Waiting-Period and Benefit-Limit Calculation Workshop

Why Waiting Periods and Benefit Limits Need Careful Timing

A disability income benefit may be payable only after an insured has remained disabled for an elimination period. Even if the insured is disabled at the start of that period, recovery or death can occur before benefits become payable. Once eligibility begins, a separate maximum benefit period may restrict how long payments continue. A correct valuation must track both clocks: the wait from disability onset and the limit on benefits after eligibility.

The calculations below value benefits conditional on a disability beginning at time 0. They use constant transition intensities while the insured is disabled and a constant force of interest. There is no relapse after recovery, and death and recovery both end the claim. For a contract valued from policy issue, an appropriate disability-incidence probability or onset-time calculation must also be applied; it is not included in these conditional claim values.

Definition: The elimination period is the time $m$ after disability onset during which no disability income benefit is payable. In this tutorial, the maximum benefit period $n$ starts when the elimination period ends. Thus, if the insured remains eligible, benefits can be paid only during the interval from $m$ to $m+n$ after disability onset. A contract that instead measures its maximum period from disability onset has a different endpoint, which must be modelled explicitly.

Model and Continuous-Benefit Formula

Assume that, while disabled, recovery occurs with constant intensity $\rho$ per year and death occurs with constant intensity $\mu_d$ per year. These are competing exits from the disabled state: the first exit ends the claim. The total exit intensity is $q=\rho+\mu_d$, so the time until recovery or death has an exponential distribution with parameter $q$. Let $\delta$ be the force of interest, and let $b$ be the continuous annual benefit rate, paid only while the insured is alive and disabled and after the elimination period.

Under these assumptions, the probability the insured remains disabled at time $t$ after onset is $e^{-qt}$. A payment at time $t$ has discount factor $e^{-\delta t}$. Therefore, the expected present value (EPV) of continuous benefits, with a maximum period $n$ measured from eligibility, is:

\operatorname{EPV} =b\int_m^{m+n}e^{-\delta t}e^{-qt}\,dt =\frac{b}{\delta+q}\left(e^{-(\delta+q)m}-e^{-(\delta+q)(m+n)}\right).

The formula applies to a continuous benefit rate under the stated constant-intensity assumptions. It does not directly value monthly payments, benefits that continue after recovery or death, or a limit measured from disability onset without adjusting the endpoint. In those cases, change the payment times, state-contingent payments, or interval of integration as appropriate.

Why the Formula Includes the Waiting Period

The integrand is the expected discounted payment rate at time $t$. A payment is made at that time only if the insured has not recovered or died; that probability is $e^{-qt}$. Discounting that payment to time 0 multiplies it by $e^{-\delta t}$. Multiplying gives $e^{-(\delta+q)t}$, and integrating the benefit rate over the payable interval gives the EPV.

Equivalently, split the calculation at eligibility. The EPV at time $m$, conditional on being disabled then, is $b(1-e^{-(\delta+q)n})/(\delta+q)$. To value it at time 0, multiply by $e^{-\delta m}$, the discount factor to time $m$, and by $e^{-qm}$, the probability of remaining disabled through the wait. Their product is $e^{-(\delta+q)m}$. This explains why surviving the wait and discounting the wait are distinct effects, even though they combine in the formula.

Worked Example: Continuous Benefits After a Three-Month Wait

A hypothetical policy pays a continuous benefit at an annual rate of $12,000 after a four-month elimination period. The maximum benefit period is eight months, measured from eligibility. A disability begins at time 0. Assume a force of interest $\delta=0.05$, constant recovery intensity $\rho=0.07$, and constant disabled-state death intensity $\mu_d=0.03$, all per year. The recovery and death waiting times are exponential, with the stated competing intensities. Find the conditional EPV of benefits.

Method. Use the continuous-benefit formula, with $m=4/12=1/3$ year and $n=8/12=2/3$ year. The combined discount and exit rate is $\delta+q=0.05+0.07+0.03=0.15$.

The payable interval runs from $m=1/3$ to $m+n=1$ year after disability onset. Thus:

\begin{aligned} \operatorname{EPV} &=\frac{12{,}000}{0.15} \left(e^{-0.15(1/3)}-e^{-0.15(1)}\right)\\ &=80{,}000\left(e^{-0.05}-e^{-0.15}\right)\\ &=80{,}000(0.95122942-0.86070798)\\ &=80{,}000(0.09052145)\\ &=\$7{,}241.72. \end{aligned}

The arithmetic is rounded only at the final step, to the nearest cent. As a check on the waiting-period interpretation, the probability of remaining disabled for four months is $e^{-0.10(1/3)}=e^{-0.033333\ldots}$, approximately $0.9672$. The formula also discounts those four months; it does not treat the benefit as if it began at time 0.

Common mistake avoided. Starting the integration at eligibility but omitting the wait factor gives $80{,}000(1-e^{-0.10})=\$7{,}613.01$. That is the value at disability onset of a hypothetical benefit available immediately for eight months; it ignores both discounting and claim survival during the four-month wait. The correct value is $7,241.72.

Monthly Payments and Maximum Payment Counts

For benefits paid in arrears at regular intervals, value only the scheduled payments that fall within the maximum benefit period. If payments are monthly, the first payment is one month after eligibility under a convention that pays after each completed month of eligible disability. A payment is made only if the insured remains disabled and alive at its payment time.

With monthly payments $B$ at times $t_j$, the EPV is the sum of each payment multiplied by its discount factor and the probability the insured is still disabled then:

\operatorname{EPV}=\sum_j B e^{-(\delta+q)t_j}.

This sum is not generally equal to a continuous-benefit integral over the same interval. In particular, a six-month maximum period with monthly payments in arrears permits six scheduled payments, at one through six months after eligibility, not a payment at the eligibility instant.

Worked Example: Six Monthly Payments After a Three-Month Wait

A hypothetical policy pays $1,000 at the end of each eligible month, after a three-month elimination period, for at most six monthly payments. Disability begins at time 0. Assume $\delta=0.06$ per year, recovery intensity $\rho=0.12$ per year, and disabled-state death intensity $\mu_d=0.03$ per year. Recovery and death have constant exponential waiting times and are competing exits. Find the conditional EPV.

Method. The total exit intensity is $q=0.12+0.03=0.15$, so $\delta+q=0.21$ per year. The first payment is at month 4, the end of the first eligible month. Six payments occur at months 4 through 9, or at times $4/12,\ldots,9/12$ years.

\begin{aligned} \operatorname{EPV} &=1{,}000\sum_{j=4}^{9}e^{-0.21j/12}\\ &=1{,}000\left(e^{-0.07}+e^{-0.0875}+e^{-0.105} +e^{-0.1225}+e^{-0.14}+e^{-0.1575}\right)\\ &=1{,}000(0.93239382+0.91621887+0.90032452 +0.88470590+0.86935824+0.85427683)\\ &=1{,}000(5.35727818)\\ &=\$5{,}357.28. \end{aligned}

The displayed exponent uses the combined force $0.21$, and each term reflects survival in the disabled state through that payment time. The result is rounded only at the final step, to the nearest cent.

Common mistake avoided. Paying at months 3 through 8 would put the first payment at the instant eligibility begins. That contradicts the stated end-of-month payment convention. The six scheduled payment times are months 4 through 9.

Limits Measured From Onset or From Eligibility

A benefit limit must be anchored to the date specified by the contract. If a 12-month maximum starts at eligibility after a six-month elimination period, the last possible benefit time is 18 months after disability onset. If the same 12-month limit starts at onset, the benefit window after eligibility is only six months. The time unit of every period should be converted before setting the integration endpoint or payment dates.

Worked Example: Converting a Limit Measured From Disability Onset

A hypothetical policy pays a continuous annual rate of $15,000 after a six-month elimination period. Its maximum benefit duration is 18 months from disability onset, not from eligibility. A disability begins at time 0. Assume a force of interest $\delta=0.04$ per year, recovery intensity $\rho=0.06$ per year, and disabled-state death intensity $\mu_d=0.02$ per year. Recovery and death are constant-intensity competing exits. Find the conditional EPV.

Method. Convert the elimination period to $m=6/12=0.5$ years and the contract endpoint to $T=18/12=1.5$ years. Because the maximum period is measured from onset, benefits can be paid only from $m=0.5$ through $T=1.5$, a one-year payable interval. The combined rate is $\delta+\rho+\mu_d=0.04+0.06+0.02=0.12$.

\begin{aligned} \operatorname{EPV} &=\frac{15{,}000}{0.12} \left(e^{-0.12(0.5)}-e^{-0.12(1.5)}\right)\\ &=125{,}000\left(e^{-0.06}-e^{-0.18}\right)\\ &=125{,}000(0.94176453-0.83527021)\\ &=125{,}000(0.10649432)\\ &=\$13{,}311.79. \end{aligned}

The final value is rounded to the nearest cent. If the 18-month limit had instead started at eligibility, the endpoint would be $0.5+1.5=2.0$ years, and this would be a different contract with a larger payable interval. Using that endpoint here would overstate the benefit value.

A Reliable Calculation Sequence

A short sequence of checks helps prevent timing and state errors in disability income valuations:

Identify disability onset and the wait.
Set time 0 at onset for a conditional claim value, and convert the elimination period into the same time unit as the intensities.

Specify exits from the disabled state.
Add the recovery and death intensities to obtain $q$. Confirm that both events terminate benefits under the model.

Set the actual payable window.
Determine whether the maximum period begins at eligibility or at onset, then identify the last possible payment time.

Match the valuation to payment timing.
Integrate a continuous benefit rate over its payable interval, or sum only the scheduled discrete payments.

Check survival, discounting, and units.
Use the combined rate $\delta+q$ for continuously valued payments, and verify that month-to-year conversions and payment dates match the contract.

Common Traps and Their Corrections

Ignoring recovery or death during the wait. The fact that disability began does not guarantee eligibility. Include the probability of remaining disabled until each payment time.
Confusing survival with discounting. Survival is governed by $q=\rho+\mu_d$; discounting is governed by $\delta$. They combine in the EPV integrand but represent different assumptions.
Starting the cap at the wrong date. Read whether the maximum benefit period runs from onset or from eligibility. Do not assume the two meanings are interchangeable.
Paying at eligibility under an arrears convention. If the first payment follows a completed month of eligible disability, it is one month after eligibility, not at the eligibility date.
Mixing monthly and annual units. Convert months to years when using annual intensities and an annual force of interest; otherwise the exponent is incorrect.

Key takeaway: For a continuous benefit after an elimination period, value payments only while the insured is still disabled and within the contract’s stated benefit window. For scheduled payments, use the actual payment dates. Always establish whether the maximum period begins at disability onset or at eligibility.

Check Your Understanding

For each problem, assume constant recovery and death intensities while disabled, with recovery and death as competing exits. Round money to the nearest cent and probabilities to four decimal places.

A hypothetical policy pays continuously at $9,000 per year after a three-month wait, for at most nine months from eligibility. Let $\delta=0.04$, $\rho=0.08$, and $\mu_d=0.02$, all per year. What is the conditional EPV? (Answer: $6,187.32)
A policy pays $800 at the end of each eligible month, for at most four payments, following a two-month elimination period. Let $\delta=0.03$, $\rho=0.09$, and $\mu_d=0.03$ per year. What is the conditional EPV? (Answer: $3,025.26)
During a six-month elimination period, recovery intensity is $0.12$ and death intensity is $0.04$ per year. What are the probabilities of recovery before six months, death before six months, and remaining disabled at six months? (Answer: $0.0577$, $0.0192$, and $0.9231$, respectively.)
A policy pays continuously at $12,000 per year after a three-month wait, with a one-year maximum benefit period measured from eligibility. Let $\delta=0.05$, $\rho=0.10$, and $\mu_d=0.05$ per year. What is the conditional EPV? (Answer: $10,345.72)
A hypothetical policy pays continuously at $15,000 per year after a six-month wait. Its maximum benefit duration is 18 months from onset. Let $\delta=0.04$, $\rho=0.06$, and $\mu_d=0.02$ per year. What is the conditional EPV? (Answer: $13,311.79)

What You'll Learn