150 policies enters a duration. Ordinary profit is $4 per policy, including terminating policies. Twelve percent surrender with an incremental profit of $6 each, and 4% die with an incremental loss of $25 each. Find the signature entry. (Answer: $558)
A hypothetical cohort of 100 policies earns ordinary profit of $3 per policy in a duration. Twenty percent surrender at its end, each with an incremental loss of $10; all others continue. Find the entry and the expected number continuing. (Answer: entry $100; 80 continue)
Fifty policies enter a final duration. Ordinary profit is $8 per policy. Ten percent die with an incremental loss of $30 each, and all others mature with an incremental profit of $14 each. Find the signature entry. (Answer: $880)
A hypothetical cohort of 60 policies earns ordinary profit of $5 in duration 1. At its end, 25% surrender with an incremental profit of $4 each, and the rest continue. In duration 2, each continuing policy earns ordinary profit of $7 and matures with an incremental profit of $10. With end-of-year profit emergence and an effective annual interest rate of 5%, find both entries and their time-zero present value. (Answer: signature $(\$360,\ \$765)$; present value $1,036.73 to the nearest cent)
Forty policies enter a duration and each has ordinary profit of $6. Twenty-five percent mature at the end, with a stated total profit of $20 each, inclusive of ordinary profit. The other policies receive only ordinary profit. Find the correct signature entry. (Answer: $380)

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