Sensitivity to Disabled-State Mortality

Why Disabled-State Mortality Matters

A disability claim can end through recovery or death. If the assumed mortality intensity while disabled changes, the expected time spent on claim changes even when the recovery intensity is fixed. That affects the duration of benefit payments and, in turn, the expected cost of a disability-income contract. In a model that allows a disabled person to recover and later become disabled again, the mortality assumption also changes the value of benefits over the whole contract.

The comparisons in this tutorial hold the incidence and recovery intensities fixed and vary only disabled-state mortality. They are model sensitivities: they show the mathematical effect of changing one assumption while leaving the others unchanged, not a prediction that mortality changes independently in practice. The formulas use constant transition intensities and continuous benefit payments; they do not directly apply to discrete monthly payments, changing intensities, or a different set of permitted transitions.

Model and notation: A hypothetical contract has an active state $A$ and a disabled state $D$. While active, the insured becomes disabled at constant intensity $\alpha$ and dies at intensity $\mu_A$. While disabled, the insured recovers to active at intensity $\gamma$ and dies at intensity $\mu_D$. Death is absorbing. The force of interest is constant at $\delta$, and a continuous benefit is paid at rate $b$ per year while disabled. Intensities are nonnegative; unless an edge case is being discussed, take $\alpha>0$ and $\gamma>0$.

One Disabled Spell: Duration and Competing Terminations

Consider a single spell that starts when the insured enters state $D$. While in that state, recovery and death are competing exits. With constant intensities, the time $T$ until the first exit has an exponential distribution with rate $\gamma+\mu_D$. Consequently, its mean is $1/(\gamma+\mu_D)$. The probability that the first exit is death is $\mu_D/(\gamma+\mu_D)$; the probability that it is recovery is $\gamma/(\gamma+\mu_D)$.

E[T]=\frac{1}{\gamma+\mu_D},\qquad P(\text{death before recovery})=\frac{\mu_D}{\gamma+\mu_D},\qquad P(\text{recovery before death})=\frac{\gamma}{\gamma+\mu_D}.

Increasing $\mu_D$ lowers the mean duration of a disabled spell. It does not decrease the probability of death before recovery. If $\gamma>0$, that probability increases strictly; if $\gamma=0$ and $\mu_D>0$, death before recovery already has probability one, so increasing $\mu_D$ leaves it unchanged. In this tutorial’s main two-way model, $\gamma>0$.

For a continuous benefit rate $b$, the expected undiscounted benefit paid during one spell is $bE[T]=b/(\gamma+\mu_D)$. If the force of interest is $\delta$, the present value of benefits during that spell, conditional on starting disabled, is instead $b/(\delta+\gamma+\mu_D)$. Discounting adds another way in which later payments receive less weight; it does not change the probability of recovery versus death.

Worked Example: A Single Claim Under Two Mortality Assumptions

Method. Use the competing-intensity formulas for one spell and distinguish the undiscounted benefit total from its discounted present value. A hypothetical policy pays $b=\$12{,}000$ per year continuously while disabled. The annual recovery intensity is $\gamma=0.40$, the annual force of interest is $\delta=0.05$, and disabled-state mortality is compared at $\mu_D=0.10$ and $\mu_D=0.30$. All intensities and the force of interest are per year.

At $\mu_D=0.10$, the total claim-termination intensity is $0.40+0.10=0.50$. Thus $E[T]=1/0.50=2.00$ years. The probability of death before recovery is $0.10/0.50=0.2000$, and the recovery probability is $0.40/0.50=0.8000$. The expected undiscounted benefit is $\$12{,}000/0.50=\$24{,}000.00$. Its discounted value is $\$12{,}000/(0.05+0.40+0.10)=\$12{,}000/0.55=\$21{,}818.18$.

At $\mu_D=0.30$, the termination intensity is $0.40+0.30=0.70$. The mean duration is $1/0.70=1.428571\ldots$ years, and the death-before-recovery probability is $0.30/0.70=0.428571\ldots$. The recovery probability is $0.40/0.70=0.571429\ldots$. The expected undiscounted benefit is $\$12{,}000/0.70=\$17{,}142.86$, and the discounted value is $\$12{,}000/(0.05+0.40+0.30)=\$16{,}000.00$.

Rounded only at the final step, money is shown to the nearest cent and probabilities to four decimal places. The higher mortality assumption shortens the expected claim and reduces its benefit value. Common mistake avoided: using $1/\mu_D$ as the expected claim duration ignores recovery; the relevant termination intensity is $\gamma+\mu_D$.

Recurrent Disability: Benefit Values From Each State

A recurrent model permits transitions $A\to D$ and $D\to A$, in addition to death from either living state. Let $V_A$ and $V_D$ be the expected present values of future disability benefits when the contract starts active and disabled, respectively. Benefits are paid continuously at rate $b$ only in state $D$; there are no premiums, expenses, or other payments in these benefit values.

For compactness, define $C=\delta+\alpha+\mu_A$, $B=\delta+\gamma+\mu_D$, and $\Delta=CB-\alpha\gamma$. The constant-intensity model gives the equations $CV_A=\alpha V_D$ and $BV_D=b+\gamma V_A$. The first equation reflects that no benefit is paid while active and that a transition to disability occurs at intensity $\alpha$. The second includes the continuous benefit flow while disabled and the possibility of recovery.

V_A=\frac{\alpha b}{\Delta},\qquad V_D=\frac{Cb}{\Delta},\qquad \Delta=(\delta+\alpha+\mu_A)(\delta+\gamma+\mu_D)-\alpha\gamma.

Here is the justification for solving those equations. Substitute $V_D=(b+\gamma V_A)/B$ into $CV_A=\alpha V_D$, multiply by $B$, and collect the terms involving $V_A$: $CBV_A=\alpha b+\alpha\gamma V_A$, so $(CB-\alpha\gamma)V_A=\alpha b$. Substitution then gives $V_D=Cb/\Delta$.

The denominator requires care. For nonnegative parameters, $\Delta=\delta^2+\delta(\alpha+\gamma+\mu_A+\mu_D)+\alpha\mu_D+\mu_A\gamma+\mu_A\mu_D$, so $\Delta\geq 0$. Under $\alpha>0$ and $\gamma>0$, it is strictly positive if $\delta>0$ or either mortality intensity is positive. If $\delta=0$ and $\mu_A=\mu_D=0$, then $\Delta=0$: the model has no discounting or death, and the perpetual disability benefit has no finite value. The displayed finite-value formulas do not apply in that case. More generally, if transition intensities are allowed to be zero, positivity must be checked from the full expression for $\Delta$, rather than inferred from $C>0$ and $B>0$ alone.

How Benefit Values Change With Disabled-State Mortality

When $\Delta>0$, $C$ does not depend on $\mu_D$, while $\partial\Delta/\partial\mu_D=C$. Differentiating the value from the active state therefore gives $\partial V_A/\partial\mu_D=-\alpha bC/\Delta^2$. This derivative is nonpositive. It is strictly negative when $\alpha b>0$ (and $\Delta>0$); if $\alpha=0$ or $b=0$, it is zero. Starting disabled, $\partial V_D/\partial\mu_D=-bC^2/\Delta^2$, also nonpositive and strictly negative when $b>0$ and $C>0$. These statements hold with the other model assumptions fixed.

Worked Example: Comparing Recurrent Benefit Values and Premiums

Method. Solve the two-state value equations, then use the equivalence principle for a continuous premium paid only while active. A hypothetical contract pays a continuous disability benefit of $b=\$12{,}000$ per year and a continuous premium at rate $P$ while active. Take $\alpha=0.10$, $\gamma=0.50$, $\mu_A=0.02$, and $\delta=0.04$, all per year. Compare $\mu_D=0.10$ with $\mu_D=0.20$, starting from state $A$.

For both assumptions, $C=0.04+0.10+0.02=0.16$ and $\alpha b=0.10(\$12{,}000)=\$1{,}200$. At $\mu_D=0.10$, $B=0.04+0.50+0.10=0.64$, and $\Delta=(0.16)(0.64)-(0.10)(0.50)=0.1024-0.0500=0.0524$. Thus $V_A=\$1{,}200/0.0524=\$22{,}900.76$ and $V_D=(0.16)(\$12{,}000)/0.0524=\$36{,}641.22$.

At $\mu_D=0.20$, $B=0.04+0.50+0.20=0.74$, and $\Delta=(0.16)(0.74)-0.0500=0.1184-0.0500=0.0684$. Therefore $V_A=\$1{,}200/0.0684=\$17{,}543.86$, and $V_D=\$1{,}920/0.0684=\$28{,}070.18$.

To calculate the equivalence premium, let $W_A$ be the expected present value of time spent active, starting active. Its state equations are $CW_A-\alpha W_D=1$ and $BW_D-\gamma W_A=0$, because the reward rate is one in $A$ and zero in $D$. Solving gives $W_A=B/\Delta$. Equating premium and benefit values, $PW_A=V_A$, yields $P=\alpha b/B$. Thus the premium is $\$1{,}200/0.64=\$1{,}875.00$ at $\mu_D=0.10$, and $\$1{,}200/0.74=\$1{,}621.62$ at $\mu_D=0.20$, rounded to cents. As a check, the active-time values are $0.64/0.0524=12.2137$ and $0.74/0.0684=10.8187$ years in present-value units; multiplying by the corresponding unrounded premiums gives $V_A$ in each case. Common mistake avoided: using claim duration alone to calculate the recurrent-contract premium ignores both recurrence and the time premiums are payable. The premium here follows from the ratio of total expected present values.

Premium Sensitivity and an Assumption Threshold

Under the same recurrent model, the equivalence premium rate for a contract starting active and charging premiums only while active is $P=\alpha b/(\delta+\gamma+\mu_D)$. Holding incidence, recovery, benefit rate, and interest fixed, its derivative is $\partial P/\partial\mu_D=-\alpha b/(\delta+\gamma+\mu_D)^2$. It is nonpositive and is strictly negative when $\alpha b>0$. The mortality assumption changes both benefit values and the expected present value of active premium-paying time; the formula accounts for both effects.

Worked Example: Finding the Mortality Assumption for a Premium Limit

Method. Use the equivalence premium formula and solve the inequality for $\mu_D$, then verify the boundary value. A hypothetical contract pays $b=\$12{,}000$ per year continuously while disabled. Premiums are paid continuously while active. Let $\alpha=0.10$, $\gamma=0.50$, $\mu_A=0.02$, and $\delta=0.04$, all per year. The premium rate must not exceed $\$1{,}500$ per year. Find the smallest disabled-state mortality intensity that meets the limit.

The premium condition is $\$1{,}200/(0.04+0.50+\mu_D)\leq \$1{,}500$. Since the denominator is positive, this is equivalent to $0.54+\mu_D\geq 1{,}200/1{,}500=0.80$, or $\mu_D\geq0.26$ per year. At the boundary, $B=0.80$, so $P=\$1{,}200/0.80=\$1{,}500.00$. The denominator for the benefit value is $\Delta=(0.16)(0.80)-0.05=0.078$, which is positive. The resulting active-start benefit value is $\$1{,}200/0.078=\$15{,}384.62$, and $W_A=0.80/0.078=10.256410\ldots$; multiplying the unrounded $W_A$ by $\$1{,}500$ gives the same benefit value. Thus the boundary satisfies both the premium constraint and the equivalence check. Common mistake avoided: reversing the inequality when solving for the mortality intensity would incorrectly imply that lower disabled-state mortality produces the lower premium.

Common Exam Traps and Interpretation

Using mortality alone as the claim-ending rate. Recovery also ends a disabled spell, so the total intensity is $\gamma+\mu_D$.
Calling the death probability strictly increasing without checking recovery. It increases strictly in $\mu_D$ when $\gamma>0$; if recovery is impossible and death is certain, that probability remains one.
Claiming a strictly negative derivative without checking its numerator. For $V_A$, the derivative is nonpositive and is strictly negative only when $\alpha b>0$ and $\Delta>0$.
Using a finite-value formula when $\Delta=0$. Check the determinant condition before dividing by it. Positive $C$ and $B$ separately do not, in general, prove that $CB-\alpha\gamma$ is positive.
Mixing payment rates and time units. If $b$, $\delta$, and all intensities are annual, the formulas produce annual-rate premiums and present values in the corresponding monetary units. Convert monthly assumptions consistently before using the continuous-time formulas.
Interpreting an isolated sensitivity as a full model forecast. The comparison holds $\alpha$, $\gamma$, $\mu_A$, $b$, and $\delta$ fixed. If these assumptions also change, the one-variable derivative does not describe the total change.

Key takeaway: With recovery intensity fixed, higher disabled-state mortality shortens expected claim duration and does not reduce the probability of death before recovery. In the constant-intensity recurrent model, it also lowers the present value of disability benefits and the equivalence premium payable while active, subject to the stated conditions for strict decreases and finite values.

Check Your Understanding

For each hypothetical problem, use annual intensities and rates unless stated otherwise. Round money to cents and probabilities to four decimal places.

A single disabled spell has recovery intensity $\gamma=0.25$ and mortality intensity $\mu_D=0.15$. Find its mean duration and the probability of death before recovery. (Answer: $2.50$ years; $0.3750$)
A single claim pays continuously at $\$18{,}000$ per year, with $\delta=0.03$, $\gamma=0.27$, and $\mu_D=0.18$. Find the discounted value of the benefits conditional on starting disabled. (Answer: $\$37{,}500.00$)
A recurrent model has $\alpha=0.08$, $b=\$15{,}000$ per year, $\gamma=0.32$, $\delta=0.04$, and $\mu_D=0.14$. Premiums are continuous and payable only while active. Find the equivalence premium rate. (Answer: $\$2{,}400.00$ per year)
A recurrent model has $\alpha=0.12$, $\gamma=0.48$, $\mu_A=0.02$, $\delta=0.03$, $\mu_D=0.17$, and $b=\$10{,}000$ per year. Find the active-start present value of benefits. (Answer: $\$20{,}689.66$)
A recurrent contract has $\alpha=0.09$, $b=\$20{,}000$ per year, $\delta=0.03$, and $\gamma=0.47$. Premiums are payable only while active and must not exceed $\$2{,}400$ per year. Find the smallest $\mu_D$ that meets the constraint. (Answer: $0.25$ per year)

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