Reconciling State-Contingent Cash-Flow Valuations

Why Reconcile a Valuation?

A state-contingent cash-flow value can be calculated in several ways. One approach lists possible state paths and their payments. Another weights payments by the probabilities of being in the relevant states. A third uses transition matrices to calculate those probabilities. When the model, payment rules, and timing are identical, these methods must give the same expected present value.

A mismatch is useful evidence: it often points to a payment made at the wrong date, an eligibility condition omitted from a state probability, or a premium and benefit with inconsistent signs. The goal is not to prefer one method in every situation, but to use one calculation to check another.

Definition: Let the contract have states $1,\ldots,m$, an initial state distribution row vector $\boldsymbol{\alpha}$, and transition matrices $P_1,\ldots,P_n$. The entry $P_k(r,s)$ is the probability of moving from state $r$ at time $k-1$ to state $s$ at time $k$. If $\boldsymbol{c}_k$ is a column vector of signed payments at time $k$, with positive values defined as outflows from the insurer, then the expected present value of occupancy-based payments is $$ \operatorname{EPV}=\sum_{k=1}^{n} v^k\boldsymbol{\alpha}P_1P_2\cdots P_k\boldsymbol{c}_k, $$ where $v=1/(1+i)$ for effective annual interest rate $i$, and payments are made at the stated integer times. A payment at time zero is added separately without discounting.

Three Ways to Calculate the Same Value

For an occupancy-based payment at time $k$, the probability distribution across states is $\boldsymbol{\alpha}P_1\cdots P_k$. Multiplying this row vector by the payment vector gives the expected payment at time $k$. Discounting and summing over payment dates gives the matrix calculation in the definition.

The direct method instead considers complete state paths. For each path, multiply its probability by its discounted payments; then add across paths. A conditional calculation groups paths using conditional probabilities. For example, the probability of being in state $D$ at time 2 can be found by summing over the possible states at time 1:

P(S_2=D)=\sum_r P(S_1=r)P(S_2=D\mid S_1=r).

This is the law of total probability. It is the reason that a matrix product and a sum over intermediate states agree: both account for each possible route to the payment state, with its probability.

Worked Example: Occupancy Payments by Paths and Matrices

A hypothetical two-state contract has states $A$ and $D$. The insured starts in $A$. At each annual step, the transition probabilities are $P(A\to A)=0.8$, $P(A\to D)=0.2$, $P(D\to A)=0.3$, and $P(D\to D)=0.7$. The effective annual interest rate is $5\%$. The insurer pays \$500 at time 1 if the state is $D$, and \$700 at time 2 if the state is $D$. There are no other payments. Reconcile a direct path calculation and a matrix calculation.

Method and assumptions. Payments depend only on the state at their payment date. The transitions are annual and follow the stated matrix at both steps. Use $v=1/1.05=0.95238095$ for calculation, retaining precision until the final amount.

Direct path calculation. At time 1, the path $A\to D$ has probability $0.2$, so the expected time-1 payment is $0.2(500)=\$100$. The time-2 paths ending in $D$ are $A\to A\to D$, with probability $0.8(0.2)=0.16$, and $A\to D\to D$, with probability $0.2(0.7)=0.14$. Their total probability is $0.16+0.14=0.30$. Thus the expected time-2 payment is $0.30(700)=\$210$.

Discount each payment at its own date:

\operatorname{EPV}_{\text{direct}} =\frac{100}{1.05}+\frac{210}{1.05^2} =95.238095+190.476190 =285.714286.

Matrix calculation. In state order $A,D$, the initial distribution and transition matrix are

\boldsymbol{\alpha}=(1,0),\qquad P=\begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix}.

The time-1 distribution is $\boldsymbol{\alpha}P=(0.8,0.2)$. The time-2 distribution is

\boldsymbol{\alpha}P^2 =(0.8,0.2) \begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix} =(0.70,0.30).

The payment vectors are $\boldsymbol{c}_1=(0,500)^{\mathsf T}$ and $\boldsymbol{c}_2=(0,700)^{\mathsf T}$. Therefore,

\operatorname{EPV}_{\text{matrix}} =v(0.8,0.2)\boldsymbol{c}_1 +v^2(0.70,0.30)\boldsymbol{c}_2 =v(100)+v^2(210) =285.714286.

Both calculations give \$285.71, rounded to the nearest cent. The check is that the path probabilities ending in $D$ at time 2 sum to the matrix-calculated occupancy probability, $0.30$. A common mistake is to discount both payments by one year; the time-2 payment must be discounted by $v^2$.

Reconciliation When Eligibility Depends on History

An occupancy probability answers whether the contract is in a state at a particular time. It does not, by itself, answer whether a rule involving earlier states has been satisfied. For example, a payment might require two consecutive periods in state $D$. The probability of $D$ at time 2 includes paths that reached $D$ only at time 2, even though those paths do not meet the two-period condition.

In that situation, a direct path calculation or a conditional calculation that retains the relevant history is valid. A matrix calculation is also possible if the model states are expanded to record that history, but simply multiplying the original occupancy probability by the payment amount is not valid. This is a matter of matching the state representation to the payment rule.

Worked Example: A Consecutive-State Eligibility Rule

Use the transition model in the previous example: the contract starts in $A$, and $P(A\to D)=0.2$, $P(D\to D)=0.7$, with the other transition probabilities as stated there. The effective annual interest rate is $5\%$. A hypothetical benefit of \$600 is payable at time 2 only if the insured was in $D$ at both time 1 and time 2. Find its expected present value.

Method and assumptions. The benefit is paid once, at time 2, and eligibility requires the specific two-period path $A\to D\to D$. No other path qualifies.

Direct path calculation. The probability of the qualifying path is

P(A\to D\to D)=P(A\to D)P(D\to D)=0.2(0.7)=0.14.

The expected payment at time 2 is $0.14(\$600)=\$84$. Discounting it for two years gives

\operatorname{EPV}=\frac{84}{1.05^2} =\frac{84}{1.1025} =76.190476.

Conditional calculation. The probability of being in $D$ at time 1 is $0.2$, and conditional on that state, the probability of being in $D$ at time 2 is $0.7$. Thus $P(S_1=D,S_2=D)=0.2(0.7)=0.14$, giving the same expected payment and present value. The answer is \$76.19, rounded to the nearest cent.

Common mistake and correction. Using $P(S_2=D)=0.30$ would produce $600(0.30)/1.05^2=\$163.27$. That calculation incorrectly pays on every path ending in $D$, including $A\to A\to D$, which fails the time-1 eligibility condition. The correct probability is the joint probability of $D$ at both dates, $0.14$, not the time-2 occupancy probability, $0.30$.

Transition Payments and Cash-Flow Signs

Some cash flows are triggered by a transition rather than by occupancy. For a payment $b_k(r,s)$ made at time $k$ when the transition from state $r$ to state $s$ occurs, its expected payment is calculated by summing over transitions:

\sum_r \sum_s P(S_{k-1}=r)P_k(r,s)b_k(r,s).

This expression follows by partitioning all possible outcomes at time $k$ into the mutually exclusive transitions $(r,s)$. Each transition contributes its probability multiplied by its payment. The payment vector for occupancy states cannot generally replace this transition-specific calculation.

Signs must also be consistent across methods. In this tutorial, a positive amount is an insurer outflow, such as a benefit or expense; an inflow, such as a premium, is negative. One may choose the opposite convention, but then every payment and the reported net value must use it consistently. The expected present value is signed: it can be negative when expected inflows exceed expected outflows.

Worked Example: Checking a Signed Transition Cash Flow

A hypothetical two-state contract has states $A$ and $D$, starts in $A$, and uses $P(A\to A)=0.8$, $P(A\to D)=0.2$, $P(D\to A)=0.3$, and $P(D\to D)=0.7$ for each annual step. The effective annual interest rate is $10\%$. At the end of each of two years, the insurer receives a premium of \$120 and incurs an expense of \$20, regardless of the transition. It also pays a benefit of \$500 if and only if the transition is $A\to D$. Calculate the expected present value of insurer outflows, taking benefits and expenses as positive and premiums as negative.

Method and assumptions. All three cash-flow items are paid at the end of the transition year. Thus the net outflow is $-120+20=-\$100$ on transitions other than $A\to D$, and $-120+20+500=\$400$ on $A\to D$.

First-year calculation. The transition is $A\to A$ with probability $0.8$ and $A\to D$ with probability $0.2$. The expected net outflow is

$$ 0.8(-100)+0.2(400)=-80+80=0. $$

Second-year calculation. At time 1 the state probabilities are $P(S_1=A)=0.8$ and $P(S_1=D)=0.2$. Starting in $A$, the expected net outflow for the next transition is $0.8(-100)+0.2(400)=0$. Starting in $D$, neither possible transition triggers the benefit, so its expected net outflow is $-\$100$. Weighting these conditional values gives $0.8(0)+0.2(-100)=-\$20$.

The present value is therefore

\operatorname{EPV} =\frac{0}{1.10}+\frac{-20}{1.10^2} =-16.528926.

Direct check at time 2. The paths and their probabilities are $A\to A\to A$: $0.64$; $A\to A\to D$: $0.16$; $A\to D\to A$: $0.06$; and $A\to D\to D$: $0.14$. Their time-2 net outflows are respectively $-100, 400, -100, -100$. The expected time-2 outflow is

$$ 0.64(-100)+0.16(400)+0.06(-100)+0.14(-100) =-64+64-6-14=-20. $$

The direct check agrees with the conditional calculation. The expected present value is -\$16.53, rounded to the nearest cent. The negative result means expected discounted inflows exceed expected discounted outflows under the stated sign convention. A common mistake is to record the premium as a positive insurer outflow; doing so reverses its economic meaning and fails this reconciliation check.

A Practical Reconciliation Procedure

When two valuations disagree, check the model and the cash-flow rule before repeating arithmetic. The following sequence separates the main sources of error.

Fix the sign convention.
State whether positive amounts are outflows or inflows, and apply that convention to benefits, premiums, and expenses alike.

Write the payment trigger.
Specify whether each cash flow depends on occupancy at a date, a transition during a period, or a condition involving multiple dates.

Attach each payment to a date.
Record whether it is paid at time zero or at the end of a later period, then use the matching discount factor.

Calculate probabilities at the required level.
Use occupancy probabilities for occupancy triggers, transition probabilities for transition triggers, and joint or conditional probabilities for history-dependent rules.

Compare expected payments before discounting.
Check each payment date separately. If those expected amounts agree, compare the discounted sums; this helps isolate a timing or discounting error.

Common Reconciliation Traps

A clean comparison requires the same assumptions and contract definition in every method. In particular, do not change from an end-of-period payment in one calculation to an advance payment in another. A benefit payable on entry into a state is not the same as one payable merely for occupying that state at a later date. And an eligibility rule that depends on prior states must be represented in the probability calculation, rather than inferred from a single marginal occupancy probability.

A useful final check is to calculate the expected cash flow at each date without discounting. Then confirm the time index of every discount factor and the sign of every item. Direct path sums may be longer, but they make triggers and signs visible; conditional and matrix calculations are often more compact. Agreement between appropriately specified methods is a strong check that the valuation is internally consistent.

Check Your Understanding

For each problem, use the stated sign convention and payment dates.

A contract starts in $A$, has states $A,D$, and transition probabilities $P(A\to A)=0.75$, $P(A\to D)=0.25$, $P(D\to A)=0.20$, $P(D\to D)=0.80$ at each annual step. At $i=4\%$, it pays \$400 in $D$ at time 1 and \$600 in $D$ at time 2. Find the expected present value. (Answer: \$311.11)
A contract starts in $A$, with $P(A\to D)=0.30$ and $P(D\to D)=0.60$. At $i=5\%$, it pays \$1,000 at time 2 only if it is in $D$ at both times 1 and 2. Find the expected present value. (Answer: \$163.27)
A contract starts in $A$. At the end of year 1, it pays \$700 if and only if the transition is $A\to D$, which has probability $0.10$. At $i=3\%$, find the expected present value. (Answer: \$67.96)
A contract starts in $X$ and has states $X,Y$. At each annual step, $P(X\to X)=0.60$, $P(X\to Y)=0.40$, $P(Y\to X)=0.25$, and $P(Y\to Y)=0.75$. At each year-end, the net outflow is \$170 for $X\to Y$, and $-\$30$ for every other transition. At $i=5\%$, calculate the expected present value over two years. (Answer: \$63.95)

What You'll Learn