State-Dependent Disability Benefit Amounts

Why Payment Amounts Need to Match the State

A disability contract can make different payments in different policy states. For example, a premium may be due only while the policyholder is active, while a disability benefit is paid only while the policyholder is disabled. The amount of that benefit may also depend on how long the current claim has lasted. To value the contract correctly, each cash flow must be matched to the state and time during which it is payable.

The central task is to combine a payment rate with the probability that the relevant state is occupied at each future time, then discount the expected payment. A benefit rate payable during the first year of a claim is not the same as a rate payable during the first year from policy issue. If disability begins later, the claim-duration clock starts at disability onset.

Definition: A state-dependent payment rate is a cash-flow rate that is paid only while the contract occupies a specified policy state. A claim-duration-dependent benefit rate is a rate payable in the disabled state whose amount depends on the elapsed time since the current disability spell began. A premium payable only while active is an active-state payment rate, not a payment due throughout the policy term.

Model and Valuation Principle

Use a continuous-time model with three states: active, disabled, and dead. Let $\alpha$ be the constant intensity of transition from active to disabled, $\mu_A$ the active-to-dead intensity, $\rho$ the disabled-to-active recovery intensity, and $\mu_D$ the disabled-to-dead intensity. Death is an absorbing state. Let $\delta$ be the force of interest, and let $v(t)=e^{-\delta t}$ be the discount factor to time 0.

Unless a problem states otherwise, the constant-intensity calculations below assume that these transition intensities do not vary with time or duration. The examples that use a claim-duration tier also assume no recovery, so there is at most one disability spell. If recovery is possible and a new spell starts, its claim-duration clock resets to zero; the valuation must then account for the possible sequence of spells.

Let $b(s)$ denote the continuous benefit rate at claim duration $s$, and let $p$ denote a continuous premium rate payable while active. In a more general model, rates can also depend on calendar time or policy state. The expected present value (EPV) of a continuous payment is the expected value of the integral of its discounted payment rate over the times when it is due.

\operatorname{EPV} = E\left[\int_0^\infty e^{-\delta t} \left(p\,\mathbf{1}_{\{J_t=A\}}+ b(t-T)\,\mathbf{1}_{\{J_t=D\}}\right)\,dt\right],

Here $J_t$ is the policy state at time $t$, $T$ is the start time of the current disability spell, and the disabled-state term is included only during that spell. If benefits depend on claim duration, the amount paid at time $t$ while disabled is $b(t-T)$, not $b(t)$.

Valuing a Benefit by Claim Duration

First consider a policyholder already disabled at time 0, with claim duration zero. Suppose there is no recovery, the disabled-state mortality intensity is $\mu_D$, and a benefit rate $b(s)$ is paid continuously while the policyholder remains alive and disabled. The probability of remaining in the disabled state through duration $s$ is $e^{-\mu_Ds}$. Thus each benefit rate is weighted by both survival and discounting.

\operatorname{EPV}_D = \int_0^\infty b(s)e^{-(\delta+\mu_D)s}\,ds.

This formula follows directly from expected cash flows. At duration $s$, the expected payment rate is $b(s)$ times the probability $e^{-\mu_Ds}$ that payment is still due; discounting that expected rate multiplies it by $e^{-\delta s}$. Integrating over time gives the EPV. If the benefit rate is constant, $b(s)=b$, the formula reduces to $b/(\delta+\mu_D)$. It does not apply unchanged if recovery is possible or the mortality intensity varies with duration.

Worked Example: A Benefit That Changes After One Claim Year

A hypothetical policyholder is already disabled at claim duration zero. There is no recovery, disabled-state mortality intensity is $0.16$ per year, and the force of interest is $0.04$ per year. A continuous benefit is paid at $\$5{,}000$ per year during the first claim year and $\$8{,}000$ per year thereafter. Find its EPV.

Method. Split the benefit integral at the claim-duration threshold of one year. The combined discount and mortality force is $\delta+\mu_D=0.04+0.16=0.20$.

\begin{aligned} \operatorname{EPV}_D &=5000\int_0^1 e^{-0.20s}\,ds +8000\int_1^\infty e^{-0.20s}\,ds\\ &=5000\frac{1-e^{-0.20}}{0.20} +8000\frac{e^{-0.20}}{0.20}. \end{aligned}

Using $e^{-0.20}=0.8187307531$, the first-year part is $5000(1-0.8187307531)/0.20=\$4{,}531.73$. The later part is $8000(0.8187307531)/0.20=\$32{,}749.23$. The total is $\$4{,}531.73+\$32{,}749.23=\$37{,}280.96$, rounded to the nearest cent.

Check and common mistake. The later benefit is payable for an unbounded period, but mortality and discounting make its EPV finite. A common error is to use a short finite benefit period or to treat the policyholder as active at time 0 and multiply by an incidence probability. The stated starting state is already disabled, so neither adjustment is appropriate.

Disability Onset After Policy Issue

If the policyholder starts active and disability can begin later, the onset time must also be valued. With constant intensities and no recovery, the probability of remaining active through time $t$ is $e^{-(\alpha+\mu_A)t}$. The density of disability onset at time $t$ is therefore $\alpha e^{-(\alpha+\mu_A)t}$. Including discounting, the EPV factor for the onset event is $\alpha/(\delta+\alpha+\mu_A)$.

After onset, a claim-duration-dependent benefit is valued from duration zero, then multiplied by the discounted onset factor. This product works here because the intensities are constant and the disability benefit pattern depends on claim duration, not on the calendar time of onset. If intensities or benefit rates vary with calendar time, onset time generally cannot be separated in this way.

\operatorname{EPV}_{A\to D} = \frac{\alpha}{\delta+\alpha+\mu_A} \int_0^\infty b(s)e^{-(\delta+\mu_D)s}\,ds \qquad \text{(no recovery; constant intensities).}

Worked Example: Tiered Disability Benefits From Active Status

A hypothetical contract starts active. The disability incidence intensity is $0.10$ per year, the active mortality intensity is $0.10$, and the disabled mortality intensity is $0.10$. There is no recovery, and the force of interest is $0.10$. A continuous benefit pays $\$4{,}000$ per year for the first two claim years and $\$6{,}000$ per year thereafter. Find the benefit EPV from active status.

Method. First value the benefit conditional on disability onset at time 0, using claim duration in the benefit tiers. Then multiply by the discounted probability factor for disability onset from the active state.

The combined discount and disabled mortality force is $0.10+0.10=0.20$. The value at disability onset is

\begin{aligned} V_D &=4000\int_0^2 e^{-0.20s}\,ds+ 6000\int_2^\infty e^{-0.20s}\,ds\\ &=4000\frac{1-e^{-0.40}}{0.20} +6000\frac{e^{-0.40}}{0.20}. \end{aligned}

Using $e^{-0.40}=0.6703200460$, the first part is $4000(1-0.6703200460)/0.20=\$6{,}593.60$. The later part is $6000(0.6703200460)/0.20=\$20{,}109.60$. Thus $V_D=\$26{,}703.20$.

The discounted disability-onset factor is \[ \frac{\alpha}{\delta+\alpha+\mu_A} =\frac{0.10}{0.10+0.10+0.10} =\frac{1}{3}. \] Therefore the benefit EPV from active status is $\frac{1}{3}(\$26{,}703.20)=\$8{,}901.07$, rounded to the nearest cent.

Check and common mistake. The onset factor is not $0.10/(0.10+0.10)$: discounting while active must be included along with disability incidence and active mortality. Omitting discounting before onset would overstate the value.

Premiums Payable Only While Active

A continuous premium at rate $p$ is paid only while the policyholder is active. In the constant-intensity model with no recovery, the probability of still being active at time $t$ is $e^{-(\alpha+\mu_A)t}$. Its EPV from active status is therefore $p/(\delta+\alpha+\mu_A)$. A premium is not paid after disability onset under this assumption, nor after death.

If benefits and premiums are both continuous, the net expected cost can be found by valuing each stream separately. For an equivalence premium, set the EPV of premiums equal to the EPV of benefits, provided the contract has no other cash flows or expenses. This is an expected-value pricing condition, not a statement that the two random present values are equal for every policyholder.

Worked Example: Equivalence Premium While Active

A hypothetical contract starts active. Disability incidence intensity is $0.10$ per year, active mortality intensity is $0.05$, and disabled mortality intensity is $0.10$. The force of interest is $0.05$. There is no recovery. A continuous benefit of $\$9{,}000$ per year is paid while disabled, and a continuous premium at rate $p$ is paid while active. Find the equivalence premium rate.

Method. Calculate the benefit EPV from active status, calculate the EPV factor for a unit premium payable while active, and equate the two.

For the disabled benefit, the combined discount and disabled mortality force is $0.05+0.10=0.15$. The EPV conditional on disability onset is $\$9{,}000/0.15=\$60{,}000$. The discounted onset factor from active status is $0.10/(0.05+0.10+0.05)=0.10/0.20=0.5$. Thus the benefit EPV from active status is $0.5(\$60{,}000)=\$30{,}000$.

The active premium factor per dollar of annual rate is $1/(0.05+0.10+0.05)=5$ years. Hence the premium EPV is $5p$. Equating EPVs gives $5p=\$30{,}000$, so $p=\$6{,}000$ per year.

Check and common mistake. At $p=\$6{,}000$, the premium EPV is $5(\$6{,}000)=\$30{,}000$, matching the benefit EPV. Treating the premium as payable while disabled as well would incorrectly use a longer payment period and a different premium factor.

A Reliable Calculation Sequence

When state and claim duration both affect payments, organize the calculation before substituting numbers. The following sequence separates the timing of state transitions from the amount paid in each state.

State the payment conditions.
Identify which state triggers each benefit or premium, whether payment is continuous or discrete, and whether benefit duration is measured from policy issue or claim onset.

Write the relevant transition assumptions.
Specify incidence, mortality, recovery, and any assumptions about whether intensities are constant or vary with time or duration.

Value each payment stream.
Integrate its rate against the probability of occupying the required state and the applicable discount factor. Split the integral wherever the payment rate changes.

Combine the EPVs and check timing.
Add benefit values or equate premium and benefit EPVs as required. Confirm that a claim-duration threshold begins at disability onset and that active-only premiums stop when active status ends.

Common Traps and Interpretation

The main difficulty is often not integration but identifying the correct clock and state. A tier described as “after the first claim year” changes at $s=1$, where $s$ is elapsed time since disability began. If onset occurs at time $T$, that change occurs at calendar time $T+1$, not at time 1 after policy issue. Similarly, an active premium is weighted by active-state occupancy, not by the probability of being alive in either active or disabled status.

A useful check is to separate a value into its components: the probability and timing of entering the payment state, followed by the value of payments conditional on entry. For a policyholder already in that state, the entry factor is absent. For a claim-duration benefit, each tier must use its own interval of claim duration and the appropriate survival and discount factor. These checks are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: Value every cash flow only over the times when its required policy state is occupied. For claim-duration benefits, start the benefit clock at disability onset; for active-only premiums, stop the premium stream when active status ends.

Check Your Understanding

For each problem, state the relevant timing assumption and calculate the requested value.

A policyholder is already disabled at claim duration zero. There is no recovery, $\delta=0.03$, and disabled mortality intensity is $0.07$. A continuous benefit is $\$3{,}000$ per year for the first 1.5 claim years and $\$4{,}500$ per year thereafter. Find the benefit EPV. (Answer: $\$42{,}910.62$)
A contract starts active, with incidence intensity $0.12$, active mortality intensity $0.08$, disabled mortality intensity $0.10$, and $\delta=0.05$. There is no recovery. The continuous benefit is $\$5{,}000$ per year during the first claim year and $\$7{,}000$ per year thereafter. Find the EPV from active status. (Answer: $\$21{,}508.53$)
A contract starts active, has incidence intensity $0.10$, active mortality intensity $0.05$, disabled mortality intensity $0.10$, and $\delta=0.05$. There is no recovery. A continuous disabled benefit is $\$9{,}000$ per year, and premiums are paid continuously only while active. Find the equivalence premium rate. (Answer: $\$6{,}000$ per year)
A contract starts active, with incidence intensity $0.20$, active mortality intensity $0.10$, disabled mortality intensity $0.10$, and $\delta=0.10$. There is no recovery. A continuous benefit is $\$2{,}000$ per year during the first claim year and $\$3{,}000$ per year thereafter. Find its EPV from active status. (Answer: $\$7{,}046.83$)

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