Why the Basis of a Profit Figure Matters
A profit figure for a particular policy duration can answer two different questions. It may describe the expected profit for a policy that has reached that duration, or it may describe the expected contribution to profit per policy originally issued. These figures are not generally equal: some policies may have terminated before the duration begins, so they cannot contribute that duration’s profit.
The conversion is a probability weighting. The relevant probability is the chance that a policy is in force at the start of the duration, accounting for any earlier death or lapse decrements in the model. This distinction is useful when reading a profit signature or translating in-force results into expected profit on an issue basis.
The word “unconditional” here means unconditional on being in force at duration \(k\); it does not mean that every policy earns the same profit. A policy that terminates before duration \(k\) may have no profit assigned to duration \(k\), while a policy still in force has a duration-\(k\) profit that may depend on events during that duration.
From Conditional Profit to Profit per Policy Issued
Write \(w_k=P(I_k)\) for the probability that a policy issued is in force at the start of duration \(k\). If a policy terminated before that point has no duration-\(k\) profit, the conversion is
This equation weights the expected profit for an in-force policy by the chance that a policy issued is in force to earn it. At issue, \(w_0=1\), so \(\pi_0=g_0\). At later durations, the two quantities differ whenever there is a positive probability of termination before that duration.
For a simple annual model, suppose that during duration \(k\), death and lapse are mutually exclusive decrements among policies in force at its start. Let \(q_k^{(d)}\) and \(q_k^{(l)}\) be their respective probabilities during the duration. If death and lapse are the only ways to leave the in-force group, the probability of remaining in force to the next duration is
The probabilities in this recurrence are conditional on being in force at the start of duration \(k\). For example, \(q_k^{(d)}\) and \(q_k^{(l)}\) must not be added if the model specifies overlapping events rather than mutually exclusive causes; in that case, use the probability of remaining in force obtained from the model’s joint probabilities. If there are other decrements, they must also be reflected in the probability of remaining in force.
Why the Weighting Formula Holds
Let \(Z_k\) denote the duration-\(k\) profit for a policy issued, taking \(Z_k=0\) if the policy is not in force at the start of duration \(k\). Conditional on being in force, the expected duration profit is \(g_k\). The law of total expectation, conditioning on whether \(I_k\) occurs, gives
This justification also identifies the assumption behind the simple conversion: there is no duration-\(k\) profit assigned to a policy that was already out of force at the start of that duration. If a profit definition assigns later payments or adjustments to a terminated policy, those amounts must be included under the appropriate event in the expected profit calculation; they cannot be silently treated as zero.
Worked Calculations
Worked Example: Weighting Profits Through Death and Lapse
A hypothetical contract has three annual durations, numbered 0, 1, and 2. At the start of duration 0, every issued policy is in force. Among policies in force at the start of each duration, the annual death and lapse probabilities are mutually exclusive. They are, respectively, \((0.02,0.08)\) in duration 0, \((0.03,0.07)\) in duration 1, and \((0.04,0.06)\) in duration 2. The expected profits conditional on being in force at the start of the durations are \(120\), \(-40\), and \(200\) dollars. Find the expected profit per policy issued for each duration.
Method. First calculate the probability of reaching each duration. Then multiply that probability by the corresponding conditional expected profit. The conditional profits are given as expected profits for an in-force policy, so no additional death or lapse weighting is applied to them before this conversion.
At duration 0, the in-force probability is \(w_0=1\). The probability of remaining in force to duration 1 is \(1-0.02-0.08=0.90\), so \(w_1=1(0.90)=0.90\).
The probability of remaining in force through duration 1 is \(1-0.03-0.07=0.90\). Therefore the probability of reaching duration 2 is \(w_2=0.90(0.90)=0.81\).
The expected profits per policy issued are \(\pi_0=1(120)=120\) dollars, \(\pi_1=0.90(-40)=-36\) dollars, and \(\pi_2=0.81(200)=162\) dollars.
Check. The weights \(1\), \(0.90\), and \(0.81\) are the probabilities of being in force at the starts of durations 0, 1, and 2. The negative duration-1 profit remains negative after weighting. The result is \(120\), \(-36\), and \(162\) dollars per policy issued. The common mistake avoided is using the duration-1 lapse and death probabilities to weight duration 1; those probabilities determine who reaches duration 2.
Worked Example: Calculating Conditional Profit Before Weighting
A hypothetical policy has an annual profit outcome for each policy in force at the start of a duration. In duration 0, 92% remain in force throughout the year and earn \(50\) dollars; 8% lapse and produce a profit of \(300\) dollars in that duration. In duration 1, among policies in force at its start, 85% remain in force and earn \(40\) dollars, 10% lapse and produce \(100\) dollars, and 5% die and produce \(-500\) dollars. In duration 2, the corresponding outcomes are: 90% remain in force and earn \(80\) dollars, 6% lapse and produce \(120\) dollars, and 4% die and produce \(-700\) dollars. Calculate the conditional expected profit in each duration and the expected profit per policy issued for duration 1 and duration 2.
Method. Each conditional duration profit is the probability-weighted average of that duration’s outcomes, conditional on being in force at its start. Then weight it by the probability of reaching that duration. The lapse and death profits are included in the conditional profit for the duration in which the termination occurs.
For duration 0, the conditional expected profit is \(0.92(50)+0.08(300)=46+24=70\) dollars. All policies are in force at issue, so the expected profit per policy issued for duration 0 is also \(70\) dollars.
The probability of reaching duration 1 is the duration-0 persistence probability, \(w_1=0.92\). The duration-1 conditional expected profit is \(0.85(40)+0.10(100)+0.05(-500)=34+10-25=19\) dollars. Thus the expected duration-1 profit per policy issued is \(\pi_1=0.92(19)=17.48\) dollars.
To reach duration 2, a policy must remain in force through duration 0 and duration 1. The persistence probability in duration 1 is \(0.85\), so \(w_2=0.92(0.85)=0.782\). The duration-2 conditional expected profit is \(0.90(80)+0.06(120)+0.04(-700)=72+7.20-28=51.20\) dollars. Therefore \(\pi_2=0.782(51.20)=40.0384\) dollars, or \(40.04\) dollars to the nearest cent.
Check. The duration-1 and duration-2 weights are \(0.92\) and \(0.782\), the probabilities of reaching those durations. The termination outcomes are already part of their respective conditional duration profits, so they are not added again as separate duration contributions. The common mistake avoided is treating the conditional profit as if every policy issued reached the duration.
Worked Example: Recovering Conditional Profit From Cohort Results
A hypothetical cohort contains 2,000 issued policies. The expected numbers in force at the starts of durations 0, 1, and 2 are 2,000, 1,800, and 1,350. The expected aggregate profits assigned to those durations are \(24{,}000\), \(-13{,}500\), and \(6{,}750\) dollars. Find the conditional expected profit per in-force policy and the expected profit per policy issued for each duration.
Method. Divide each expected aggregate duration profit by the expected number in force at that duration’s start to recover the conditional expected profit per in-force policy. Divide by the number issued to obtain the expected duration profit per policy issued.
The conditional profits are \(24{,}000/2{,}000=12\) dollars in duration 0, \(-13{,}500/1{,}800=-7.50\) dollars in duration 1, and \(6{,}750/1{,}350=5\) dollars in duration 2.
The in-force weights are \(w_0=2{,}000/2{,}000=1\), \(w_1=1{,}800/2{,}000=0.90\), and \(w_2=1{,}350/2{,}000=0.675\). Consequently, the expected profits per policy issued are \(1(12)=12\) dollars, \(0.90(-7.50)=-6.75\) dollars, and \(0.675(5)=3.375\) dollars.
Check. Multiplying each per-issued result by 2,000 gives aggregate expected profits of \(24{,}000\), \(-13{,}500\), and \(6{,}750\) dollars, as stated. The common mistake avoided is dividing the aggregate profit by policies issued when the requested result is conditional profit per policy in force.
Reading and Applying the Conversion Correctly
A duration’s conditional profit is useful for understanding the expected experience of the policies that reach that duration. The per-issued figure answers a different question: how much that duration contributes, on average, for each original policy issued. As the in-force probability falls, the per-issued figure is attenuated relative to the conditional figure. Its sign is unchanged when the weight is positive, but its magnitude is reduced.
When working with a sequence of durations, calculate each weight at the start of the duration in question. For example, the weight for duration 2 includes the persistence probabilities for durations 0 and 1, but not the probability of persisting through duration 2. That final persistence probability affects the chance of reaching duration 3.
Identify the start and end of each annual duration, and identify whether the profit given is conditional on being in force at its start.
Start with \(w_0=1\), then apply the appropriate probability of remaining in force for each completed duration.
Use \(\pi_k=w_k g_k\) for expected profit per policy issued, or \(g_k=\pi_k/w_k\) when \(w_k>0\).
Confirm that termination profits are assigned to the intended duration and are neither omitted nor counted twice.
If \(w_k=0\), then no issued policy can reach duration \(k\), and the per-issued expected profit for that duration is zero under the stated convention. Division by \(w_k\) is not possible, so a conditional profit for that unreachable duration cannot be recovered from its per-issued figure. Also check whether the supplied probability is already an issue-basis weight: applying it again would weight the profit twice.
Check Your Understanding
For each question, distinguish the in-force basis from the policy-issued basis and use the probability of reaching the duration being considered.
- A policy has conditional expected profits of \(25\), \(-10\), and \(40\) dollars in durations 0, 1, and 2. The probabilities of remaining in force through durations 0 and 1 are \(0.96\) and \(0.75\). Find the expected profit per policy issued for each duration. (Answer: \(25\), \(-9.60\), and \(28.80\) dollars.)
- A cohort has 800 policies issued. The probability of remaining in force through duration 0 is \(0.90\), and the conditional expected profit in duration 1 is \(30\) dollars. Find the expected duration-1 profit per policy issued and the expected aggregate duration-1 profit for the cohort. (Answer: \(27\) dollars per policy issued and \(21{,}600\) dollars in aggregate.)
- At duration 0, death and lapse probabilities are mutually exclusive and equal to \(0.02\) and \(0.13\). The conditional expected profit is \(-50\) dollars. Find the expected profit per policy issued for duration 0 and the probability of reaching duration 1. (Answer: \(-42.50\) dollars and \(0.85\).)
- A cohort has 900 policies issued and 540 expected policies in force at the start of duration 2. Its expected aggregate profit for duration 2 is \(8{,}100\) dollars. Find the conditional expected profit per in-force policy and the expected profit per policy issued. (Answer: \(15\) dollars and \(9\) dollars.)
- The probability of reaching a duration is zero, and the expected profit per policy issued assigned to that duration is zero. Can the conditional expected profit for an in-force policy at that duration be recovered from these data? (Answer: No; division by the zero in-force weight is undefined, and no policy reaches the duration.)