Deferred and Waiting-Period Benefits

Why a Waiting Period Changes the Benefit Value

A benefit described as “payable after a two-year waiting period” is not fully specified until we know what must happen during those two years. For a disability benefit, the contract might require continuous disability from claim onset until the waiting period ends. Recovery before then can prevent eligibility, and a later disability episode may start a new waiting period. In that case, valuing the benefit as if it were merely postponed by two years overstates its expected value.

The key is to track both the policy state and the waiting-period status. A person who is disabled but has not yet satisfied the waiting period is not in the same benefit state as a person who is disabled and eligible to receive benefits. The calculations below focus on benefit values; premiums and expenses are excluded unless explicitly stated.

For a continuous-disability condition, it is useful to distinguish the state \(D_W\), disabled and still waiting, from \(D_E\), disabled and eligible. Recovery and death can terminate eligibility or prevent it from being reached. A waiting period measured from the onset of disability is therefore different from a fixed payment date measured from policy issue.

Prospective Value While Still Waiting

Consider a hypothetical contract with a continuous benefit paid at rate \(b\) per year once eligibility begins. Suppose the insured is currently disabled and has \(r\) years remaining in the waiting period. Assume that recovery and death are competing transitions with constant intensities \(\rho_W\) and \(\mu_W\) while waiting, and that either event ends the current disability claim. Write \(q_W=\rho_W+\mu_W\). Once eligible, assume the corresponding total termination intensity is the constant \(q_E\). Let the force of interest be the constant \(\delta\).

There is no benefit during the remaining waiting time. To reach eligibility, the insured must remain disabled for all \(r\) years, with probability \(e^{-q_Wr}\). Conditional on reaching eligibility, the expected present value at that time of a continuous benefit paid until recovery or death is \(b/(\delta+q_E)\). Discounting that value back through the waiting period gives:

This formula assumes constant transition intensities, a benefit that starts as soon as the waiting period is completed, and continuous payments that cease at recovery or death. It is a prospective value conditional on being disabled and waiting now; it is not automatically the value at policy issue. To calculate an issue-date value, the probability and timing of entering the waiting state must also be included.

Why the Waiting-Period Formula Holds

The formula follows by conditioning on whether the insured reaches eligibility. While waiting, the total rate of recovery or death is \(q_W\). Under constant competing intensities, the probability of no such transition for \(r\) years is \(e^{-q_Wr}\). The force of interest discounts a payment at time \(r\) by \(e^{-\delta r}\). Thus the combined factor for both remaining disabled and discounting is \(e^{-(q_W+\delta)r}\).

At eligibility, consider a small interval of length \(dt\). Conditional on being eligible at its start, the probability of still being eligible at time \(t\) is \(e^{-q_Et}\), and the discount factor is \(e^{-\delta t}\). The expected value of the continuous payments from that point is therefore:

Multiplying the value at eligibility by the probability-and-discount factor for reaching eligibility gives the stated prospective value. The argument also shows why using \(q_E\) during the waiting period, or omitting \(q_W\) altogether, can produce an incorrect result.

Discrete Payments and State Eligibility

For annual benefits, the same state distinction applies, but the payment timing must also be specified. Suppose a benefit is paid at the end of each year only if the insured is eligible at that payment date. If qualification requires two complete years of continuous disability, the first possible payment is at the end of year 2. Under a simple annual model with probability \(p\) of remaining disabled in each year, the probability of remaining continuously disabled through year \(k\) is \(p^k\). With annual effective interest rate \(i\), the value of a level payment \(B\) at each eligible year-end is \(B\sum_{k=2}^{\infty}p^kv^k\), where \(v=1/(1+i)\), provided \(pv<1\).

This annual model is a discrete-time assumption, not a conversion of continuous transition intensities. It treats the disabled-or-terminated transition as occurring over each annual interval and makes payment eligibility depend on the state at each year-end. The contract’s timing convention determines whether a payment is due exactly when the waiting period ends.

Residual Waiting Time and Changing Intensities

A reserve during a disability episode may be calculated after part of the waiting period has already elapsed. Use the remaining waiting time, not the original full waiting period. If the claimant has completed eight months of a twelve-month waiting period and the contract requires continuous disability, then the residual waiting time is four months. Rates expressed per year must be applied to \(4/12\) of a year, not to four years or to the original twelve months.

Transition intensities may also vary with time or with waiting status. If the total rate of recovery or death at time \(t\) during the remaining wait is \(q_W(t)\), the probability of remaining in the waiting state for \(r\) years is \(\exp(-\int_0^r q_W(t)\,dt)\). If the force of interest is also time-varying, its discount factor over the wait is \(\exp(-\int_0^r\delta(t)\,dt)\). These factors multiply the eligible-state value when the post-qualification benefit model is separate and correctly specified.

Common Errors and Model Checks

Waiting-period calculations are especially sensitive to the contract’s definition of qualification. Before calculating, state whether the clock starts at policy issue or claim onset; whether the claimant must remain in one state continuously; what recovery does to the clock; and when the first payment is due. These are model assumptions, not details that can safely be inferred from the word “deferred.”

• Discounting without allowing for failure to qualify. Multiplying an eligible-state value only by \(e^{-\delta r}\) assumes everyone reaches eligibility. For continuous-disability qualification, include the probability of remaining disabled during the wait.
• Using the wrong transition rate. Waiting-state and eligible-state termination intensities can differ. Use each rate over the period in which it applies.
• Starting payments too early. In a discrete model, a two-year waiting period does not ordinarily permit a payment at the end of year 1. Check the precise payment-date condition.
• Mixing time units. A rate per year must be multiplied by time in years. Convert months to a fraction of a year when using annual rates.
• Confusing benefit deferral with eligibility. If the benefit is merely postponed but remains payable regardless of intervening state changes, a continuous-disability survival factor is not appropriate.

A useful check is to compare the waiting-state value with the corresponding value for someone already eligible. Under the assumptions used above, the waiting-state value is the eligible-state value multiplied by factors no greater than one, so it cannot be larger. Also check that a longer waiting period or higher waiting-state termination intensity does not increase the value, all else equal.