Decomposing Expected Costs by Disability Spell

Why Decompose Claim Costs by Onset?

A disability contract’s expected cost can be viewed in two complementary ways. One is to find the probability of being disabled at each payment time and value the resulting benefit. The other is to group benefits by the disability spell that generated them: identify when each spell begins, then value its payments conditional on that onset. The second view makes waiting periods, maximum durations, recovery, death, and recurrence easier to track.

The decomposition does not assume that every disabled person has only one claim. A person may recover, return to the active state, and later experience another onset. Each onset starts a new spell if the contract’s terms treat it as a new claim. The active-state probability at each time reflects earlier recovery and therefore determines the opportunity for subsequent onsets.

Definition: A disability onset is a transition from the active state into the disabled state. A spell is the period in the disabled state following one onset, ending at recovery, death, or the contract’s maximum benefit duration. A later transition from active to disabled is a new spell.

Onset-Time Decomposition in Continuous Time

Consider a hypothetical contract with active, disabled, and dead states. Let $\alpha(t)$ be the intensity of transition from active to disabled at time $t$, and let $p_A(t)$ be the probability that the person is active at time $t$. Conditional on being active at $t$, the probability of an onset during a short interval of length $dt$ is $\alpha(t)dt+o(dt)$. Thus the probability mass of onsets near $t$ is $p_A(t)\alpha(t)dt$.

Let $\gamma(t)$ and $\mu_D(t)$ be the recovery and death intensities while disabled. Assume these intensities depend on calendar time but not on the duration of the current spell. Given an onset at time $t$, the probability the spell is still in force $u$ time units later is

{}_u p_D(t)=\exp\left(-\int_0^u \left[\gamma(t+r)+\mu_D(t+r)\right]\,dr\right).

This is the probability that neither recovery nor death has occurred during the spell’s first $u$ time units. Suppose a continuous benefit is paid at rate $b(t+u)$ while disabled, after a waiting period $w$, and no more than $m$ time units after onset. For a contract ending at time $T$, the expected present value of benefits generated by an onset at $t$, conditional on that onset, is

C(t)=\int_w^{\min(m,T-t)} e^{-\delta(t+u)} b(t+u)\,{}_u p_D(t)\,du,

where an integral with upper limit below $w$ is zero. The factor $e^{-\delta(t+u)}$ discounts a payment at time $t+u$ to time 0. If there is no waiting period, take $w=0$. If the contract pays a different amount by claim duration, that amount can be represented by allowing $b$ to depend on both calendar time and duration.

Theorem: If the person starts active and the expected claim cost is the sum of payments attributable to all disability spells, then

\operatorname{EPV} = \int_0^T p_A(t)\alpha(t)C(t)\,dt.

Here $C(t)$ is the conditional expected present value of a spell beginning at $t$. If claim intensity or claim cost also depends on the person’s history or spell duration, the conditional cost must reflect that information; the displayed expression applies directly when the stated active-state and spell assumptions suffice.

Why the Decomposition Holds

The argument is conditional expectation applied to each onset, followed by addition across onsets. In a short interval around $t$, the chance of being active and then experiencing an onset is $p_A(t)\alpha(t)dt+o(dt)$. Given that onset, the expected discounted cost attributable to the resulting spell is $C(t)$. The expected contribution from onsets in that interval is therefore $p_A(t)\alpha(t)C(t)dt$, up to a term that vanishes as the interval shrinks.

Adding these contributions over time gives the integral. This addition is valid even when spells recur: expected total cost is the sum of expected costs assigned to each spell. The recurrence is reflected in $p_A(t)$, since only people who are active at $t$ can have a new onset. There is no requirement that different spells be independent.

For annual models, the same reasoning becomes a sum. Let $q_k$ be the probability of transition from active to disabled during period $k$, conditional on being active at its start, and let $p_A(k-1)$ be the active-state probability at that start. The onset probability mass for period $k$ is $p_A(k-1)q_k$. A spell’s conditional cost is calculated using the subsequent probability of remaining disabled and the payments allowed by the waiting period, maximum duration, and contract horizon.

Worked Examples in an Annual Model

The following examples use a discrete-time Markov model. At each annual transition, the person moves between active, disabled, or dead states. A benefit is paid at the transition time if the contract says it is payable in the resulting disabled state. The annual timing convention is part of each example; it avoids implying that onset and payment occur at arbitrary times within a year.

Worked Example: Separating New Onsets from Continuing Claims

A hypothetical two-year contract starts with an active person. At each annual transition, an active person becomes disabled with probability $0.20$ and otherwise remains active. A disabled person remains disabled with probability $0.50$, recovers with probability $0.30$, and dies with probability $0.20$. The contract pays $\$1{,}000$ at each year-end when disabled, including the year of onset. The effective annual interest rate is $i=5\%$. Find the expected present value of benefits.

Method. Calculate onset probability in each year, then value each onset cohort’s payments conditional on persistence. The discount factor is $v=1/1.05=0.952381$, so $v^2=0.907029$.

In year 1, the person starts active, so the onset probability is $1(0.20)=0.20$. The expected year-1 benefit is $0.20(\$1{,}000)=\$200$. Of that onset cohort, the probability of still being disabled at the end of year 2 is $0.50$, giving an additional expected year-2 benefit of $0.20(0.50)(\$1{,}000)=\$100$.

The probability of being active at the start of year 2 is $0.80$, because the only year-1 active-to-disabled transition has probability $0.20$. The year-2 onset probability is therefore $0.80(0.20)=0.16$, producing an expected year-2 benefit of $0.16(\$1{,}000)=\$160$.

Discount the three contributions separately:

\operatorname{EPV} =200v+100v^2+160v^2 =200(0.952381)+260(0.907029) =\$426.30.

As a check, the probability of being disabled at the end of year 2 is $0.20(0.50)+0.16=0.26$. Thus expected year-2 benefits are $0.26(\$1{,}000)=\$260$, matching the two year-2 spell contributions. The common mistake is to treat all disabled people in year 2 as new onsets; the $0.10$ probability mass continuing from year 1 is a continuation, not a new onset.

Worked Example: Waiting Period and Maximum Claim Duration

A hypothetical three-year contract starts active. In each year, an active person becomes disabled with probability $0.25$ and otherwise stays active. A disabled person remains disabled at the next annual transition with probability $0.60$; otherwise the person recovers or dies and the spell ends. The benefit is $\$1{,}200$ per year, but the first full year of each spell is unpaid. The maximum spell duration is three annual periods, so at most the second and third spell years can be paid. The effective annual interest rate is $4\%$. Find the expected present value.

Method. Work by onset cohort. The discount factors are $v=1/1.04=0.961538$, $v^2=0.924556$, and $v^3=0.888996$.

An onset in year 1 has probability $0.25$. It generates a payment in year 2 only if the spell persists, with probability $0.60$. Its expected year-2 payment is $0.25(0.60)(\$1{,}200)=\$180$. To generate a year-3 payment, the spell must persist through two transitions, with probability $0.60^2=0.36$. Its expected year-3 payment is $0.25(0.36)(\$1{,}200)=\$108$.

The probability of being active at the start of year 2 is $0.75$, so the year-2 onset probability is $0.75(0.25)=0.1875$. This spell’s first year is unpaid. It produces a year-3 payment only if it persists through the transition, giving expected payment $0.1875(0.60)(\$1{,}200)=\$135$. A year-3 onset has no payable year remaining within the contract.

\operatorname{EPV} =180v^2+(108+135)v^3 =180(0.924556)+243(0.888996) =\$382.45.

The year-3 expected payment is $\$243$: $\$108$ from the year-1 onset cohort and $\$135$ from the year-2 cohort. The maximum duration is applied to each spell, not to the whole contract. A common mistake is to pay the year-1 cohort in its onset year despite the waiting period; the first payment is one full year later and requires persistence.

Worked Example: Recurrence After Recovery

A hypothetical three-year contract starts active. At each annual transition, the probabilities are: active to disabled $0.20$, active to active $0.80$; disabled to disabled $0.50$, disabled to active $0.30$, and disabled to dead $0.20$. Each new spell produces exactly one $\$500$ payment at its onset transition; continuation of that spell produces no additional payment. Recovery allows a later spell to qualify for another payment. The effective annual interest rate is $5\%$. Find the expected present value.

Method. Find the active-state probability before each transition, then multiply by the onset probability $0.20$. At the start of year 1, the active probability is $1$, so the onset probability is $0.20$.

At the end of year 1, the active probability is $0.80$. At the end of year 2, it is the sum of those who stayed active and those who recovered: $0.80(0.80)+0.20(0.30)=0.64+0.06=0.70$. Thus the year-2 onset probability is $0.80(0.20)=0.16$, and the year-3 onset probability is $0.70(0.20)=0.14$.

Using $v=0.952381$, $v^2=0.907029$, and $v^3=0.863838$, the expected present value is

\operatorname{EPV} =500\left(0.20v+0.16v^2+0.14v^3\right) =\$228.27.

The year-3 active probability includes recovery from the disabled state, so it permits recurrent onsets. The check is that the onset probabilities are $0.20$, $0.16$, and $0.14$, each no greater than the corresponding active-state probability. The common mistake is to count a person who remains disabled as having a second onset; only a transition from active to disabled starts a new spell.

Common Checks and Modeling Limits

A useful check is to compare the spell decomposition with a direct state-occupancy calculation. At each payment time, sum the probabilities of being disabled from all onset cohorts that can still generate a payment. Multiply by the benefit, discount, and add over payment times. The result should match the onset-based valuation, as in the first example.

The decomposition also makes product conditions visible. For each onset, identify the waiting period, remaining contract term, maximum spell duration, and whether recovery ends the claim. If transition probabilities depend on claim duration, the disabled state alone may not contain enough information; use duration-specific persistence probabilities or augment the model with duration states. A benefit limit that applies per spell must not be confused with one that applies once over the entire contract.

Key takeaway: Expected claim cost is the sum of onset probability mass times conditional expected spell cost. Use active-state occupancy to determine when new spells can begin, and use disabled-state persistence and contract terms to value each spell’s payments.

Check Your Understanding

For each problem, use the stated annual timing convention and round any dollar amount to the nearest cent.

A two-year contract starts active. Each year an active person becomes disabled with probability $0.10$; otherwise the person remains active. A disabled person persists with probability $0.50$. Benefits of $\$800$ are paid at each year-end when disabled, including the onset year. If $i=4\%$, find the expected present value. (Answer: $\$180.47$)
A two-year contract starts active. The annual onset probability is $0.20$, and active people who do not have an onset remain active. A disabled person persists with probability $0.75$. The first full year of each spell is unpaid; the benefit is $\$1{,}000$ per year thereafter. There is no interest. Find the expected benefit cost. (Answer: $\$150.00$)
A two-year contract starts active. Each year, active-to-disabled probability is $0.30$ and active-to-active probability is $0.70$. Each spell generates one $\$600$ payment at onset, and recovery returns a disabled person to active. There is no interest. Find the expected total cost, regardless of the recovery probability. (Answer: $\$306.00$)
A three-year contract starts active. Active-to-disabled probability is $0.20$, and active people without an onset remain active. A disabled person persists with probability $0.50$. The first spell year is unpaid, and payments of $\$1{,}000$ are made in the second and third spell years only. There is no interest. Find the expected benefit cost. (Answer: $\$230.00$)
In a continuous-time model, write the onset-time density at time $t$ in terms of active-state probability and active-to-disabled intensity. (Answer: $p_A(t)\alpha(t)$ per unit time.)

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