Solving Reserve Tables from Terminal Conditions

Why Terminal Conditions Let Us Work Backward

A reserve table is often calculated from the end of a contract term toward its beginning. At maturity, the future obligations may be known directly: perhaps no liability remains after a final benefit is paid, or a specified reserve is assigned to each state at the terminal duration. Once those terminal values are set, the one-step reserve recursion determines the value one period earlier. Repeating that calculation produces the earlier state-specific reserves.

The direction of calculation matters. At each step, use the transition probabilities and cash flows for that particular period, and use the reserve in the state reached at the end of the period. A terminal reserve is not automatically the same thing as a benefit paid at maturity; the timing convention must keep those amounts separate.

Definition: In a discrete-time multiple-state model, let $V_{t,i}$ be the prospective reserve at time $t$, conditional on being in state $i$. Let $p_{t,ij}$ be the probability of being in state $j$ at time $t+1$, given state $i$ at time $t$. Let $c_{t,ij}$ be the net cash outgo paid at the end of the period if the transition is from $i$ to $j$. Benefits are positive outgo and premiums received are negative outgo. For effective annual interest rate $i$, the one-period recursion is $$ V_{t,i}=v\sum_j p_{t,ij}\bigl(c_{t,ij}+V_{t+1,j}\bigr), \qquad v=\frac{1}{1+i}. $$

This convention treats $V_{t,i}$ as the value immediately before the period begins and $V_{t+1,j}$ as the reserve after the period-end cash flow has been paid. Thus, the cash flow $c_{t,ij}$ and the next reserve $V_{t+1,j}$ are both measured at time $t+1$, and their sum is discounted once. A premium or benefit paid at another time must be moved to the appropriate valuation time; it cannot be inserted into this formula unchanged.

For a finite horizon ending at time $n$, specify a terminal reserve $V_{n,j}$ for each possible state $j$. Then calculate $V_{n-1,i}$, followed by $V_{n-2,i}$, and continue until the desired duration is reached. The transition probabilities and cash flows may vary by duration.

Why the Backward Recursion Holds

The recursion follows from conditioning on the state reached at the end of the period. Conditional on a transition from $i$ to $j$, the insurer pays the period-end net outgo $c_{t,ij}$ and then holds a reserve $V_{t+1,j}$. Both together are a time-$t+1$ amount. The probability of that transition is $p_{t,ij}$, so the expected time-$t+1$ amount is the probability-weighted sum over destination states. Discounting that expectation one year gives the reserve at time $t$.

\begin{aligned} V_{t,i} &=v\,E\!\left[c_{t,I_t,I_{t+1}}+V_{t+1,I_{t+1}}\mid I_t=i\right]\\ &=v\sum_j p_{t,ij}\bigl(c_{t,ij}+V_{t+1,j}\bigr). \end{aligned}

Here $I_t$ denotes the state occupied at time $t$. This argument assumes the listed transition probabilities describe the one-period state changes and that the cash flow is paid at the end of the period conditional on that transition. If cash flows occur continuously, at the beginning of the period, or at several different times, this discrete end-of-period formula requires adjustment.

In matrix form, let $P_t=(p_{t,ij})$, and define $g_{t,i}=\sum_jp_{t,ij}c_{t,ij}$, the expected period-end net outgo conditional on starting in state $i$. Then

\boldsymbol V_t=v\bigl(P_t\boldsymbol V_{t+1}+\boldsymbol g_t\bigr).

The matrix form emphasizes that the reserve calculation is a backward projection: the known vector $\boldsymbol V_{t+1}$ is used to obtain $\boldsymbol V_t$. A reserve may be negative if expected future premiums exceed expected future benefits and expenses. That result is not, by itself, evidence of an arithmetic error.

Backward Calculation in Practice

Set the terminal vector.
Write a reserve for every state at the final duration, using the stated valuation point and benefit timing.

Choose the period being calculated.
Use the transition probabilities and cash flows applicable from time $t$ to time $t+1$, not automatically those from another duration.

Calculate each starting-state reserve.
For each state $i$, combine the period-end cash flow and destination-state reserve for every possible transition, weight by its probability, and discount once.

Repeat and check.
Continue backward. Substitute a calculated reserve into the forward recursion to confirm that it reproduces the expected period-end amount.

Worked Example: One-Year Reserve With a Premium and Disability Benefit

A hypothetical one-year contract has active and disabled states. At time 1, the specified terminal reserves are $V_{1,A}=0$ and $V_{1,D}=\$100$. The annual effective interest rate is 5%, so $v=1/1.05$. There is no mortality in this simplified model. Conditional on the state at time 0, transitions have categorical distributions: an active policy remains active with probability 0.80 and becomes disabled with probability 0.20; a disabled policy recovers to active with probability 0.25 and remains disabled with probability 0.75. At the end of the year, a $\$50$ premium is received only if the policy remains active, and a $\$30$ disability payment is made if the policy started disabled, regardless of its destination state. No other cash flows occur.

Method. Apply the one-period backward recursion, representing the premium as negative outgo and the disability payment as positive outgo. The terminal reserves apply after the end-of-year cash flows.

For an active policy, the active-to-active transition has $c_{AA}=-\$50$; the active-to-disabled transition has $c_{AD}=0$. Therefore,

\begin{aligned} V_{0,A} &=\frac{0.80(-50+0)+0.20(0+100)}{1.05}\\ &=\frac{-40+20}{1.05} =\frac{-20}{1.05} =-\$19.0476\ldots \end{aligned}

For a disabled policy, the $\$30$ payment applies on either transition. Thus $c_{DA}=c_{DD}=\$30$:

\begin{aligned} V_{0,D} &=\frac{0.25(30+0)+0.75(30+100)}{1.05}\\ &=\frac{7.50+97.50}{1.05} =\frac{105}{1.05} =\$100.00. \end{aligned}

The active-state reserve is negative because, under these hypothetical assumptions, the expected premium outweighs the weighted terminal liability. To check the disabled-state result, its expected end-of-year amount is $0.25(\$30)+0.75(\$130)=\$105$. Accumulating $\$100$ for one year at 5% also gives $\$100(1.05)=\$105$. The values are consistent. The common mistake avoided here is applying the disability payment only to policies that remain disabled, despite the problem specifying payment for every policy that started disabled.

Worked Example: Two-Year Table With Time-Varying Transitions

Consider a hypothetical contract with active $A$, disabled $D$, and dead $Z$ states. The annual effective interest rate is 10%, so $v=1/1.10$. Each period's state change follows the categorical transition probabilities stated below, conditional on the starting state. Death is absorbing, and there are no mortality transitions other than those explicitly shown. At time 2, the terminal reserves are $V_{2,A}=\$120$, $V_{2,D}=\$80$, and $V_{2,Z}=0$. All stated cash flows are paid at the end of the relevant year.

From time 1 to time 2, an active policy has transition probabilities $0.70$ to $A$, $0.20$ to $D$, and $0.10$ to $Z$. The corresponding net outgo amounts are $-\$20$, $\$10$, and $\$100$. From $D$, the probabilities to $A,D,Z$ are $0.10,0.70,0.20$, and the corresponding outgo amounts are $\$0,\$20,\$100$. Death remains in $Z$ with probability 1 and has no further outgo.

First calculate the time-1 active reserve:

\begin{aligned} V_{1,A} &=\frac{0.70(120-20)+0.20(80+10)+0.10(0+100)}{1.10}\\ &=\frac{0.70(100)+0.20(90)+0.10(100)}{1.10}\\ &=\frac{70+18+10}{1.10} =\frac{98}{1.10} =\$89.0909\ldots \end{aligned}

For the disabled state:

\begin{aligned} V_{1,D} &=\frac{0.10(120+0)+0.70(80+20)+0.20(0+100)}{1.10}\\ &=\frac{12+70+20}{1.10} =\frac{102}{1.10} =\$92.7272\ldots \end{aligned}

Since death is absorbing and has no future payments, $V_{1,Z}=0$. The transitions and cash flows from time 0 to time 1 are different: from $A$, probabilities to $A,D,Z$ are $0.80,0.10,0.10$, with outgo $-\$20,\$10,\$100$; from $D$, the probabilities are $0.20,0.60,0.20$, with outgo $\$0,\$20,\$100$. Therefore,

\begin{aligned} V_{0,A} &=\frac{0.80(89.0909\ldots-20)+0.10(92.7272\ldots+10)+0.10(0+100)}{1.10}\\ &=\frac{0.80(69.0909\ldots)+0.10(102.7272\ldots)+10}{1.10}\\ &=\frac{55.2727\ldots+10.2727\ldots+10}{1.10}\\ &=\$68.6776\ldots, \\[4pt] V_{0,D} &=\frac{0.20(89.0909\ldots)+0.60(92.7272\ldots+20)+0.20(0+100)}{1.10}\\ &=\frac{17.8181\ldots+67.6363\ldots+20}{1.10}\\ &=\$95.8677\ldots. \end{aligned}

Rounded to the nearest cent, $V_{1,A}=\$89.09$, $V_{1,D}=\$92.73$, $V_{1,Z}=\$0.00$, $V_{0,A}=\$68.68$, and $V_{0,D}=\$95.87$. As a check, the unrounded numerator for $V_{0,A}$ is approximately $\$75.5454$; multiplying the calculated $V_{0,A}$ by 1.10 returns that amount. The key trap is reusing the time-1 transition probabilities at time 0 even though the problem specifies duration-dependent probabilities.

Worked Example: A Termination State and a Specified Maturity Reserve

A hypothetical two-year contract has active state $A$ and terminated state $T$. The terminated state is absorbing. There is no mortality in this example, and state changes in each year follow the specified categorical probabilities conditional on the current state. The annual effective interest rate is 25%, giving $v=1/1.25=0.80$. At time 2, the specified terminal reserves are $V_{2,A}=\$50$ and $V_{2,T}=0$. The zero reserve in state $T$ means no obligations remain after termination; it does not eliminate a termination payment made when the transition into $T$ occurs.

From time 1 to time 2, an active policy remains active with probability 0.50 and terminates with probability 0.50. A $\$40$ termination benefit is paid at the end of the year on transition to $T$; there are no other period-1 cash flows. Hence,

\begin{aligned} V_{1,A} &=0.80\left[0.50(0+50)+0.50(40+0)\right]\\ &=0.80(25+20) =\$36. \end{aligned}

At time 0, the probabilities from $A$ are 0.75 to $A$ and 0.25 to $T$. A $\$10$ premium is received at year-end if the policy remains active, and a $\$20$ termination payment is made if it terminates. Thus,

\begin{aligned} V_{0,A} &=0.80\left[0.75(-10+36)+0.25(20+0)\right]\\ &=0.80\left[0.75(26)+0.25(20)\right]\\ &=0.80(19.50+5.00) =\$19.60. \end{aligned}

The terminated-state reserve is zero at each duration after termination, because the contract has no continuing obligations there. The entry payment remains in the transition cash flow. The check is $V_{0,A}(1.25)=\$24.50$, equal to the expected year-end amount $0.75(\$26)+0.25(\$20)=\$24.50$. A common error is to omit the $\$20$ payment because the terminal reserve in state $T$ is zero; that confuses a payment on entering a state with a reserve after entering it.

Terminal Values, Timing, and Checks

A terminal condition may come from contract design, a specified maturity value, or the fact that no benefits remain after a terminal event. It must be stated for every state that can be occupied at the terminal duration. If a maturity benefit is paid at time $n$, decide whether the terminal reserve is defined immediately before or immediately after that payment. Under the convention used here, the payment belongs in the period cash flow leading to time $n$, while $V_n$ is the reserve after that payment.

An absorbing state is one that cannot be left once entered. It does not necessarily have a zero reserve: future expenses, annuity payments, or other obligations may continue while the contract is in that state. Conversely, a zero terminal reserve in an absorbing state does not mean that no payment is made on the transition into it.

A reliable calculation has three checks:

For each starting state and period, the transition probabilities sum to 1.
Each transition's net cash flow is assigned to the correct path, with premiums negative and benefit outgo positive.
After rounding only at the end, substituting the reserve into the forward relationship $V_{t,i}(1+i)=\sum_jp_{t,ij}(c_{t,ij}+V_{t+1,j})$ reproduces the weighted end-of-period amount.

These checks are especially useful when reserves are calculated in a table. Keep unrounded values between durations when possible; rounding intermediate reserves can create small discrepancies in later rows. Report monetary results to the nearest cent unless a problem specifies another unit.

Key takeaway: Start from a clearly timed terminal reserve for every state. At each earlier duration, use the transition probabilities and period cash flows for that duration, add each cash flow to the reserve in its destination state, take the conditional expectation, and discount once.

Check Your Understanding

For each question, use end-of-period cash flows and round monetary answers to the nearest cent.

A one-year hypothetical contract has states $X$ and $Y$, terminal reserves $V_{1,X}=\$40$ and $V_{1,Y}=\$10$, and an annual effective interest rate of 4%. Starting in $X$, the transition probabilities are 0.60 to $X$ and 0.40 to $Y$. The end-of-year net outgo is $-\$12$ on $X$-to-$X$ and $\$30$ on $X$-to-$Y$. Find $V_{0,X}$. (Answer: $\$31.54$)
A one-year contract has active and disabled states with terminal reserves $\$60$ and $\$20$, respectively. At 5% annual effective interest, an active policy remains active with probability 0.75 and becomes disabled with probability 0.25. The end-of-year outgo is $-\$10$ if it remains active and $\$50$ if it becomes disabled. Find the active-state reserve. (Answer: $\$52.38$)
A two-year contract has active state $A$ and absorbing termination state $T$, with no mortality. The annual effective interest rate is 10%, and terminal reserves at time 2 are $V_{2,A}=\$100$, $V_{2,T}=0$. From time 1, an active policy remains active with probability 0.80 or terminates with probability 0.20; the transition outgo is $\$0$ or $\$30$, respectively. From time 0, the probabilities are 0.50 to each state, with outgo $-\$20$ on remaining active and $\$40$ on termination. Find $V_{0,A}$. (Answer: $\$44.63$)
At a particular duration, a disabled policy has next-state probabilities 0.20 to active, 0.50 to disabled, and 0.30 to dead. The corresponding next-duration reserves are $\$80,\$40,\$0$, and the period-end outgo amounts are $\$0,\$15,\$100$. With annual effective interest of 5%, calculate the disabled-state reserve for this one period. (Answer: $\$70.00$)

What You'll Learn