Disability Incidence Stresses

Why Stress Disability Incidence?

A disability-incidence stress increases the probability that an active policyholder becomes disabled. Its financial effect is not limited to the additional claims that may result. If premiums are payable only while active, more transitions into disability can also reduce future premium receipts. The balance between these effects depends on the contract’s benefit and premium rules, the timing of cash flows, mortality, and the projection horizon.

This tutorial uses a discrete annual active-disabled-dead model to isolate those effects. The scenarios are hypothetical. All comparisons change disability incidence while holding the other assumptions and contract terms fixed, so the resulting difference is a one-factor sensitivity rather than a combined scenario.

Definition: In an annual active-disabled-dead model, let $p$ be the probability that a policyholder who is active at the start of a year is disabled and alive at its end, and let $q$ be the probability that the policyholder dies during that year. The probability of remaining active is $1-p-q$. Let $r$ be the probability that a disabled policyholder dies during a year; the probability of remaining disabled is $1-r$. Death is absorbing. This model has no transition from disabled back to active.

These are annual transition probabilities, not forces of transition. In particular, $p$ is conditional on being active at the start of the year. The model requires $p\geq 0$, $q\geq 0$, and $p+q\leq 1$, as well as $0\leq r\leq 1$. It treats disability and death as year-end states; a different within-year payment convention would require different cash-flow probabilities.

A stress can be specified multiplicatively, $p'=kp$, where $k$ is a stress factor, or additively, $p'=p+\Delta p$. In either case, check that $p'+q\leq 1$. Unless stated otherwise, $q$, $r$, the interest rate, benefits, premiums, and expenses are unchanged.

Projecting States and Cash Flows

Let $A_t$ and $D_t$ be the probabilities that a policyholder is active and disabled, respectively, at the start of year $t$. For a policyholder initially active, $A_0=1$ and $D_0=0$. Applying the transition probabilities gives

A_{t+1}=A_t(1-p-q),\qquad D_{t+1}=A_t p+D_t(1-r).

If an annual premium $P$ is collected at the start of a year only when the policyholder is active, its expected payment in year $t$ is $P A_t$. If a benefit $B$ is paid at year-end whenever the policyholder is disabled and alive at that year-end, its expected payment for that year is $B D_{t+1}$. This convention includes both new disability during the year and disabled policyholders who survive from the prior year.

With effective annual interest rate $i$, set $v=1/(1+i)$. For a projection of $n$ years, the present value of premiums is $P\sum_{t=0}^{n-1}v^t A_t$, and the present value of disability benefits is $B\sum_{t=0}^{n-1}v^{t+1}D_{t+1}$. In the examples, projected profit is premiums less benefits, with no expenses, tax, capital charge, or other cash flows. A more complete profit test would include those items explicitly.

Why the Incidence Stress Changes Both Sides of the Projection

The state recursions explain the two main directions of impact. Increasing $p$, with $q$ fixed, reduces the probability of remaining active in the next year, because $A_{t+1}=A_t(1-p-q)$. It also increases the probability of entering the disabled state, because the new-incidence component of $D_{t+1}$ is $A_t p$. Disability benefits may therefore rise while future active-only premiums fall.

This is a projection effect, not a universal claim that every product’s profit must decrease by a particular amount. The outcome depends on benefit levels, premium rules, projection length, and other contract cash flows. A disability benefit may, for example, be offset by other payments under a contract; such terms must be represented rather than assumed away.

Set the reference assumptions.
Record baseline incidence, mortality, interest, premium timing, benefit timing, and the projection horizon.

Apply the incidence stress.
Calculate the stressed probability and check that it is valid alongside the unchanged active mortality probability.

Reproject states and cash flows.
Use the same recurrence and payment conventions under both scenarios; change only the stated assumption.

Compare results consistently.
Calculate the change from unrounded scenario values, then round the reported amounts.

Worked Example: One-Year Benefit Outgo Under an Incidence Stress

A hypothetical one-year contract starts with one active policyholder. A premium of $500 is collected immediately. At the end of the year, a $10,000 disability benefit is paid if the policyholder is disabled and alive. The annual effective interest rate is 5%. In the reference scenario, $p=0.04$ and $q=0.01$; in the stress, $p'=0.06$, with $q$ unchanged. There is no death benefit or expense. Calculate the present value of profit in each scenario and the change.

Method. Apply the annual transition probabilities once, discount the expected year-end benefit by $v=1/1.05=20/21$, and subtract it from the immediate premium. Both probability sets are valid: $0.04+0.01=0.05\leq1$ and $0.06+0.01=0.07\leq1$.

In the reference scenario, the probability of a disability benefit is $0.04$, so expected benefit outgo at year-end is $10{,}000(0.04)=\$400$. Its present value is $400(20/21)=\$8{,}000/21=\$380.9524$. Thus, profit is $500-8{,}000/21=\$119.0476$, or $119.05 to the nearest cent.

In the stressed scenario, expected benefit outgo at year-end is $10{,}000(0.06)=\$600$. Its present value is $600(20/21)=\$12{,}000/21=\$571.4286$. Profit is $500-12{,}000/21=-\$71.4286$, or -$71.43 to the nearest cent.

The change, calculated from the unrounded scenario values, is $-\$71.428571\ldots-\$119.047619\ldots=-\$190.476190\ldots$, or -$190.48. The result is consistent with the $200 increase in expected year-end benefit outgo, discounted by $20/21$.

Common mistake avoided: Do not treat the increased disability probability as a benefit paid immediately. The benefit is paid at year-end here, so its expected amount must be discounted.

Worked Example: Two-Year Profit Includes Lost Active Premiums

A hypothetical contract starts with one active policyholder. It collects a $1,000 premium at the start of each year if the policyholder is active then, and pays $5,000 at each year-end if the policyholder is disabled and alive then. The effective annual interest rate is 5%. The active death probability is $q=0.05$, and the disabled death probability is $r=0.10$, each year. Compare a reference incidence probability $p=0.10$ with a stress $p'=0.20$. There are no other cash flows.

Method. Project $A_1,D_1,D_2$ under each incidence probability. Premiums are paid at times 0 and 1 when active at the start of those years; benefits are paid at times 1 and 2 according to the disabled probabilities at those year-ends. Here $v=20/21$ and $v^2=400/441$.

In the reference scenario, $A_1=1(1-0.10-0.05)=0.85$ and $D_1=0.10$. For the second year, $D_2=0.85(0.10)+0.10(1-0.10)=0.085+0.090=0.175$.

The present value of premiums is $1{,}000+v(1{,}000)(0.85)=1{,}000+809.523810=\$1{,}809.523810$. The present value of benefits is $5{,}000[v(0.10)+v^2(0.175)]$. The two discounted probability terms are $0.10(20/21)=0.095238095$ and $0.175(400/441)=0.158730159$, giving benefits of $5{,}000(0.253968254)=\$1{,}269.841270$. Profit is therefore $\$539.682540$, or $539.68.

Under stress, $A_1=1(1-0.20-0.05)=0.75$ and $D_1=0.20$. Then $D_2=0.75(0.20)+0.20(0.90)=0.150+0.180=0.330$. The present value of premiums is $1{,}000+v(1{,}000)(0.75)=\$1{,}714.285714$. The benefit terms are $0.20v=0.190476190$ and $0.330v^2=0.299319728$, so the present value of benefits is $5{,}000(0.489795918)=\$2{,}448.979592$. Stressed profit is $-\$734.693878$, or -$734.69.

Subtracting the unrounded profits gives a change of $-\$734.693878-\$539.682540=-\$1{,}274.376418$, or -$1,274.38 to the nearest cent. The displayed rounded profits subtract to -$1,274.37; that one-cent difference is solely due to rounding the two scenario results separately. The incidence stress increases disability benefit outgo and also reduces the probability of collecting the second premium.

Common mistake avoided: Do not calculate the second premium as certain. It is conditional on being active at the start of year 2, whose probability falls from 0.85 to 0.75 under stress.

Prospective Reserves Under Stressed Incidence

A prospective net reserve is the expected present value of future benefits less the expected present value of future premiums, conditional on the policyholder’s state at the valuation time. The cash-flow timing matters: a premium due immediately is not discounted, while a benefit due at year-end is discounted.

Result: For an active policyholder immediately before a one-year premium $P$ is due, with benefit $B$ paid at year-end if disability occurs and the policyholder is alive, the net reserve is $V_A=v p B-P$, assuming no other cash flows. This conditional reserve uses the incidence probability for the coming year.

Worked Example: One-Year Active-State Reserve

A hypothetical contract has one year remaining. Immediately before the premium is due, the policyholder is active. A premium of $400 is due now. A $10,000 benefit is paid at year-end if the policyholder is disabled and alive. The effective annual interest rate is 5%, and active mortality is $q=0.02$. Compare $p=0.06$ with $p'=0.10$; all other assumptions are unchanged. There are no other cash flows.

Method. Calculate the prospective net reserve conditional on being active, using $v=20/21$. Both transition sets are feasible: $0.06+0.02\leq1$ and $0.10+0.02\leq1$. Death does not generate a payment in this contract, so the expected disability benefit is $pB$, not $(p+q)B$.

At $p=0.06$, the expected year-end benefit is $10{,}000(0.06)=\$600$. Its present value is $600(20/21)=\$571.428571$. Subtracting the premium due now gives $V_A=\$571.428571-\$400=\$171.428571$, or $171.43.

At $p'=0.10$, the expected year-end benefit is $10{,}000(0.10)=\$1{,}000$. Its present value is $1{,}000(20/21)=\$952.380952$. Thus, the stressed reserve is $\$952.380952-\$400=\$552.380952$, or $552.38. The increase, calculated before rounding, is $0.04(10{,}000)(20/21)=\$380.952381$, or $380.95.

Check. The only changed assumption is the four-percentage-point increase in incidence. The corresponding increase in expected benefit present value is exactly the reserve increase, since the premium and all other cash flows are unchanged.

Common mistake avoided: The reserve is conditional on being active at the valuation time. Do not multiply the one-year benefit by the probability of being active at the start of the year; that probability is already implicit in the conditioning.

Common Exam Traps

Using the wrong probability base. Apply incidence to active lives, not to all lives in the original group. In the recurrence, new disabilities are $A_t p$.
Ignoring competing active mortality. The probability of remaining active is $1-p-q$, not $1-p$, when death and disability are mutually exclusive annual outcomes.
Breaking the probability model. A stressed $p'$ must satisfy $p'+q\leq1$. If it does not, the proposed stress cannot be used as an annual transition probability without reformulating the scenario.
Keeping active-only premiums fixed in expectation. The premium amount may be unchanged, but expected future premiums fall when fewer policyholders remain active.
Mixing payment timings. State explicitly whether premiums are due at the start or end of a year and when benefits are paid. Discount each cash flow from its actual payment time.
Rounding before comparing. Calculate the stressed-minus-reference change from unrounded values, then round the change. Rounded scenario amounts may not subtract to the rounded change.

Disability-incidence sensitivity is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central exam discipline is to separate the incidence assumption from the transition model and cash-flow timing: project the states first, attach the contract’s payments to those states, and compare the resulting values on a consistent basis.

Check Your Understanding

For each problem, use the stated annual transition and payment conventions. Round monetary answers to the nearest cent.

A one-year hypothetical policy starts active. It collects $600 immediately and pays $8,000 at year-end if disability and survival occur. The effective annual interest rate is 4%, and active mortality is $q=0.02$. Compare incidence $p=0.08$ with $p'=0.11$. Find the change in present value of profit. (Answer: -$230.77)
A two-year hypothetical policy pays a $900 premium at the start of year 2 only if the policyholder is active then. Initially active, the policy has annual $q=0.03$, $p=0.12$ in the reference case, and $p'=0.18$ in the stress. At 5% effective interest, find the change in the present value of the year-2 premium receipts. (Answer: -$51.43)
A one-year contract is valued immediately before a $500 premium is due to an active policyholder. It pays $12,000 at year-end if disability and survival occur. At 4% effective interest, compare $p=0.05$ and $p'=0.09$, with $q=0.02$ fixed. Find the increase in the active-state prospective net reserve. (Answer: $461.54)
A two-year hypothetical contract starts with one active policyholder, collects $500 at the start of each year if active, and pays $4,000 at each year-end if disabled and alive. There is no mortality, recovery, or interest. Compare $p=0.10$ with $p'=0.20$. For each case, calculate the two-year profit and then the stressed-minus-reference change. (Answer: Reference profit -$210.00; stressed profit -$1,340.00; change -$1,130.00)

What You'll Learn