Weighting State Profits by Transition Probabilities

Why State Probabilities Matter in a Profit Test

In a multistate contract, annual profit can differ depending on whether a policyholder is active, disabled, or dead, and on which transition occurs during the year. A profit amount conditional on one state or transition is not yet the expected profit for the policy at the start of the year. It must be weighted by the probability that the relevant state or transition occurs.

The key is to keep two probability questions separate. Occupancy probabilities describe the state at a particular time. Transition probabilities describe the state at the end of an interval, conditional on the state at the beginning. Combining them gives the probability of a particular state-to-state transition, which can then be used to weight its profit.

Definition: Let the states be active, disabled, and dead. At the beginning of year $t$, let $\alpha_i^{(t)}$ be the probability of occupying state $i$. Let $p_{ij}$ be the probability of being in state $j$ at the end of the year, given state $i$ at its beginning. Then $\alpha_i^{(t)}p_{ij}$ is the probability that the year begins in state $i$ and ends in state $j$. If $g_{ij}$ is the profit for that transition, the expected annual profit is $\sum_i\sum_j \alpha_i^{(t)}p_{ij}g_{ij}$.

The transition probabilities form a matrix $P$, with rows indexed by the state at the beginning of the year and columns indexed by the state at the end. Each row sums to 1. If occupancy probabilities are written as a row vector, the next year's beginning occupancy is $\alpha^{(t+1)}=\alpha^{(t)}P$. The occupancy at the beginning of the year being valued is the vector to use for that year's profit—not the vector after applying the transition matrix again.

Two Ways to Specify Annual Profit

A profit test may give profit conditional on the state at the beginning of the year, or it may give profit conditional on both the beginning and ending states. These are different levels of detail. If the annual profit depends only on the beginning state $i$, write its conditional expected value as $m_i$. Then expected profit is $\sum_i \alpha_i^{(t)}m_i$. The transition probabilities have already been averaged over in $m_i$, so multiplying by them again would count them twice.

If profit varies by transition, use $g_{ij}$ and weight each amount by the joint probability $\alpha_i^{(t)}p_{ij}$. More generally, if the profit for a transition is random, $g_{ij}$ in the expectation formula means its conditional expected value given that transition. This approach does not require the profit to be constant on every path within a transition.

\begin{aligned} E[\text{profit in year }t] &=\sum_i \alpha_i^{(t)}m_i &&\text{(profit conditional on beginning state)}\\ &=\sum_i\sum_j \alpha_i^{(t)}p_{ij}g_{ij} &&\text{(profit conditional on beginning and ending states).} \end{aligned}

Why the Weighting Formula Holds

The formula follows by dividing the possible outcomes into mutually exclusive state-transition events. The event “begin in $i$, end in $j$” has probability $\alpha_i^{(t)}p_{ij}$. Its contribution to expected profit is that probability multiplied by the profit conditional on the event. Adding across all possible $i,j$ accounts for every outcome.

Find the beginning occupancy.
Use the probabilities of each state at the start of the year being valued.

Find each joint transition probability.
For a transition from $i$ to $j$, multiply the beginning occupancy $\alpha_i^{(t)}$ by the conditional transition probability $p_{ij}$.

Weight and add the profits.
Multiply each transition profit by its joint probability, then sum over all transitions.

P(\text{begin in }i\text{ and end in }j) =\alpha_i^{(t)}p_{ij}, \qquad E[\text{profit}] =\sum_i\sum_j P(\text{begin in }i,\text{ end in }j)\,g_{ij}.

This is a one-year calculation. For a later year, first obtain the occupancy at the beginning of that year by applying the transition matrix once for each intervening year. In particular, starting entirely active in year 1 does not mean the contract is entirely active at the start of year 2; nor does it mean the start-of-year-2 occupancy should be advanced by another transition before valuing year 2.

Worked Example: Beginning-of-Year Occupancy Versus a Second Transition

A hypothetical contract has active, disabled, and dead states. Each year, the transition probabilities from active are $(0.70,0.25,0.05)$, in that order. The annual profit conditional on the beginning state is $\$12$ if active, $-\$8$ if disabled, and $\$0$ if dead. The policyholder is active at the beginning of year 1. Find the expected profit in years 1 and 2.

Method. First find the occupancy at the beginning of each year. Then weight the state-conditional profit amounts by that occupancy. The transition row is used to update occupancy, not applied an extra time before valuing the year.

At the beginning of year 1, occupancy is $(1,0,0)$. Therefore:

E[\text{profit in year 1}] =1(12)+0(-8)+0(0)=\$12.00.

The beginning-of-year-2 occupancy is the year-1 beginning occupancy multiplied by one transition matrix. Since the policyholder starts year 1 active, this is the active transition row:

\alpha^{(2)}=(0.70,0.25,0.05).

Consequently, expected year-2 profit is:

E[\text{profit in year 2}] =0.70(12)+0.25(-8)+0.05(0) =8.40-2.00+0 =\$6.40.

As a check, the occupancy probabilities sum to $0.70+0.25+0.05=1$, and the expected profit is between the lowest and highest state-conditional profits, $-\$8$ and $\$12$. Common mistake avoided: applying the transition matrix again would give occupancy after two transitions, not the occupancy at the beginning of year 2.

When Profit Depends on the Transition

Some contracts have different cash flows according to the transition during the year. For example, a recovery from disability may generate a different expense or benefit than remaining disabled. In that case, a single profit amount for each beginning state may not retain enough information. Use transition-specific profits and weight each one by its joint probability.

Worked Example: Weighting Transition-Specific Profits

At the beginning of a year, a hypothetical insured population has occupancy probabilities $(0.60,0.30,0.10)$ in the active, disabled, and dead states. The transition rows are active: $(0.80,0.15,0.05)$; disabled: $(0.10,0.70,0.20)$; and dead: $(0,0,1)$. Profits in dollars for transitions from active are $(10,-6,2)$, and for transitions from disabled are $(4,-3,-1)$. Profit from dead to dead is $0$. Calculate expected annual profit.

Method. For each beginning state, first calculate its conditional expected profit by weighting the transition profits with that row of transition probabilities. Then weight those conditional means by beginning occupancy.

Conditional on beginning active:

0.80(10)+0.15(-6)+0.05(2) =8.00-0.90+0.10 =\$7.20.

Conditional on beginning disabled:

0.10(4)+0.70(-3)+0.20(-1) =0.40-2.10-0.20 =-\$1.90.

Conditional on beginning dead, the only possible transition is dead to dead, giving expected profit $0$. Weight these conditional means by the beginning occupancy:

\begin{aligned} E[\text{profit}] &=0.60(7.20)+0.30(-1.90)+0.10(0)\\ &=4.32-0.57+0\\ &=\$3.75. \end{aligned}

As a check, the total probability of the beginning states is $0.60+0.30+0.10=1$. The result is also the direct sum of transition contributions: the active row contributes $0.60(7.20)=\$4.32$, and the disabled row contributes $0.30(-1.90)=-\$0.57$. Common mistake avoided: weighting a transition profit by $p_{ij}$ alone omits the probability of being in its beginning state.

Including a Reserve Movement

A profit amount may include both the year's net cash flow and an adjustment for the reserve held for future obligations. To make the calculation unambiguous, use the following convention: reserves are measured at the beginning and end of the year; beginning reserves accumulate at effective annual interest rate $i$; and $C_{ij}$ is the net external cash flow for the year, expressed at year-end with inflows positive and outflows negative. If the reserve at the beginning is $V_i$ and at the end is $V_j$, define the year-end profit for transition $i$ to $j$ by:

g_{ij}=(1+i)V_i+C_{ij}-V_j.

The opening reserve plus the year's net cash flow is the amount available at year-end under this convention. The closing reserve is the amount retained for future obligations. Their difference is profit. This formula is conditional on the stated timing and sign convention; if cash flows or reserves are valued at different times, they must first be put on a consistent basis. It is not a general formula for every profit-test timing convention.

Worked Example: Weighting Profits That Include Reserve Movements

A hypothetical one-year contract has active, disabled, and dead states. The effective annual interest rate is $i=5\%$. At the start of the year, occupancy is $(0.80,0.20,0)$, and the reserves at the beginning of the year are $V_A=\$100$, $V_D=\$160$, and $V_{\text{dead}}=\$0$. At year-end, reserves are $\$110$ in active, $\$150$ in disabled, and $\$0$ in dead. Transition probabilities are active: $(0.75,0.20,0.05)$; disabled: $(0.10,0.70,0.20)$; dead: $(0,0,1)$. The year-end net cash flows $C_{ij}$, in dollars, are active-origin transitions $(8,-20,-95)$ and disabled-origin transitions $(5,-10,-170)$; dead to dead has cash flow $0$. Calculate expected annual profit.

Method. Calculate each transition profit from the stated reserve convention, then average by transition probabilities and beginning occupancy. All cash flows are already specified at year-end.

For transitions from active, the accumulated opening reserve is $1.05(\$100)=\$105$. Thus:

\begin{aligned} g_{AA}&=105+8-110=\$3,\\ g_{AD}&=105-20-150=-\$65,\\ g_{A,\text{dead}}&=105-95-0=\$10. \end{aligned}

For transitions from disabled, the accumulated opening reserve is $1.05(\$160)=\$168$. Thus:

\begin{aligned} g_{DA}&=168+5-110=\$63,\\ g_{DD}&=168-10-150=\$8,\\ g_{D,\text{dead}}&=168-170-0=-\$2. \end{aligned}

The conditional expected profit given active at the start is:

0.75(3)+0.20(-65)+0.05(10) =2.25-13.00+0.50 =-\$10.25.

The conditional expected profit given disabled at the start is:

0.10(63)+0.70(8)+0.20(-2) =6.30+5.60-0.40 =\$11.50.

Finally, weight these conditional means by the beginning occupancy:

0.80(-10.25)+0.20(11.50)+0(0) =-8.20+2.30 =-\$5.90.

The transition rows sum to 1, and the beginning occupancy probabilities sum to 1. Common mistake avoided: using the closing reserve as a profit addition would reverse its role. Under this convention, the closing liability is subtracted, while the accumulated opening reserve is added.

Common Traps and Practical Checks

Before accepting an expected-profit calculation, check that probabilities refer to the correct time and condition, and that the cash-flow amounts use a consistent profit convention. These checks are especially useful when a problem gives several years of state transitions alongside a separate table of state-specific profits.

Advancing occupancy too far. Beginning-of-year-$t$ occupancy is used to weight year $t$. Apply one transition matrix to obtain beginning-of-year-$t+1$ occupancy, not an additional transition before calculating year $t$'s profit.
Omitting occupancy. A transition probability $p_{ij}$ is conditional on beginning in state $i$. Its unconditional weight is $\alpha_i^{(t)}p_{ij}$.
Using transition probabilities twice. If a provided profit is already the expected profit conditional on the beginning state, weight it by occupancy only. Do not re-average it over the transition probabilities.
Mixing transition order. State the order of rows and columns before multiplying a matrix or reading a profit table. A misplaced disabled-state column can materially change the result.
Ignoring time and signs in reserve adjustments. State whether cash flows are positive for inflows or outflows, and put reserves and cash flows on the same valuation date before combining them.

This calculation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. Its central discipline is to identify exactly what each profit amount is conditional on, then apply only the probability weights that remain to be applied.

Check Your Understanding

For each problem, identify the occupancy at the start of the year being valued before applying the relevant profit weights.

A hypothetical contract starts year 1 active. Beginning-state profits are $\$9$ active, $-\$5$ disabled, and $\$0$ dead. The active transition row is $(0.60,0.30,0.10)$. Find expected profit in year 2. (Answer: $\$3.90$)
Beginning occupancy is $(0.50,0.40,0.10)$. Transition rows are active $(0.80,0.10,0.10)$, disabled $(0.25,0.60,0.15)$, and dead $(0,0,1)$. Transition profits are active-origin $(7,-4,1)$, disabled-origin $(3,-2,-1)$, and dead-origin $0$. Find expected profit. (Answer: $\$2.41$)
A contract starts year 1 active. The transition rows are active $(0.70,0.25,0.05)$, disabled $(0.10,0.70,0.20)$, and dead $(0,0,1)$. Beginning-state profits are $\$12$, $-\$8$, and $\$0$. Find expected profit in year 3. (Answer: $\$3.38$)
Under the convention $g=(1+i)V_i+C-V_j$, the annual effective interest rate is $4\%$, beginning reserve is $\$80$, year-end net cash flow is $\$15$, and closing reserve is $\$90$. Find the profit conditional on this transition. (Answer: $\$8.20$)
Beginning occupancy is $(0.60,0.40,0)$. Transition rows are active $(0.75,0.20,0.05)$, disabled $(0.10,0.80,0.10)$, and dead $(0,0,1)$. Transition profits are active-origin $(10,-5,0)$, disabled-origin $(4,-2,0)$, and dead-origin $0$. Find expected annual profit. (Answer: $\$3.42$)

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