Recovery and Morbidity-Transition Stresses

Why Recovery and Morbidity Transitions Matter

Disability incidence determines how many people enter a disabled state, but it does not determine how long they remain there or how their condition changes. Once a claim has started, recovery, disabled-state mortality, and transitions between morbidity states can all change projected benefit payments. A stress analysis separates these effects by changing one assumption at a time while holding the other model inputs fixed.

The direction of the effect depends on the product and the cash flows. For a hypothetical income benefit that is paid only while a person is disabled, faster recovery will generally reduce expected claim payments. Higher disabled-state mortality can also reduce those payments if death ends the benefit and no death benefit is included. Neither conclusion automatically describes the effect on every contract: a death benefit, premium waiver, or other payment would need to be included in the model.

Definition: A recovery stress changes the probability of transition from a disabled state to an active state. A disabled-mortality stress changes the probability of transition from a disabled state to death. A morbidity-transition stress changes the probability of movement between disabled states, such as progression from mild to severe disability. To isolate each stress, keep the remaining transition probabilities, benefits, interest assumptions, and starting population unchanged unless the scenario explicitly says otherwise.

The examples use an annual discrete-time Markov model. At each year-end, each person occupies exactly one state. A transition probability is conditional on the person's state at the start of that year. Benefits in the examples are paid at year-end if the person occupies a specified disabled state then; death and active states have no disability payment. The interest rate is effective annual $i=5\%$, so $v=1/(1+i)=1/1.05$. These timing and benefit assumptions are part of each calculation, not universal features of disability products.

State Occupancy and the Reserve Recursion

Let $P_{ij}$ be the probability of moving from state $i$ at the start of a year to state $j$ at year-end. Each row of the transition matrix sums to 1. Let $b_j$ be the benefit paid at year-end when the ending state is $j$. If $R_j^{(n-1)}$ is the value, at that year-end, of the remaining benefits over the next $n-1$ years given state $j$, then the value at the start of the year is

R_i^{(n)}=v\sum_j P_{ij}\left(b_j+R_j^{(n-1)}\right),\qquad R_j^{(0)}=0.

This recursion follows by conditioning on the state reached at year-end. Conditional on transition $i\to j$, the insurer pays $b_j$ then and has a further value $R_j^{(n-1)}$ at that time. Their sum is discounted one year and weighted by $P_{ij}$. A state with no further disability payments has reserve zero for this benefit component. The formula relies on the stated end-of-year payment timing; a benefit paid at the start of a year or continuously during the year requires a different cash-flow valuation.

Specify states and timing.
Define which states generate benefits, when transitions occur, and when payments are made.

Set a reference basis.
Record the transition probabilities, benefit amounts, discount rate, and initial state or population.

Apply one stress.
Change only the selected recovery, mortality, or morbidity-transition assumption; adjust other probabilities in the affected row so it still sums to 1.

Reproject and compare.
Calculate state occupancy and discounted cash flows on both bases, then interpret the difference in context.

Worked Examples: Isolating the Sources of Change

Worked Example: A Recovery Stress on a Disability Income Claim

A hypothetical claimant is disabled at time 0. For each of the next two years, the recovery probability is 20%, disabled-state mortality is 5%, and the remaining probability is continued disability. A benefit of $1,000 is paid at each year-end if the claimant is disabled at that year-end. A stress raises recovery to 30%, leaving mortality unchanged. Calculate the two-year claim value and expected disabled years under both assumptions. A hypothetical contract has a time-0 premium of $1,400 and an expense of $75.

Method. In this one-disabled-state model, the probability of remaining disabled for one year is $s=1-r-m$, where $r$ is recovery and $m$ is disabled-state mortality. The probability of occupying the disabled state at the end of year 2 is $s^2$, because the claimant must remain disabled through both annual transitions.

Reference basis. With $r=0.20$ and $m=0.05$, $s=1-0.20-0.05=0.75$. The year-1 benefit has probability 0.75 of being paid, and the year-2 benefit has probability $0.75^2=0.5625$. Thus

\text{Claim value}=1000(0.75)v+1000(0.5625)v^2 =\frac{750}{1.05}+\frac{562.50}{1.1025} =714.2857+510.2041 =\$1{,}224.49.

The expected number of year-end disabled-state occupancies over the two years is $0.75+0.5625=1.3125$ years. This is a two-year truncated measure, not an uncapped lifetime claim duration.

Recovery stress. With $r=0.30$ and mortality still 0.05, $s=1-0.30-0.05=0.65$. The stressed value is

1000(0.65)v+1000(0.65^2)v^2 =\frac{650}{1.05}+\frac{422.50}{1.1025} =619.0476+383.2199 =\$1{,}002.27.

Expected disabled-state occupancies fall to $0.65+0.4225=1.0725$ years, a decrease of $1.0725-1.3125=-0.2400$ years. The claim value falls by $\$1{,}002.27-\$1{,}224.49=-\$222.22$, using unrounded values before rounding the difference. The time-0 profit on the stated premium and expense basis is $1400-75-1224.4898=\$100.51$ on the reference basis and $1400-75-1002.2676=\$322.73$ under stress. The recovery stress raises this simplified projected profit by $\$222.22$.

Check and common mistake. In each transition row, recovery, mortality, and continued disability sum to 1. A common error is to use the recovery probability as the year-1 payment probability. Payment requires remaining disabled, so it is the persistence probability, 0.75 or 0.65, that weights the benefit.

Worked Example: Increasing Disabled-State Mortality

Use the same hypothetical claimant, $1,000 annual year-end disability benefit, two-year horizon, 5% effective annual interest rate, and reference transition probabilities as above. Now increase disabled-state mortality from 5% to 10%, while holding recovery at 20%. Find the stressed claim value, the change in expected disabled years, and the effect on the simplified profit.

Method. Recalculate the persistence probability after changing mortality, then value each year's benefit using the probability of still being disabled at that year-end. The reference value is $\$1{,}224.49$, with $1.3125$ expected disabled-state occupancies over the two years.

Under stress, persistence is $1-0.20-0.10=0.70$. The year-1 and year-2 probabilities of being disabled are therefore $0.70$ and $0.70^2=0.49$. The claim value is

1000(0.70)v+1000(0.49)v^2 =\frac{700}{1.05}+\frac{490}{1.1025} =666.6667+444.4444 =\$1{,}111.11.

Expected disabled-state occupancies are $0.70+0.49=1.19$ years, a change of $1.19-1.3125=-0.1225$ years. The claim value changes by $\$1{,}111.11-\$1{,}224.49=-\$113.38$. Keeping the premium at $1,400 and expense at $75 gives stressed profit $1400-75-1111.1111=\$213.89$, an increase of $\$113.38$ over the reference profit of $\$100.51$.

Check and common mistake. The stressed row sums to $0.20+0.10+0.70=1$. This model has no death benefit and ends all disability payments at death; adding a death benefit could change the direction of the total contract-value effect. It is incorrect to conclude from this disability-benefit calculation alone that higher mortality is favorable for every product or for solvency overall.

Worked Example: Stressing Progression From Mild to Severe Disability

A hypothetical claimant starts in mild disability, $D_1$. The other states are severe disability, $D_2$, active, and dead. At each year-end, a mild-state benefit of $1,000 or a severe-state benefit of $2,000 is paid if the claimant occupies that state. The two-year transition probabilities are constant. On the reference basis, the transition probabilities from $D_1$ to active, $D_1$, $D_2$, and dead are 0.20, 0.55, 0.20, and 0.05, respectively. From $D_2$, the corresponding probabilities to active, $D_1$, $D_2$, and dead are 0.10, 0.10, 0.72, and 0.08. A progression stress raises the $D_1$-to-$D_2$ probability to 0.30 and reduces the $D_1$-to-$D_1$ probability to 0.45; all other probabilities remain unchanged. Calculate the two-year claim value on each basis.

Method. First find the probability of each benefit-paying state at each year-end. Then multiply each state probability by its benefit, discount each year's expected payment, and add the present values. Active and dead generate no benefit in this example.

On the reference basis, year-1 probabilities are $P(D_1)=0.55$ and $P(D_2)=0.20$. The expected year-1 payment is $0.55(1000)+0.20(2000)=\$950$. For year 2, the mild-state probability is $0.55(0.55)+0.20(0.10)=0.3225$. The severe-state probability is $0.55(0.20)+0.20(0.72)=0.2540$. The expected year-2 payment is $0.3225(1000)+0.2540(2000)=\$830.50$. Therefore

\text{Reference value}=\frac{950}{1.05}+\frac{830.50}{1.1025} =904.7619+753.2880 =\$1{,}658.05.

Under stress, year-1 probabilities are $P(D_1)=0.45$ and $P(D_2)=0.30$, giving expected payment $0.45(1000)+0.30(2000)=\$1{,}050$. At year 2, $P(D_1)=0.45(0.45)+0.30(0.10)=0.2325$, and $P(D_2)=0.45(0.30)+0.30(0.72)=0.3510$. The expected year-2 payment is $0.2325(1000)+0.3510(2000)=\$934.50$. Hence

\text{Stressed value}=\frac{1050}{1.05}+\frac{934.50}{1.1025} =1000.0000+847.6190 =\$1{,}847.62.

The progression stress increases the two-year claim value by $\$1{,}847.62-\$1{,}658.05=\$189.57$. This increase reflects both a higher severe-state probability in year 1 and the resulting effect on year-2 occupancy. Both affected transition rows still sum to 1: the stressed $D_1$ row is $0.20+0.45+0.30+0.05=1$, while the $D_2$ row is unchanged and sums to 1.

Check and common mistake. A common error is to increase the progression probability without reducing another probability in the same row. That would make the row exceed 1 and would not define a valid transition model. Here progression is increased by 0.10 at the expense of remaining in mild disability.

Interpreting Stress Results

The examples illustrate why the stress must be identified precisely. A recovery stress changes exits from disability to active status. A disabled-mortality stress changes exits to death. A morbidity-transition stress can move claimants into a state with a different benefit amount and a different future transition pattern. Each change can affect both current-period payments and later state occupancy.

For a projected reserve or profit test, use the same starting population, cash-flow definition, discount basis, and projection horizon on the reference and stressed bases. In a model with premiums, expenses, or benefits triggered by several states, include each cash flow in the state-dependent projection before interpreting the net result. The examples' profit calculation is deliberately limited to a fixed time-0 premium less a fixed expense and the disability-benefit value; it is not a complete capital or solvency calculation.

A useful review is to check each transition row, verify the payment timing, and compare state probabilities before comparing totals. If a stress affects more than one probability, state explicitly which probability is offset so the row remains a valid probability distribution. A higher recovery assumption does not mean that every future payment is reduced by the same percentage: its impact compounds through later occupancy probabilities and can interact with other states' transition rules.

Key takeaway: Separate stresses by changing one transition assumption at a time. Recalculate state occupancy first, then value the resulting cash flows with consistent timing and discounting. Interpret the direction of a claim-value or profit change only in light of the benefits and payments included in the model.

Check Your Understanding

Use the stated annual transition assumptions, end-of-year payment timing, and effective annual interest rate in each question.

A claimant is disabled at time 0. Over one year, recovery is 25%, disabled-state mortality is 5%, and a $800 benefit is paid if the claimant remains disabled at year-end. At $i=4\%$, calculate the present value of the benefit. (Answer: $538.46)
A claimant remains disabled with probability 0.80 each year for two years; recovery and mortality are mutually exclusive exits, and the annual benefit is $900 if disabled at year-end. At $i=5\%$, calculate the two-year benefit value. (Answer: $1,208.16)
From mild disability, the probabilities of recovery, remaining mild, progressing to severe, and death are 0.15, 0.50, 0.30, and 0.05. Mild and severe year-end benefits are $600 and $1,500. At $i=5\%$, find the one-year benefit value. (Answer: $714.29)
A one-year transition from mild disability has probabilities 0.20 to active, 0.60 to remain mild, 0.15 to severe, and 0.05 to death. A stress raises progression to severe to 0.25 and reduces remaining mild to 0.50. If mild and severe benefits are $1,000 and $2,000, respectively, at $i=5\%$, by how much does the present value increase? (Answer: $95.24)
For a two-year claim with recovery probabilities 0.20 on the reference basis and 0.30 under stress, disabled mortality 0.05 on both bases, and a disabled benefit of $1,000 at each year-end, how much does expected disabled-state occupancy over the two years change under stress? (Answer: a decrease of 0.2400 years)

What You'll Learn