Separating Expected Profit from Conditional Profit

Why the Conditioning Matters

A policy’s profit depends not only on its starting state but also on the transition that occurs during the period. An active policy that remains active may produce a different profit from one that becomes disabled; a policy already in a benefit-paying state may have yet another set of possible outcomes. The expected profit for a policy in one specified state is therefore not automatically the expected profit for the whole portfolio.

The distinction is between conditional profit, calculated given the policy’s occupied state, and portfolio-level expected profit, which combines those conditional results using the proportion of policies in each starting state. The same transition-contingent profit information can also be conditioned on the ending state, but that requires different probabilities.

Definition: Consider a one-period product with states $A$ and $B$. Let $p_{ij}=P(\text{ending in state }j\mid\text{starting in state }i)$, and let $\pi_{ij}$ be the period-end profit when the transition is $i\to j$. The expected profit conditional on starting in state $i$ is $g_i=\sum_j p_{ij}\pi_{ij}$. If the proportions of the portfolio starting in states $A$ and $B$ are $w_A$ and $w_B$, respectively, portfolio expected profit per policy is $w_Ag_A+w_Bg_B$, where $w_A+w_B=1$.

Here, $\pi_{ij}$ is the profit on the specified transition, not merely the benefit paid on that transition. It is assumed to include all relevant premium, expense, benefit, investment, and reserve-change components for the period under the stated profit definition. If the problem supplies those components separately, calculate the transition profit first using the stated sign convention. The method below then weights the resulting profits by probabilities; it does not alter their cash-flow timing or sign.

From State-Conditional Profit to Portfolio Profit

For a policy known to start in state $A$, only the probabilities in the $A$ row of the transition matrix apply. The conditional expected profit is the weighted average of the profits on transitions out of $A$. The same calculation uses the $B$ row for a policy known to start in $B$. To obtain a portfolio average, weight these two conditional expectations by the portfolio’s starting-state mix.

g_A=p_{AA}\pi_{AA}+p_{AB}\pi_{AB},\qquad g_B=p_{BA}\pi_{BA}+p_{BB}\pi_{BB}

If there are $N$ policies, with $N_A$ starting in $A$ and $N_B$ starting in $B$, then expected aggregate profit is $N_Ag_A+N_Bg_B$. This follows from adding the expected profits of the policies in each group. Independence between policies is not required for this expectation calculation; linearity of expectation is sufficient. Independence would matter for questions about the variance or distribution of aggregate profit.

The portfolio weights must describe the population at the time the conditional profits apply. They are not necessarily the transition probabilities for either state. A transition probability answers, “Given the current state, where does this policy go?” A portfolio weight answers, “What proportion of policies are currently in each starting state?”

Worked Example: One-Period Profit by Starting State

A hypothetical two-state income product has an active state $A$ and a benefit-paying state $B$. Consider a one-year period. Each transition profit is paid at year-end and is stated in dollars per policy, net of all other period cash flows and reserve changes. The transition probabilities and profits are:

From $A$, the policy remains in $A$ with probability $0.80$, producing profit of $40; it moves to $B$ with probability $0.20$, producing profit of $-$160$. From $B$, it moves to $A$ with probability $0.25$, producing profit of $70; it remains in $B$ with probability $0.75$, producing profit of $-$20$. At the start of the year, 90% of policies are in $A$ and 10% are in $B$. Calculate each starting-state conditional expectation and the portfolio expected profit per policy.

Method: condition first on the starting state, then weight by the starting-state mix. For a policy starting in $A$, the two possible profits are $40 and $-$160$, so

g_A=0.80(40)+0.20(-160)=32-32=\$0.00.

For a policy starting in $B$,

g_B=0.25(70)+0.75(-20)=17.50-15.00=\$2.50.

The portfolio expected profit per policy is

0.90g_A+0.10g_B=0.90(0.00)+0.10(2.50)=\$0.25.

Thus, for 1,000 policies with the stated starting mix, expected aggregate profit is $1{,}000(0.25)=\$250.00$. As a check, 900 policies start in $A$, contributing expected profit $900(0.00)=\$0.00$, while 100 start in $B$, contributing $100(2.50)=\$250.00$. Common mistake: averaging $g_A$ and $g_B$ equally would give $1.25 per policy, as if half the portfolio started in each state. The supplied portfolio mix is 90% and 10%, so equal weighting is incorrect.

Why the Weighting Formula Holds

The portfolio formula is an application of the law of total expectation. Let $G$ be a policy’s period profit, and let $S$ denote its starting state. The starting states $A$ and $B$ form a mutually exclusive and exhaustive partition, so the unconditional expectation is the sum of the conditional expectations multiplied by the probabilities of the conditioning events.

Partition by starting state.
The policy starts in exactly one of $A$ or $B$, so $P(S=A)=w_A$, $P(S=B)=w_B$, and $w_A+w_B=1$.

Use the conditional profit within each group.
By definition, $E[G\mid S=A]=g_A$ and $E[G\mid S=B]=g_B$.

Sum across the partition.
The law of total expectation gives $E[G]=P(S=A)E[G\mid S=A]+P(S=B)E[G\mid S=B]=w_Ag_A+w_Bg_B$.

This justification also shows why a conditional result should not be presented as the portfolio result without qualification. $g_A$, for example, describes a policy known to start in $A$. It describes the portfolio only if all policies start in $A$, or if the question specifically asks for the $A$-state subgroup.

Conditioning on the Ending State Is Different

The phrase “profit for policies in state $B$” can be ambiguous. It may mean policies starting in $B$, or policies ending in $B$. The first uses a row of the transition matrix. The second combines transitions from every possible starting state that lead to $B$, weighted by their probabilities in the portfolio.

If the starting-state proportions are $w_A,w_B$, the probability of ending in $B$ is $w_Ap_{AB}+w_Bp_{BB}$. The expected profit conditional on ending in $B$ is the total probability-weighted profit on transitions into $B$, divided by that ending-state probability, provided the probability is positive.

Worked Example: Profit Conditional on Ending in the Benefit State

A hypothetical product has states $A$ and $B$. At the start of one year, 70% of policies are in $A$ and 30% are in $B$. The transition probabilities are $p_{AA}=0.90$, $p_{AB}=0.10$, $p_{BA}=0.20$, and $p_{BB}=0.80$. Year-end transition profits per policy are $\pi_{AA}=\$30$, $\pi_{AB}=-\$170$, $\pi_{BA}=\$50$, and $\pi_{BB}=-\$10$. Calculate the portfolio expected profit per policy and the expected profit conditional on ending in $B$.

First calculate profit conditional on the starting state:

g_A=0.90(30)+0.10(-170)=27-17=\$10, $$ $$ g_B=0.20(50)+0.80(-10)=10-8=\$2.

The portfolio expected profit per policy is therefore

0.70(10)+0.30(2)=7+0.6=\$7.60.

To condition on ending in $B$, calculate the probability of each route into $B$, including the starting-state proportion:

P(\text{end in }B)=0.70(0.10)+0.30(0.80)=0.07+0.24=0.31.

The probability-weighted profit from those routes is

0.70(0.10)(-170)+0.30(0.80)(-10)=-11.90-2.40=-\$14.30.

Thus the expected profit conditional on ending in $B$ is

\frac{-14.30}{0.31}=-\$46.1290\ldots\approx-\$46.13.

As a check, the probability of ending in $A$ is $0.70(0.90)+0.30(0.20)=0.69$. The probability-weighted profit on routes ending in $A$ is $0.70(0.90)(30)+0.30(0.20)(50)=18.90+3.00=\$21.90$, giving conditional profit $21.90/0.69=\$31.7391\ldots$. Recombining by ending state gives $0.31(-46.1290\ldots)+0.69(31.7391\ldots)=-14.30+21.90=\$7.60$, matching the portfolio result. Common mistake: using $g_B=\$2$ as the expected profit for policies ending in $B$ conditions on the wrong event; $g_B$ conditions on starting in $B$.

Expected Profit Across Multiple Years

For a multi-year projection, state proportions change as transitions occur. In each year, first use the state distribution at the start of that year to calculate expected profit for that year. Then update the state distribution using the transition probabilities. If profits occur at year-end, discount each year’s expected profit from its payment time to the valuation date. This procedure calculates expected discounted profit; it does not assert that the realized profit in each year equals its expectation.

For a fixed transition model, the state distribution update is obtained by summing the probabilities of all routes into each destination state. With $w_A^{(t)}$ and $w_B^{(t)}$ as the proportions in each state at the start of year $t$, the next year’s starting proportions are

w_A^{(t+1)}=w_A^{(t)}p_{AA}+w_B^{(t)}p_{BA},\qquad w_B^{(t+1)}=w_A^{(t)}p_{AB}+w_B^{(t)}p_{BB}.

Worked Example: Two-Year Expected Discounted Profit

A hypothetical two-state product has states $A$ and $B$. At the start of year 1, all 200 policies are in $A$. In each year, the probabilities are $p_{AA}=0.80$, $p_{AB}=0.20$, $p_{BA}=0.30$, and $p_{BB}=0.70$. For every year, transition profits are $\pi_{AA}=\$20$, $\pi_{AB}=-\$80$, $\pi_{BA}=\$40$, and $\pi_{BB}=-\$10$, paid at year-end. Assume the same transition and profit assumptions apply in both years, no policies enter or leave the model, and the effective annual discount rate is $i=4\%$. Calculate the expected present value of aggregate profit over the two years.

Method: calculate conditional profit by starting state, update the state counts, then discount each year’s expected aggregate profit. In state $A$,

g_A=0.80(20)+0.20(-80)=16-16=\$0.

In state $B$,

g_B=0.30(40)+0.70(-10)=12-7=\$5.

All 200 policies start year 1 in $A$, so expected aggregate profit for year 1 is $200(0)=\$0$. At the end of year 1, expected policy counts are $200(0.80)=160$ in $A$ and $200(0.20)=40$ in $B$. These expected counts sum to 200, as required.

At the start of year 2, expected aggregate profit is

160g_A+40g_B=160(0)+40(5)=\$200.

The year-1 profit is discounted by $v=1/(1.04)$, and the year-2 profit by $v^2=1/(1.04)^2$. Therefore,

E[\text{present value}] =0(1.04)^{-1}+200(1.04)^{-2} =\frac{200}{1.0816} =\$184.91.

The result is rounded to the nearest cent. As a check, the undiscounted expected aggregate profit is $200, all of it expected at the end of year 2; discounting that amount for two years at 4% gives the same $184.91. Common mistake: carrying the initial all-$A$ distribution into year 2 would incorrectly give expected profit of zero in that year. The year-2 calculation must use the state distribution after year-1 transitions.

Exam Checks and Common Traps

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a profit problem, identify exactly what is being conditioned on before selecting weights. A reliable sequence is to write the starting-state conditional expectations, identify the relevant portfolio distribution, and then aggregate. For multiple years, update the state mix before calculating later-year expectations, and apply discount factors according to when profit is paid.

Using transition probabilities as portfolio weights. Transition probabilities apply within a given starting state. Portfolio weights describe the distribution of starting states.
Mixing up starting and ending states. “Given start in $B$” and “given end in $B$” generally produce different conditional expectations.
Ignoring the profit sign. A transition with a claim may have negative net profit, but use the specified transition profit rather than replacing it with the claim amount.
Using the wrong year’s state mix. Later-year expected profit uses the state distribution after prior transitions.
Discounting at the wrong time. A year-end profit at time $t$ is discounted by $v^t$, not $v^{t-1}$.

Key takeaway: First condition on a known starting state and average over the transitions out of it. Then weight those conditional expectations by the actual starting-state distribution. To condition on an ending state, instead combine all incoming routes and divide by the probability of ending in that state.

Check Your Understanding

For each question, distinguish the conditioning event from the portfolio mix before calculating.

A one-year product has states $A$ and $B$. From $A$, the probabilities of ending in $A$ and $B$ are 0.75 and 0.25, with profits $24 and $-$36. From \(B$, the probabilities are 0.40 and 0.60, with profits $54 and $-$6. The starting mix is 80% \(A$, 20% $B$. Find each starting-state conditional expected profit and the portfolio expected profit per policy. (Answer: $g_A=\$9.00$, $g_B=\$18.00$, portfolio $=\$10.80$ per policy.)
A portfolio has 60 policies in $A$ and 40 in $B$. From $A$, the transition probabilities are 0.90 to $A$ and 0.10 to $B$, with profits $15 and $-$85. From \(B$, they are 0.25 to $A$ and 0.75 to $B$, with profits $45 and \(-$5. Find expected aggregate profit for the year. (Answer: $600.00.)
Half a portfolio starts in each of $A$ and $B$. The probabilities are $p_{AA}=0.80$, $p_{AB}=0.20$, $p_{BA}=0.10$, and $p_{BB}=0.90$. Transition profits are $\pi_{AA}=\$10$, $\pi_{AB}=-\$40$, $\pi_{BA}=\$30$, and $\pi_{BB}=-\$10$. Find the expected profit conditional on ending in $B$. (Answer: $-\$15.45$ per policy, rounded to the nearest cent.)
All 200 policies start in $A$. In each year, $p_{AA}=0.80$, $p_{AB}=0.20$, $p_{BA}=0.30$, and $p_{BB}=0.70$; transition profits are $20, $-$80, $40, and \(-$10 for \(AA,AB,BA,BB$, respectively. Profits are paid at year-end, and the effective annual discount rate is 4%. Find the expected present value of two years of aggregate profit. (Answer: $184.91.)

What You'll Learn