Deriving the One-Step Reserve Recursion

Why a One-Step Reserve Recursion Works

A reserve at a valuation time represents the amount needed, on the chosen assumptions, to meet future net outgo. In a multi-state contract, that future depends on which state the policy reaches next. The one-step recursion makes this dependence explicit: for each possible next state, combine the cash flow paid during the period with the reserve required after the period, weight by the transition probability, and discount the result to the current valuation time.

The timing convention matters. The recursion below assumes that the reserve is measured just after any cash flows due at time $t$, and that the net cash flow for the next period is paid at time $t+1$. This convention lets us compare all amounts on the same date before discounting.

Definition: Suppose a policy can occupy states $1,\ldots,m$. Let $V_i(t)$ be the reserve at time $t$, conditional on the policy being in state $i$ just after time-$t$ cash flows. Let $p_{ij}(t)$ be the probability that the policy is in state $j$ at time $t+1$, given that it is in state $i$ at time $t$. Let $C_{ij}(t)$ be the net outgo at time $t+1$ on the transition from $i$ to $j$: benefits and expenses less premiums, with a positive value meaning net outgo. If the effective annual interest rate is $i$, the one-step reserve recursion is

V_i(t)=\frac{1}{1+i}\sum_{j=1}^{m}p_{ij}(t)\bigl(C_{ij}(t)+V_j(t+1)\bigr).

The probabilities in row $i$ sum to one across the possible next states. A transition with zero probability contributes nothing. The reserve in a terminal state can be set to zero if no future payments remain; a payment due at termination must still be included in the relevant $C_{ij}(t)$.

The cash-flow quantity $C_{ij}(t)$ is defined for the full transition over the period. It can combine a premium, an expense, and a benefit when the contract’s payment convention says those amounts are due at the period end. If amounts are actually paid at different times, they must first be converted to a consistent valuation date; the displayed formula cannot be used unchanged with cash flows from different dates.

Deriving the Recursion by Conditioning

The derivation uses the reserve as a fund available at time $t$. If the policy is in state $i$, the fund $V_i(t)$ accumulates for one year at rate $i$. At time $t+1$, the accumulated fund must meet the net outgo for the period and leave the reserve appropriate to the state then occupied. Since the next state is uncertain at time $t$, the required amount at $t+1$ is conditional on that state.

Accumulate the current reserve.
At time $t+1$, the amount developed from the time-$t$ reserve is $(1+i)V_i(t)$.

Write the requirement for each next state.
If the transition is from $i$ to $j$, the fund at time $t+1$ must cover $C_{ij}(t)$ and leave $V_j(t+1)$. The total requirement is $C_{ij}(t)+V_j(t+1)$.

Take the conditional expectation.
Given state $i$ at time $t$, transition $i$ to $j$ occurs with probability $p_{ij}(t)$. Thus the expected requirement at time $t+1$ is $\sum_j p_{ij}(t)[C_{ij}(t)+V_j(t+1)]$.

Equate and discount.
Set the accumulated current reserve equal to that expected requirement, then divide by $1+i$ to obtain the recursion at time $t$.

In symbols, the balance equation is $(1+i)V_i(t)=\sum_j p_{ij}(t)[C_{ij}(t)+V_j(t+1)]$. Dividing both sides by $1+i$ gives the displayed formula. This also explains why the cash flow and the next reserve appear together inside the same weighted sum: both are required at the end of the period, and both depend on the transition.

Worked Applications

Worked Example: A Reserve with Three Possible Transitions

Consider a hypothetical one-year policy with active, disabled, and dead states, denoted $A,D,M$. The annual effective interest rate is $5\%$. A policy active at time $t$ has transition probabilities $p_{AA}=0.80$, $p_{AD}=0.15$, and $p_{AM}=0.05$. The end-of-year net outgo is $C_{AA}=\$1{,}000$, $C_{AD}=\$6{,}000$, and $C_{AM}=\$20{,}000$. At time $t+1$, the reserves are $V_A=\$400$, $V_D=\$1{,}200$, and $V_M=\$0$. Find the reserve at time $t$, conditional on being active.

Method. Apply the recursion to each possible transition. For each row, add the transition-specific end-of-year outgo to the reserve in the destination state, multiply by the transition probability, and discount the total one year.

V_A(t)=\frac{0.80(1{,}000+400)+0.15(6{,}000+1{,}200)+0.05(20{,}000+0)}{1.05}.

The three weighted contributions are $0.80(1{,}400)=1{,}120$, $0.15(7{,}200)=1{,}080$, and $0.05(20{,}000)=1{,}000$. Their sum is $\$3{,}200$, so $V_A(t)=3{,}200/1.05=\$3{,}047.6190\ldots$. Rounded to the nearest cent, the reserve is \$3,047.62.

Check. Accumulating this reserve for one year gives $3{,}047.6190\ldots(1.05)=\$3{,}200$, which equals the probability-weighted requirement at year end. The common mistake avoided here is to discount only the future reserves while leaving the year-end benefits undiscounted.

Worked Example: Premiums as Negative Net Outgo

A hypothetical contract begins the year in state $A$. The effective annual interest rate is $4\%$. The next-state probabilities are $0.90$ for $A$, $0.08$ for $D$, and $0.02$ for death $M$. Under the specified end-of-year payment convention, the net outgo amounts, after allowing for any premiums due, are $C_{AA}=-\$800$, $C_{AD}=\$4{,}200$, and $C_{AM}=\$12{,}000$. The time-$t+1$ reserves are $\$500$ in $A$, $\$1{,}500$ in $D$, and zero in $M$. Calculate $V_A(t)$.

Method. Use the same conditional recursion, keeping the negative net outgo in the active-to-active outcome. A negative $C_{AA}$ means that the premium exceeds benefits and expenses for that transition; it is not a negative reserve.

V_A(t)=\frac{0.90(-800+500)+0.08(4{,}200+1{,}500)+0.02(12{,}000+0)}{1.04}.

The transition totals are $-\$300$, $\$5{,}700$, and $\$12{,}000$. Their probability-weighted contributions are $-\$270$, $\$456$, and $\$240$, respectively. The sum is $-270+456+240=\$426$. Therefore $V_A(t)=426/1.04=\$409.6153\ldots$, or \$409.62 to the nearest cent.

Check. Accumulation gives $\$409.6153\ldots(1.04)=\$426$, matching the weighted end-of-year amount. A common mistake is to treat a premium as positive outgo. With the stated sign convention, premiums reduce $C_{ij}$, so the active-to-active net outgo is $-\$800$, not $+\$800$.

Worked Example: Recursing Back Through Two Durations

A hypothetical two-year policy has active state $A$, disabled state $D$, and death state $M$. The annual effective interest rate is $10\%$, and all cash flows are end-of-year net outgo. At time $1$, the transition probabilities from $A$ to $A,D,M$ are $0.70,0.20,0.10$, with corresponding outgo amounts $\$300,\$5{,}000,\$15{,}000$. From $D$, the probabilities to $A,D,M$ are $0.10,0.80,0.10$, with outgo amounts $\$1{,}000,\$4{,}000,\$10{,}000$. All reserves at time $2$ are zero. At time $0$, an active policy has probabilities $0.80,0.15,0.05$ to $A,D,M$, with outgo amounts $\$200,\$1{,}000,\$10{,}000$. Find its time-0 reserve.

Method. Work backward. First calculate the time-1 reserves using the zero terminal reserves, then use those results as the next-state reserves in the time-0 recursion.

For an active policy at time $1$, the weighted requirement at time $2$ is $0.70(300)+0.20(5{,}000)+0.10(15{,}000)=210+1{,}000+1{,}500=\$2{,}710$. Hence $V_A(1)=2{,}710/1.10=\$2{,}463.6364\ldots$.

For a disabled policy at time $1$, the corresponding amount is $0.10(1{,}000)+0.80(4{,}000)+0.10(10{,}000)=100+3{,}200+1{,}000=\$4{,}300$. Hence $V_D(1)=4{,}300/1.10=\$3{,}909.0909\ldots$. The death-state reserve is zero.

Now calculate the time-0 requirement conditional on state $A$. The active transition contributes $0.80(200+2{,}463.6364\ldots)=\$2{,}130.9091\ldots$. The disabled transition contributes $0.15(1{,}000+3{,}909.0909\ldots)=\$736.3636\ldots$. The death transition contributes $0.05(10{,}000+0)=\$500$. The total is $\$3{,}367.2727\ldots$, so $V_A(0)=3{,}367.2727\ldots/1.10=\$3{,}061.1570\ldots$. Rounded to the nearest cent, the reserve is \$3,061.16.

Check. Accumulating the time-0 reserve gives approximately $\$3{,}367.27$, the sum of the three weighted transition requirements. The common mistake avoided is to use zero reserves at time $1$ merely because the terminal reserves at time $2$ are zero.

Interpretation, Assumptions, and Common Traps

This recursion is a one-period balance relation under the specified transition probabilities, interest rate, and cash-flow timing. It does not require that the next state be known at time $t$; instead, it averages the time-$t+1$ requirements conditional on the current state. Nor does it imply that each outcome’s reserve is individually equal to the current reserve accumulated with interest. The equality is between the accumulated current reserve and the probability-weighted requirement.

Use the following checks whenever applying the formula:

Check the time unit. The transition probabilities and interest accumulation must refer to the same period. For a one-year step, use the annual effective rate; for a different period, use the matching accumulation factor.
Check the sign. Benefits and expenses are positive net outgo; premiums reduce net outgo.
Check the destination state. The reserve after the period is $V_j(t+1)$ for the state actually reached, not the reserve in the starting state.
Check payment timing. The displayed recursion places all amounts inside the sum at time $t+1$. Cash flows paid at another time need appropriate adjustment before they can be combined this way.
Check terminal treatment. A zero terminal reserve does not erase a benefit payable at termination; include that payment in the period’s net outgo.

The recursion is useful because it turns a state-contingent valuation into a systematic backward calculation. Once the terminal values and each period’s transition probabilities and net outgo are specified, reserves can be calculated state by state and duration by duration. This is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus, where careful conditioning, valuation dates, and state-specific cash flows are essential.

Check Your Understanding

For each question, use the convention that net outgo and the next reserve are valued at the end of the period unless stated otherwise.

A one-year model has effective annual interest $5\%$. From state $A$, the probabilities to $A,D,M$ are $0.75,0.20,0.05$. The respective net outgo amounts are $\$100,\$4{,}000,\$12{,}000$, and the next reserves are $\$300,\$1{,}000,\$0$. Find $V_A(t)$. (Answer: \$1,809.52)
At $4\%$ effective annual interest, a policy in state $A$ transitions to $A,D,M$ with probabilities $0.60,0.30,0.10$. Net outgo is $-\$500,\$2{,}500,\$9{,}000$, and next reserves are $\$700,\$2{,}000,\$0$. Calculate $V_A(t)$. (Answer: \$2,278.85)
At $5\%$ effective annual interest, a policy in state $D$ transitions to $A,D,M$ with probabilities $0.25,0.65,0.10$. The transition net outgo amounts are $\$800,\$3{,}000,\$10{,}000$, and next reserves are $\$1{,}200,\$2{,}500,\$0$. Find $V_D(t)$. (Answer: \$4,833.33)
At $10\%$ effective annual interest, the time-1 reserves are $\$1{,}000$ in $A$ and $\$3{,}000$ in $D$. From $A$ at time 0, the probabilities to $A,D$ are $0.80,0.20$, and the corresponding end-of-year net outgo is $\$200,\$1{,}000$. Find the time-0 reserve. (Answer: \$1,600.00)
An analyst writes $V_i(t)=\sum_jp_{ij}(t)C_{ij}(t)+v\sum_jp_{ij}(t)V_j(t+1)$, where $v=1/(1+i)$, for cash flows and reserves both payable at time $t+1$. State the error and give the corrected expression. (Answer: The end-of-period outgo must also be discounted; $V_i(t)=v\sum_jp_{ij}(t)[C_{ij}(t)+V_j(t+1)]$.)

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