State-Contingent Profit Signatures

Why State-Contingent Weighting Is Needed

A policy can occupy different states during its term. A disability income contract, for example, might be active, disabled, or terminated by death. The profit associated with a duration can differ substantially by state: premiums may be received while active, disability benefits may be paid while disabled, and a death benefit may be paid on termination. A single profit amount cannot represent all these possibilities unless it has already averaged over them.

A profit signature records profit by duration. When profits depend on policy state, the entry for a duration must combine the profit conditional on each relevant state with the probability of being in that state. The probability used must match how the conditional profit was defined: it may refer to the state at the start of the duration, the state at the end, or a particular transition between states.

Definition: Let $J_k$ denote the policy state at time $k$, and let $\Pi_k$ be profit assigned to duration $k$. If $g_{kj}=E[\Pi_k\mid J_k=j]$ is the expected duration-$k$ profit conditional on state $j$ at the start of that duration, the unconditional duration entry is $$ G_k=E[\Pi_k]=\sum_j P(J_k=j)g_{kj}. $$ If the profit is instead defined conditional on the state at the end of the duration, use $P(J_{k+1}=j)$ with $E[\Pi_k\mid J_{k+1}=j]$. These are different conditionings and should not be interchanged.

The state probabilities are measured for the portfolio or policy cohort being studied. For a single policy issued in state $a$, they are obtained from the initial state and the transition probabilities. For a group of policies, they may instead be given as proportions in each state. In either case, they must refer to the correct duration boundary and sum to one across an exhaustive set of states.

State Probabilities and Conditional Profits

Suppose a finite-state model has transition probabilities $p_{jl}=P(J_{k+1}=l\mid J_k=j)$, with the same transition structure applying during the duration in question. If the state distribution at time $k$ is $\boldsymbol{\pi}_k$, then the state distribution at time $k+1$ is obtained by multiplying by the transition matrix. In row-vector notation,

\boldsymbol{\pi}_{k+1}=\boldsymbol{\pi}_k\mathbf{P}.

This probability propagation is separate from the profit calculation. Once the relevant state probabilities are known, the profit entry is an expectation: each conditional profit is weighted by the probability of the event on which it is conditioned.

Transition-specific form: If $c_{k,jl}=E[\Pi_k\mid J_k=j,J_{k+1}=l]$, then $$ G_k=\sum_j\sum_l P(J_k=j)\,p_{jl}\,c_{k,jl}. $$ This form applies when profit is specified for each start-to-end transition. It makes the joint probability of that transition explicit.

The conditional start-state profit $g_{kj}$ can be obtained from transition-specific profits by averaging over the possible ending states:

g_{kj}=\sum_l p_{jl}c_{k,jl}.

Consequently, weighting $g_{kj}$ by the start-state probabilities gives the same result as weighting every transition-specific profit by its joint probability. This equivalence is useful as a calculation check.

Worked Examples

Worked Example: Weighting Profits by the Start State

A hypothetical disability income policy has three states: active $A$, disabled $D$, and dead $Z$. The annual effective interest rate is $i=4\%$. Each living state has an annual mortality probability of $0.02$; death is absorbing. Conditional on being alive at the next year, the transition probabilities are $P(A\to A)=0.80/0.98$, $P(A\to D)=0.18/0.98$, $P(D\to A)=0.10/0.98$, and $P(D\to D)=0.88/0.98$. Thus the unconditional one-year transition probabilities from $A$ are $0.80,0.18,0.02$, and from $D$ are $0.10,0.88,0.02$. Starting from $A$ at time 0, calculate the expected profit for duration 2 if the conditional profits given the state at the start of that duration are $120$ dollars in $A$, $-300$ dollars in $D$, and $0$ dollars in $Z$. Assume these profits already include the duration's premiums, expenses, benefits, and reserve movement under the stated assumptions.

Method. First find the state probabilities at time 1, the start of duration 2. The initial state is $A$, so after one year the probabilities are

P(J_1=A)=0.80,\qquad P(J_1=D)=0.18,\qquad P(J_1=Z)=0.02.

These probabilities sum to $0.80+0.18+0.02=1.00$. Apply the start-state weighting formula:

G_2=(0.80)(120)+(0.18)(-300)+(0.02)(0) =96-54+0=\$42.

The expected duration-2 profit is $\$42$. As a check, the positive active-state contribution exceeds the negative disabled-state contribution by $96-54=42$, and the dead-state contribution is zero. The common mistake this calculation avoids is weighting the duration-2 profits by probabilities at time 2: the stated conditional profits are given the state at the start of duration 2, which is time 1.

Worked Example: Profits That Depend on the Transition

A hypothetical long-term care contract has states active $A$, disabled $D$, and dead $Z$, with $Z$ absorbing. The annual effective interest rate is $i=5\%$, and the annual mortality probability from either living state is $0.02$. The unconditional transition probabilities are $P(A\to A)=0.85$, $P(A\to D)=0.13$, $P(A\to Z)=0.02$, $P(D\to A)=0.20$, $P(D\to D)=0.78$, and $P(D\to Z)=0.02$. At the start of the duration, the state probabilities are $0.70$ in $A$, $0.20$ in $D$, and $0.10$ in $Z$. The transition-specific duration profits, in dollars, are: $A\to A:80$, $A\to D:-220$, $A\to Z:-500$, $D\to A:40$, $D\to D:-60$, $D\to Z:-300$, and $Z\to Z:0$. Find the expected duration profit.

Method. First average the transition profits within each starting state. Conditional on starting in $A$,

g_A=(0.85)(80)+(0.13)(-220)+(0.02)(-500) =68-28.60-10=\$29.40.

Conditional on starting in $D$,

g_D=(0.20)(40)+(0.78)(-60)+(0.02)(-300) =8-46.80-6=-\$44.80.

A policy already in $Z$ remains there and has profit $g_Z=0$. Weight these conditional profits by the probabilities at the start:

G=(0.70)(29.40)+(0.20)(-44.80)+(0.10)(0) =20.58-8.96=\$11.62.

The expected duration profit is $\$11.62$. As a check using joint transition probabilities, the nonzero contributions are $0.595(80)$, $0.091(-220)$, $0.014(-500)$, $0.040(40)$, $0.156(-60)$, and $0.004(-300)$. Their sum is $47.60-20.02-7+1.60-9.36-1.20=11.62$. The common mistake avoided is weighting a transition profit by only its starting-state probability; a particular transition also requires its conditional transition probability.

Worked Example: Weighting by the End State

A hypothetical policy has states active $A$, disabled $D$, and dead $Z$. The annual effective interest rate is $i=3\%$, and annual mortality is $0.02$ from either living state. Use the same unconditional living-state transitions as in the preceding example: from $A$, the probabilities to $A,D,Z$ are $0.85,0.13,0.02$; from $D$, they are $0.20,0.78,0.02$; $Z$ is absorbing. At the start of a duration, the probabilities in $A,D,Z$ are $0.60,0.30,0.10$. Suppose the duration profit is specified by the state at the end of the duration: it is $\$50$ if the policy ends in $A$, $-\$100$ if it ends in $D$, and $-\$200$ if it ends in $Z$. Calculate the expected duration profit.

Method. Find the end-state probabilities by summing the probability of each route into that state. The probability of ending in $A$ is

P(J_{k+1}=A)=(0.60)(0.85)+(0.30)(0.20)=0.51+0.06=0.57.

The probability of ending in $D$ is

P(J_{k+1}=D)=(0.60)(0.13)+(0.30)(0.78)=0.078+0.234=0.312.

Death can occur from either living state, and policies already dead remain dead:

P(J_{k+1}=Z)=(0.60)(0.02)+(0.30)(0.02)+0.10 =0.012+0.006+0.10=0.118.

The end-state probabilities sum to $0.57+0.312+0.118=1$. Therefore,

G=(0.57)(50)+(0.312)(-100)+(0.118)(-200) =28.50-31.20-23.60=-\$26.30.

The expected profit is $-\$26.30$. The calculation uses end-state probabilities because the profits are defined conditional on the state at the end. It would be incorrect to multiply these profits by the start-state probabilities $0.60,0.30,0.10$.

Why the Weighting Formula Holds

The state events at a given boundary are mutually exclusive and collectively exhaustive: exactly one state is occupied. This allows the expectation to be split according to those events. For start-state-conditioned profits, partition the event space into $J_k=j$. The contribution from each part is its probability multiplied by the conditional expectation on that part:

E[\Pi_k] =\sum_j E[\Pi_k\,\mathbf{1}_{\{J_k=j\}}] =\sum_j P(J_k=j)E[\Pi_k\mid J_k=j].

The second equality follows from the definition of conditional expectation on an event with positive probability. A state with probability zero contributes zero and does not require a conditional expectation to be evaluated. The same partition argument applies to end states or to the mutually exclusive transition events $\{J_k=j,J_{k+1}=l\}$. Thus the probability weight and conditional profit must always refer to the same event.

Identify the conditioning event.
Determine whether each profit is conditional on the start state, end state, or a specific transition.

Find matching probabilities.
Use the distribution at that boundary, or the joint transition probabilities if profits are transition-specific.

Form the expectation.
Multiply each conditional profit by its matching probability and add the contributions.

Check the probability accounting.
Confirm that the relevant state probabilities, or transition probabilities from each state, sum to one.

Common Exam Trap: Mixing the Conditioning Boundary

Suppose the profit in a duration is conditional on being active at its start, but a calculation multiplies that profit by the probability of being active at the end. This silently changes the conditioning event. The resulting product does not, in general, represent any part of the expected profit.

For example, in the first worked example, the active-state conditional profit for duration 2 is $\$120$, and the probability of being active at the start of that duration is $0.80$. Its correct contribution is $(0.80)(120)=\$96$. The probability of being active at the end of the duration, given the stated model, is $(0.80)(0.80)+(0.18)(0.10)=0.658$. The incorrect contribution would be $(0.658)(120)=\$78.96$, because it pairs a start-state-conditioned profit with an end-state probability. The correction is to use the state probability at time 1, or to redefine the profit as conditional on the end state and calculate that conditional expectation consistently.

A related error is to weight a transition-specific profit using only $P(J_k=j)$, omitting $P(J_{k+1}=l\mid J_k=j)$. If the profit is attached to $j\to l$, its weight is the joint probability $P(J_k=j)p_{jl}$, not just $P(J_k=j)$.

Key takeaway: A state-contingent profit-signature entry is an expectation, not a list of conditional profits. Pair every profit with the probability of exactly the state or transition on which that profit is conditioned, and add the resulting weighted contributions.

Check Your Understanding

For each problem, identify the appropriate state boundary or transition conditioning before calculating.

At the start of a duration, a cohort is $75\%$ active and $25\%$ disabled. Conditional profits given these start states are $\$40$ and $-\$80$, respectively. What is the unconditional duration profit? (Answer: $\$10$)
A policy starts a duration with probabilities $0.50$ active, $0.40$ disabled, and $0.10$ dead. From active, the next-state probabilities to active, disabled, and dead are $0.90,0.08,0.02$; from disabled they are $0.25,0.73,0.02$; dead is absorbing. Profit is determined by end state: $\$20$ if active, $-\$50$ if disabled, and $-\$100$ if dead. Find the expected duration profit. (Answer: $-\$17.40$)
A duration begins $60\%$ active and $40\%$ disabled. The only possible transitions are $A\to A$ with probability $0.90$, $A\to Z$ with probability $0.10$, $D\to D$ with probability $0.80$, and $D\to Z$ with probability $0.20$, where $Z$ is dead. Transition profits are $\$30,-\$170,-\$20,-\$120$, respectively. Find the expected duration profit. (Answer: $-\$10$)
A policy starts in $A$. Each year, the transitions from $A$ to $A,D,Z$ have probabilities $0.80,0.18,0.02$, and from $D$ to $A,D,Z$ have probabilities $0.10,0.88,0.02$; $Z$ is absorbing. Find the duration-2 profit if profits conditional on the start state of that duration are $\$25$ in $A$, $-\$50$ in $D$, and zero in $Z$. (Answer: $\$11$)
A cohort starts in $A$. From $A$, the annual probabilities to $A,D,Z$ are $0.80,0.18,0.02$; from $D$, they are $0.10,0.88,0.02$; $Z$ is absorbing. For duration 1, conditional start-state profits in $A,D,Z$ are $\$10,-\$40,\$0$. For duration 2, they are $\$20,-\$30,\$0$. Find the two unconditional duration entries. (Answer: Duration 1: $\$10$; duration 2: $\$10.60$)

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