Recursive Valuation Across Payment Dates

Why Value a Contract Backward?

For a finite-horizon contract, the value at an early payment date depends on the benefits that may be paid at later dates and on the states the insured may occupy. A direct calculation can list every possible path through the state model. That approach becomes cumbersome as the number of dates and states increases. A backward recursion instead starts at the final date, where the remaining value is known, and works toward the valuation date.

This method uses the same ingredients as state-contingent valuation from earlier material: transition probabilities, payment timing, and discounting. Its advantage is organizational. At each date, it combines the cash flow due then with the discounted expected value of what remains. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Consider payment dates $t_0<t_1<\cdots<t_N$ and a finite state space. Let $V_k(i)$ be the expected present value, measured at date $t_k$, of all payments at dates $t_k,\ldots,t_N$, conditional on the contract being in state $i$ at date $t_k$. Let $c_k(i)$ be the payment at date $t_k$ conditional on state $i$. For $k<N$, let $p_{ij}^{(k)}$ be the probability of being in state $j$ at $t_{k+1}$, given state $i$ at $t_k$. If a transition-contingent payment $b_{ij}^{(k)}$ is paid at $t_{k+1}$ when the transition is from $i$ to $j$, it is valued as a payment at that next date.

Assume that the stated transition probabilities are appropriate conditional probabilities for the contract’s state information, that the payment rules and interest rate are specified, and that the horizon is finite. The model may vary by interval; the probabilities need not be identical at every date. If the state definition does not contain enough information to determine the future transition probabilities or payments, the recursion must use a more informative state description.

With an effective annual interest rate $i$, write $v=1/(1+i)$. For a time interval of $\Delta_k=t_{k+1}-t_k$ years, the discount factor is $q_k=v^{\Delta_k}$. For annual dates, $q_k=v$. The recursion is

V_N(i)=c_N(i), \qquad V_k(i)=c_k(i)+q_k\sum_j p_{ij}^{(k)} \left(b_{ij}^{(k)}+V_{k+1}(j)\right), \quad k=N-1,\ldots,0.

If there is no transition-contingent payment during an interval, set $b_{ij}^{(k)}=0$. If the contract has an uncertain initial state with probabilities $\pi_i$ at date $t_0$, its value at $t_0$ is $\sum_i \pi_iV_0(i)$. This initial-state average is not an extra discounting step: the $V_0(i)$ values are already measured at $t_0$.

Why the Backward Recursion Holds

Conditional on state $i$ at $t_k$, the payment $c_k(i)$ is due at that date. The next date’s state is $j$ with probability $p_{ij}^{(k)}$. Given that transition, the payments at $t_{k+1}$ consist of the transition payment $b_{ij}^{(k)}$ and the value $V_{k+1}(j)$ of all payments from that date onward. Their expected value at $t_{k+1}$ is discounted to $t_k$ by $q_k$.

Applying the law of total expectation over the possible next states gives

\begin{aligned} V_k(i) &=c_k(i)+q_k\,E\!\left[b_{i,J}^{(k)}+V_{k+1}(J)\mid \text{state }i\text{ at }t_k\right]\\ &=c_k(i)+q_k\sum_j p_{ij}^{(k)} \left(b_{ij}^{(k)}+V_{k+1}(j)\right). \end{aligned}

At the final date there are no later payments in the contract, so the remaining value is just the payment due there: $V_N(i)=c_N(i)$. This terminal condition starts the backward calculation.

Set the terminal values.
At the last date, assign the payment for each possible state, so $V_N(i)=c_N(i)$.

Move back one date.
For each current state, average the next-date transition payment plus next-date value using that row of transition probabilities.

Discount and add the current payment.
Multiply the expected next-date amount by the interval discount factor, then add $c_k(i)$.

Continue to the valuation date.
Repeat for each earlier date. If the initial state is uncertain, average the date-zero conditional values using the initial-state probabilities.

Worked Valuations

Worked Example: Averaging Values Across Two Possible States

A hypothetical contract is valued at date 1, when the insured is in one of two states, A or D. The probabilities are $P(A)=0.8$ and $P(D)=0.2$. Interest is $i=5\%$ effective annually. At date 1, the contract pays $100 in state A and $300 in state D. The state persists with certainty until date 2, when it pays a further $41 in A or $90 in D. Find the expected value at date 1.

Method. Work backward from date 2. Since the state persists with certainty, the one-year transition probability from each state back to itself is 1, and the probability of moving to the other state is 0. The discount factor is

v=\frac{1}{1.05}=0.952380952\ldots

At date 2, the terminal values are $V_2(A)=41$ and $V_2(D)=90$. At date 1, the values conditional on state are

\begin{aligned} V_1(A)&=100+v(1\cdot 41+0\cdot 90) =100+\frac{41}{1.05} =139.047619\ldots,\\ V_1(D)&=300+v(0\cdot 41+1\cdot 90) =300+\frac{90}{1.05} =385.714286\ldots. \end{aligned}

Average these conditional values using the date 1 state probabilities. This produces a value measured at date 1, so no additional discount is applied:

\begin{aligned} E[V_1] &=0.8(139.047619\ldots)+0.2(385.714286\ldots)\\ &=111.238095\ldots+77.142857\ldots\\ &=188.380952\ldots\approx \$188.38. \end{aligned}

Check. The two probabilities sum to 1, and the weighted value lies between the two state-specific values. The calculation avoids the common mistake of discounting this date 1 average again, which would incorrectly report a date 0 value instead.

Worked Example: A Two-Interval Contract With State Payments

A hypothetical contract has states A and D and payment dates $t_0,t_1,t_2$, one year apart. Interest is $i=10\%$ effective annually. The state at $t_0$ is A. Between $t_0$ and $t_1$, the probabilities from A are 0.8 to A and 0.2 to D. At $t_1$, the contract pays $100 in D and nothing in A. For the interval from $t_1$ to $t_2$, the transition probabilities are: from A, 0.7 to A and 0.3 to D; from D, 0.2 to A and 0.8 to D. At $t_2$, the contract pays $500 in D and nothing in A. Find the value at $t_0$.

Method. Start with the terminal values $V_2(A)=0$ and $V_2(D)=500$. The discount factor for each annual interval is $v=1/1.10=0.9090909\ldots$. At $t_1$, use the transition probabilities for the $t_1$-to-$t_2$ interval:

\begin{aligned} V_1(A)&=0+v(0.7\cdot 0+0.3\cdot 500) =\frac{150}{1.10} =136.363636\ldots,\\ V_1(D)&=100+v(0.2\cdot 0+0.8\cdot 500) =100+\frac{400}{1.10} =463.636364\ldots. \end{aligned}

The $100 payment is included in $V_1(D)$ because it is due at $t_1$. Now use the probabilities for the earlier interval, $t_0$-to-$t_1$. There is no payment at $t_0$:

\begin{aligned} V_0(A) &=v\left(0.8V_1(A)+0.2V_1(D)\right)\\ &=\frac{1}{1.10} \left(0.8(136.363636\ldots)+0.2(463.636364\ldots)\right)\\ &=\frac{1}{1.10}(109.090909\ldots+92.727273\ldots)\\ &=183.471074\ldots\approx \$183.47. \end{aligned}

Check and trap. If the discount factor is omitted when calculating $V_1(A)$ and $V_1(D)$, the incorrect values are $150$ and $500$; this leads to an incorrect $V_0=(0.8(150)+0.2(500))/1.10=\$200.00$. Each one-year interval must be discounted exactly once. The correct recursion discounts the $t_1$-to-$t_2$ values in $V_1$, then discounts the $t_0$-to-$t_1$ expectation in $V_0$.

Worked Example: An Irregular Interval and a Transition Benefit

A hypothetical contract is valued at $t_0=0$, $t_1=1$, and $t_2=3$ years. Its states are A, C, and D. Interest is $i=4\%$ effective annually. From A at $t_0$, the probabilities at $t_1$ are 0.75 for A and 0.25 for C, with zero probability of D. At $t_1$, the contract pays $80 if the state is C and nothing otherwise. Over the two-year interval from $t_1$ to $t_2$, the transition probabilities are: from A, 0.6 to A, 0.3 to C, and 0.1 to D; from C, 0.1 to A, 0.7 to C, and 0.2 to D; from D, 0 to A, 0 to C, and 1 to D. At $t_2$, it pays $600 in C and nothing in A or D. A transition payment of $50 is also paid at $t_2$ if the state at $t_2$ is C, regardless of the state at $t_1$. Find the value at $t_0$.

Method. First combine the transition payment with the terminal state payment when the next state is C. Thus, the total amount at $t_2$ is $650 in C and zero in A or D. The two-year discount factor is

v^2=\frac{1}{(1.04)^2}=\frac{1}{1.0816}.

At $t_2$, $V_2(A)=0$, $V_2(C)=600$, and $V_2(D)=0$; the $50 transition benefit is included in the interval calculation because it depends on the transition into C. At $t_1$, the expected $t_2$ amount before discounting is $0.3(650)=195$ from A, $0.7(650)=455$ from C, and 0 from D. Therefore,

\begin{aligned} V_1(A)&=0+\frac{195}{1.0816} =180.288462\ldots,\\ V_1(C)&=80+\frac{455}{1.0816} =80+420.673077\ldots =500.673077\ldots,\\ V_1(D)&=0+\frac{0}{1.0816}=0. \end{aligned}

Now discount one year from $t_1$ to $t_0$ and apply the stated probabilities from A:

\begin{aligned} V_0(A) &=\frac{1}{1.04} \left(0.75V_1(A)+0.25V_1(C)+0V_1(D)\right)\\ &=\frac{1}{1.04} \left(0.75(180.288462\ldots)+0.25(500.673077\ldots)\right)\\ &=\frac{135.216346\ldots+125.168269\ldots}{1.04}\\ &=250.369822\ldots\approx \$250.37. \end{aligned}

Check. The transition probabilities in each row sum to 1, and the two-year interval uses $v^2$, not $v$. The $50 payment is included once, for transitions into C at $t_2$, rather than being added again as a separate unconditional payment.

Interpretation and Common Traps

The recursion produces a separate value for each state at each payment date. The state-specific values are conditional expected present values, not amounts that every policyholder will receive. A realized contract follows one path through the states and produces its associated payments; the recursion averages across possible future paths under the specified probabilities.

The method is useful because it avoids listing every complete path. Each step requires only the next-date state values, the appropriate transition probabilities, the payments due, and the interval discount factor. Before using it, verify that the payment timing is explicit and that the state captures the information needed to apply the transition and payment rules.

Discounting the wrong number of times. A value at $t_{k+1}$ must be discounted once to $t_k$. Do not discount a value again if it has already been expressed at the current valuation date.
Using the wrong interval’s probabilities. In the backward step from $t_k$, use probabilities for the transition from $t_k$ to $t_{k+1}$, not probabilities from an earlier or later interval.
Dropping a payment or counting it twice. Include a payment in the date and condition specified by its rule. A transition payment at the next date is included inside the transition expectation; a state payment at the current date is added as $c_k(i)$.
Using an annual discount for a longer interval. For a two-year interval at effective annual rate $i$, use $v^2$, not $v$.
Averaging the initial state at the wrong date. Weight the conditional values by the state probabilities at the valuation date. If those conditional values are already measured at that date, do not apply an extra discount.

Key takeaway: Set the terminal values first. Then, at each earlier date, add the current state payment to the discounted expected amount of the next transition payment and next-date value. Keep the time index, transition probabilities, and discount factor aligned with the interval being valued.

Check Your Understanding

For each problem, work backward from the final payment date and report the requested value to the nearest cent.

A one-state contract pays $100 at year 1 and $200 at year 2. Interest is $i=5\%$ effective annually. What is its value at year 0? (Answer: $276.64)
At a valuation date, a policyholder is in state A with probability 0.6 and state D with probability 0.4. The conditional contract values at that same date are $120 in A and $300 in D. What is the expected value at the valuation date? (Answer: $192.00)
A contract is in state A now. One year from now, the probability of state C is 0.2 and the probability of state A is 0.8. It pays $400 at that date only in C. There are no other payments, and $i=4\%$ effective annually. Find the value now. (Answer: $76.92)
A contract has states A and D and dates 0, 1, and 2 years. Interest is $i=5\%$ effective annually. At each annual interval, the transition probabilities from A are 0.75 to A and 0.25 to D; from D they are 0.1 to A and 0.9 to D. At year 1 it pays $100 in D only; at year 2 it pays $500 in D only. The initial state is A. Find the value at year 0. (Answer: $210.88)
A contract is in state A now. Two years from now, it pays $300 if the state is C, and nothing otherwise. The probability of being in C at that date is 0.4. There are no interim payments, and $i=5\%$ effective annually. Find the value now. (Answer: $108.84)

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