Exposure Bases For Annual Profit

Why the Exposure Basis Matters

An annual profit amount can describe different things. It may be the expected profit from a policy given that the policy is in force at the start of the year, or it may be the expected contribution from one policy in the original issue cohort, including the possibility that the policy has already left the portfolio. These measures answer different questions and should not be interchanged.

The distinction matters when a profit vector is used to describe successive policy years. A policy that has terminated cannot generate a later year's profit, so the original cohort's expected contribution in a later year depends both on the profit conditional on being in force and on the probability of still being in force. The exposure probability is the conversion factor.

Definition: For policy year $k+1$, let $I_k$ be the indicator that a policy is in force at the start of the year, and let $e_k=P(I_k=1)$ be its start-of-year in-force probability. Let $G_k$ be the profit generated during that year, defined for a policy that is in force at its start. The conditional annual profit is $g_k=E[G_k\mid I_k=1]$. The expected profit per original policy for that year is $e_k g_k$, provided profit is zero for a policy that is no longer in force.

The index $k$ counts completed policy years: year $k+1$ runs from time $k$ to time $k+1$. Thus $e_0=1$ for a newly issued policy. In a life-only model, $e_k={}_kp_x$. If lapse or other termination is possible, $e_k$ must reflect all ways a policy can leave the in-force portfolio. It is not automatically a mortality survival probability.

The conditional profit $g_k$ is an expectation over the possible events during the year, conditional on being in force at its start. It can include profits arising under different outcomes, such as survival, death, or lapse, if those outcomes are part of the stated annual profit measure. The exposure conversion does not calculate those event-contingent profits; it weights their conditional expectation by the chance that the policy reaches the start of the year.

The Conversion and Its Justification

The conversion follows directly from conditioning on whether the policy is in force at the start of the year. Let the profit from the year, viewed from the original policy, be zero when $I_k=0$, and equal $G_k$ when $I_k=1$. Its expectation is:

E[I_kG_k] =P(I_k=1)E[G_k\mid I_k=1] +P(I_k=0)E[I_kG_k\mid I_k=0] =e_k g_k.

The second term is zero because $I_k=0$ makes $I_kG_k=0$. This is the reason for the multiplier: only the fraction of original policies expected to be in force at the start of the year can generate that year's conditional profit. No assumption that $G_k$ is independent of in-force status is needed; the expectation $g_k$ is explicitly conditional on that status.

Theorem: If $g_k$ is the expected annual profit conditional on being in force at time $k$, then the expected profit in year $k+1$ per original policy is $e_k g_k$. For $N$ identical original policies, the expected total profit for that year is $N e_k g_k$. If the profit amount is measured at the end of year $k+1$, its expected present value at issue, under effective annual interest rate $i$, is $v^{k+1}e_k g_k$, where $v=1/(1+i)$.

The discount factor in the last expression is used only when the aim is an issue-date present value of an end-of-year profit. If the profit vector is already stated on a present-value basis, it should not be discounted a second time. Likewise, if a problem asks for the expected annual profit at the end of a policy year, the exposure conversion is $e_k g_k$, without an additional discount factor.

Worked Examples

Worked Example: Mortality Exposure in a Later Policy Year

A hypothetical term policy has annual effective interest rate $i=4\%$. There is no lapse, and mortality is the only decrement. The annual death probabilities are $q_0=0.02$, $q_1=0.03$, and $q_2=0.04$. During policy year 3, the end-of-year profit is $480 if the policyholder survives the year and \(-$1,520 if death occurs during the year. Find the expected profit per policy in force at the start of year 3, the expected year-3 profit per original policy, and the issue-date present value of that expected profit. All stated profits are measured at the end of year 3.

Method and assumptions. The event during year 3 is binary, with death probability $0.04$ and survival probability $1-0.04=0.96$, conditional on being in force at the start of that year. The annual profit figures are already specified on the $i=4\%$ valuation basis; no additional interest calculation is needed to obtain them.

First find the probability of reaching the start of year 3. The policy must survive both earlier years:

$$ e_2=(1-q_0)(1-q_1)=0.98(0.97)=0.9506. $$

The expected profit conditional on being in force at the start of year 3 is:

g_2=(0.96)(480)+(0.04)(-1{,}520) =460.80-60.80 =\$400.00.

Therefore, the expected year-3 profit per original policy is:

e_2g_2=0.9506(400)=\$380.24.

For an issue-date present value, year-3 end-of-year profit is discounted three years. Since $v=1/1.04$,

v^3e_2g_2=\frac{380.24}{1.04^3} =\frac{380.24}{1.124864} =\$338.03.

The result is rounded to the nearest cent only at the final step. A common mistake is to multiply by $0.96$, the probability of surviving year 3, as the exposure factor. That gives $0.96(400)=\$384.00$, but it omits deaths in years 1 and 2. The correct start-of-year-3 exposure is $0.9506$, based on survival through the two completed years.

Worked Example: Combining Mortality and Lapse Exposure

A hypothetical block contains 2,000 original policies. For each policy, the annual effective interest rate is $i=4\%$. In year 1, the probabilities of death and lapse are 0.02 and 0.03, respectively. In year 2, conditional on being in force at its start, those probabilities are 0.04 and 0.06. Death and lapse are mutually exclusive exits, and there is no re-entry. In year 3, the probabilities of death, lapse, and remaining in force through the year are 0.02, 0.05, and 0.93. The end-of-year-3 profits for those respective outcomes are $-\$950$, $\$50$, and $\$130$. Find the conditional year-3 profit and the expected year-3 profit per original policy and for the block.

Method and assumptions. In each year, the specified probabilities form a categorical distribution over mutually exclusive outcomes. In-force exposure at the start of year 3 depends on avoiding both death and lapse in each of the two earlier years. The interest rate is $4\%$; the end-of-year profits are supplied on that basis.

The probability of remaining in force through year 1 is $1-0.02-0.03=0.95$. For year 2, it is $1-0.04-0.06=0.90$. Hence:

$$ e_2=(0.95)(0.90)=0.855. $$

The expected year-3 profit conditional on being in force at its start is found by weighting each outcome's profit by its conditional probability:

g_2=(0.02)(-950)+(0.05)(50)+(0.93)(130) =-19+2.50+120.90 =\$104.40.

Thus the expected year-3 profit per original policy is:

e_2g_2=(0.855)(104.40)=\$89.262.

The expected number of policies in force at the start of year 3 is $2{,}000(0.855)=1{,}710$. Therefore the expected total year-3 profit for the block is:

1{,}710(104.40)=\$178{,}524.00, \qquad 2{,}000(89.262)=\$178{,}524.00.

The two calculations agree, which checks the exposure conversion. The per-original-policy amount is not the conditional amount of $\$104.40$: it is smaller because some original policies terminate before year 3.

Worked Example: Different Exposure by Policy Group

A hypothetical portfolio is issued with 60% of policies in group A and 40% in group B. The annual effective interest rate is $i=5\%$. In each of the first two years, group A has death probability 0.02 and lapse probability 0.08; group B has death probability 0.10 and lapse probability 0.40. Death and lapse are mutually exclusive, and there is no re-entry. Conditional on being in force at the start of year 3, expected year-3 profit is $\$200$ for group A and $-\$100$ for group B. Find the portfolio's year-3 expected profit per original policy and its conditional profit among policies in force at the start of year 3.

Method and assumptions. The two groups have different decrement probabilities, so calculate each group's exposure before combining them. The group-specific conditional profits are the expected year-end amounts on the stated $i=5\%$ basis.

In each of the first two years, group A's probability of remaining in force is $1-0.02-0.08=0.90$. Group B's is $1-0.10-0.40=0.50$. Therefore:

e_{2,A}=0.90^2=0.81, \qquad e_{2,B}=0.50^2=0.25.

Per original policy in the full portfolio, group A contributes an expected $0.60(0.81)(200)=\$97.20$, while group B contributes $0.40(0.25)(-100)=-\$10.00$. Thus:

E[\text{year-3 profit per original policy}] =97.20-10.00 =\$87.20.

The overall probability of being in force at the start of year 3 is:

$$ e_2=0.60(0.81)+0.40(0.25)=0.486+0.100=0.586. $$

Consequently, the conditional expected profit among policies in force at that time is:

g_2=\frac{87.20}{0.586}=\$148.81

rounded to the nearest cent. Multiplying back gives $0.586(148.81)$, which is approximately $\$87.20$; the small difference before rounding reflects rounding $g_2$ to cents. The exact conditional mean is $87.20/0.586$.

The common mistake is to average the two conditional profits using the original issue proportions first: $0.60(200)+0.40(-100)=\$80$. Multiplying that by the overall exposure gives only $0.586(80)=\$46.88$. This is wrong because the group proportions among policies still in force are no longer 60% and 40%. The correct calculation weights each group's profit by both its original proportion and its own in-force probability.

Exposure, Aggregation, and Timing

For a homogeneous group of $N$ original policies, $Ne_k$ is the expected number in force at the start of year $k+1$, and $Ne_kg_k$ is the expected profit for that year. The count $Ne_k$ need not be an integer: it is an expectation, not a prediction that a fractional policy exists. The conversion also works when $e_k$ is obtained from a multiple-decrement model, provided it is the probability of being in force at the correct time.

For a portfolio with groups $j=1,\ldots,m$, let $w_j$ be group $j$'s proportion at issue, $e_{k,j}$ its in-force probability, and $g_{k,j}$ its conditional annual profit. The expected profit per original policy in the whole portfolio is:

\sum_{j=1}^{m}w_j e_{k,j}g_{k,j}.

If an overall conditional profit is required, divide this amount by the overall in-force probability $\sum_jw_je_{k,j}$, provided that probability is positive. This division automatically gives more weight to groups that are more likely to remain in force. It is not generally correct to average group conditional profits using the original issue proportions.

Keep the time basis explicit. Exposure for profit generated during year $k+1$ is measured at the start of that year. For a policy that must survive to the end of a year to receive a particular benefit, the benefit probability may involve end-of-year survival; that does not change the exposure base for the entire year's profit. The conditional annual profit already accounts for the possible events during the year.

Common Exam Traps

Using end-of-year survival as the exposure. For a year’s profit conditional on being in force at its start, use the probability of reaching the start of that year. End-of-year survival may be relevant to a particular cash flow, but it is not the start-of-year exposure.
Ignoring earlier exits. For a later policy year, multiply the appropriate persistency probabilities across all earlier years. Include lapse and other termination causes as well as mortality when the model includes them.
Mixing conditional and original-policy amounts. A conditional profit vector entry is not yet an expected amount per original policy. Multiply by the corresponding in-force probability exactly once.
Using the wrong group weights. If group persistency differs, the mix among in-force policies changes over time. Calculate group contributions first, then aggregate.
Discounting inconsistently. Apply discounting only when converting an amount at a future time to present value. Do not discount a present-value profit entry again.

Key takeaway: Identify the year and the exposure basis before multiplying. For a year’s profit $g_k$ conditional on being in force at the start, the expected profit per original policy is $e_kg_k$, where $e_k$ is the probability of being in force at that start-of-year point.

Check Your Understanding

For each problem, distinguish the start-of-year exposure from the conditional profit. Unless stated otherwise, profits are measured at the end of the year and are not discounted.

A hypothetical policy has no lapse and annual death probabilities of 0.03 in year 1 and 0.04 in year 2. Conditional on being in force at the start of year 3, its expected year-3 profit is $\$150$. What is the expected year-3 profit per original policy? (Answer: $\$139.68$)
A hypothetical block has annual effective interest rate $i=4\%$. In year 1 the death and lapse probabilities are 0.02 and 0.04. In year 2 they are 0.05 and 0.05. In year 3, conditional on being in force at its start, death, lapse, and persistence probabilities are 0.04, 0.06, and 0.90, with corresponding profits $-\$500$, $\$40$, and $\$120$. Find the year-3 expected profit per original policy. (Answer: $\$76.48$)
A portfolio is 70% group A and 30% group B at issue. Their probabilities of being in force at the start of a specified year are 0.80 and 0.40, and their respective conditional profits for that year are $\$50$ and $\$200$. Find both the expected profit per original policy and the conditional expected profit among those in force. (Answer: $\$52.00$ per original policy; $\$76.47$ per policy in force)
A policy has annual effective interest rate $i=5\%$, no lapse, and annual death probabilities 0.03, 0.05, and 0.10 in years 1, 2, and 3. Its conditional year-4 profit is $\$240$, payable at the end of year 4. Find the issue-date present value of the expected year-4 profit per original policy. (Answer: $\$163.75$)
A homogeneous group has 5,000 original policies. The probability of being in force at the start of a specified year is 0.72, and conditional expected profit for that year is $-\$25$ per policy in force. Find the expected number in force and the expected profit for the original group. (Answer: 3,600 in force; $-\$90,000$)

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