Introduction
A multi-state contract is valued by following both where the policy is and when a cash flow is due. A premium may be payable only while the insured is active; an income benefit may require disability to have lasted through a specified period; and a benefit for the period just completed may still be payable at an anniversary when recovery occurs. If any of those conditions are attached to the wrong state or date, the cash-flow model no longer represents the contract.
This tutorial develops a practical mapping from contract wording to state-contingent cash flows and reserve dates. The examples use hypothetical annual-anniversary models. Their timing conventions are stipulated explicitly; a real contract must be modelled using its own terms.
Set the Timeline Before Assigning Amounts
For an anniversary-based model, begin by marking the contract dates \(t=0,1,2,\ldots\). State the ordering of events at each date. In the examples below, the state at the anniversary is established first; any transition benefit due at that date is then paid; and scheduled premiums or expenses due at that date are assessed according to the contract wording. Benefits paid in arrears are based on the state over the interval that has just ended, not automatically on the state newly established at the anniversary.
A useful mapping records four items for each amount:
- Trigger: a state, a transition, or completion of a specified duration.
- Amount: the dollar amount or formula for the payment.
- Date: the exact time the amount is due.
- Condition: whether the condition is assessed at the date or over the preceding interval.
In particular, a state label alone may not be sufficient. If benefits are deferred, increase with duration, or stop after a maximum number of installments, the model must also track disability duration. Conversely, a cash flow triggered by entry to disability is attached to the transition into that state, not to every date spent disabled.
Worked Example: Mapping a Three-Year Disability Contract
Consider a hypothetical three-year contract with states active (\(A\)) and disabled (\(D\)); the contract ends at \(t=3\). The insured is active at \(t=0\), enters disability at \(t=1\), remains disabled through the intervals \([1,2]\) and \([2,3]\), and the term then ends. A premium of $500 is due at \(t=0,1,2\) only if the insured is active at that anniversary. An issue expense of $60 is due at \(t=0\), and a renewal expense of $20 is due at \(t=1,2\) only if active. A lump sum of $300 is due on entry to disability. Disability income of $800 is paid in arrears at an anniversary for each full year spent disabled in the preceding interval.
Method. Lay out the state and the interval first, then attach each payment to its trigger and date. At \(t=0\), the insured is active, so the $500 premium and $60 issue expense are due. No disability benefit is due.
At \(t=1\), the state changes from active to disabled. The $300 transition lump sum is due because this is the date of entry to disability. The premium and renewal expense are not due: both require active status at the anniversary after the state is established.
At \(t=2\), the insured has been disabled throughout \([1,2]\). The first $800 arrears payment is therefore due. The insured is disabled at \(t=2\), so neither a premium nor a renewal expense is due.
At \(t=3\), the insured has been disabled throughout \([2,3]\), so a second $800 payment is due for that completed interval. The term ends at this date; there is no premium due for a new year of coverage.
The resulting timeline is:
| Date | State or event | Premium received | Expense paid | Benefit paid |
|---|---|---|---|---|
| \(t=0\) | Active; issue | $500 | $60 | $0 |
| \(t=1\) | Entry to disability | $0 | $0 | $300 |
| \(t=2\) | Disabled over \([1,2]\) | $0 | $0 | $800 |
| \(t=3\) | Disabled over \([2,3]\); term ends | $0 | $0 | $800 |
As a path-level check, total premiums are $500, total expenses are $60, and total benefits are \(300+800+800=\)$1,900. The $300 lump sum is counted once, at entry; the two income payments each correspond to a different completed disability year. This avoids counting the onset benefit again simply because the insured remains disabled.
Why State and Duration Both Matter
A convenient formal representation uses an indicator for each state. Suppose a payment \(b\) is due at date \(t\) exactly when the insured is disabled at that date. Its amount on a particular path is \(b\,\mathbf{1}_{\{S_t=D\}}\), where the indicator is one in state \(D\) and zero otherwise. If payment instead requires disability for at least two completed years, the condition becomes \(b\,\mathbf{1}_{\{S_t=D,\ k_t\geq 2\}}\). This representation makes the contract condition visible in the cash-flow formula.
The time of payment is a separate part of the mapping. If an amount \(b\) is paid at date \(u\), its value at an earlier date \(t\), under effective annual interest rate \(i\), is \(b v^{u-t}\), where \(v=1/(1+i)\). The exponent is the number of years from the valuation date to payment. This follows because an amount invested at \(t\) accumulates by \((1+i)^{u-t}\) to date \(u\); the amount at \(t\) that accumulates to \(b\) must therefore be \(b/(1+i)^{u-t}\).
Thus the date and condition work together: the discounted amount for the disability benefit in the preceding interval is not the same cash flow as a benefit conditional on the state at the next anniversary. The contract wording determines which condition applies.
Worked Example: Applying a Waiting Period and a Payment Limit
A hypothetical contract has anniversaries \(t=0,1,2,3,4\). The insured is active at \(t=0\), enters disability at \(t=1\), and remains disabled through \(t=4\). A $400 premium is due at \(t=0\) if active. Disability income is paid in arrears at $700 per installment: the first full year of disability is a waiting year, and the contract pays for at most the next two completed years of disability. There is no onset lump sum.
Method. Count completed years disabled, rather than treating every anniversary in state \(D\) as a payable date. At \(t=1\), disability has just begun, so \(k_1=0\). At \(t=2\), one full disabled year, \([1,2]\), has been completed; it is the waiting year, so no benefit is due. At \(t=3\), two full disabled years have been completed. The interval \([2,3]\) is the first payable year, so $700 is due. At \(t=4\), three full disabled years have been completed; \([3,4]\) is the second payable year, so another $700 is due. The two-installment maximum has now been reached.
| Date | Completed disability years | Installment due | Reason |
|---|---|---|---|
| \(t=0\) | 0 | $0 | Active; premium of $400 is due |
| \(t=1\) | 0 | $0 | Disability has just begun |
| \(t=2\) | 1 | $0 | First year is the waiting year |
| \(t=3\) | 2 | $700 | First payable year completed |
| \(t=4\) | 3 | $700 | Second and final payable year completed |
The total benefit is \(2(700)=\)$1,400, and the total premium on this path is $400. No benefit is due at \(t=2\), even though the insured is disabled at that date, because the completed interval is the waiting year. No third installment is due at \(t=5\), because the payment limit has been reached. The common error is to pay at \(t=2\) merely because the state is disabled; the correction is to test both the completed duration and the payment limit.
Reserve Dates Must Match the Cash-Flow Schedule
A reserve is tied to a specific instant. A prospective reserve immediately before an anniversary's transactions includes cash flows due at that anniversary among the future cash flows. A reserve immediately after those transactions excludes them. The difference is not a change in the contract; it is a change in the valuation instant.
For clarity, use the common net prospective convention: the reserve is the expected present value of future benefits and expenses less the expected present value of future premiums. All amounts and the valuation basis must be specified. A reserve conditional on a current state is calculated using future cash flows conditional on that state. Do not include a premium or expense as future if it has already been paid at the valuation instant.
Worked Example: Valuing Immediately Before or After an Anniversary
At \(t=1\), a hypothetical one-year contract is active. A premium of $50 and an expense of $10 are both due at \(t=1\). At \(t=2\), a benefit of $1,000 is paid if the insured is disabled. The probability of being disabled at \(t=2\), conditional on being active at \(t=1\), is \(0.10\). Assume an effective annual interest rate of \(i=5\%\), no other cash flows, and the net prospective reserve convention just stated.
Method. First value the contingent benefit one year from \(t=1\). Its expected amount at \(t=2\) is
The benefit's present value at \(t=1\) is \(100v=100/1.05=\)$95.2381. Immediately before the \(t=1\) transactions, both the premium and expense are still future amounts. Therefore,
Immediately after those transactions, the premium and expense are no longer future cash flows. The remaining prospective value is
Rounded to cents, the before-transaction reserve is $55.24 and the after-transaction reserve is $95.24. The difference is $40.00, equal to the premium less the expense collected and incurred at \(t=1\). As a timing check, the $1,000 benefit is one year after \(t=1\), so its expected present value uses one discount period.
Common Timing and Mapping Errors
A frequent mistake is to attach a payment due for the interval just completed to the state established at the end of that interval. If recovery occurs at \(t=3\), for example, a benefit for disability throughout \([2,3]\) may still be due at \(t=3\). Whether it is due depends on the contract's arrears condition, not solely on the state recorded after the transition at \(t=3\).
Another common error is discounting from the wrong reserve date. In the preceding example, the benefit is due at \(t=2\) and the reserve is valued at \(t=1\). The incorrect calculation \(100/1.05^2=\)$90.70 discounts for two years, although only one year separates those dates. The correct calculation is \(100/1.05=\)$95.24. Always subtract the valuation date from the payment date to determine the discount period.
Before completing a valuation, check that every cash flow has a stated trigger and date, that waiting periods and limits use completed duration consistently, and that the reserve instant is identified as before or after any transactions at that date. These checks are especially useful when a contract combines active-state premiums, transition benefits, and arrears income.
Check Your Understanding
For each hypothetical contract, identify the relevant payment dates and apply the stated timing convention.
- An active insured enters disability at \(t=2\) and remains disabled through \(t=4\). A $900 benefit is paid in arrears for each completed disabled year, with no waiting period. At which anniversaries are benefits due, and what is the total? (Answer: At \(t=3\) and \(t=4\); total $1,800.)
- A premium of $120 is due at \(t=3\) only if the insured is active immediately after the state at \(t=3\) is established. The insured transitions from disabled to active at that anniversary. Is the premium due under this wording? (Answer: Yes, $120 is due at \(t=3\).)
- A disability benefit pays $500 in arrears after a two-year waiting period for each completed disabled year after the waiting period. Disability begins at \(t=0\) and continues through \(t=4\). How many installments are paid by \(t=4\), and what is their total? (Answer: Two installments, at \(t=3\) and \(t=4\); total $1,000.)
- A $600 benefit is due at \(t=5\), and the effective annual interest rate is \(4\%\). Find its value at \(t=3\), rounded to cents. (Answer: \(600/(1.04)^2=\)$554.73.)
- At \(t=2\), a reserve is valued immediately before a $70 premium and a $15 expense, both due at that date. The reserve immediately after those transactions is $310, with no other cash flow at \(t=2\). Under the net prospective convention, what is the reserve immediately before the transactions? (Answer: \(310+15-70=\)$255.)