Morbidity Changes Across Disability States

Why Morbidity Changes Need State-Specific Analysis

In an income protection contract, morbidity affects more than the likelihood of a claim starting. The rate at which active lives become disabled changes expected benefit outgo, while the rate at which disabled lives recover changes both the duration of benefit payments and the chance that premiums resume. These effects apply to different states and should not be combined into a single adjustment.

The analysis below uses a hypothetical contract with three states: active $A$, disabled $D$, and dead $X$. Benefits and premiums are specified in each example. The distinction between a transition intensity and an annual transition probability matters: an intensity is an instantaneous rate, whereas a transition probability describes movement over a stated interval. For annual calculations, the examples give annual transition probabilities directly.

Definition: In the three-state model, disability incidence is the transition $A\to D$, with intensity $\alpha$; recovery is the transition $D\to A$, with intensity $\rho$. Death intensities from the active and disabled states are $\mu_A$ and $\mu_D$. The dead state is absorbing. A morbidity assumption change changes one or more transition rates or probabilities involving $A$ and $D$; its valuation effect depends on the benefits and premiums attached to the states.

For a continuous-time Markov model, the intensity matrix, in the order $A,D,X$, is

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

Over a short interval of length $h$, the probability of an $A\to D$ transition is approximately $\alpha h$, and the probability of a $D\to A$ transition is approximately $\rho h$. For a longer interval, transition probabilities depend on the whole matrix $Q$; in general, an intensity is not itself an annual probability.

State Reserves and Morbidity in a Continuous Model

Let $V_A(t)$, $V_D(t)$, and $V_X(t)$ denote prospective reserves at time $t$, conditional on the current state. Define a reserve as the present value of future benefit outgo less future premium income. Suppose the force of interest is $\delta$, premiums are paid continuously at rate $p$ while active, disability benefits are paid continuously at rate $b$ while disabled, and there are no payments on transitions. Assume constant intensities over the period considered. The dead-state reserve is zero.

\begin{aligned} V_A'(t)&=\delta V_A(t)+p+\alpha\bigl(V_A(t)-V_D(t)\bigr) +\mu_A\bigl(V_A(t)-V_X(t)\bigr),\\ V_D'(t)&=\delta V_D(t)-b+\rho\bigl(V_D(t)-V_A(t)\bigr) +\mu_D\bigl(V_D(t)-V_X(t)\bigr),\\ V_X(t)&=0. \end{aligned}

These equations apply under the stated continuous-payment and no-transition-payment assumptions. A benefit payable on disability onset, recovery, or death must be included as a transition payment; it cannot be omitted while using these equations unchanged. At a contract termination time $T$, if there is no terminal payment, the boundary conditions are $V_A(T)=V_D(T)=V_X(T)=0$.

Why the transition terms appear: Over a short interval $h$, a reserve in state $A$ is exposed to a probability approximately $\alpha h$ of ending the interval in $D$. That transition replaces the continuation reserve $V_A$ by $V_D$, giving a change approximately $\alpha h(V_D-V_A)$ in expected future liability. Discounting and the premium flow must also be allowed for. Rearranging the short-interval reserve balance gives the term $\alpha(V_A-V_D)$ in $V_A'$. The recovery term follows in the same way, with the states reversed.

The signs give a useful check. If $V_D$ exceeds $V_A$, increasing incidence tends to raise the active-state liability through $\alpha(V_A-V_D)$. If recovery moves a disabled policyholder toward a lower-reserve active state, increasing $\rho$ tends to reduce the disabled-state liability through $\rho(V_D-V_A)$. These are local effects in the differential equations, not universal conclusions about every reserve comparison: changing a transition rate also changes future time spent in each state.

Annual Prospective Reserves for Profit Tests

Annual profit projections commonly use state transition probabilities over each policy year. Let $p_{ij}$ be the probability of ending the year in state $j$, given state $i$ at the start. Let $P_i$ be a premium paid at the start of the year in state $i$, and let $B_{ij}$ be a benefit paid at year-end for a transition from $i$ to $j$. With $v=1/(1+i)$, the reserve just before the start-of-year premium is

V_t(i)=-P_i+v\sum_j p_{ij}\bigl(B_{ij}+V_{t+1}(j)\bigr).

This is a prospective value: the premium is an inflow and is therefore subtracted from the expected future net outgo. The equation also makes the morbidity effect explicit. Changing incidence alters the active-state row of transition probabilities; changing recovery alters the disabled-state row. Both changes can affect a reserve, but through different conditional pathways.

Worked Example: Higher Disability Incidence and the Active-State Reserve

A hypothetical one-year contract pays a $12,000 benefit at year-end if the policyholder is disabled then. An active policyholder pays a $2,000 premium at the start of the year. The annual effective interest rate is 5%. Under the valuation assumption, an active policyholder has probabilities $0.80$ of remaining active, $0.15$ of becoming disabled, and $0.05$ of dying. Find the reserve from the active state. Then repeat if disability incidence rises so the probabilities are $0.70$, $0.25$, and $0.05$, respectively. There is no terminal reserve after the year.

Method. Apply the one-year prospective reserve equation with a year-end disability benefit and a start-of-year premium. The discount factor is $v=1/1.05=0.95238095$.

Under the original assumption, expected benefit at year-end is $0.15(12{,}000)=\$1{,}800$. Thus

V_0(A)=-2{,}000+\frac{1{,}800}{1.05} =-2{,}000+1{,}714.2857 =-\$285.7143.

Under higher incidence, expected benefit is $0.25(12{,}000)=\$3{,}000$, so

V_0^{\mathrm{high}}(A)=-2{,}000+\frac{3{,}000}{1.05} =-2{,}000+2{,}857.1429 =\$857.1429.

The reserve rises by $\$857.1429-(-\$285.7143)=\$1{,}142.8572$, or $\$1{,}142.86$ to the nearest cent. As a check, the increase equals the present value of the additional expected benefit: $(0.25-0.15)(12{,}000)/1.05=\$1{,}142.8571$, with the small difference in the displayed subtraction due to rounding. The calculation holds the premium and all other assumptions fixed.

Worked Example: Higher Recovery and a Two-Year Disabled-State Reserve

Consider a two-year hypothetical contract with a $12,000 benefit at the end of each year if the policyholder is disabled at that year-end. An active policyholder pays a $2,000 premium at the start of each year. The annual effective interest rate is 5%. At the start of the final year, the active-state transition probabilities are $0.80$ to active, $0.15$ to disabled, and $0.05$ to dead. For a disabled policyholder, the original probabilities are $0.40$ to active, $0.50$ to disabled, and $0.10$ to dead. Now increase recovery so the disabled-state probabilities become $0.60$, $0.30$, and $0.10$. Calculate the reserve at the start of the first year, conditional on being disabled.

Method. First value the remaining year in each state. Then use those state reserves and the first-year transition probabilities to value the two-year liability from state $D$. Set all reserves after the second year to zero.

At the start of the final year, before the premium, the active-state reserve is

V_1(A)=-2{,}000+\frac{0.15(12{,}000)}{1.05} =-2{,}000+1{,}714.2857=-\$285.7143.

With the original recovery assumption, the disabled-state reserve for that final year is

V_1(D)=\frac{0.50(12{,}000)}{1.05} =\$5{,}714.2857.

With the higher recovery assumption, it becomes

V_1^{\mathrm{high}}(D)=\frac{0.30(12{,}000)}{1.05} =\$3{,}428.5714.

At the start of the first year, a disabled policyholder receives the year-one benefit if still disabled at year-end and then has the corresponding year-two reserve. Under the original probabilities, the amount inside the discounting step is

$$ 0.50(12{,}000)+0.40(-285.7143)+0.50(5{,}714.2857) =6{,}000-114.2857+2{,}857.1429 =8{,}742.8572. $$

The resulting reserve is $V_0(D)=8{,}742.8572/1.05=\$8{,}326.53$. Under higher recovery, the corresponding calculation is

\begin{aligned} V_0^{\mathrm{high}}(D) &=\frac{0.30(12{,}000)+0.60(-285.7143)+0.30(3{,}428.5714)}{1.05}\\ &=\frac{3{,}600-171.4286+1{,}028.5714}{1.05}\\ &=\frac{4{,}457.1428}{1.05} =\$4{,}244.90. \end{aligned}

The reserve falls by $\$8{,}326.53-\$4{,}244.90=\$4{,}081.63$. The decrease reflects fewer year-end disability benefits and more transitions to the active state, whose final-year reserve is negative because of the premium. As a check, the changed disabled-state row sums to $0.60+0.30+0.10=1$, and the active-state row also sums to 1.

From State Reserves to a Profit Signature

A profit test compares actual-period cash flows and the change in reserves. Under a convention in which the opening reserve is held for a full year, the premium is paid at the start, and benefits and closing reserves are measured at year-end, conditional expected profit from initial state $i$ is

\Pi_i=(V_0(i)+P_i)(1+i) -\sum_j p_{ij}\bigl(B_{ij}+V_1(j)\bigr).

The first term is the accumulated opening reserve plus premium. The second is expected benefit outgo plus the closing reserve. For a profit test on a cohort, the state-conditional profits are weighted by the probabilities or proportions in each state at the start of the year. Repeating this by year produces a profit signature; discounting its entries gives the NPV under the specified discount basis.

The reserve basis and the experience basis must be distinguished. A reserve may be calculated using one morbidity assumption, while projected cash flows use another. If both bases are identical and the reserve recursion is applied consistently, expected profit for the period is zero before considering items outside that recursion. A difference in morbidity assumptions can therefore change profit emergence even when the product cash flows themselves have not changed.

Worked Example: Morbidity Deviation in a One-Year Profit Signature

A hypothetical in-force group has 90% of policies active and 10% disabled at the start of a one-year period. The valuation basis uses the assumptions from the first worked example: active-to-disabled probability $0.15$, disabled-to-disabled probability $0.50$, and 5% annual interest. The premium is $2,000 for active policies, the year-end disability benefit is $12,000, and terminal reserves are zero. Actual projected morbidity instead has active-to-disabled probability $0.25$ and disabled-to-disabled probability $0.30$. Calculate the expected year-one profit and its NPV when the profit is paid at year-end and discounted at 5%.

Method. Use valuation-basis opening reserves, calculate profit separately by initial state using actual expected benefits, and weight the results by the initial state proportions.

The valuation-basis reserves are $V_0(A)=-\$285.7143$ and $V_0(D)=\$5{,}714.2857$. For an active policy, the accumulated opening reserve and premium are

(-285.7143+2{,}000)(1.05)=\$1{,}800.

Actual expected benefit is $0.25(12{,}000)=\$3{,}000$, so active-state profit is $1{,}800-3{,}000=-\$1{,}200$. For a disabled policy, the accumulated reserve is $5{,}714.2857(1.05)=\$6{,}000$. Actual expected benefit is $0.30(12{,}000)=\$3{,}600$, so disabled-state profit is $6{,}000-3{,}600=\$2{,}400$.

The year-one profit signature entry is

\Pi_1=0.90(-1{,}200)+0.10(2{,}400) =-1{,}080+240=-\$840.

Discounting the year-end profit gives $\operatorname{NPV}=-840/1.05=-\$800.00$. As a check, the weighted calculation uses the stated 90% and 10% starting proportions, which sum to 1. The negative incidence effect and positive recovery effect partly offset, but the net expected profit remains negative.

Common Mistake: Reversing the Recovery Probability

In the second example, a recovery probability of $0.60$ means a transition from disabled to active. It does not mean that $60\%$ remain disabled. The correct disabled persistence probability is $0.30$, so the final-year disabled reserve is $0.30(12{,}000)/1.05=\$3{,}428.57$. A mistaken calculation using $0.60$ as the disabled persistence probability would give $0.60(12{,}000)/1.05=\$6{,}857.14$. That reverses the direction of the assumption change and materially overstates the liability.

A related trap is to change incidence and recovery in the same row of a transition matrix. Incidence is read from the active-state row; recovery is read from the disabled-state row. Each row must still sum to 1, including the probability of remaining in the current state and any probability of death.

Key takeaway: Trace a morbidity change from its originating state to its destination state, then recalculate the state-specific reserve before aggregating profits. The direction and size of the effect depend on future benefits, premiums, transition probabilities, interest, and the reserve basis. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the stated payment timing and reserve convention in each problem. Amount answers are rounded to the nearest cent.

A one-year contract has a $10,000 year-end benefit if disabled, a $1,500 premium paid at the start if active, annual interest of 4%, and active-state probabilities $0.83$ active, $0.12$ disabled, and $0.05$ dead. There is no terminal reserve. Find the active-state opening reserve. (Answer: $-\$346.15$)
A disabled policyholder has a one-year probability of $0.35$ of remaining disabled, $0.55$ of recovering, and $0.10$ of dying. A $9,000 benefit is paid at year-end if disabled, with 5% annual interest and no premium or terminal reserve. Find the disabled-state reserve. (Answer: $\$3{,}000.00$)
A two-year contract pays $8,000 at each year-end if disabled and charges a $1,000 start-of-year premium if active. Interest is 4%. In the final year, active transition probabilities are $0.85$ active, $0.10$ disabled, and $0.05$ dead; disabled transition probabilities are $0.50$ active, $0.40$ disabled, and $0.10$ dead. Find the start-of-first-year reserve conditional on being disabled. (Answer: $\$4{,}149.41$)
A one-year profit test uses valuation-basis reserves of $-\$247.62$ when active and $\$4{,}761.90$ when disabled. The active premium is $1,200, the benefit is $10,000 if disabled at year-end, and interest is 5%. Actual probabilities of ending disabled are $0.16$ from active and $0.40$ from disabled. If 75% of the cohort starts active and 25% disabled, find the expected year-one profit. (Answer: $-\$200.00$)
In an annual three-state model, a disabled-state transition row is $0.55$ to active, $0.35$ to disabled, and $0.10$ to dead. State which probability represents recovery and which represents remaining disabled. (Answer: Recovery is $0.55$; remaining disabled is $0.35$.)

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