Common Timing Errors in State-Contingent Valuation

Why Timing Errors Change a Valuation

A state-contingent cash flow depends not only on its amount, but also on which event makes it payable and when that event is observed. A benefit payable because a person is disabled at the end of a year is not necessarily the same benefit as one payable when the person moves from active to disabled during that year. The destination state may be the same, but the contractual event differs.

A second source of error is discounting from the wrong date. A value calculated at time 1 is not yet a time-0 value: it must be discounted one year to time 0. Conversely, if a cash flow at time 3 is already discounted three years to time 0, it must not be discounted again. A reliable valuation therefore begins by writing down the payment date, the relevant state or transition event, and the valuation date.

Definition: Let $J_k$ denote the state occupied immediately after the transition at time $t_k$. An occupancy payment at $t_k$ is payable according to $J_k$. A transition payment at $t_k$ is payable according to the ordered pair $(J_{k-1},J_k)$, such as a transition from active to disabled during $(t_{k-1},t_k]$. The payment amount and event must be read from the contract; one type cannot be substituted for the other.

For the calculations below, assume a deterministic effective annual interest rate $i$, annual payment dates, and a time-0 valuation unless another valuation date is explicitly stated. Write $v=1/(1+i)$. A payment at time $t_k=k$ is discounted to time 0 by $v^k$. The state probabilities and transition probabilities are supplied as part of each hypothetical model; no particular transition-rate model is assumed.

Write the Event Before the Formula

Suppose $b_j(k)$ is the amount payable at time $k$ if the state then occupied is $j$, and let $P(J_k=j)$ be the probability of that event. For a transition payment, let $c_{ab}(k)$ be the amount payable if the state changes from $a$ at time $k-1$ to $b$ at time $k$. The probability needed is the joint probability of that ordered transition, $P(J_{k-1}=a,J_k=b)$, not simply the probability of occupying $b$ at time $k$.

\text{Time-0 expected present value} = \sum_k v^k \sum_j b_j(k)P(J_k=j) + \sum_k v^k \sum_{a,b} c_{ab}(k)P(J_{k-1}=a,J_k=b).

The sums can include whichever occupancy and transition payments the contract specifies. They do not imply that every possible payment is made at every date. A payment amount is zero for events that do not trigger a benefit. If a contract pays only once on first entry into a state, repeated transitions must also be handled according to that payment rule; an unrestricted sum over all entries would not represent a first-entry benefit.

Assumptions and limits: The displayed valuation is for payments at stated discrete dates, fixed benefit amounts, and a deterministic discount rate. It values expected present amounts by event probabilities. It does not determine the contract’s payment rule, and it does not apply unchanged to a payment made at an unstated time or to a benefit whose amount varies with the path unless that variation is included in the cash-flow specification.

Why the Event-Weighted Formula Holds

For a single payment at time $k$, define an indicator $I_E$ equal to 1 if the event $E$ that triggers payment occurs and 0 otherwise. If the benefit is $B$, its time-0 present amount is $v^k B I_E$. Since $E[I_E]=P(E)$, the expected present amount is $v^k B P(E)$. Applying this argument separately to each occupancy and transition payment, then using additivity of expectation, gives the valuation formula above.

$$ E[v^k B I_E]=v^k B E[I_E]=v^k B P(E). $$

The indicator identifies the key timing distinction. For an occupancy benefit, $E$ is the event that the specified state is occupied at the payment date. For a transition benefit, $E$ is the event that the specified origin-to-destination transition occurred in the relevant interval. The discount factor depends on the payment date in either case, not on the date at which the state was entered unless that is also the payment date.

Worked Example: Occupancy Benefit at Two Payment Dates

Method. Find the probability of being in the benefit state at each payment date, then discount each payment from its own date. A hypothetical contract starts in state A at time 0 and pays $1,000 at the end of each of years 1 and 2 if the person is then in state D. The annual effective interest rate is 5%. In each year, the transition probabilities are $P(A\to A)=0.8$, $P(A\to D)=0.2$, $P(D\to A)=0.4$, and $P(D\to D)=0.6$.

At time 1, the person starts in A, so $P(J_1=D)=0.2$. At time 2, state D can be reached either by being in A at time 1 and moving to D, or by being in D at time 1 and remaining there:

$$ P(J_2=D)=(0.8)(0.2)+(0.2)(0.6)=0.16+0.12=0.28. $$

The discount factors are $v=1/1.05=0.95238095$ and $v^2=1/1.05^2=0.90702948$. Thus the time-0 expected present value is

$$ 1000(0.2)v+1000(0.28)v^2 =190.47619+253.96825 =444.44444. $$

Rounded to the nearest cent, the value is $444.44. The calculation uses state occupancy at the payment date, not the probability of entering D during that year. As a check, each probability is between 0 and 1, and each payment is discounted for one or two years as appropriate. Common mistake avoided: treating the year-2 payment as payable only on a new A-to-D transition, even though the contract pays to anyone in D at year-end.

Worked Example: Transition Benefit Is Not Occupancy Benefit

Method. Calculate the joint probability of the specified transition in each interval. A hypothetical benefit pays $6,000 at the end of each year in which the person moves from A to D during that year. The person starts in A, the annual transition probabilities are $P(A\to A)=0.8$, $P(A\to D)=0.2$, $P(D\to A)=0.4$, and $P(D\to D)=0.6$, and the annual effective interest rate is 5%.

For year 1, the probability of A-to-D is $0.2$, since the initial state is A. For year 2, the transition must again start in A. The person is in A at time 1 with probability $0.8$, and then moves to D with probability $0.2$. Therefore the joint year-2 probability is

$$ P(J_1=A,J_2=D)=(0.8)(0.2)=0.16. $$

The time-0 value is

$$ 6000(0.2)v+6000(0.16)v^2 =1200(0.95238095)+960(0.90702948) =1142.85714+870.74830 =2013.60544. $$

Rounded to the nearest cent, the expected present value is $2,013.61. The year-2 probability is not $P(J_2=D)=0.28$: that occupancy probability also includes people who were already in D at time 1 and stayed there. Those people did not make an A-to-D transition in year 2. Using $0.28$ would give $6000(0.2v+0.28v^2)=$2,666.67$, an incorrect value for this transition-only benefit. The corrected calculation follows the ordered transition specified by the contract.

Worked Example: Conditional Value and Discount Origin

Method. Value the future payment at the conditioning date first, then move that value back to time 0 exactly once. A hypothetical contract starts in A and pays $5,000 at time 3 if the person is in D both at time 1 and at time 3. The annual effective interest rate is 5%. The same transition probabilities apply in each year: from A, the probabilities of A and D next year are 0.8 and 0.2; from D, they are 0.4 and 0.6.

At time 1, the probability of being in D is $0.2$. Conditional on being in D at time 1, the probability of being in D two years later is the sum of the two possible two-step paths ending in D:

P(J_3=D\mid J_1=D) =(0.6)(0.6)+(0.4)(0.2) =0.36+0.08 =0.44.

The payment is two years after the conditioning date $t=1$. Its expected value at time 1, conditional on $J_1=D$, is therefore

$$ 5000(0.44)v^2 =2200(0.90702948) =1995.46485. $$

To obtain the time-0 contribution, multiply by the probability of $J_1=D$ and discount one year from time 1:

$$ (0.2)v(1995.46485) =0.2(0.95238095)(1995.46485) =380.08854. $$

Equivalently, the joint probability of $J_1=D$ and $J_3=D$ is $(0.2)(0.44)=0.088$, and the payment is three years from time 0:

$$ 5000(0.088)v^3 =440(0.86383760) =380.08854. $$

Both routes give $380.09 to the nearest cent. The check is that the first route discounts two years from time 1 and then one more year to time 0, giving the same three-year discount as the direct route. Common mistake avoided: discounting the conditional time-1 value by $v^3$ again, which counts the time from 1 to 3 twice.

A Practical Timing Check

Before calculating, translate each payment clause into an event and a date. The following sequence separates the tasks and helps expose errors before they are carried through a longer valuation.

Identify the payment date.
Write the actual time of payment, such as year-end 2. Do not substitute the date of onset or another transition unless the contract pays then.

State the triggering event.
Use a state at the payment date for an occupancy benefit, or an ordered origin-and-destination pair over an interval for a transition benefit.

Calculate the required probability.
For a transition payment, include the probability of being in the origin state at the start of the interval as well as the transition probability.

Discount from the valuation date.
If valuing at time 0, a payment at time $k$ receives $v^k$. If the value is already at a later date, discount only the remaining time to the requested date.

Check the event and timing in words.
Confirm that the probability represents exactly the payment rule and that the total discount period equals the time between valuation and payment.

Common Traps and Why the Distinction Matters

The most frequent error is replacing “is in state D at time $k$” with “moves into D during year $k$.” Occupancy probability includes people already in D who remain there; transition probability does not. The reverse error is also possible: a transition benefit cannot generally be valued using only the probability of the destination state.

A second trap is using a probability conditional on a state but treating the resulting value as unconditional. If a value is calculated conditional on $J_1=D$, it must be multiplied by $P(J_1=D)$ when forming an unconditional time-0 expected value. A third trap is applying a discount factor that does not match the valuation origin. Writing “payment at $t=3$, valued at $t=0$” beside the cash flow is a simple control.

These distinctions matter because a small timing or event error can change both the probability and the discount factor. An occupancy event may be more likely than a particular transition, and a later payment is worth less at an earlier valuation date. State-contingent cash-flow valuation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus; precise event and date definitions make the arithmetic auditable as well as correct.

Key takeaway: For every cash flow, record three items before valuing it: what event triggers payment, when payment occurs, and the date from which it is discounted. Occupancy and transition events are different, and conditional values must be moved between dates only for the elapsed time.

Check Your Understanding

For each hypothetical contract, identify the relevant event and use the stated valuation date and interest rate.

A contract starts in A and pays $2,500 at the ends of years 1 and 2 if the person is in D then. The annual effective interest rate is 4%, and the probabilities of D at those dates are 0.30 and 0.24. Find the time-0 expected present value. (Answer: $1,275.89)
A contract starts in A and pays $4,000 at the end of each year in which an A-to-D transition occurs. Each year, $P(A\to A)=0.7$, $P(A\to D)=0.3$, $P(D\to A)=0.5$, and $P(D\to D)=0.5$. The annual effective interest rate is 6%. Find the time-0 value of payments at the ends of years 1 and 2. (Answer: $1,879.67)
At time 1, a person is in D with probability 0.4. Conditional on being in D at time 1, the probability of being in D at time 2 is 0.75. A $1,000 occupancy payment is made at time 2 if in D. At an annual effective interest rate of 5%, find the time-0 expected present value. (Answer: $272.11)
A contract starts in A. Each year $P(A\to A)=0.9$, $P(A\to D)=0.1$, $P(D\to A)=0.2$, and $P(D\to D)=0.8$. At time 2 it pays $1,000 if in D, plus $3,000 if an A-to-D transition occurred during year 2. At an annual effective interest rate of 5%, find the time-0 expected present value. (Answer: $399.09)

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