Annual Profit Vectors Per Policy In Force

Two Different Bases for Annual Profit

An annual profit amount is meaningful only when its denominator is clear. A value of $60 might mean expected profit for each policy active at the start of a particular year, or it might mean expected profit per policy originally issued, including policies that have since exited. Those are different measures: the first is conditional on being in force, while the second spreads expected profit over the original issue group.

The annual profit vector uses the first basis. Its entries describe expected profit in each policy year per policy in force at the start of that year. To express the same annual profit on an original-issue basis, weight each entry by the probability that a policy is still in force at that year's start.

Definition: For a policy with a specified model and assumptions, the annual profit vector per policy in force is $(g_1,g_2,\ldots,g_n)$, where $g_k$ is the expected profit during policy year $k$, conditional on the policy being in force at the start of that year. Profit is measured per policy in force at that start. The vector itself does not apply a survival or persistence weight for reaching year $k$.

Policy year 1 begins at issue, so an initially issued policy is in force at the start of year 1. Reaching the start of year 2 requires the policy to remain in force through year 1; reaching the start of year $k$ requires it to remain in force through the preceding $k-1$ years. Let ${}_{k-1}p^{\mathrm{IF}}$ denote the probability of being in force at the start of year $k$, given issue. The superscript here indicates persistence in force, which may reflect all modeled exits, not just death.

Key relation: If $g_k$ is the conditional expected profit per policy in force at the start of year $k$, then the expected profit in year $k$ per policy originally issued is $$ {}_{k-1}p^{\mathrm{IF}}\,g_k. $$ This is an annual amount on an original-issue basis. It is not the definition of the $k$th entry of the annual profit vector.

Why the Persistence Weight Appears

The conversion follows from conditioning on whether the policy has reached the start of the year. Let $I_k$ equal 1 if the policy is in force at the start of year $k$, and 0 otherwise. Let $Z_k$ be its profit attributed to year $k$, with $Z_k=0$ if it was not in force at the start of that year. Conditional on $I_k=1$, the expected profit is $g_k$. Conditional on $I_k=0$, the attributed year-$k$ profit is zero. The law of total expectation therefore gives:

\begin{aligned} E[Z_k] &=P(I_k=1)E[Z_k\mid I_k=1]+P(I_k=0)E[Z_k\mid I_k=0]\\ &={}_{k-1}p^{\mathrm{IF}}g_k+(1-{}_{k-1}p^{\mathrm{IF}})\cdot 0\\ &={}_{k-1}p^{\mathrm{IF}}g_k. \end{aligned}

This argument does not require the profit within the year to be deterministic. It requires $g_k$ to be the appropriate conditional expected profit for policies in force at that year's start, and the year-$k$ profit for policies not in force at that start to be excluded from this attribution. A payment arising from an exit during year $k$, for example, can be included in $g_k$ if the annual profit model attributes it to policies that were in force at the start of that year.

Identify the denominator.
Determine whether the stated annual profit is per policy in force at the start of that year or per policy originally issued.

Find the year-start persistence probability.
For year $k$, use the probability of reaching the start of year $k$, accumulating persistence through years 1 to $k-1$.

Convert only when needed.
Multiply the conditional in-force profit by the year-start probability to obtain expected profit per original issue.

Worked Applications

Worked Example: Converting a Three-Year Vector

A hypothetical three-year contract has annual profit vector $(120,-30,80)$ dollars per policy in force at the start of each year. A policy remains in force through year 1 with probability $0.90$, and, conditional on reaching year 2, remains in force through year 2 with probability $0.80$. There are no other exits. Assume the conditional annual profit amounts are deterministic, so their conditional distributions are concentrated at the stated values. No discounting is performed; equivalently, use $i=0$. Find the expected annual profits per policy originally issued.

Method. First calculate the probability of reaching each year start, then multiply the corresponding in-force vector entry by that probability.

At the start of year 1, the policy is in force with probability 1. At the start of year 2, the probability is $0.90$. At the start of year 3, it must persist through both earlier years:

{}_{2}p^{\mathrm{IF}}=0.90(0.80)=0.72.

Thus the original-issue-basis annual amounts are:

\begin{aligned} \text{Year 1: }&1(120)=120,\\ \text{Year 2: }&0.90(-30)=-27,\\ \text{Year 3: }&0.72(80)=57.60. \end{aligned}

The annual amounts per original issue are therefore $(120,-27,57.60)$ dollars. As a check, 100 originally issued policies have expected year-2 profit $100(-27)=-\$2{,}700$, the same as 90 expected year-2 policies in force times $-\$30$ each. The common mistake is to call $-\$27$ the year-2 entry of the in-force vector; the correct in-force entry is $-\$30$, while $-\$27$ is the converted original-issue amount.

Worked Example: Scaling from a Portfolio Count

A hypothetical insurer issues 1,000 identical contracts. Each contract remains in force through year 1 with probability $0.95$. The expected profit during year 2 is \$70 per policy in force at the start of that year. The year-1 expected profit is \$50 per policy in force at the start of year 1. Assume independent persistence outcomes across contracts, no other exits, and deterministic conditional profit amounts; the number reaching year 2 is therefore binomial with parameters $1{,}000$ and $0.95$. No discounting is used, so $i=0$. Calculate the year-2 expected profit for the portfolio and per policy originally issued.

The expected number in force at the start of year 2 is:

E[N_2]=1{,}000(0.95)=950.

Each of these expected year-2 in-force policies contributes an expected \$70 under the stated conditional profit assumption. Hence the portfolio's expected year-2 profit is:

950(70)=\$66{,}500.

Dividing by the 1,000 policies originally issued gives:

\frac{\$66{,}500}{1{,}000}=\$66.50

per original issue. The year-2 annual profit vector entry remains \$70 per policy in force. The answer \$66.50 has a different denominator. As a check, the year-1 portfolio expected profit is $1{,}000(\$50)=\$50{,}000$, or \$50 per original issue, because every policy starts year 1 in force.

Worked Example: Combining Death and Lapse

A hypothetical policy has annual death probability $0.02$ and annual lapse probability $0.08$, conditional on being in force at the start of the year. These exits are mutually exclusive within the annual model, and a policy that experiences neither remains in force. The annual profit vector per policy in force is $(40,100,-20)$ dollars. Assume these conditional profits are deterministic, the exit probabilities are the same in each year, and $i=0$ because no discounting is required. Find the expected profit per original issue in each of the three years.

The probability of remaining in force for one year, conditional on being in force at its start, is:

$$ 1-0.02-0.08=0.90. $$

The probabilities of being in force at the starts of years 1, 2, and 3 are consequently $1$, $0.90$, and $0.90^2=0.81$. Weight each vector entry by its corresponding probability:

\begin{aligned} \text{Year 1: }&1(40)=\$40.00,\\ \text{Year 2: }&0.90(100)=\$90.00,\\ \text{Year 3: }&0.81(-20)=-\$16.20. \end{aligned}

The original-issue-basis annual amounts are $(\$40.00,\$90.00,-\$16.20)$. The year-3 weight is cumulative: it is the probability of avoiding both death and lapse in each of the first two years. Using $0.90$ rather than $0.81$ in year 3 would incorrectly treat the year-3 start as requiring persistence through only one year.

Interpreting the Vector Carefully

The annual profit vector is not a vector of cumulative profits, present values, or expected profits per original issue. Each entry is an annual conditional expectation on the year-start in-force basis. For a vector $(g_1,\ldots,g_n)$, the corresponding original-issue annual amounts are $(g_1,{}_{1}p^{\mathrm{IF}}g_2,\ldots,{}_{n-1}p^{\mathrm{IF}}g_n)$. If a later calculation requires a total across years, those annual amounts can be combined using the method appropriate to that calculation; the conversion itself does not discount or accumulate them.

The persistence probability is the probability of being in force at the year start, not the probability of remaining in force for that entire year. In particular, an exit during year $k$ does not remove the policy from the start-of-year exposure for $g_k$. Any profit consequences attributed to that year's exposure belong in the conditional expected profit for year $k$, according to the annual model.

The distinction matters when comparing products or years. A later-year in-force profit can be larger than an earlier-year amount while the corresponding expected profit per original issue is smaller, because fewer policies reach that later year. Conversely, a negative conditional profit can have a smaller magnitude per original issue when only a fraction of the original group reaches the year.

Common Mistakes and the Role of Selection

Mixing denominators. Do not label a persistence-weighted amount as profit per policy in force. State the basis beside every vector or reported annual amount.
Using a one-year probability instead of a cumulative one. The year-$k$ weight is the probability of reaching its start after $k-1$ years, which generally involves multiplying the earlier conditional persistence probabilities.
Using the year-end persistence probability. Profit for a year defined per policy in force at that year's start is weighted by the probability at the start, not the probability of still being in force at year-end.
Assuming the conditional vector is automatically constant across a changing pool. If policy characteristics affect profit or exit rates, the mix of policies still in force can change over time.

The conversion identity remains valid when the in-force pool is heterogeneous, provided $g_k$ is defined as the conditional average expected profit among policies that are actually in force at the start of year $k$. That conditional average may differ by year because of selection. If instead a model reports separate vectors for different policy types, apply the persistence weighting to each type on its own basis before combining results.

Keeping the two bases distinct is a basic control in profit testing and is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. A reliable calculation labels the conditional per-in-force vector first, identifies the relevant year-start persistence probabilities second, and converts only when an original-issue denominator is required.

Check Your Understanding

For each problem, distinguish the in-force-basis amount from the original-issue-basis amount where requested. Assume no discounting unless a problem states otherwise.

A policy has year-1 in-force profit \$75 and year-2 in-force profit \$30. It persists through year 1 with probability $0.92$, with no other exits. What is the year-2 expected profit per original issue? (Answer: \$27.60.)
A hypothetical contract has annual lapse probabilities $0.10$ in year 1 and $0.20$ in year 2, conditional on being in force at the start of the relevant year. Its three-year in-force profit vector is $(100,-25,60)$. Find the three original-issue-basis annual amounts. (Answer: $(\$100.00,-\$22.50,\$43.20)$.)
A group of 2,400 identical policies has probability $0.96$ of remaining in force through year 1. Year-2 expected profit is \$125 per policy in force at the start of year 2. Find the expected number in force then, the portfolio's expected year-2 profit, and the year-2 amount per original issue. Assume independent persistence across policies and deterministic conditional profit. (Answer: 2,304 policies; \$288,000; \$120 per original issue.)
In each year, a policy in force at the start has mutually exclusive death probability $0.03$ and lapse probability $0.07$. Its three-year in-force profit vector is $(20,-10,80)$. Find the original-issue-basis annual amounts. (Answer: $(\$20.00,-\$9.00,\$64.80)$.)
A cohort has 100 policies of type A and 100 of type B at issue. By the start of year 2, $90$ type A and $50$ type B policies are expected to remain. The year-2 expected profit per in-force policy is \$40 for type A and \$100 for type B. Find the year-2 conditional average per policy in force and the expected year-2 profit per policy originally issued. (Answer: $\$61.43$ per policy in force; $\$43.00$ per original issue.)

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