Valuing Benefits After a Waiting Period

Why a Waiting Period Changes the Valuation

A disability income contract may require a person to remain disabled for a specified period before benefits become payable. This elimination period is not merely a delay in sending a payment: if the person recovers or dies before it ends, the spell does not qualify. Even after the waiting period has ended, a later payment is generally due only if the person is still disabled on that payment date. Valuation must therefore track both where the person is and how long the current disability spell has lasted.

The distinction matters because the probability of being disabled at a date includes people whose disability began recently. Those people may not yet satisfy the elimination period. The value of benefits is based on the probability of being disabled and eligible, not simply the probability of occupying the disabled state.

Definition: An elimination period of length $w$ requires a disability spell to persist continuously for at least $w$ time units before benefits become payable. Under a contract that pays at specified dates, a benefit at date $t$ is payable only if the insured is alive and disabled at $t$, and the current uninterrupted disability spell began no later than $t-w$. A recovery or death ends the spell; a subsequent disability starts a new spell and a new elimination period.

Discrete Payment Dates and Claim Eligibility

First use a discrete-time model with annual state observations. At each integer time, an insured is active ($A$), disabled ($D$), or dead ($X$). Death is absorbing. A transition from $A$ to $D$ starts a disability spell; a transition from $D$ to $A$ is recovery. The transition probabilities describe the state at the next observation date. This convention does not model the exact time of a transition within a year.

Suppose the insured is active at time 0, and a benefit is paid at integer time $n$ if the insured is alive, disabled, and has remained continuously disabled for at least $k$ full annual intervals. Write $p_A(j)$ for the probability of being active at time $j$. For a time-homogeneous model, let $q_{AD}$ be the one-year probability of transition from active to disabled and $p_{DD}$ the one-year probability of remaining disabled.

P(\text{eligible at }n) = \sum_{m=1}^{n-k}p_A(m-1)\,q_{AD}\,(p_{DD})^{n-m}, \qquad n\ge k+1.

Here $m$ is the date at which the spell starts: the insured moves from active at $m-1$ to disabled at $m$. The spell must then remain in $D$ through payment date $n$. Its duration is $n-m$ full intervals, so eligibility requires $n-m\ge k$. If $n\le k$, there has not been enough time for a disability beginning after time 0 to complete the waiting period before payment.

The probability $p_A(m-1)$ is important. It includes all paths that leave the insured active at that date, including paths involving a prior disability followed by recovery. Thus the formula allows a recovered person to have a later, potentially qualifying spell. A prior spell does not make the new spell eligible.

Payment-date valuation: If $b_n$ is the benefit paid at time $n$, $i$ is the effective annual interest rate, and $E_n$ is the event that the insured is eligible at that date, the expected present value of benefits at specified dates is $$ \operatorname{EPV}=\sum_n b_n(1+i)^{-n}P(E_n). $$ Only dates at which payment is contractually due belong in the sum. Eligibility is evaluated separately at each such date.

Why the Eligibility Formula Holds

For a payment at time $n$, split the event of eligibility into disjoint possibilities according to the spell's start date $m$. A qualifying spell must start while the insured is active, so the probability of an active state at $m-1$ followed by a transition to disability at $m$ is $p_A(m-1)q_{AD}$. From $m$ through $n$, the insured must remain disabled for $n-m$ consecutive intervals, with probability $(p_{DD})^{n-m}$. Multiplying gives the probability for that particular start date. Summing over all possible start dates with $m\le n-k$ gives the stated expression. The cases are disjoint because one continuous spell cannot have two different start dates.

For a non-homogeneous model, use the relevant transition probability for each interval instead of a power. For example, the persistence factor is the product of the disabled-to-disabled probabilities from the interval after onset through the payment date. The same reasoning applies: identify the onset date, require the transition into disability, and then require uninterrupted disabled status.

Worked Example: One-Year Elimination Period and a Single Payment

A hypothetical annual disability contract pays $12,000 at time 2 if the insured qualifies. The insured is active at time 0. Each year, the transition probabilities from active are $p_{AA}=0.88$, $q_{AD}=0.10$, and $p_{AX}=0.02$; from disabled they are $p_{DA}=0.15$, $p_{DD}=0.80$, and $p_{DX}=0.05$. Death is absorbing. The elimination period is one year, and the effective annual interest rate is 5%. Find the expected present value of the payment.

Method. Enumerate the disability start dates that leave enough time to complete the one-year waiting period by time 2. The only possible start is $m=1$. The insured must transition from active to disabled in year 1 and remain disabled in year 2.

The probability of that path is

P(E_2)=p_A(0)q_{AD}p_{DD} =1(0.10)(0.80)=0.08.

The discount factor to time 2 is

v^2=(1.05)^{-2}=\frac{1}{1.1025}=0.907029\ldots.

Therefore,

\operatorname{EPV}=12{,}000(0.08)(1.05)^{-2} =960(0.907029\ldots)=870.748\ldots.

Rounded to the nearest cent, the expected present value is $870.75. The result is consistent with the model: the $12,000 payment has an 8% probability of being made, and is discounted for two years. A common mistake is to use the probability of being disabled at time 2 without checking when the spell began.

Valuing More Than One Payment Date

For a stream of benefits, eligibility must be re-evaluated at each payment date. A spell can qualify at one date and fail to qualify later because of recovery or death. Conversely, a person who has recovered may start a new spell that qualifies at a later date. The cash flows are valued date by date; the same probability of eligibility should not be reused for every installment.

Worked Example: Multiple Payments After a One-Year Wait

A hypothetical contract pays $1,000 at times 2 and 3, provided the insured is alive and disabled with a disability spell of at least one full year. The insured is active at time 0. Annual transition probabilities are $p_{AA}=0.88$, $q_{AD}=0.10$, $p_{AX}=0.02$, $p_{DA}=0.15$, $p_{DD}=0.80$, and $p_{DX}=0.05$, with death absorbing. The effective annual interest rate is 5%. Find the expected present value.

At time 2. The only qualifying onset is at time 1, followed by one year in $D$:

$$ P(E_2)=1(0.10)(0.80)=0.08. $$

At time 3. A qualifying spell can start at time 1 or time 2. For a start at time 1, the insured must remain disabled for two intervals. For a start at time 2, the insured must be active at time 1, transition to disabled, and remain disabled through time 3. Thus

$$ P(E_3) =(1)(0.10)(0.80)^2+(0.88)(0.10)(0.80) =0.064+0.0704=0.1344. $$

The second term uses $p_A(1)=0.88$, the probability of being active at time 1. The probability of being disabled at time 1 is not used as a starting probability for a new spell.

Discount each expected installment at its own payment time:

\begin{aligned} \operatorname{EPV} &=1{,}000(0.08)(1.05)^{-2} +1{,}000(0.1344)(1.05)^{-3}\\ &=1{,}000(0.072562\ldots+0.116100\ldots)\\ &=\$188.66. \end{aligned}

The answer is rounded to the nearest cent. This method avoids treating the first eligible payment as proof that later payments are also due: each installment requires the insured to remain eligible on its own date.

Recovery, Death, and Later Claim Onsets

Recovery and death affect eligibility in different ways. Death makes future payments impossible. Recovery ends the current spell, but returns the insured to the active state, from which another disability may begin. To calculate the chance of a later onset, the active-state probability must include recovered paths.

For example, under a one-year transition model,

p_A(j)=p_A(j-1)p_{AA}+p_D(j-1)p_{DA},

where $p_D(j-1)$ is the probability of being disabled at time $j-1$. The first term represents remaining active; the second represents recovering from disability. A death transition contributes to neither term. This state-probability recursion accounts for recovery and death without treating a recovered spell as continuous disability.

Worked Example: Recovery Can Lead to a New Qualifying Spell

A hypothetical annual contract pays $6,000 at times 2, 3, and 4 if the insured has been continuously disabled for at least one year and is alive at the payment date. The insured starts active. Annual transition probabilities are $p_{AA}=0.80$, $q_{AD}=0.15$, $p_{AX}=0.05$, $p_{DA}=0.20$, $p_{DD}=0.70$, and $p_{DX}=0.10$; death is absorbing. The effective annual interest rate is 4%. Find the expected present value of these three payments.

First find the active-state probabilities needed for later spell starts. At time 1, $p_A(1)=0.80$ and $p_D(1)=0.15$. Therefore,

p_A(2)=p_A(1)p_{AA}+p_D(1)p_{DA} =(0.80)(0.80)+(0.15)(0.20)=0.67.

The value $0.67$ includes the $0.03$ probability of being disabled at time 1 and recovering in the next interval. It also reflects that transitions to death do not return to the active state.

At time 2, a spell must start at time 1 and remain disabled for one interval:

$$ P(E_2)=(1)(0.15)(0.70)=0.105. $$

At time 3, the spell can start at time 1 or 2:

$$ P(E_3)=(1)(0.15)(0.70)^2+(0.80)(0.15)(0.70) =0.0735+0.084=0.1575. $$

At time 4, it can start at time 1, 2, or 3:

\begin{aligned} P(E_4) &=(1)(0.15)(0.70)^3+(0.80)(0.15)(0.70)^2+(0.67)(0.15)(0.70)\\ &=0.05145+0.0588+0.07035=0.1806. \end{aligned}

The expected present value is

\begin{aligned} \operatorname{EPV} &=6{,}000\left[0.105(1.04)^{-2} +0.1575(1.04)^{-3} +0.1806(1.04)^{-4}\right]\\ &=6{,}000(0.097078\ldots+0.140174\ldots+0.154378\ldots)\\ &=\$2{,}349.78. \end{aligned}

The result is rounded to the nearest cent. This calculation counts a recovered person as active for a possible new spell, but never carries the duration of the earlier spell into that new claim.

Continuous-Time Interpretation

In a continuous-time disability model, let $\mu_{AD}(s)$ be the disability-onset intensity from the active state at time $s$. Let $\mu_{DA}(u)$ and $\mu_{DX}(u)$ be the recovery and death intensities from the disabled state at time $u$. Assume the process has the Markov property, starts active, and the only way to begin a disability spell is a transition from active to disabled. Write $p_A(s)$ for the probability of being active at time $s$.

For a payment at time $t$ and elimination period $w$, the probability of eligibility is

P(E_t) = \int_0^{t-w} p_A(s)\mu_{AD}(s) \exp\left(-\int_s^t[\mu_{DA}(u)+\mu_{DX}(u)]\,du\right)\,ds, \qquad t\ge w.

The integrand is the probability density of starting a spell at time $s$, multiplied by the probability of remaining disabled and alive continuously through time $t$. The upper limit $t-w$ restricts onset to dates early enough for the waiting period to finish. As in the discrete formula, $p_A(s)$ includes paths that recovered from earlier spells. The exponential persistence factor excludes any path with recovery or death during the current spell. If a payment occurs at time $t$, the benefit amount is multiplied by $P(E_t)$ and the appropriate discount factor.

This continuous-time expression applies to the stated transition structure and payment condition. It is not a substitute for a contract that pays retroactively after the waiting period, has a different recurrence rule, or uses a more detailed disability classification. Such terms change the event whose probability must be valued.

Common Mistake: Occupancy Is Not Eligibility

A common error is to multiply the benefit by the probability of being disabled at the payment date. That probability includes both long-standing claims and recent onsets. It can therefore include spells that have not yet completed the elimination period. For instance, in the first worked example, a disability onset at time 2 could contribute to disability occupancy at time 2, but could not satisfy a one-year waiting period.

Exam check: For each payment date, ask two separate questions: is the insured disabled and alive at that date, and did the current spell begin early enough to complete the elimination period without recovery? Only paths satisfying both conditions generate a benefit.

Check Your Understanding

For each problem, assume transitions occur at annual dates, death is absorbing, and benefits are paid only if the insured is alive, disabled, and has completed the stated continuous-disability waiting period. Round present values to the nearest cent.

An insured starts active. The annual probabilities are $q_{AD}=0.12$ and $p_{DD}=0.75$. A $5,000 benefit is due at time 2 after a one-year elimination period. The interest rate is 0%. Find its expected present value. (Answer: $450.00)
An insured starts active, with $p_{AA}=0.90$, $q_{AD}=0.08$, and $p_{DD}=0.80$. A $2,000 benefit is due at times 2 and 3 after a one-year elimination period. The effective annual interest rate is 5%. Find the expected present value. (Answer: $304.07)
An insured starts active. The annual transition probabilities include $q_{AD}=0.10$ and $p_{DD}=0.90$. A $10,000 benefit is due at time 3 after a two-year elimination period. The interest rate is 0%. Find the expected present value. (Answer: $810.00)
An insured starts active. Each year, $p_{AA}=0.70$, $q_{AD}=0.20$, $p_{AX}=0.10$, $p_{DA}=0.25$, and $p_{DD}=0.65$, with the remaining disabled-state probability assigned to death. A $4,000 benefit is due at time 3 after a one-year elimination period. The interest rate is 0%. Find the expected present value. (Answer: $702.00)
In a continuous-time model, a disability spell begins at time $s=0.5$. The disabled-state recovery intensity is $0.20$ per year and the disabled-state mortality intensity is $0.10$ per year, both constant. What is the probability that this spell remains continuously disabled and alive through time $t=2$? (Answer: $e^{-0.45}=0.6376$)

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