Expected Profit with Survival Weighting

Why Annual Profits Need Survival Weights

A profit signature records annual profit amounts on a per-policy basis, but an issued policy contributes a profit in a later year only if it is still in force when that year begins. Multiplying each annual profit by the probability of being in force at the start of its duration converts the conditional, per-policy figures into expected profits per original policy. This distinction matters whether profits are positive or negative: a later loss is also less likely to occur if fewer contracts remain in force.

The timing convention is central. The annual profit for duration $k$ is understood to be the expected profit for that policy year, conditional on the policy being in force at its start. It can reflect events during that year, such as death claims, provided those events are already included in the stated conditional profit. The weight for that duration is therefore the probability of being in force at the start, not the probability of surviving to its end.

Definition: Let $R_k$ be the probability that a policy is in force at the start of duration $k$, where duration 1 starts at issue. Let $\pi_k$ be the expected profit for duration $k$, conditional on being in force at that duration’s start. The expected profit attributable to one original policy in duration $k$ is $R_k\pi_k$. In particular, $R_1=1$. With mortality as the only exit and a life aged $x$ at issue, $R_k={}_{k-1}p_x$.

If mortality and lapse both remove policies, $R_k$ is the probability of remaining in force under the specified decrement assumptions. It need not be a pure survival probability. Use the in-force probabilities on the same basis as the profit signature: a profit calculated conditional on being in force at the start of the year must not be multiplied by a probability that already conditions on some additional event unless the profit definition calls for it.

From a Profit Signature to Expected Profit

For $n$ policy years, the expected cumulative profit per original policy, without discounting, is the sum of the survival-weighted annual profits. The calculation uses linearity of expectation; it does not require profits in different years to be independent.

E[\text{cumulative profit per original policy}] =\sum_{k=1}^{n}R_k\pi_k.

If a portfolio has $N$ identical policies at issue and the assumptions apply equally to each one, its expected aggregate profit is $N$ times the expected profit per original policy. This is an expectation, not a prediction that the actual number of surviving policies will equal a fractional expected count.

When each duration’s profit is valued at the end of that policy year and the effective annual interest rate is $i$, its present value at issue is discounted by $v^k$, where $v=1/(1+i)$. Thus the expected present value per original policy is $\sum_{k=1}^{n}v^kR_k\pi_k$. The timing must be changed if the profit is paid at a different time; for example, a profit at the start of duration $k$ is not discounted by $v^k$.

Why the Weight Is the Probability at the Start

Let $I_k$ equal 1 if the policy is in force at the start of duration $k$, and 0 otherwise. Let $Y_k$ denote the profit for that duration when the policy is in force at its start. The profit attributable to the original policy in that year is $I_kY_k$: it is zero if the policy was already out of force, and otherwise it is the duration’s profit. By conditioning on whether the policy is in force,

E[I_kY_k] =P(I_k=1)E[Y_k\mid I_k=1] +P(I_k=0)E[I_kY_k\mid I_k=0] =R_k\pi_k+0 =R_k\pi_k.

The event $I_k=1$ means the contract has reached the start of the policy year. Death or another exit during that year does not change that start-of-year event. Its effect on the year’s profit belongs in $E[Y_k\mid I_k=1]$, if relevant to the profit-test definition. This is why using a through-the-year survival probability as the weight can omit profit associated with events during the year.

Set the duration timing.
Identify when each annual profit is measured and whether it is conditional on being in force at that year’s start.

Find each start-of-duration probability.
For mortality alone, multiply the relevant annual survival probabilities up to the start of the duration. Include other exits if the model specifies them.

Weight and aggregate.
Multiply each conditional profit by its start-of-duration in-force probability, then sum. Apply discount factors only if a present value is requested.

Worked Calculations

Worked Example: A Three-Year Profit Signature

A hypothetical contract has no lapses. Conditional on being in force at the start of a year, its annual mortality probabilities are $q_1=0.10$, $q_2=0.20$, and $q_3=0.25$. Assume mortality outcomes follow these conditional Bernoulli probabilities; the stated annual profit is the expected profit conditional on being in force at the start of that year. The profit signature is $\pi_1=100$, $\pi_2=200$, and $\pi_3=-50$ dollars. Find expected cumulative profit per issued policy, undiscounted. The interest rate is $i=0\%$.

Method. Calculate the probability of reaching each duration’s start, multiply it by that duration’s conditional profit, and add the resulting expected profits.

At issue, $R_1=1$. The probability of being in force at the start of year 2 is the probability of surviving year 1: $R_2=1(1-q_1)=0.90$. The probability of being in force at the start of year 3 is $R_3=0.90(1-q_2)=0.90(0.80)=0.72$. The third-year mortality probability does not affect whether the policy reaches the start of year 3.

The weighted contributions are $R_1\pi_1=1(100)=100$, $R_2\pi_2=0.90(200)=180$, and $R_3\pi_3=0.72(-50)=-36$, all in dollars. Therefore, $100+180-36=244$. Expected cumulative profit is $244 per issued policy.

Check. The negative third-year profit reduces the expected total, and its magnitude is reduced from $50 to $36 because only 72% of policies are expected to be in force at the start of that year. The common mistake is to weight a year’s profit by survival through that same year rather than to use the probability at its start.

Worked Example: Discounted Profit with Mortality Weighting

A hypothetical two-year policy has no lapses and mortality probabilities $q_1=0.04$ and $q_2=0.10$, conditional on being in force at the start of each year. Its conditional expected year-end profits are $\pi_1=\$1{,}200$ and $\pi_2=\$300$. Assume an effective annual interest rate of $i=4\%$, with both profits occurring at their respective year ends. Find the expected present value per issued policy.

Method. Weight each conditional profit by its start-of-year in-force probability, then discount that expected year-end amount. The start probabilities are $R_1=1$ and $R_2=1-q_1=0.96$. The value of $q_2$ does not affect the probability of reaching the start of year 2.

The expected year-end profit for year 1 per original policy is $1(\$1{,}200)=\$1{,}200$. For year 2 it is $0.96(\$300)=\$288$. Since $v=1/1.04=25/26$, the two discounted contributions are $\$1{,}200/1.04=\$15{,}000/13$, and $\$288/(1.04)^2=\$45{,}000/169$.

Adding the exact expressions gives $\$15{,}000/13+\$45{,}000/169$, or approximately $\$1{,}420.118343$. Rounded only at the final step, the expected present value is $1,420.12 per issued policy.

Check. The year-2 profit is first reduced from $300 to an expected $288 per original policy because of mortality during year 1; it is then discounted two years. Discounting $300 and applying the 0.96 weight afterward gives the same result here because both are multiplicative, but omitting the survival weight would overstate expected present value.

Worked Example: Aggregating Two Mortality Groups

A hypothetical portfolio has 600 policies in group A and 400 in group B. There are no lapses. Group A has annual mortality probability $q=0.02$ in each of two years; group B has annual mortality probability $q=0.10$ in each year. Within each group, each policy’s annual death outcome is Bernoulli with the stated conditional probability. Conditional expected profits per policy in force at the start of the year are $50 in year 1 and $100 in year 2 for both groups. Profits are undiscounted, so take $i=0\%$. Find the expected aggregate profit over the two years.

Method. Calculate expected in-force counts at the start of each duration separately for the two groups, multiply by the corresponding conditional profit, and add across groups and years.

All 1,000 policies are in force at the start of year 1, so expected year-1 profit is $(600+400)(\$50)=\$50{,}000$. At the start of year 2, group A has an expected $600(0.98)=588$ policies in force, and group B has an expected $400(0.90)=360$. The total expected in-force count is $588+360=948$.

Expected year-2 profit is $948(\$100)=\$94{,}800$. The expected aggregate two-year profit is therefore $\$50{,}000+\$94{,}800=$ $144,800.

Check. The group-specific year-2 expected counts sum to 948, which is less than the initial 1,000. Applying one portfolio-wide survival rate without accounting for the groups’ different mortality assumptions would not in general produce this result.

Common Mistakes and Scope

The most frequent error is using ${}_kp_x$ to weight the profit for duration $k$. Under mortality-only assumptions, that probability describes survival through $k$ years, whereas duration $k$ starts after $k-1$ years. Its start-of-duration weight is ${}_{k-1}p_x$. For multiple decrements, the equivalent weight is the probability of remaining in force to the start of that duration under the applicable decrement model.

For example, if a year’s profit is defined as the expected result for a policy in force at the start of the year, replacing the start probability with an end-of-year survival probability can incorrectly suppress that year’s profit when a death occurs during the year. Conversely, if the stated amount is explicitly payable only on survival to year end, its conditional definition must reflect that condition. Establish what the annual profit represents before choosing the weight.

Another error is to confuse expected profit per policy in force with expected profit per policy originally issued. The former is the conditional quantity $\pi_k$; the latter’s contribution for duration $k$ is $R_k\pi_k$. For groups with different decrement assumptions, calculate the weights by group before aggregating. A single average survival rate is safe only if it correctly reproduces the group-weighted expected in-force counts at each duration.

Key takeaway: Match the probability to the timing of the profit. If $\pi_k$ is conditional on being in force at the start of duration $k$, the expected contribution per original policy is $R_k\pi_k$, where $R_k$ is the probability of being in force at that start.

Check Your Understanding

For each problem, use the stated in-force timing and show the weights before combining profits.

A hypothetical two-year contract has no lapses, annual mortality probabilities $q_1=0.08$ and $q_2=0.20$, and conditional expected profits of $75 and -$25. Find expected undiscounted cumulative profit per issued policy. (Answer: $52)
A hypothetical three-year contract has no lapses and mortality probabilities $0.10$, $0.25$, and $0.20$ in years 1, 2, and 3. Its conditional expected annual profits are $40, $50, and $100. Find expected undiscounted cumulative profit per issued policy. (Answer: $152.50)
A hypothetical two-year contract has no lapses, mortality probabilities $q_1=0.10$ and $q_2=0.20$, conditional expected year-end profits of $500 and -$100, and $i=5\%$. Find expected present value per issued policy. (Answer: $394.56)
A portfolio has 200 policies with annual mortality probability 0.05 and 300 with annual mortality probability 0.20, with no lapses. Each policy’s conditional expected profits are $10 in year 1 and $30 in year 2. At $i=0\%$, find expected aggregate profit over the two years. (Answer: $17,900)
A hypothetical three-year contract has no lapses, mortality probabilities $0.10$, $0.20$, and $0.25$, and conditional expected profits of $100, -$40, and $80. Find expected undiscounted cumulative profit per issued policy, and state the probability used to weight year 3. (Answer: $121.60; year-3 weight $0.72$)

What You'll Learn