Scaling Per-Policy Profits To A Cohort

Introduction

A profit entry stated per policy is not yet a cohort result. To estimate the total expected profit for a group, the entry must be matched to the number of policies to which it applies. In particular, a per-policy profit conditional on being in force at the start of a year is multiplied by the expected number in force at that point—not automatically by the number originally issued.

The key is to align three things: the profit basis (conditional or unconditional), the exposure count (original or currently in force), and the time point (usually the start of the policy year). This scaling is simple arithmetic once those conventions are clear, but mismatches can materially overstate or understate a cohort result.

Definition: Let $N_0$ be the number of policies originally issued. Let $p_t$ be the probability, or expected proportion, of those policies in force at the start of policy year $t$. The expected start-of-year exposure is $E_t=N_0p_t$. If $g_t$ is the expected annual profit per policy conditional on being in force at the start of year $t$, the expected cohort profit for that year is $E_tg_t=N_0p_tg_t$.

The profit entry $g_t$ may already reflect premiums, expenses, benefits, investment income, and other items included in the annual profit definition. For this tutorial, treat that entry as given: the task is to scale it to the cohort. In particular, if it is conditional on beginning the year in force, any event probabilities used to calculate that conditional annual entry belong inside $g_t$; the proportion $p_t$ separately accounts for how many original policies are expected to reach the start of that year.

From Original Policies to Start-of-Year Exposure

If an in-force proportion is given directly, multiply it by the original cohort size. If the proportion is instead built from several successive conditional proportions, multiply those proportions to obtain the fraction of the original cohort expected to remain in force. For example, if $0.90$ of issued policies reach the start of year 2 and $0.80$ of those reach the start of year 3, the year-3 fraction is $0.90(0.80)=0.72$.

This calculation concerns exposure at the start of the year. The per-policy annual profit entry can include expected cash flows arising during that year, including event-contingent items. Do not additionally multiply by the probability of surviving the whole year unless the profit entry is specifically defined only for policies that survive to year-end.

Identify the profit basis.
Determine whether the per-policy entry is conditional on being in force at the start of the year or is already unconditional per original policy.

Find the matching exposure.
For a conditional entry, calculate the expected number in force at the stated start-of-year point.

Multiply and preserve the sign.
Multiply exposure by the conditional per-policy profit. Positive entries add to the total; negative entries reduce it.

Check the units and timing.
The result is a cohort profit for one policy year, not a per-policy amount or a profit accumulated across multiple years.

Why the Scaling Formula Holds

The formula follows from adding the profit contributions of individual policies. For policy $j$, let $I_j$ equal 1 if it is in force at the start of the year and 0 otherwise. Let $Z_j$ be its annual profit conditional on starting in force. Its contribution to the cohort is $I_jZ_j$: it contributes no policy-year profit if it is not in force at the start, and otherwise contributes its annual profit.

If all $N_0$ policies have the same start-of-year in-force probability $p_t$ and the same conditional expected profit $g_t$, then the expected total is

\begin{aligned} E\left[\sum_{j=1}^{N_0} I_jZ_j\right] &=\sum_{j=1}^{N_0}E[I_jZ_j]\\ &=\sum_{j=1}^{N_0}P(I_j=1)E[Z_j\mid I_j=1]\\ &=N_0p_tg_t. \end{aligned}

The second equality uses the fact that $I_jZ_j$ is zero when $I_j=0$. No assumption that different policies are independent is needed for this expectation: expectation is additive. If policies have different in-force probabilities or different conditional profit entries, add their expected contributions individually instead.

Worked Example: Scaling a Conditional Profit Entry

A hypothetical insurer issued 1,200 identical policies. At the start of policy year 2, the expected in-force proportion is 0.85. The year-2 annual profit entry is $42.60 per policy, conditional on being in force at the start of that year. Calculate the expected year-2 cohort profit.

Assumptions and method. Take the effective annual interest rate used in the supplied profit entry to be $i=4.0\%$. Mortality and lapse effects on reaching the start of year 2 are summarized by the stipulated proportion 0.85; no separate mortality distribution is needed for this scaling calculation. Model each policy's start-of-year status as Bernoulli with parameter 0.85, and use the supplied conditional expected profit of $42.60.

First calculate the expected number in force:

E_2=1{,}200(0.85)=1{,}020.

Then multiply that exposure by the conditional annual profit:

\text{Expected year-2 cohort profit} =1{,}020(\$42.60)=\$43{,}452.00.

The result is positive because the per-policy entry is positive. Check: the expected exposure is less than the 1,200 policies originally issued, as required by an in-force proportion below 1. This method avoids multiplying the conditional entry by the original issue count.

Worked Example: Combining Successive In-Force Proportions

A hypothetical cohort contains 2,000 policies at issue. Of the original policies, 90% are expected to reach the start of year 2. Of those then in force, 80% are expected to reach the start of year 3. The year-3 conditional annual profit entry is $-\$15.25$ per policy in force at the start of the year. Find the expected year-3 cohort profit.

Assumptions and method. Use $i=4.0\%$ as the effective annual interest rate underlying the supplied profit entry. Mortality and lapse are represented together by the two specified conditional in-force proportions. Each transition is modeled as a Bernoulli survival-to-next-start indicator, conditional on being in force at the preceding start; the conditional expected year-3 profit is $-\$15.25$.

The year-3 in-force fraction is the product of the sequential proportions:

$$ p_3=0.90(0.80)=0.72. $$

Thus the expected exposure at the start of year 3 is

E_3=2{,}000(0.72)=1{,}440.

The cohort profit is

1{,}440(-\$15.25)=-\$21{,}960.00.

Check: the negative sign is retained, so the cohort has an expected loss of $21,960.00 for year 3. Using 80% of the original 2,000 would be wrong because the 80% applies only to the policies expected to reach the start of year 2.

Different Groups May Require Different Scaling

A single cohort-wide multiplication is appropriate only when the policies share the same relevant exposure proportion and conditional profit entry. If groups differ—for example, by product design, underwriting class, or policy duration—calculate the expected contribution for each group and then add the results. A weighted average profit can be used as a shortcut only if its weights reflect the expected in-force counts, not the original group sizes.

Worked Example: Aggregating Two Policy Groups

A hypothetical insurer has two groups of policies at issue. Group A has 900 policies, of which 80% are expected to be in force at the start of year 4; its conditional year-4 annual profit is $26.40 per policy. Group B has 400 policies, of which 90% are expected to be in force at that time; its conditional year-4 annual profit is $-\$8.75$ per policy. Calculate the total expected year-4 cohort profit.

Assumptions and method. The profit entries are conditional expected amounts calculated using an effective annual interest rate $i=4.0\%$. Each group's start-of-year status is modeled as Bernoulli with its stated in-force probability. Any mortality and lapse effects are summarized in those proportions; no separate mortality assumption is needed for the aggregation.

For Group A, the expected exposure and profit are

E_{4,A}=900(0.80)=720, \qquad 720(\$26.40)=\$19{,}008.00.

For Group B, they are

E_{4,B}=400(0.90)=360, \qquad 360(-\$8.75)=-\$3{,}150.00.

Add the group contributions:

\$19{,}008.00-\$3{,}150.00=\$15{,}858.00.

The expected year-4 total is therefore $15,858.00. Check: the expected total exposure is $720+360=1{,}080$ policies, but there is no single valid per-policy entry to multiply by 1,080 unless that entry is weighted to reflect the two groups' different exposures and profits.

Common Mistakes and Checks

A useful distinction is whether an entry is conditional on being in force or already includes the probability of being in force. If $g_t$ is conditional, multiply it by $N_0p_t$. If an entry $h_t$ is already the expected profit per original policy, then it has already incorporated in-force status, so the cohort expectation is $N_0h_t$. Multiplying $h_t$ by $p_t$ again would count the in-force adjustment twice.

The following incorrect calculation illustrates the problem. Suppose 1,000 policies were issued, 70% are expected to be in force at the start of the year, and the stated entry is $20 expected profit per original policy, already including the in-force weighting. The incorrect calculation is $1{,}000(0.70)(\$20)=\$14{,}000$. The correct calculation is $1{,}000(\$20)=\$20{,}000$, because $20 is already unconditional per original policy. By contrast, if $20 were conditional on starting in force, then $1{,}000(0.70)(\$20)=\$14{,}000$ would be correct.

Also distinguish an expected exposure from an observed count. If 700 policies are expected to be in force, the expected total based on a conditional entry is found using 700. If the actual count observed is 684, an actual-period aggregation should use 684 and the appropriate realized per-policy profits. An expected count need not be a whole number; it represents an expectation, not a literal fractional policy.

Match the time point. Use the proportion in force at the start of the policy year when the entry is a start-of-year conditional annual profit.
Multiply conditional proportions in sequence. A transition proportion applies to the policies in force at that transition's starting point.
Keep group calculations separate when needed. Use each group's own exposure and conditional profit before adding totals.
Preserve signs and units. A negative per-policy entry produces a negative cohort contribution; a per-policy annual amount becomes a total for the same year after scaling.

Key takeaway: For a conditional per-policy annual profit entry, expected cohort profit equals expected start-of-year exposure multiplied by that entry. Confirm the entry's conditional basis before applying an in-force proportion, and apply each proportion at the correct point in time.

Check Your Understanding

For each problem, identify the applicable start-of-year exposure before scaling the annual profit entry.

A hypothetical cohort has 1,500 policies at issue. A proportion of 0.92 is expected to be in force at the start of year 2. The conditional year-2 profit entry is $31.25 per policy. Find the expected cohort profit. (Answer: $43,125.00)
A hypothetical cohort has 8,000 policies at issue. The conditional proportions reaching the starts of years 2 and 3 are 0.95 and 0.90, respectively. The conditional year-3 entry is $-\$4.80$ per policy. Find the expected year-3 cohort profit. (Answer: $-\$32,832.00$)
Group A has 450 policies, an in-force proportion of 0.80 at the relevant year start, and a conditional profit of $18.60. Group B has 250 policies, an in-force proportion of 0.72, and a conditional profit of $-\$11.50$. Find the combined expected cohort profit. (Answer: $4,626.00)
A cohort of 1,200 policies has 0.90 expected to reach the start of year 2; 0.75 of those are expected to reach the start of year 3. The conditional year-3 profit entry is $52.40. Find the expected start-of-year-3 exposure and the cohort profit. (Answer: 810 policies; $42,444.00)
A cohort has 1,000 original policies and an in-force proportion of 0.80 at the start of a year. The stated profit entry is $15.75 per original policy, already unconditional with respect to being in force. Find the expected cohort profit and state whether to apply the 0.80 again. (Answer: $15,750.00; do not apply the 0.80 again)

What You'll Learn