Elimination Periods and Deferred Disability Benefits

Why an Elimination Period Changes the Value

A disability income benefit may be payable only if disability continues for a specified period after onset. That waiting period is the elimination period. It is not simply a delay in payment: a claimant who recovers or dies before the period ends receives no benefit under the contract considered here. The value therefore depends both on when benefits can begin and on the probability of remaining eligible until then.

The central valuation task is to keep distinct three parts of the calculation: the chance that disability begins, the chance that the insured remains disabled through the elimination period, and the discounted value of payments after commencement. Treating the elimination period as interest discount alone overstates the benefit value whenever recovery or disabled-state death can occur.

Definition: An elimination period of length $\tau$ requires the insured to remain continuously disabled for $\tau$ time units after disability onset before income benefits commence. In this tutorial, recovery or death during that period ends the claim without payment. Once benefits commence, they are paid continuously only while the insured remains disabled, and recovery or death ends them.

Model and Assumptions

Let the states be active, disabled, and dead. While active, disability begins with constant intensity $\alpha$, and death occurs with constant intensity $\mu_A$. While disabled, recovery occurs with constant intensity $\rho$, and death occurs with constant intensity $\mu_D$. The force of interest is constant at $\delta$. All intensities and $\delta$ are measured per year, and time is measured in years.

The calculations below assume constant transition intensities during the periods considered, a continuous benefit rate $b$ per year, and no benefit before the elimination period is complete. Benefits cease on recovery or death. Unless a finite benefit term is specified, the benefit continues indefinitely while disabled. For the active-state examples, the claim considered is the first disability claim; after recovery, the contract terminates for purposes of this calculation. This avoids including a new disability claim after recovery, which requires a recurrent-disability model.

For constant competing exit intensities, the probability of avoiding both exits for $t$ years is exponential. Thus, from disability onset, the probability of remaining disabled to time $t$ is $e^{-(\rho+\mu_D)t}$. Interest discounting over the same period contributes another factor, $e^{-\delta t}$. Both factors matter during the elimination period as well as during payment.

Key result: For an insured already disabled at claim onset, the value at onset of a continuous benefit rate $b$, commencing after an elimination period $\tau$ and payable indefinitely while disabled, is $$ V_D(0)=b\int_{\tau}^{\infty}e^{-(\delta+\rho+\mu_D)t}\,dt =\frac{b e^{-(\delta+\rho+\mu_D)\tau}}{\delta+\rho+\mu_D}. $$ This result requires $\delta+\rho+\mu_D>0$ and the stated constant-intensity assumptions.

Why the Deferred-Value Formula Holds

A small payment $b\,dt$ at time $t$ is made only if the insured has remained disabled to that time. Its expected present value at claim onset is therefore the payment amount multiplied by the probability of remaining disabled and by the discount factor:

b\,dt\; e^{-(\rho+\mu_D)t}e^{-\delta t} =b e^{-(\delta+\rho+\mu_D)t}\,dt.

Payments are possible from time $\tau$ onward, so adding their expected present values gives the integral from $\tau$ to infinity. Integrating the exponential gives the displayed result. In particular, the probability of qualifying through the elimination period is $e^{-(\rho+\mu_D)\tau}$, while discounting over that period is $e^{-\delta\tau}$. Their product is $e^{-(\delta+\rho+\mu_D)\tau}$. Omitting either survival or discounting changes the value.

Worked Example: Deferred Income for an Insured Already Disabled

A hypothetical insured is disabled at claim onset. The force of interest is $\delta=0.04$, the recovery intensity is $\rho=0.06$, and the disabled-state death intensity is $\mu_D=0.02$, all per year. A continuous benefit of $b=\$12{,}000$ per year begins after a half-year elimination period and continues indefinitely while disabled. Find its value at claim onset.

Method. Apply the deferred continuous-benefit formula using the total rate $\delta+\rho+\mu_D$. The total is $$ 0.04+0.06+0.02=0.12. $$

The half-year deferral contributes the factor $$ e^{-0.12(0.5)}=e^{-0.06}=0.9417645336. $$ The value of the benefit stream, before applying this deferral factor, is $$ \frac{12{,}000}{0.12}=\$100{,}000. $$ Therefore, $$ V_D(0)=100{,}000(0.9417645336)=\$94{,}176.45336. $$

Rounded to the nearest cent, the value is \$94,176.45. The method includes both interest and the chance of recovery or death before and during payments. A common mistake is to use only $e^{-\delta\tau}$ for the waiting period; that would ignore that the insured may cease to qualify before benefits commence.

Adding Disability Incidence Before the Claim

When valuation begins with an active insured, the claim-onset value is not the value at issue. Disability must first occur before active-state death. Under the constant-intensity assumptions, the probability of still being active at time $s$ is $e^{-(\alpha+\mu_A)s}$. Hence the density of disability incidence at time $s$ is $\alpha e^{-(\alpha+\mu_A)s}$. Discounting that incidence time to issue adds $e^{-\delta s}$.

If $V_D(0)$ is the value at disability onset of the benefits attached to that claim, the issue value is the integral of the discounted incidence density multiplied by $V_D(0)$. For indefinite continuous benefits, this gives:

V_A(0)=\int_0^\infty \alpha e^{-(\delta+\alpha+\mu_A)s}V_D(0)\,ds =\frac{\alpha}{\delta+\alpha+\mu_A}V_D(0).

This formula values one possible first claim while active. It does not add new claims following recovery, and it does not apply unchanged if intensities vary with time or duration. With time-varying intensities, the transition probabilities and discounting must instead be integrated over the relevant paths.

Worked Example: Issue Value with a Half-Year Elimination Period

A hypothetical active insured has disability incidence intensity $\alpha=0.04$ and active-state death intensity $\mu_A=0.01$. If disabled, recovery has intensity $\rho=0.08$ and death has intensity $\mu_D=0.02$. The force of interest is $\delta=0.05$. A continuous benefit of $\$15{,}000$ per year begins after a half-year elimination period and continues indefinitely while disabled. Find the issue value.

Step 1: Value the benefit at disability onset. The total disabled-state rate including interest is $$ \delta+\rho+\mu_D=0.05+0.08+0.02=0.15. $$ The elimination-period factor is $$ e^{-0.15(0.5)}=e^{-0.075}=0.9277434863. $$ The deferred claim value is therefore $$ V_D(0)=\frac{15{,}000}{0.15}(0.9277434863) =100{,}000(0.9277434863) =\$92{,}774.34863. $$

Step 2: Account for the time to disability incidence. The active-state denominator, including interest, is $$ \delta+\alpha+\mu_A=0.05+0.04+0.01=0.10, $$ so the incidence factor is $$ \frac{\alpha}{\delta+\alpha+\mu_A} =\frac{0.04}{0.10}=0.4. $$

Step 3: Calculate and check the issue value. $$ V_A(0)=0.4(92{,}774.34863)=\$37{,}109.73945. $$ Rounded to the nearest cent, the issue value is \$37,109.74. The factor $0.4$ accounts for the possibility of active-state death or discounting before disability incidence; the claim value separately accounts for disabled-state survival and discounting after incidence.

Benefits with a Fixed Maximum Duration

Some hypothetical contracts stop benefits at a specified time after disability onset, even if the insured remains disabled. If benefits end $T$ years after onset and $T>\tau$, the integral is over the interval from $\tau$ to $T$, not from $\tau$ to infinity. Under constant intensities, the value at claim onset is

V_D(0)=b\int_{\tau}^{T}e^{-(\delta+\rho+\mu_D)t}\,dt =\frac{b\left(e^{-(\delta+\rho+\mu_D)\tau}-e^{-(\delta+\rho+\mu_D)T}\right)} {\delta+\rho+\mu_D}.

If $T\leq\tau$, no benefit can be paid and the value is zero. The integration variable is time since disability onset: replacing it with time since benefit commencement without shifting the limits would misstate the discounting and survival periods.

Worked Example: Active-State Value with a Benefit End Date

A hypothetical active insured has disability incidence intensity $\alpha=0.06$, active-state death intensity $\mu_A=0.02$, and force of interest $\delta=0.04$. If disabled, recovery intensity is $\rho=0.06$ and death intensity is $\mu_D=0.02$. A continuous benefit of $\$18{,}000$ per year begins six months after disability onset and ends two and a half years after onset. Find the issue value.

Step 1: Value payments from claim onset. The combined rate for discounting and disabled-state exits is $$ \delta+\rho+\mu_D=0.04+0.06+0.02=0.12. $$ The benefit interval is from $\tau=0.5$ to $T=2.5$, so its discounted, survival-weighted duration is $$ \int_{0.5}^{2.5}e^{-0.12t}\,dt =\frac{e^{-0.12(0.5)}-e^{-0.12(2.5)}}{0.12} =\frac{e^{-0.06}-e^{-0.30}}{0.12}. $$ Using $e^{-0.06}=0.9417645336$ and $e^{-0.30}=0.7408182207$, this is $$ \frac{0.9417645336-0.7408182207}{0.12} =\frac{0.2009463129}{0.12} =1.6745526075. $$ Thus the value at disability onset is $$ V_D(0)=18{,}000(1.6745526075)=\$30{,}141.94694. $$

Step 2: Include active-state incidence. The active-state denominator is $$ \delta+\alpha+\mu_A=0.04+0.06+0.02=0.12, $$ giving an incidence factor of $$ \frac{0.06}{0.12}=0.5. $$ The issue value is consequently $$ V_A(0)=0.5(30{,}141.94694)=\$15{,}070.97347. $$ Rounded to the nearest cent, the answer is \$15,070.97. The finite benefit term is applied at claim onset before multiplying by the active-state incidence factor; the end date does not mean that payments continue for two and a half years after the elimination period.

Common Mistake: Treating the Wait as Interest-Only Deferral

Consider the second worked example. The disabled-state total rate is $0.15$, so survival and discounting through the half-year elimination period together contribute $e^{-0.075}=0.9277434863$. The correct value at disability onset is $\$92{,}774.35$, and the issue value is $\$37{,}109.74$.

The incorrect interest-only calculation would use $e^{-0.05(0.5)}=e^{-0.025}=0.9753099120$, giving a claim value of $$ \frac{15{,}000}{0.15}(0.9753099120)=\$97{,}530.99. $$ Multiplying by the same active-state incidence factor gives an incorrect issue value of $0.4(\$97{,}530.99)=\$39{,}012.40$. The error is that this calculation discounts for the waiting period but does not require the insured to remain disabled during it. Recovery and disabled-state death must be reflected in the qualification probability as well as in the post-commencement payment stream.

The continuous-payment formulas also should not be transferred mechanically to a contract paying monthly or annually in arrears. For discrete payments, each payment date must be valued using the probability of being eligible on that date and the appropriate discount factor. The elimination-period condition still applies, but the cash-flow timing differs.

Key takeaway: For a deferred disability income claim, begin payments only after the insured has survived and remained disabled through the elimination period. Combine discounting with disabled-state exit intensities over every time payments are conditional on continued disability; then, if valuation begins while active, separately account for incidence and active-state competing exits.

Check Your Understanding

For these original problems, assume constant intensities, a constant force of interest, and continuous benefits payable only while disabled. Round dollar answers to the nearest cent.

An insured is disabled at claim onset. The force of interest is $0.02$, recovery intensity is $0.08$, and disabled-state death intensity is $0.02$. A continuous benefit of $\$10{,}000$ per year begins after a three-month elimination period and continues indefinitely. Find the value at claim onset. (Answer: $\$80{,}870.46$)
An active insured has disability incidence intensity $0.03$, active-state death intensity $0.02$, and force of interest $0.05$. If disabled, recovery intensity is $0.05$ and death intensity is $0.03$. A $\$12{,}000$-per-year continuous benefit begins after one year and continues indefinitely. Find the issue value for a first claim. (Answer: $\$24{,}316.49$)
An insured is disabled at claim onset. The force of interest is $0.04$, recovery intensity is $0.06$, and disabled-state death intensity is $0.02$. A continuous benefit of $\$20{,}000$ per year is payable from six months to two and a half years after onset. Find the value at claim onset. (Answer: $\$33{,}491.05$)
An active insured has disability incidence intensity $0.05$, active-state death intensity $0.01$, and force of interest $0.04$. If disabled, recovery intensity is $0.07$ and death intensity is $0.03$. A $\$12{,}000$-per-year continuous benefit begins after a half-year elimination period and continues indefinitely. Find the issue value for a first claim. (Answer: $\$39{,}959.74$)

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