Conditional Versus Unconditional Reserves

Why the Conditioning Matters

A multi-state model can assign a different prospective reserve to a policy in each possible current state. An active policy might have one reserve, a disabled policy another, and a policy that has terminated with all obligations settled no further reserve. These state-specific amounts answer a conditional question: what is the expected value of future net liabilities, given the policy’s current state?

A different question arises when considering a cohort whose members occupy different states at the valuation time. Its average reserve per original policy is a probability-weighted average of the state-specific reserves. If the question concerns only policies still in force, the average must instead be conditioned on being in one of the in-force states. The two averages generally differ.

Definition: Let $J_t$ denote the policy’s state at valuation time $t$, and let $V_j(t)$ be the prospective reserve conditional on $J_t=j$. If $p_j(t)=P(J_t=j)$, the unconditional average reserve per original policy is $$ E[V_{J_t}(t)] = \sum_j p_j(t)V_j(t). $$ If $\mathcal{I}$ is the set of in-force states, the average reserve conditional on being in force is $$ E[V_{J_t}(t)\mid J_t\in\mathcal{I}] =\frac{\sum_{j\in\mathcal{I}}p_j(t)V_j(t)} {\sum_{j\in\mathcal{I}}p_j(t)}, $$ provided $P(J_t\in\mathcal{I})>0$. A state after termination, such as death after all benefits have been settled, is assigned zero reserve when calculating the average per original policy.

The state probabilities must refer to the same valuation time and population as the reserves. The reserve $V_j(t)$ is already conditional on state $j$; the probability $p_j(t)$ describes how often that state occurs in the cohort. Do not multiply a state-specific reserve by a probability that is itself conditional on a different event unless that is the quantity being calculated.

From State-Conditional Values to Cohort Averages

The unconditional average keeps the original cohort as its denominator, including policies that have left the in-force states. If the reserve after termination is zero, those policies contribute zero to the weighted sum. The in-force conditional average changes the denominator: it divides by the probability of being in any in-force state. It is not the unconditional average divided by the probability of one selected state.

For a cohort of $N$ policies, the expected aggregate reserve is $N$ times the unconditional average per original policy. Equivalently, if $N_j$ is the number in state $j$, the aggregate is $\sum_j N_jV_j(t)$. The average among in-force policies instead divides that aggregate by the expected number in force, $N\sum_{j\in\mathcal{I}}p_j(t)$, when using expected state counts.

Worked Example: Three States and Two Averages

At a valuation time, a hypothetical cohort has probabilities $0.70$ of being active, $0.20$ of being disabled, and $0.10$ of having died with all benefits settled. On a valuation basis using an effective annual interest rate of $4\%$ and the applicable future state-transition and mortality assumptions, the state-conditional reserves are $ \$8{,}000$ active, $ \$20{,}000$ disabled, and $ \$0$ after settled death. Find the unconditional average per original policy and the average conditional on being in force. The supplied reserves are the state-specific valuation results; the task is to combine them with the stated cohort probabilities.

Method. First multiply each state reserve by its probability. Then, for the in-force average, divide the total contribution from active and disabled policies by the probability of being in either of those states.

The unconditional average per original policy is $$ 0.70(8{,}000)+0.20(20{,}000)+0.10(0) =5{,}600+4{,}000+0 =\$9{,}600. $$

The probability of being in force is $$ 0.70+0.20=0.90. $$ Thus the average conditional on being in force is $$ \frac{0.70(8{,}000)+0.20(20{,}000)}{0.90} =\frac{9{,}600}{0.90} =\$10{,}666.666\ldots. $$ Rounded to the nearest cent, the in-force average is $\$10{,}666.67$.

Check. Multiplying the in-force average by the in-force probability gives $0.90(10{,}666.666\ldots)=9{,}600$, matching the in-force contribution to the unconditional average. The calculation avoids treating the dead-state probability as if it were part of the in-force denominator.

Why Probability Weighting Works

The state at valuation time determines which conditional reserve applies. Define a random variable $R_t$ to equal $V_j(t)$ when the policy is in state $j$. The events $J_t=j$ are mutually exclusive and exhaustive when all possible states, including termination, are included. Splitting the expectation of $R_t$ across these events gives

E[R_t] =\sum_j E[R_t\mid J_t=j]P(J_t=j) =\sum_j V_j(t)p_j(t).

The last equality holds because $R_t=V_j(t)$ whenever $J_t=j$, so its conditional expectation in that state is $V_j(t)$. To condition on being in force, apply the definition of conditional expectation: divide the expected reserve contribution from in-force states by the probability of being in those states. This explains both the weighted sum and the denominator in the in-force formula.

This calculation averages reserves across states; it does not recalculate the prospective reserve within each state. The underlying reserves must already be valued on a consistent basis and at the same time point. The interest rate and future transition assumptions used to determine those reserves are distinct from the current-state probabilities used to average them.

Worked Example: Obtain the State Probabilities First

A hypothetical cohort begins a year with $0.80$ of its policies active and $0.20$ disabled. During the year, the transition probabilities are as follows: an active policy remains active with probability $0.80$, becomes disabled with probability $0.15$, and dies with probability $0.05$; a disabled policy becomes active with probability $0.10$, remains disabled with probability $0.75$, and dies with probability $0.15$. Death is absorbing and all benefits are settled at death. At the valuation time, state-conditional reserves are $\$6{,}000$ active, $\$18{,}000$ disabled, and $\$0$ dead. These reserves use an effective annual interest rate of $4\%$ and the stated future mortality and transition basis. Find both averages.

Method. Use the law of total probability to find the state distribution at the valuation time. The initial states are mutually exclusive, so each final-state probability is the sum of the paths leading to that state.

The active probability is $$ 0.80(0.80)+0.20(0.10)=0.64+0.02=0.66. $$ The disabled probability is $$ 0.80(0.15)+0.20(0.75)=0.12+0.15=0.27. $$ The dead probability is $$ 0.80(0.05)+0.20(0.15)=0.04+0.03=0.07. $$ The probabilities sum to $0.66+0.27+0.07=1.00$, as required.

The unconditional average reserve per original policy is $$ 0.66(6{,}000)+0.27(18{,}000)+0.07(0) =3{,}960+4{,}860 =\$8{,}820. $$ The in-force probability is $0.66+0.27=0.93$. Therefore, the in-force conditional average is $$ \frac{8{,}820}{0.93}=\$9{,}483.870967\ldots. $$ Rounded to the nearest cent, it is $\$9{,}483.87$.

Check. The active and disabled probabilities are the only probabilities in the in-force denominator. Their weighted reserve contribution is $\$8{,}820$; dividing the conditional average $\$9{,}483.870967\ldots$ by neither the active probability alone nor the total probability of one initial state is appropriate.

Expected Aggregate Reserves and Interpretation

A per-policy average and an aggregate reserve have different units. If a hypothetical cohort has $1{,}000$ policies with state probabilities $0.72$ active, $0.18$ disabled, and $0.10$ terminated, and conditional reserves of $\$3{,}500$, $\$12{,}500$, and zero, its unconditional average is

0.72(3{,}500)+0.18(12{,}500)=2{,}520+2{,}250=\$4{,}770 \quad\text{per original policy}.

The expected aggregate reserve is $1{,}000(4{,}770)=\$4{,}770{,}000$. The expected number in force is $1{,}000(0.72+0.18)=900$, so the average conditional on being in force is $\$4{,}770{,}000/900=\$5{,}300$. This agrees with $\$4{,}770/0.90=\$5{,}300$.

These are expectations based on the state probabilities, not statements that a fractional number of policies is observed. Actual state counts in a finite cohort may differ from their expected values. If actual counts are supplied, calculate the actual aggregate from those counts and reserves, then divide by the actual number of policies in the relevant population.

Common Exam Trap: The Wrong Conditioning Denominator

In the three-state example, one incorrect calculation is

\frac{0.70(8{,}000)+0.20(20{,}000)}{0.80} =\frac{9{,}600}{0.80} =\$12{,}000.

This divides by the active probability alone while retaining the disabled policies’ reserve contribution in the numerator. The numerator includes both active and disabled states, so its matching denominator must include both: $0.70+0.20=0.90$. The corrected in-force average is $9{,}600/0.90=\$10{,}666.67$, not $\$12{,}000$.

Another error is to take the simple arithmetic mean of the state reserves. That would give $(8{,}000+20{,}000+0)/3$, which treats the three states as equally likely. Unless their probabilities are equal, this is not the cohort average. Always pair each state reserve with its probability before summing.

Key takeaway: Use $\sum_j p_j(t)V_j(t)$ for the average per original policy. For the average conditional on being in force, retain only in-force contributions in the numerator and divide by the sum of their probabilities. Match the numerator and denominator to the same population.

Check Your Understanding

For each problem, identify the relevant state probabilities before averaging. Treat any stated terminated state as having zero reserve.

At valuation, a hypothetical cohort has probabilities $0.60$ active, $0.25$ disabled, and $0.15$ dead. The state reserves are $\$5{,}000$, $\$17{,}000$, and zero, respectively. Find the unconditional average and the average conditional on being in force. (Answer: $\$7{,}250$ per original policy; $\$8{,}529.41$ conditional on being in force.)
A cohort starts with probabilities $0.70$ active and $0.30$ disabled. In one year, an active policy has probabilities $0.85$, $0.10$, and $0.05$ of ending active, disabled, and dead; a disabled policy has probabilities $0.05$, $0.80$, and $0.15$ of ending active, disabled, and dead. At the valuation time the reserves are $\$7{,}000$ active, $\$15{,}000$ disabled, and zero dead. Find the state probabilities and both averages. (Answer: probabilities $0.61, 0.31, 0.08$; unconditional $\$8{,}920$; in-force conditional $\$9{,}695.65$.)
A hypothetical cohort has probabilities $0.45$ active, $0.35$ disabled, and $0.20$ dead, with corresponding state reserves of $\$12{,}000$, $\$9{,}000$, and zero. Find the average per original policy and the average among in-force policies. (Answer: $\$8{,}550$ per original policy; $\$10{,}687.50$ among in-force policies.)
A group of $2{,}000$ hypothetical policies has state probabilities $0.72$ active, $0.18$ disabled, and $0.10$ terminated. The reserves are $\$3{,}500$ active, $\$12{,}500$ disabled, and zero terminated. Find the expected aggregate reserve, the expected number in force, and the average reserve per in-force policy. (Answer: $\$9{,}540{,}000$; $1{,}800$; $\$5{,}300$.)
A cohort starts with probabilities $0.60$ active and $0.40$ disabled. Over the next year, active policies have probabilities $0.90$, $0.06$, and $0.04$ of ending active, disabled, and dead; disabled policies have probabilities $0.12$, $0.73$, and $0.15$ of ending active, disabled, and dead. At valuation, the reserves are $\$8{,}000$ active, $\$24{,}000$ disabled, and zero dead. Calculate the final state distribution, the unconditional average, and the in-force conditional average. (Answer: probabilities $0.588, 0.328, 0.084$; unconditional $\$12{,}576$; in-force conditional $\$13{,}729.26$.)

What You'll Learn