Disability Income with Incidence, Recovery, and Death

Disability Income as a Multiple-State Model

A disability income contract may pay while a person is disabled, stop payments on recovery, and end coverage on death. Its expected cost depends not only on the chance of becoming disabled, but also on how long the person remains disabled, the possibility of recovery, and mortality in each state. A multiple-state model organizes these events so that each payment is valued according to the state or transition that triggers it.

The model below has three states: active $A$, disabled $D$, and dead $X$. The covered person starts in $A$ or $D$. A transition from $A$ to $D$ is incidence, a transition from $D$ to $A$ is recovery, and death can occur from either living state. Benefits and premiums must be matched to the state or transition that makes them payable.

Definition: A continuous-time Markov disability model has transition intensities $q_{ij}$, where $q_{ij}$ is the instantaneous rate of transition from state $i$ to state $j$, conditional on the current state. Here, $\lambda$ is the incidence intensity $A\to D$, $\rho$ is the recovery intensity $D\to A$, $\mu_A$ is the mortality intensity while active, and $\mu_D$ is the mortality intensity while disabled. Death is absorbing.

Unless stated otherwise, assume the intensities are constant over time, the process is Markov, and at most one transition occurs at an instant. The force of interest $\delta$ is constant and positive. Continuous benefits accrue only while the specified living state is occupied; transition benefits are paid at the instant of the specified transition. No payment is made after death. In this three-state model, recovery returns the person to the active state, from which another incidence transition is possible under the model. Product rules that restrict later claims require additional states or transition rules.

Generator and Discounted State Occupancy

Use row vectors for state probabilities. In the order $A,D,X$, the generator is

Q= \begin{pmatrix} -(\lambda+\mu_A)&\lambda&\mu_A\\ \rho&-(\rho+\mu_D)&\mu_D\\ 0&0&0 \end{pmatrix}.

The upper-left $2\times2$ block, denoted $T$, describes transitions among the living states. The dead state is excluded from $T$ because the person cannot return to a living state. If the initial state is $i$, let $p_i(t)$ be the row vector of probabilities of occupying $A$ and $D$ at time $t$. Its discounted expected time in each living state is

z_i=\int_0^\infty e^{-\delta t}p_i(t)\,dt.

The components of $z_i$ have units of discounted years. A continuous payment at rate $b_A$ while active and $b_D$ while disabled therefore has expected present value $z_{i,A}b_A+z_{i,D}b_D$. A payment at a transition is valued differently: for example, a lump sum $c$ on incidence has expected present value $\lambda z_{i,A}c$.

Result: For a constant generator and constant force of interest, discounted living-state occupancy satisfies $z_i=e_i(\delta I-T)^{-1}$, where $e_i$ is the row vector selecting the initial living state. The inverse exists under the stated positive-discount and eventual-absorption assumptions. For a transition $r\to s$ with intensity $q_{rs}$ and payment $c_{rs}$ at that transition, the expected present value is $z_{i,r}q_{rs}c_{rs}$.

Why the Occupancy Formula Works

For the transient living states, the Kolmogorov forward equation is $p_i'(t)=p_i(t)T$, with $p_i(0)=e_i$. Integrate the derivative of the discounted probability row vector:

\frac{d}{dt}\big(e^{-\delta t}p_i(t)\big) =e^{-\delta t}p_i(t)(T-\delta I).

Integrating from zero to infinity gives $-e_i=z_i(T-\delta I)$. The terminal term is zero because discounting and eventual exit make $e^{-\delta t}p_i(t)$ tend to zero. Rearranging yields $z_i(\delta I-T)=e_i$, hence $z_i=e_i(\delta I-T)^{-1}$. For a transition $r\to s$, conditional on occupying $r$ at time $t$, the probability of that transition in a short interval of length $dt$ is $q_{rs}dt+o(dt)$. Thus its discounted expected payment is the integral of $e^{-\delta t}p_{i,r}(t)q_{rs}c_{rs}$, which equals $z_{i,r}q_{rs}c_{rs}$.

Worked Valuations

The following examples use hypothetical contracts and assumptions. All intensities and the force of interest are annual. The same basic calculation can value continuous disability income, transition lump sums, and premiums, but each cash flow must be attached to its correct state or transition.

Worked Example: Disability Income Plus Incidence and Death Benefits

A hypothetical policyholder starts active. Assume constant incidence intensity $\lambda=0.10$, active mortality intensity $\mu_A=0.10$, recovery intensity $\rho=0.20$, disabled mortality intensity $\mu_D=0.10$, and force of interest $\delta=0.10$. The policy pays \$12,000 per year continuously while disabled, \$5,000 on each incidence transition, and \$20,000 on death from either living state. Calculate the expected present value of benefits.

Method. Find discounted state occupancy from the transient generator, then multiply each occupancy by the appropriate payment rate or transition intensity. This avoids treating an incidence intensity as if it were a probability.

Step 1: Form the transient generator. The active-state exit intensity is $0.10+0.10=0.20$; the disabled-state exit intensity is $0.20+0.10=0.30$. Therefore,

T= \begin{pmatrix} -0.20&0.10\\ 0.20&-0.30 \end{pmatrix}, \qquad \delta I-T= \begin{pmatrix} 0.30&-0.10\\ -0.20&0.40 \end{pmatrix}.

Step 2: Invert and select the initial state. The determinant is $0.30(0.40)-(-0.10)(-0.20)=0.12-0.02=0.10$. Hence

(\delta I-T)^{-1} =\frac{1}{0.10} \begin{pmatrix} 0.40&0.10\\ 0.20&0.30 \end{pmatrix} = \begin{pmatrix} 4&1\\ 2&3 \end{pmatrix}.

Starting active selects the first row, so the discounted expected times are $z_A=4$ active years and $z_D=1$ disabled year.

Step 3: Value each benefit. The continuous disability income has present value $\$12{,}000(1)=\$12{,}000$. The discounted expected number of incidence transitions is $\lambda z_A=0.10(4)=0.40$, giving incidence benefits of $\$5{,}000(0.40)=\$2{,}000$. Discounted expected deaths from active and disabled states are $0.10(4)=0.40$ and $0.10(1)=0.10$, respectively. The death benefit present value is $\$20{,}000(0.40+0.10)=\$10{,}000$.

Answer and check. Total expected present value is $\$12{,}000+\$2{,}000+\$10{,}000=\boxed{\$24{,}000}$. The death payment uses both state-specific mortality flows, while the continuous income uses only disabled occupancy.

Worked Example: Equivalence Premium Paid While Active

Use the same hypothetical intensities and force of interest as in the preceding example, with an active initial state. A policy pays \$10,000 per year continuously while disabled and \$8,000 on death from either living state. It charges a continuous premium at rate $P$ per year only while active; premiums stop during disability and do not resume after death. Find the equivalence premium rate.

Method. Set the expected present value of premiums equal to the expected present value of benefits. The occupancy factors remain $z_A=4$ and $z_D=1$, because the benefit and premium cash flows do not alter the assumed transition model.

The disability income present value is $\$10{,}000(1)=\$10{,}000$. Discounted expected deaths are $0.10(4)+0.10(1)=0.50$, so the death benefit present value is $\$8{,}000(0.50)=\$4{,}000$. Total benefit present value is therefore $\$14{,}000$.

Premiums accrue only during active occupancy, so their expected present value is $4P$. Equivalence requires $4P=\$14{,}000$, giving $P=\$14{,}000/4=\boxed{\$3{,}500\text{ per year}}$. As a check, the premium present value at this rate is $4(\$3{,}500)=\$14{,}000$, equal to the benefit present value.

Worked Example: Initial Disability and a Recovery Payment

A hypothetical contract begins with the policyholder already disabled. Assume $\lambda=0.10$, $\mu_A=0.10$, $\rho=0.20$, $\mu_D=0.10$, and $\delta=0.10$, all constant per year. It pays \$6,000 per year continuously while disabled, \$1,500 at each recovery transition, and \$12,000 at death from either living state. Find the expected present value of benefits.

Method. Use the disabled-start row of the inverse matrix, then separately value disabled-state payments, recovery transitions, and death transitions.

From the matrix calculation, the disabled-start row is $(z_A,z_D)=(2,3)$. Thus the continuous disability income is $\$6{,}000(3)=\$18{,}000$. The discounted expected number of recoveries is $\rho z_D=0.20(3)=0.60$, so the recovery payment present value is $\$1{,}500(0.60)=\$900$.

The discounted expected number of deaths is $0.10(2)+0.10(3)=0.20+0.30=0.50$. The death benefit present value is $\$12{,}000(0.50)=\$6{,}000$. Therefore, total expected present value is $\$18{,}000+\$900+\$6{,}000=\boxed{\$24{,}900}$. The calculation checks because each payment is attached to the state or transition that triggers it; in particular, the recovery payment is based on disabled occupancy, not active occupancy.

Common Exam Trap: An Intensity Is Not an Event Probability

An intensity is a rate conditional on currently occupying its source state. It is not, by itself, the probability of ever making that transition over a contract. For example, in the first worked example, the incorrect calculation for the expected present value of the \$5,000 incidence benefit might be $\$5{,}000(0.10)=\$500$, treating the incidence intensity $0.10$ as a probability. That calculation ignores the duration and discounted occupancy of the active state.

The correct calculation is $\$5{,}000\lambda z_A=\$5{,}000(0.10)(4)=\$2{,}000$. This is a discounted expected payment count, not necessarily a probability: if the model permits return to the active state and another incidence transition, the expected number of such transitions can reflect multiple episodes. For a finite-period probability of a transition, use the relevant state-transition probabilities or solve the corresponding Kolmogorov equations; do not substitute an intensity directly.

Key takeaway: Value a state-contingent continuous payment using discounted expected occupancy of that state. Value a transition payment using discounted expected transition count, calculated as source-state occupancy times transition intensity. Apply the same distinction to premiums and benefits.

Check Your Understanding

For Questions 1–4, assume the constant intensities $\lambda=0.10$, $\mu_A=0.10$, $\rho=0.20$, $\mu_D=0.10$, and force of interest $\delta=0.10$, unless a question states otherwise. Use the three-state model and the transition-payment conventions developed above.

A policyholder starts active. The policy pays \$9,000 per year continuously while disabled and \$2,000 at each incidence transition. Find the expected present value of these benefits. (Answer: \$9,800)
A policyholder starts active and receives a \$15,000 lump sum at death from either living state. Find the expected present value of the death benefit. (Answer: \$7,500)
A policyholder starts disabled. A contract pays \$4,000 per year continuously while disabled and \$500 at each recovery. Premiums of $P$ per year are paid continuously while active only. Find the equivalence premium rate. (Answer: \$6,150 per year)
Initial state is active with probability $0.60$ and disabled with probability $0.40$. A hypothetical policy pays \$7,000 per year continuously while disabled and \$10,000 at death from either living state. Find its expected present value of benefits. (Answer: \$17,600)

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