Irregular Payment Dates in State Models

Why Irregular Payment Dates Need Careful Matching

A state-contingent contract may pay at dates that are neither equally spaced nor aligned with whole years. At each payment date, two questions must be answered separately: what event makes the payment due, and when is the payment made? The first determines the probability-weighted amount; the second determines the discount factor.

For example, a payment may be due at a review date if the insured is disabled then, or it may be due at that date only if a transition into disability occurred during the preceding interval. Those rules use different probabilities. In either case, discounting uses the time from issue to the actual payment date, not the length of the interval used to calculate the probability.

Definition: Let $0<t_1<t_2<\cdots<t_n$ be payment dates measured in years from the valuation date. A payment $B_j$ at time $t_j$ is state-contingent if its amount or whether it is paid depends on the state or transition history by that date. Its expected present value is the sum, over payment dates, of the expected payment at each date multiplied by the discount factor from that date to the valuation date.

Separate the Payment Rule From the Discounting

Assume a deterministic effective annual interest rate $i$, with $v=(1+i)^{-1}$. A payment $B_j$ made at time $t_j$, including a fractional number of years, has present value $v^{t_j}B_j$. If the payment amount is random because it depends on the contract state, then its expected present value is $v^{t_j}E[B_j]$. Linearity of expectation gives the total expected present value:

\operatorname{EPV}=\sum_{j=1}^{n}v^{t_j}E[B_j].

For an occupancy benefit of amount $b_{j,r}$ paid at date $t_j$ when the process is in state $r$, the expected payment is found by summing over eligible states. If the state at time $t_j$ is $Z(t_j)$, then:

E[B_j]=\sum_{r\in\mathcal{E}_j}b_{j,r}P(Z(t_j)=r), \qquad \operatorname{EPV}_j=v^{t_j}\sum_{r\in\mathcal{E}_j}b_{j,r}P(Z(t_j)=r).

Here $\mathcal{E}_j$ is the set of states that qualify for payment at date $t_j$. In a discrete state model, state probabilities at a later date can be obtained from the probabilities at an earlier date and the conditional transition probabilities over the intervening interval. The interval can have any positive length; it need not equal one year.

Timing rule: Use the probability that matches the contractual trigger, and discount over the full time from valuation to payment. An occupancy benefit uses the probability of being in an eligible state at the payment date. A transition benefit uses the probability of the qualifying transition under its stated timing rule. Neither probability determines the discount period.

Deriving the Date-by-Date Valuation

The valuation formula follows directly from discounting each realized payment and then taking expectations. Let the realized present value of the contract be $Y=\sum_{j=1}^{n}v^{t_j}B_j$. Because $i$ is deterministic, each $v^{t_j}$ is nonrandom. Therefore:

E[Y] =E\left[\sum_{j=1}^{n}v^{t_j}B_j\right] =\sum_{j=1}^{n}v^{t_j}E[B_j].

The last equality is linearity of expectation; it does not require payments at different dates to be independent. If a payment depends on which state applies, the law of total expectation gives $E[B_j]$ by multiplying each possible payment by the probability of its qualifying state or event and summing. This is why a valuation can be assembled date by date even when the cash flows are dependent across time.

List the actual payment dates.
Measure every date from the valuation time, in the same time unit used for the interest rate.

Write the trigger at each date.
Decide whether the payment depends on occupancy at that date, a transition during an interval, or another stated condition.

Find the expected payment.
Use state probabilities or a probability of the qualifying transition, accounting for any required prior state.

Discount from the payment date.
Multiply by $v^{t_j}$, then add the present values for all payment dates.

Worked Valuations

Worked Example: Occupancy Benefits at Two Unevenly Spaced Dates

A hypothetical contract pays $500 at time $0.5$ if the insured is in eligible state $E$, and $800 at time $1.5$ if the insured is in $E$ then. The initial state is known. The state model gives $P(Z(0.5)=E)=0.20$ and $P(Z(1.5)=E)=0.35$. These probabilities are supplied by the model and need not be reconstructed here. The effective annual interest rate is $i=10\%$. Find the expected present value. Method: value each occupancy-contingent payment at its own date. This avoids applying an interval probability or interval discount to a payment based on occupancy at a date.

At time $0.5$, the expected payment is

E[B_1]=500(0.20)=\$100.

At time $1.5$, the expected payment is

E[B_2]=800(0.35)=\$280.

The discount factors are $v^{0.5}=1.1^{-0.5}\approx0.953462589$ and $v^{1.5}=1.1^{-1.5}\approx0.866784172$. Thus:

\begin{aligned} \operatorname{EPV} &=100(0.953462589)+280(0.866784172)\\ &\approx95.346259+242.699568\\ &\approx\$338.045827. \end{aligned}

Rounded to the nearest cent, the expected present value is $338.05. As a check, the first term uses the probability of eligibility at time $0.5$, while the second uses eligibility at time $1.5$; each is discounted for its own full time from issue.

Worked Example: A Benefit Paid at the End of a Transition Interval

A hypothetical contract pays $1,000 at time $0.5$ if a specified transition has occurred by then. The probability of that event by time $0.5$ is $0.20$. If it has not occurred by $0.5$, the probability that it occurs during $(0.5,1.5]$ is $0.25$, conditional on no earlier occurrence. A single benefit is paid at the end of the interval in which the first occurrence takes place. The effective annual interest rate is $8\%$. Find the expected present value. Method: calculate the unconditional probability for each mutually exclusive payment time, then discount at that payment time. This avoids treating a conditional interval probability as unconditional.

The probability of payment at time $0.5$ is $0.20$, so the expected payment then is $1000(0.20)=\$200$. To receive the benefit at time $1.5$, there must first be no occurrence by $0.5$, followed by an occurrence during the next interval. Therefore:

P(\text{first occurrence in }(0.5,1.5]) =(1-0.20)(0.25)=0.80(0.25)=0.20.

The expected payment at time $1.5$ is also $1000(0.20)=\$200$. The discount factors are $1.08^{-0.5}\approx0.962250449$ and $1.08^{-1.5}\approx0.890972638$. Hence:

\begin{aligned} \operatorname{EPV} &=200(0.962250449)+200(0.890972638)\\ &\approx192.450090+178.194528\\ &\approx\$370.644618. \end{aligned}

Rounded to the nearest cent, the expected present value is $370.64. The two payment events are disjoint because the benefit is paid only at the end of the interval of first occurrence.

Worked Example: Updating State Probabilities Across Unequal Intervals

A hypothetical policyholder begins in active state $A$. The states are $A$ and disabled $D$. At time $0.25$, the probabilities are $P(A)=0.90$ and $P(D)=0.10$. From $0.25$ to $1$, the conditional probabilities of being in $D$ at time $1$ are $0.20$ from $A$ and $0.70$ from $D$. From time $1$ to $2.5$, the corresponding conditional probabilities of being in $D$ are $0.10$ from $A$ and $0.80$ from $D$. The contract pays $100 at time $0.25$ in $D$, pays $300 in $D$ and $50 in $A$ at time $1$, and pays $500 in $D$ at time $2.5$. The effective annual interest rate is $10\%$. Find the expected present value. Method: update occupancy probabilities interval by interval, then value each date-specific payment. This avoids using a later state probability for an earlier payment.

At time $0.25$, the expected payment is $100(0.10)=\$10$. At time $1$, the probability of disability is found by conditioning on the state at time $0.25$:

P(D\text{ at }1)=0.90(0.20)+0.10(0.70)=0.18+0.07=0.25.

Thus $P(A\text{ at }1)=1-0.25=0.75$, and the expected payment at time $1$ is

300(0.25)+50(0.75)=75+37.50=\$112.50.

At time $2.5$, condition on the state at time $1$:

P(D\text{ at }2.5)=0.75(0.10)+0.25(0.80)=0.075+0.200=0.275.

The expected payment at time $2.5$ is $500(0.275)=\$137.50$. The discount factors are $1.1^{-0.25}\approx0.976454$, $1.1^{-1}\approx0.909090909$, and $1.1^{-2.5}\approx0.787986$. Therefore:

\begin{aligned} \operatorname{EPV} &=10(0.976454)+112.50(0.909090909)+137.50(0.787986)\\ &\approx9.764540+102.272727+108.348075\\ &\approx\$220.385342. \end{aligned}

Rounded to the nearest cent, the expected present value is $220.39. The state probabilities sum to one at each payment date: $0.90+0.10=1$, $0.75+0.25=1$, and $0.725+0.275=1$. This checks the probability updates before discounting.

Common Timing Errors

Irregular schedules make it easy to confuse the period used to describe a transition with the time used to discount a payment. Consider the second worked example. A common incorrect calculation treats $0.25$, the probability of occurrence conditional on reaching time $0.5$ without occurrence, as the unconditional probability of payment at time $1.5$. It would use $1000(0.25)=\$250$ as the expected payment at time $1.5$, giving:

200(1.08^{-0.5})+250(1.08^{-1.5}) \approx192.450090+222.743160 =\$415.19.

That calculation omits the required probability $0.80$ of having no occurrence by time $0.5$. The correct unconditional probability is $0.80(0.25)=0.20$, making the second expected payment $200 and the correct EPV $370.64. A different error would discount the time-$1.5$ payment for only the interval length $1.0$. That is also wrong: the payment is made $1.5$ years after valuation, so its factor is $v^{1.5}$, regardless of which interval probability was used.

Before adding the discounted terms, it is useful to check that the contract’s payment rule is respected. A benefit paid for occupancy at a date is not automatically a benefit for entering that state during the preceding interval. Likewise, a first-transition benefit cannot be paid twice merely because the process occupies the destination state at multiple later dates.

Key takeaway: For every irregular payment date, identify the exact qualifying event, calculate its probability at the correct conditioning level, and discount the resulting expected payment from the valuation time to that actual date. Add the date-specific present values only after these steps are complete.

Check Your Understanding

For each problem, identify the payment trigger before calculating. Dollar answers are rounded to the nearest cent.

A contract pays $700 in state $D$ at time $0.5$, when $P(D)=0.12$, and $400 in $D$ at time $2$, when $P(D)=0.30$. The effective annual rate is $5\%$. Find the EPV. (Answer: $190.82)
A first-occurrence benefit of $900 is paid at time $1$ if occurrence is in $(0,1]$, or at time $2$ if it is in $(1,2]$. The first-interval probability is $0.10$; the second-interval probability is $0.20$ conditional on no first-interval occurrence. At an effective annual rate of $6\%$, find the EPV. (Answer: $229.09)
A process is in $A$ at issue. At time $0.4$, $P(D)=0.15$. From $0.4$ to $1.6$, the conditional probabilities of being in $D$ are $0.10$ from $A$ and $0.60$ from $D$. Payments are $250 in $D$ at time $0.4$ and $600 in $D$ at time $1.6$. The effective annual rate is $4\%$. Find the EPV. (Answer: $135.53)
A contract pays $400 in $D$ at time $0.5$ and $1,000 in $D$ at time $2$. At time $0.5$, $P(A)=0.70$ and $P(D)=0.30$. Between time $0.5$ and time $2$, the conditional probabilities of being in $D$ are $0.20$ from $A$ and $0.50$ from $D$. The effective annual rate is $5\%$. Find the EPV. (Answer: $380.15)

What You'll Learn