Expected Reserve Across a Population

Why Expected Reserve Is a Population Quantity

At a valuation date, two otherwise identical policies can have different reserves because their owners occupy different states. An active policy may have a modest prospective liability, while a disabled policy may have a larger one because benefits are being paid. To summarize a population whose state distribution is known but whose individual state assignments are not being used, combine each state-specific reserve with the probability of being in that state.

This calculation produces an expected reserve across the population. It is a probability-weighted average, not a replacement for the reserve of a policy whose state is already known. The distinction matters when estimating a portfolio liability from modelled state probabilities and when valuing an actual in-force list with observed states.

Definition: Suppose a policy can occupy one of $m$ mutually exclusive, exhaustive states at time $t$. Let $J(t)$ denote its state, let $p_j(t)=P(J(t)=j)$, and let $V_j(t)$ be its prospective reserve conditional on being in state $j$ at time $t$. The expected reserve per policy is $$ E[R(t)]=\sum_{j=1}^{m}p_j(t)V_j(t), $$ where $R(t)=V_{J(t)}(t)$ is the reserve associated with the policy’s actual state. The probabilities must refer to the same valuation time and state definitions as the reserves.

A reserve is a conditional prospective value: it represents the value, under the stated valuation basis, of future benefits and expenses less future premiums, given the policy’s state at the valuation date. It can be positive, zero, or negative. State occupancy probabilities describe how likely the policy is to be in each state; they do not replace the conditional reserve values.

For $N$ homogeneous policies with the same state probabilities and state-specific reserves, linearity of expectation gives

E\!\left[\sum_{k=1}^{N}R_k(t)\right] =\sum_{k=1}^{N}E[R_k(t)] =N\sum_{j=1}^{m}p_j(t)V_j(t).

Independence between policies is not required for this expected-total formula. Independence would matter for some measures of fluctuation around the expected total, but it is not needed to calculate the mean. If the policies are not homogeneous, calculate each policy’s expected reserve, or calculate it separately by homogeneous cohort, and then add the results.

Calculating the State Probabilities

The probabilities $p_j(t)$ must describe the population at the valuation date. They may be supplied directly, obtained from a model of transitions between states, or estimated for a defined cohort. For a discrete-time model, write the state probabilities as a row vector. If $P(s,t)$ is the matrix of transition probabilities from time $s$ to time $t$, then

\boldsymbol{p}(t)=\boldsymbol{p}(s)P(s,t).

Each entry in a row of $P(s,t)$ is a probability of moving from the row’s starting state to a destination state. The occupancy probabilities at a date should sum to one when the states are exhaustive. A dead or terminated state can be included in that sum even when its reserve is zero. Use the reserve appropriate to the state and valuation timing: if a termination payment is still due at the valuation instant, the reserve for that state need not be zero.

Worked Example: A Hypothetical In-Force Population at One Valuation Date

A hypothetical insurer has 1,000 identical long-term care policies from one original cohort. At the end of year 6, the model gives probabilities of 0.80 for active, 0.15 for disabled, and 0.05 for dead. On the stated valuation basis, the reserves at that date are $1,200 for an active policy, $6,000 for a disabled policy, and $0 for a dead policy because any death settlement has already been paid. Find the expected reserve per policy and for the cohort.

Method. Apply the law of total expectation by multiplying each state-specific reserve by its occupancy probability and summing.

The probabilities are exhaustive: $0.80+0.15+0.05=1.00$. The expected reserve per policy is

$$ 0.80(1{,}200)+0.15(6{,}000)+0.05(0) =960+900+0 =1{,}860. $$

Thus, the expected reserve for the 1,000-policy cohort is

$$ 1{,}000(1{,}860)=1{,}860{,}000. $$

The expected amount is therefore $1,860 per policy, or $1,860,000 for the cohort. The expected counts are 800 active, 150 disabled, and 50 dead; weighting these counts by the respective reserves gives $800(1{,}200)+150(6{,}000)+50(0)=960{,}000+900{,}000=1{,}860{,}000$, which checks the result.

Common mistake avoided. Using only the active reserve would give $1{,}000(1{,}200)=1{,}200{,}000$, incorrectly treating every policy as active. The calculation must include every state with its own occupancy probability and reserve. For a particular policy known to be disabled, the conditional reserve is $6,000, not the population average of $1,860.

Why the Weighted Average Formula Holds

The events “the policy is in state $j$ at time $t$” form a partition: exactly one occurs. On the event that state $j$ occurs, the policy’s reserve is $V_j(t)$. The expected value of a quantity that takes the value $V_j(t)$ on each event in this partition is the sum of the value on that event times its probability.

Partition by state.
Let $A_j=\{J(t)=j\}$. The events $A_1,\ldots,A_m$ are mutually exclusive, and their probabilities sum to one.

Write the reserve on each event.
On $A_j$, the policy’s reserve is $R(t)=V_j(t)$, by the definition of a state-specific reserve.

Sum the probability-weighted values.
Therefore $E[R(t)]=\sum_j E[R(t)\mathbf{1}_{A_j}]=\sum_j V_j(t)P(A_j)=\sum_j p_j(t)V_j(t)$. For a portfolio, add this expectation across policies.

This argument is the discrete form of conditioning on the policy’s state. It also explains the central distinction: if the state is observed to be $j$, the relevant conditional reserve is $V_j(t)$; if the state is uncertain and only its distribution is being used, the expected reserve averages across the possible states.

Carrying Occupancy Probabilities Across Periods

When a state distribution is not given directly at the valuation date, obtain it from the starting distribution and the applicable transition probabilities. The following example uses a hypothetical one-year transition matrix. The matrix entries are assumed to apply over both years, and all policies begin active.

Worked Example: Two-Year Occupancy and Expected Reserve

A hypothetical policy model has three states: active $A$, disabled $D$, and dead $X$. In each year, the transition probabilities are

P= \begin{pmatrix} 0.80&0.15&0.05\\ 0.20&0.70&0.10\\ 0&0&1 \end{pmatrix}.

Rows are starting states and columns are destination states, in the order $A,D,X$. All 400 policies start active. At the end of year 2, the state reserves are $1,000 for active, $5,000 for disabled, and $0 for dead. Find the expected aggregate reserve.

Method. First propagate the row vector of state probabilities for two years, then take its dot product with the year-2 state reserves.

The initial state vector is $\boldsymbol{p}(0)=(1,0,0)$. After one year,

\boldsymbol{p}(1)=(1,0,0)P=(0.80,0.15,0.05).

After the second year, multiply the year-1 probabilities by the transition matrix:

\begin{aligned} p_A(2)&=0.80(0.80)+0.15(0.20)+0.05(0)=0.64+0.03=0.67,\\ p_D(2)&=0.80(0.15)+0.15(0.70)+0.05(0)=0.12+0.105=0.225,\\ p_X(2)&=0.80(0.05)+0.15(0.10)+0.05(1)=0.04+0.015+0.05=0.105. \end{aligned}

The probabilities sum to $0.67+0.225+0.105=1.000$. The expected reserve per policy is

$$ 0.67(1{,}000)+0.225(5{,}000)+0.105(0) =670+1{,}125 =1{,}795. $$

For 400 policies, the expected aggregate reserve is $400(1{,}795)=718{,}000$, or $718,000. As a check using expected state counts, the model gives $400(0.67)=268$ active policies and $400(0.225)=90$ disabled policies. Their weighted reserves are $268(1{,}000)=268{,}000$ and $90(5{,}000)=450{,}000$; their sum is $718,000. The expected dead count is 42, with zero reserve under the stated timing assumption.

Common mistake avoided. The one-year probabilities are not the year-2 probabilities. Applying the reserve weights to $(0.80,0.15,0.05)$ would calculate the expected reserve at the end of year 1, not year 2. The transition multiplication must be performed for each elapsed period.

Mixed Cohorts and Known Individual States

A portfolio may contain groups with different state probabilities, different state-specific reserves, or both. A single portfolio-wide state distribution is not enough if the reserve for a given state also differs by group. Calculate each group’s expected reserve using that group’s own probabilities and reserves, then add the group totals.

Worked Example: Combining Two Hypothetical Cohorts

A hypothetical portfolio has two groups at the same valuation date. Group 1 contains 120 policies, with probabilities 0.90 active and 0.10 disabled; its reserves are $800 active and $6,000 disabled. Group 2 contains 80 policies, with probabilities 0.60 active and 0.40 disabled; its reserves are $1,200 active and $5,000 disabled. There are no other states. Calculate the expected total reserve and the expected reserve per policy across all 200 policies.

Method. Calculate each group’s weighted reserve per policy, multiply by its size, and sum. This preserves the different reserve bases by group.

For group 1, the expected reserve per policy is

$$ 0.90(800)+0.10(6{,}000)=720+600=1{,}320. $$

Its expected total reserve is $120(1{,}320)=158{,}400$. For group 2, the expected reserve per policy is

$$ 0.60(1{,}200)+0.40(5{,}000)=720+2{,}000=2{,}720. $$

Its expected total reserve is $80(2{,}720)=217{,}600$. Therefore, the expected total for the portfolio is

$$ 158{,}400+217{,}600=376{,}000. $$

The expected reserve per policy across the 200 policies is $376{,}000/200=1{,}880$. The calculation checks by adding the expected state-specific amounts within each group: group 1 contributes $120(0.90)(800)=86{,}400$ active and $120(0.10)(6{,}000)=72{,}000$ disabled; group 2 contributes $80(0.60)(1{,}200)=57{,}600$ active and $80(0.40)(5{,}000)=160{,}000$ disabled. These four amounts sum to $376,000.

Common mistake avoided. Averaging the two groups’ state reserves and applying one pooled state distribution can pair the wrong probabilities with the wrong reserves. Keep each cohort’s probabilities and reserves together unless the policies truly share a common basis.

Common Interpretation and Timing Traps

The expected reserve is useful when summarizing modelled liabilities for a population, but it should not be misread as the reserve of every policy. In an actual policy file, if a policy is known to be disabled, use its disabled-state reserve. The portfolio total from known states is the sum of the reserves for the actual policies, not a fresh probability average over states whose status is already observed.

For an aggregate calculation, expected state counts can be a convenient intermediate step. For homogeneous policies, the expected number in state $j$ is $Np_j(t)$, so multiplying by $V_j(t)$ and summing gives the same result as multiplying the expected per-policy reserve by $N$. Expected counts need not be whole numbers; they are averages under the model, not predictions that a fractional policy will be observed.

Always check three items before accepting a calculation:

Matching valuation time: the occupancy probabilities and reserves must refer to the same time.
Matching state and timing conventions: include every state in the probability distribution, and use its reserve under the stated settlement timing.
Matching cohort assumptions: apply one set of probabilities and state reserves only to policies for which those assumptions are appropriate.

These calculations use state-specific prospective reserves and occupancy probabilities, a combination aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The essential technique is compact, but careful interpretation determines whether the result describes one known policy or an expected amount across a population.

Key takeaway: For a population with state probabilities $p_j(t)$ and conditional state reserves $V_j(t)$, the expected reserve per policy is $\sum_j p_j(t)V_j(t)$. Multiply by the number of homogeneous policies, or calculate and add the results by cohort. A policy’s known state reserve remains conditional on that state; it is not replaced by the population average.

Check Your Understanding

For each problem, use the probabilities and reserves at the stated valuation date. Treat all monetary amounts as dollars.

A homogeneous cohort of 300 policies has probabilities 0.75 active, 0.20 disabled, and 0.05 dead. The respective state reserves are $900, $4,500, and $0. Find the expected reserve per policy and for the cohort. (Answer: $1,575 per policy; $472,500 total)
A cohort of 240 policies has probabilities 0.85 active, 0.10 disabled, and 0.05 dead. The respective reserves are $1,100, $7,000, and $0. Calculate the expected aggregate reserve. (Answer: $392,400)
Policies are currently 70% active and 30% disabled. Over one year, transitions follow the matrix with rows and columns $A,D,X$: $\begin{pmatrix}0.80&0.15&0.05\\0.20&0.70&0.10\\0&0&1\end{pmatrix}$. At year-end, state reserves are $1,500 active, $4,000 disabled, and $0 dead. Find the expected reserve per policy at year-end. (Answer: $2,190)
A cohort of 50 policies has probabilities 0.80 active and 0.20 disabled, with reserves $500 active and $3,000 disabled. A second cohort of 30 has probabilities 0.50 active and 0.50 disabled, with reserves $900 active and $2,500 disabled. Find the expected total reserve across both cohorts. (Answer: $101,000)
A policy is known to be disabled at the valuation date, and its disabled-state reserve is $4,500. Across the full cohort, the state probabilities are 0.70 active, 0.25 disabled, and 0.05 dead, with reserves $1,000 active, $4,500 disabled, and $0 dead. State the conditional reserve for this known policy and the expected reserve per policy across the cohort. (Answer: $4,500 for the known disabled policy; $1,825 per policy across the cohort)

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