Integrated Multi-State Prospective Reserve Workshop

Why Integrated Reserve Calculations Need a Consistent Convention

A multi-state reserve calculation can combine premiums, benefits, recovery, disability, and termination. The arithmetic is usually manageable; the main challenge is keeping each cash flow at the correct time and attaching it to the correct state or transition. A benefit paid on entering a state is not the same as a reserve held while in that state, and a termination payment does not imply a continuation reserve after termination.

This tutorial uses annual time steps. The examples are hypothetical contracts, and all probabilities are conditional on the policy being in the stated state at the start of the year. Interest is an effective annual rate. The approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: The prospective reserve $V_j(t)$ is the expected present value, at time $t$, of future benefits and expenses less future premiums, conditional on the contract being in state $j$ at time $t$. In this tutorial, $V_j(t)$ is measured immediately before any premium or other time-$t$ cash flow. A positive amount is a net liability; a negative amount means the expected value of future premiums exceeds that of future benefits and expenses.

For a one-year step, let $p_{jk}(t)$ be the probability of being in state $k$ at time $t+1$, given state $j$ at time $t$. Let $C_j(t)$ be the net time-$t$ outgo: benefits and expenses due then, less premiums due then. Let $b_{jk}(t+1)$ be the net outgo paid at time $t+1$ if the transition is from $j$ to $k$. The transition amount can be zero. If $i$ is the effective annual interest rate, $v=1/(1+i)$.

Reserve recursion: For a discrete-time model with the timing just specified, $$ V_j(t)=C_j(t)+v\sum_k p_{jk}(t)\bigl[b_{jk}(t+1)+V_k(t+1)\bigr]. $$ At a terminal time, set the continuation reserve to the value specified by the contract. For a state in which the contract has terminated and no further amounts are due, that value is zero. A termination benefit is included in $b_{jk}$, not in the post-termination reserve.

Why the Recursion Holds

The recursion follows by conditioning on the state at the end of the year. The time-$t$ cash flow $C_j(t)$ is certain once the current state is known. For each possible destination $k$, the end-of-year outgo is $b_{jk}(t+1)$, followed by the continuation value $V_k(t+1)$. The probability of that branch is $p_{jk}(t)$, and its value at time $t$ is discounted by $v$.

\begin{aligned} V_j(t) &=C_j(t)+\sum_k p_{jk}(t)\,v\bigl[b_{jk}(t+1)+V_k(t+1)\bigr]\\ &=C_j(t)+v\sum_k p_{jk}(t)\bigl[b_{jk}(t+1)+V_k(t+1)\bigr]. \end{aligned}

This is a one-step conditioning argument, not a continuous-time reserve equation. It applies to the annual cash-flow convention stated here; if premiums or benefits occur at different times, their present values must be placed at those times rather than inserted unchanged into this recursion.

Specify the states and terminal conditions.
Identify which states continue the contract and which have no further reserve.

Fix the timing.
List start-of-year premiums or expenses separately from year-end transition payments.

Work backward.
Calculate the reserves with one year remaining first, then substitute those values into earlier years.

Check the result.
Verify probability rows sum to one, destination reserves match their states, and termination cash flows are not carried beyond termination.

Worked Problems by State and Time

The examples use a finite horizon and annual transitions. A payment described as being made when the year ends in Disabled is included whenever the destination state is Disabled, whether the policy entered that state or remained there. A payment on transition to Exit is made only on that transition.

Worked Example: A Two-Year Contract With Disability and Exit Payments

A hypothetical two-year contract has states Active (A), Disabled (D), and Exit (X). The effective annual interest rate is 4%. A premium of $700 is due at the start of each year in A and no premium is due in D. At each year-end, $900 is paid if the destination state is D, and $3,000 is paid on transition to X. State X is absorbing, with no continuation reserve. In the final year, the transition probabilities from A are 0.70 to A, 0.20 to D, and 0.10 to X; from D they are 0.10 to A, 0.80 to D, and 0.10 to X. In the first year, from A the probabilities are 0.60, 0.30, and 0.10, respectively. Find the time-0 reserve in A.

Method and terminal values. At time 2 the contract has ended, so the continuation reserves are zero. Let $v=1/1.04=25/26$. At time 1, one year remains. In A, the immediate premium is an outgo of $-$700$, using the net-outgo convention. The only positive year-end amounts are the disability and exit payments:

\begin{aligned} V_A(1) &=-700+\frac{0.20(900)+0.10(3{,}000)}{1.04}\\ &=-700+\frac{480}{1.04} =-238.4615. \end{aligned}

In D there is no premium. Thus

\begin{aligned} V_D(1) &=\frac{0.80(900)+0.10(3{,}000)}{1.04}\\ &=\frac{1{,}020}{1.04} =980.7692. \end{aligned}

Move back to time 0. The first-year transition probabilities from A differ from the final-year probabilities. If the destination is D, both the $900 transition-contingent payment and the continuation reserve $V_D(1)$ are included. Therefore

\begin{aligned} V_A(0) &=-700+\frac{0.60V_A(1)+0.30[900+V_D(1)]+0.10(3{,}000)}{1.04}\\ &=-700+\frac{0.60(-238.4615)+0.30(1{,}880.7692)+300}{1.04}\\ &=-700+\frac{-143.0769+564.2308+300}{1.04}\\ &=-700+\frac{721.1539}{1.04}\\ &=-6.58. \end{aligned}

The reserve is negative because, under the stated assumptions, the expected present value of future premiums slightly exceeds that of future benefits. This is not by itself an arithmetic error.

Common mistake avoided: The $900 paid when the year ends in D must not replace $V_D(1)$; the contract has both the payment and the future reserve. Conversely, the $3,000 Exit payment is included, but there is no $V_X(1)$ in addition to it.

Worked Example: Recovery, Death, and State-Dependent Premiums

A hypothetical two-year income-protection contract has states Active (A), Disabled (D), and Death (M). The annual effective interest rate is 5%. A $1,000 premium is due at the start of a year in A only. At year-end, $1,200 is paid if the destination is D, and $6,000 is paid on transition to M. There is no payment for a transition from D back to A. Death is absorbing and has zero continuation reserve. In the final year, the probabilities from A are 0.80 to A, 0.15 to D, and 0.05 to M; from D they are 0.25 to A, 0.65 to D, and 0.10 to M. In the first year, from A the probabilities are 0.70, 0.20, and 0.10; from D they are 0.20, 0.70, and 0.10. Calculate the time-0 reserves in A and D.

Final-year reserves. Let $v=1/1.05=20/21$. At time 1, a policy in A pays the premium and then may make a disability or death payment:

\begin{aligned} V_A(1) &=-1{,}000+\frac{0.15(1{,}200)+0.05(6{,}000)}{1.05}\\ &=-1{,}000+\frac{480}{1.05} =-542.8571. \end{aligned}

A policy in D has no premium at time 1:

\begin{aligned} V_D(1) &=\frac{0.65(1{,}200)+0.10(6{,}000)}{1.05}\\ &=\frac{1{,}380}{1.05} =1{,}314.2857. \end{aligned}

Time-0 reserve in A. The first-year premium is due now. A transition to D includes the year-end benefit as well as the reserve in D:

\begin{aligned} V_A(0) &=-1{,}000+\frac{0.70V_A(1)+0.20[1{,}200+V_D(1)]+0.10(6{,}000)}{1.05}\\ &=-1{,}000+\frac{0.70(-542.8571)+0.20(2{,}514.2857)+600}{1.05}\\ &=-1{,}000+\frac{-380+502.8571+600}{1.05}\\ &=-1{,}000+688.4354\\ &=-311.56. \end{aligned}

Time-0 reserve in D. No premium is due in D at time 0. Recovery to A leads to $V_A(1)$; remaining Disabled leads to the $1,200 payment and $V_D(1)$:

\begin{aligned} V_D(0) &=\frac{0.20V_A(1)+0.70[1{,}200+V_D(1)]+0.10(6{,}000)}{1.05}\\ &=\frac{0.20(-542.8571)+0.70(2{,}514.2857)+600}{1.05}\\ &=\frac{-108.5714+1{,}760+600}{1.05}\\ &=2{,}144.22. \end{aligned}

The different signs and magnitudes are plausible: the Active reserve reflects future premiums, while the Disabled reserve has no premiums and includes a high probability of future disability payments. The recovery branch correctly returns to the Active reserve, including its future premium pattern.

Worked Example: Recovery and an Expected-Reserve Check

A hypothetical contract has states Active (A), Disabled (D), and Exit (X), with two years remaining and an effective annual interest rate of 25%, so $v=0.80$. A $300 premium is due at the start of a year in A only. At year-end, $500 is paid if the destination is D, and $2,000 is paid on transition to X. X has no continuation reserve. In the final year, A transitions to A, D, and X with probabilities 0.60, 0.30, and 0.10; D transitions to A, D, and X with probabilities 0.40, 0.50, and 0.10. In the first year, A transitions with probabilities 0.70, 0.20, and 0.10, while D transitions with probabilities 0.30, 0.60, and 0.10. Find both time-0 state reserves and check the time-1 expected reserve for a policy initially in A.

Work backward from the final year. With one year remaining, an Active policy has an immediate premium, and its final-year probabilities give expected year-end benefits of $0.30(500)+0.10(2{,}000)=350$:

$$ V_A(1)=-300+0.80(350)=-20. $$

For a Disabled policy, the expected year-end benefit is $0.50(500)+0.10(2{,}000)=450$, with no premium:

$$ V_D(1)=0.80(450)=360. $$

Calculate time-0 reserves. From A, a transition to D produces the $500 payment and the continuation reserve of $360; an Exit transition produces $2,000 and no continuation reserve:

\begin{aligned} V_A(0) &=-300+0.80\{0.70(-20)+0.20(500+360)+0.10(2{,}000)\}\\ &=-300+0.80(-14+172+200)\\ &=-300+286.40 =-13.60. \end{aligned}

For a policy initially in D, recovery to A leads to $V_A(1)$, while remaining Disabled leads to its payment and $V_D(1)$:

\begin{aligned} V_D(0) &=0.80\{0.30(-20)+0.60(500+360)+0.10(2{,}000)\}\\ &=0.80(-6+516+200) =568.00. \end{aligned}

Check by weighting the time-1 reserves. Starting in A at time 0, the time-1 state probabilities are 0.70 in A, 0.20 in D, and 0.10 in X. The expected time-1 continuation reserve per original policy is $0.70(-20)+0.20(360)+0.10(0)=58$. The expected year-end payment is $0.20(500)+0.10(2{,}000)=300$. Thus the time-0 recursion gives $-300+0.80(58+300)=-13.60$, agreeing with the state-specific calculation.

Common mistake shown and corrected: If the $500 payment on transition to D is omitted at time 0, the incorrect calculation is $V_A(0)=-300+0.80\{0.70(-20)+0.20(360)+0.10(2{,}000)\}=-93.60$. The correction is to use $0.20[500+V_D(1)]$, giving $-$13.60$. The transition payment and destination reserve are separate components.

Checks That Catch Integrated-Problem Errors

A reserve calculation should be reviewed as a model, not only as a final number. The following checks are useful when several states, cash flows, and time periods are involved.

Check	What to verify
Probability rows	For every state and year, transition probabilities are nonnegative and sum to 1.
Cash-flow timing	Start-of-year premiums are not discounted; year-end amounts are discounted one year in the recursion.
Destination reserve	Each transition uses the reserve belonging to its destination state and next valuation time.
Termination	A termination payment is included on the transition, while the terminated policy's continuation reserve is zero if no later amounts are due.
Population weighting	An expected reserve across states uses the state probabilities and includes a zero reserve for terminated policies when appropriate.

Do not assume every state reserve must be positive. Premium timing and state-dependent premium cessation can make a reserve negative in one state and positive in another. The sign should be interpreted using the contract cash flows and the stated valuation convention.

Key takeaway: For each state and time, identify the immediate net cash flow, enumerate the possible destination states, add each transition payment to the correct continuation reserve, weight by the conditional transition probability, and discount to the valuation time. Then verify the terminal values and state-probability weighting.

Check Your Understanding

For each problem, use the annual timing convention from this tutorial and round money to the nearest cent.

A one-year contract has states A, D, and X. At 10% effective annual interest, an A policy pays a $200 premium now. At year-end, $500 is paid if the destination is D and $1,000 on transition to X. From A, the probabilities are 0.60, 0.30, and 0.10. All terminal continuation reserves are zero. Find $V_A(0)$. (Answer: $27.27)
One year remains and a policy is in D. At 5% effective annual interest, the year-end transition probabilities to A, D, and X are 0.20, 0.70, and 0.10. A $800 payment is made if the destination is D and $3,000 on transition to X; there is no premium. Find $V_D$. (Answer: $819.05)
A two-year contract has states A, D, X and $v=0.80$. A $250 premium is due at the start of each year in A only. A $400 payment is made at year-end if the destination is D, and $1,500 on transition to X. With one year remaining, A transitions to A, D, X with probabilities 0.50, 0.30, 0.20; D transitions with probabilities 0.25, 0.65, 0.10. In the first year, an A policy transitions with probabilities 0.60, 0.25, 0.15. Find the time-0 reserve in A. (Answer: $116.88)
At time 1, the probabilities of being in A, D, and X for a policy cohort are 0.65, 0.25, and 0.10. The corresponding reserves are $-$40$, $300, and $0. Find the expected time-1 reserve per original policy and, conditional on being in A or D, the expected reserve. (Answer: $49.00 per original policy; $54.44 conditional on remaining in A or D)
One year remains, the effective annual interest rate is 4%, and a policy is in D. At year-end it transitions to A, D, or X with probabilities 0.50, 0.40, and 0.10. The year-end payment is $600 if the destination is D and $2,400 on transition to X. The next-time reserves are $V_A=100$, $V_D=500$, and $V_X=0$. There is no current premium. Find $V_D$. (Answer: $701.92)

What You'll Learn