Peak Profit, Loss Durations, and Profit Emergence

Why the Largest Duration Matters

A profit signature records how discounted expected profit is distributed across policy durations. Its largest positive entry identifies the duration contributing the greatest discounted gain; its most negative entry identifies the duration contributing the greatest discounted loss. These are useful diagnostics: a product can have gains in many years and still have one particularly loss-making duration, or a single duration may account for a large share of its positive profit emergence.

A peak in the signature is not necessarily a peak in the profit earned on each policy that is in force. The signature also reflects how many policies are expected to remain in force and how strongly each duration’s profit is discounted. Keeping these quantities separate is essential when interpreting the pattern.

Definition: Adopt an end-of-year profit convention for an $n$-duration contract. Let $\pi_k$ be expected net profit during duration $k$, conditional on the policy being in force at the start of that duration. Let $p_k$ be the probability of being in force at that start, and let $v=1/(1+i)$ for effective annual discount rate $i$. The discounted duration-$k$ profit signature entry is $z_k=v^k p_k\pi_k$. The peak gain duration is a duration with the largest positive $z_k$; the peak loss duration is a duration with the most negative $z_k$, if any entries are negative.

For a single-life contract with annual death probabilities $q_1,q_2,\ldots$, and no other way to leave the in-force population, the duration-start weights are $p_1=1$ and $p_k=\prod_{j=1}^{k-1}(1-q_j)$ for $k>1$. Thus duration $k$'s signature entry depends on three factors: the conditional expected profit, the probability the policy reaches that duration in force, and the discount factor. If the model includes lapses or other exits, the in-force probability must reflect those exits as well.

The definition depends on the timing convention. Here profits are discounted from the end of each duration, so duration $k$ uses $v^k$. A midyear convention would use a different discount timing. Do not change the timing factor without changing the convention consistently.

Why the Signature Weights Each Duration

The signature entry follows from conditioning on whether a policy is in force at the start of the duration. If it is not in force, it contributes no duration-$k$ profit. If it is in force, its expected duration-$k$ profit is $\pi_k$. Therefore the unconditional expected profit attributed to that duration is $p_k\pi_k$. Discounting this amount from the end of duration $k$ to time zero gives $v^kp_k\pi_k$.

E[\text{duration-}k\text{ profit}] = P(\text{in force at start of }k)\, E[\text{duration-}k\text{ profit}\mid \text{in force at start of }k] = p_k\pi_k, \qquad z_k=v^kp_k\pi_k.

This argument does not require the duration profit to be certain for a policy that is in force. The value $\pi_k$ may itself be an expectation over death, expenses, or other contingent cash flows within the year. What matters is that its conditioning event and timing match the stated definition. The formula also shows why a conditional profit table alone is not enough to determine the peak of the signature.

Worked Examples: Finding and Interpreting Peaks

Worked Example: Locating the Largest Gain and Loss

A hypothetical four-year protection contract has annual death probabilities $q_1=0.10$, $q_2=0.20$, $q_3=0.25$, and $q_4=0.40$, conditional on being alive at the start of each year. Assume death is the only exit, with the annual death indicator conditionally Bernoulli with the stated $q_k$. The expected net profits conditional on being in force at the start of the year are $\pi_1=-\$80$, $\pi_2=\$40$, $\pi_3=\$100$, and $\pi_4=\$60$. The effective annual interest rate is $i=5\%$, and profits emerge at year-end. Find the signature peak gain and peak loss.

Method. Calculate the probability of reaching each duration, apply the end-of-year discount factor, and compare the resulting signed entries. The conditional death distribution is specified by the four annual Bernoulli probabilities; the supplied $\pi_k$ values already incorporate the expected net profit within each duration.

The duration-start probabilities are $p_1=1$, $p_2=1(1-0.10)=0.90$, $p_3=0.90(1-0.20)=0.72$, and $p_4=0.72(1-0.25)=0.54$. The fourth-year death probability affects exits during year four, not the probability of being in force at its start.

With $v=1/1.05=0.952381$, the four signature entries are:

\begin{aligned} z_1&=(0.952381)(1)(-\$80)=-\$76.19,\\ z_2&=(0.952381)^2(0.90)(\$40)=\$32.65,\\ z_3&=(0.952381)^3(0.72)(\$100)=\$62.20,\\ z_4&=(0.952381)^4(0.54)(\$60)=\$26.66. \end{aligned}

Rounded to the nearest cent, duration 3 is the peak gain duration at $\$62.20$, and duration 1 is the peak loss duration at $-\$76.19$. The peak loss is the most negative entry, not the entry with the largest absolute value after ignoring its sign. The calculation also verifies the role of the fourth-year survival weight: only 54% of the initial cohort is expected to be in force at the start of that year.

Common mistake avoided: choosing duration 3 as both the largest gain and loss because its conditional profit has the greatest positive amount, or using $q_4$ to reduce $p_4$. Peak loss is determined from the signed signature entries, and $p_4$ includes survival only through the first three years.

Worked Example: Conditional Profit Peak Versus Signature Peak

A hypothetical three-year savings contract has conditional expected annual profits of $\$30$, $\$50$, and $\$80$ at durations 1, 2, and 3. Annual death probabilities are $q_1=0.20$, $q_2=0.50$, and $q_3=0.75$, conditional on being alive at the start of each year; death is the only exit and follows the corresponding Bernoulli probabilities. The effective annual interest rate is $4\%$, and profits emerge at year-end. Compare the duration with the highest conditional profit to the duration with the highest signature entry.

The conditional profit increases through duration 3, so duration 3 has the highest conditional profit, $\$80$. But the in-force probabilities at the start of the three durations are $p_1=1$, $p_2=1-0.20=0.80$, and $p_3=0.80(1-0.50)=0.40$. The probability of death during year 3 does not enter $p_3$.

Here $v=1/1.04=0.961538$, so:

\begin{aligned} z_1&=(0.961538)(1)(\$30)=\$28.85,\\ z_2&=(0.961538)^2(0.80)(\$50)=\$36.98,\\ z_3&=(0.961538)^3(0.40)(\$80)=\$28.45. \end{aligned}

Rounded to the nearest cent, the signature peak is at duration 2, with $\$36.98$, even though the highest conditional profit occurs at duration 3. The later profit is larger per in-force policy, but fewer policies are expected to reach that duration and its profit is discounted for longer.

Common mistake avoided: identifying the signature peak by inspecting only the conditional profit column. A peak of $\pi_k$ and a peak of $z_k$ answer different questions.

Worked Example: Reading an Early-Loss, Later-Gain Pattern

A hypothetical five-year contract has a constant annual death probability of $q=0.05$, conditional on being alive at the start of each year. Death is the only exit, with annual death indicators conditionally Bernoulli with this probability. Its conditional expected annual net profits are $-\$100$, $-\$30$, $\$50$, $\$90$, and $\$70$ by duration. Use an effective annual interest rate of $3\%$, with profits at year-end. Identify the most negative and largest positive signature entries, then interpret their locations.

The probability of reaching each duration in force is $p_1=1$, $p_2=0.95$, $p_3=0.95^2=0.9025$, $p_4=0.95^3=0.857375$, and $p_5=0.95^4=0.81450625$. The discount factor is $v=1/1.03=0.970874$. Applying $z_k=v^kp_k\pi_k$ gives:

\begin{aligned} z_1&=(0.970874)(1)(-\$100)=-\$97.09,\\ z_2&=(0.970874)^2(0.95)(-\$30)=-\$26.86,\\ z_3&=(0.970874)^3(0.9025)(\$50)=\$41.30,\\ z_4&=(0.970874)^4(0.857375)(\$90)=\$68.56,\\ z_5&=(0.970874)^5(0.81450625)(\$70)=\$49.18. \end{aligned}

Rounded to the nearest cent, duration 1 is the peak loss at $-\$97.09$, and duration 4 is the peak gain at $\$68.56$. The pattern shows a large initial loss followed by a move into positive expected profits, with the greatest discounted gain emerging in the fourth year. The fifth-year conditional profit is positive, but its signature contribution is smaller than year four’s.

This pattern is consistent with early costs or other first-year pressures followed by later profit emergence, but the signature alone does not establish the cause. To explain why a duration is loss-making, an analyst must inspect the cash flows and assumptions that produced its conditional profit.

Common mistake avoided: treating a positive late-duration profit as proof that the earlier loss has been recovered by that point. These calculations compare individual duration entries. They do not calculate cumulative profit or a payback duration.

Interpreting the Shape Without Overclaiming

A peak’s location describes when the greatest single-duration gain or loss is assigned in the chosen signature. It does not, by itself, explain why that entry has its value. For example, a low signature entry could result from a small conditional profit, a low probability of reaching that duration, or a long discount period. A negative entry indicates a negative expected contribution at that duration under the stated model; it does not necessarily mean every policy in force experiences a loss.

A useful diagnostic is to look at both the conditional profits and the signature entries. If $\pi_k$ rises but $z_k$ falls, survival weighting and discounting together outweigh the increase in conditional profit. If the two have their peaks at the same duration, the conditional profit peak is strong enough to remain the largest after those weights are applied. Neither pattern is inherently good or bad: interpretation depends on the product, the assumptions, and the question being asked.

Check the model and timing.
Confirm the profit convention, the in-force definition, the interest rate, and whether profits are discounted from year-end or another specified point.

Build the duration-start weights.
Use survival through earlier durations, and include any other modeled exits where relevant. Do not apply the current duration's death probability to the probability of reaching its start.

Calculate each signed entry.
For every duration, multiply its conditional expected profit by its in-force probability and its discount factor.

Find and describe the extremes.
Select the greatest positive entry and the most negative entry, then describe their duration and what the pattern suggests without treating it as a causal explanation.

The signature entries are components of the overall discounted profit measure, but the peak is only one component. A large positive entry does not show that the contract is profitable overall, and a large loss entry does not show that the contract is unprofitable overall. The total depends on all durations, not just their extremes. Similarly, locating the peak does not determine when earlier losses are offset by later gains; that requires a cumulative analysis.

Common exam errors include mixing conditional and unconditional profits, using the wrong survival probability, applying the discount factor for the wrong duration, and comparing absolute values when asked for the most negative entry. State what is being maximized or minimized and show enough intermediate values to make the weighting and timing transparent. This diagnostic approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: A duration’s signature entry is its conditional expected profit multiplied by the probability of being in force at the duration’s start and discounted according to the stated timing convention. Locate peaks from these signed entries, not from the conditional profit amounts alone.

Check Your Understanding

For each problem, use the stated end-of-year convention and identify the requested signature peak or peaks.

A three-year hypothetical contract has $q_1=q_2=q_3=0.10$, with death the only exit and annual death indicators conditionally Bernoulli. Conditional expected profits are $-\$20,\ \$40,\ \$50$, and $i=5\%$. Which duration has the peak gain, and which has the peak loss? (Answer: Peak gain, duration 3; peak loss, duration 1.)
A hypothetical three-year contract has $q_1=0.25$ and $q_2=0.40$, with death the only exit and the annual death indicator conditionally Bernoulli. Conditional expected profits are $\$60,-\$100,\$80$, and $i=0\%$. Find the largest signature gain and the most negative entry. (Answer: Largest gain, duration 1 at $\$60.00$; most negative entry, duration 2 at $-\$75.00$.)
A hypothetical three-year contract has constant annual death probability $q=0.20$, with death the only exit and annual death indicators conditionally Bernoulli. Conditional expected profits are $\$40,\$60,\$100$, and $i=5\%$. Calculate the three entries and identify the peak gain. (Answer: $z_1=\$38.10,\ z_2=\$43.54,\ z_3=\$55.29$; peak gain, duration 3.)
A hypothetical three-year product has $q_1=0.10,\ q_2=0.60,\ q_3=0.50$, with death the only exit and annual death indicators conditionally Bernoulli. Conditional expected profits are $\$20,\$50,\$90$, and $i=0\%$. Which duration has the highest conditional profit, and which has the highest signature entry? (Answer: Highest conditional profit, duration 3 at $\$90.00$; highest signature entry, duration 2 at $\$45.00$.)
A hypothetical three-year contract has constant annual death probability $q=0.05$, with death the only exit and annual death indicators conditionally Bernoulli. Conditional expected profits are $-\$80,-\$50,\$10$, and $i=0\%$. Identify the peak loss and the largest positive entry. (Answer: Peak loss, duration 1 at $-\$80.00$; largest positive entry, duration 3 at $\$9.03$.)

What You'll Learn