Morbidity Shocks in Profit Tests

Why Morbidity Shocks Affect More Than Claim Outgo

A higher disability incidence assumption can increase benefit payments immediately, but its effect does not stop there. When disability removes a policy from the active state, it also reduces the number of policies that can pay future active premiums or experience a new disability. If disabled policies remain covered, the shock changes the mix of states in force and therefore the reserves held for future benefits. Profit in a duration reflects these cash-flow and reserve effects together.

The key is to distinguish a change in the incidence rate from a change in exposure. Incidence is conditional on being active at the start of the period. It is not a percentage of the original cohort in every subsequent period. This tutorial uses simple annual models to make that distinction and then incorporates the changed exposure into a reserve-adjusted profit test. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A morbidity shock is a change in an assumption governing entry into, or movement through, a morbidity state such as disability. In an annual active-disabled model, $q_t$ is the probability that an active policy enters disability during year $t$. It is conditional on being active at the start of that year. A morbidity shock may affect only the current year's incidence, or it may change the assumed incidence in future years as well.

Unless stated otherwise, examples use expected policy counts, no mortality or lapse, and disability as an absorbing state: a policy that becomes disabled remains disabled through the projection. These are simplifying assumptions, not general properties of disability products. For a richer model, state counts are projected through the transition probabilities for each period; an incidence assumption applies only to the transitions from the eligible state.

Projecting Incidence and Remaining Active Exposure

Let $A_t$ be the expected number active at the start of year $t$, and let $D_t$ be the expected number disabled at that time. With no other exits, and with disability absorbing, the expected counts satisfy

\text{New disabilities in year }t=A_tq_t,\qquad A_{t+1}=A_t(1-q_t),\qquad D_{t+1}=D_t+A_tq_t.

These equations show why a one-period morbidity shock can have effects in later durations even if the incidence assumption then returns to its original level. The first year's shock changes $A_{t+1}$; the next year's incidence is applied to that changed active count. A benefit paid on disability incidence is driven by new disabilities, while a recurring benefit for all disabled policies is driven by the disabled count during the benefit period. The benefit definition must therefore be stated before projecting outgo.

Worked Example: Incidence Shock and Two-Year Exposure

A hypothetical block has 250 active policies at the start of year 1. There is no mortality, lapse, or recovery, and disability is absorbing. A lump-sum benefit of $12,000 is paid at the end of the year in which a policy becomes disabled. The baseline incidence probability is 0.04 in both years. In the shock scenario it is 0.07 in year 1 and returns to 0.04 in year 2. The effective annual interest rate is 5%. Calculate expected new disabilities, active exposure at the start of year 2, and the present value at issue of the two years' benefits.

Method. Apply each year's incidence probability to the active count at the start of that year. Discount the year 1 and year 2 benefit payments by $v$ and $v^2$, respectively, where $v=1/1.05$.

Baseline. Year 1 has $250(0.04)=10$ expected new disabilities, leaving $250-10=240$ active policies at the start of year 2. Year 2 has $240(0.04)=9.6$ expected new disabilities. Expected benefits are $10($12,000)=$120,000$ at the end of year 1 and $9.6($12,000)=$115,200$ at the end of year 2. Thus,

\operatorname{PV}_{\text{baseline}} =\frac{$120,000}{1.05}+\frac{$115,200}{1.05^2} =$114,285.7143+$104,489.7959 =$218,775.5102.

Shock scenario. Year 1 has $250(0.07)=17.5$ expected new disabilities, leaving $250-17.5=232.5$ active policies at the start of year 2. The year 2 incidence probability has returned to 0.04, so expected new disabilities then are $232.5(0.04)=9.3$, not $250(0.04)$. Expected benefits are $210,000 and $111,600, respectively. Therefore,

\operatorname{PV}_{\text{shock}} =\frac{$210,000}{1.05}+\frac{$111,600}{1.05^2} =$200,000.0000+$101,224.4898 =$301,224.4898.

The increase in the present value of these incidence benefits is $$301,224.4898-$218,775.5102=$82,448.9796$, or $82,448.98 to the nearest cent. The check is that the shock raises year 1 incidence by 7.5 expected policies, while reducing year 2 incidence by 0.3 because fewer policies remain active. The year 1 increase in outgo dominates that later reduction.

Reserves and the Profit Balance

A prospective reserve is the present value of future benefits and expenses less the present value of future premiums, conditional on the policy's current state and on a specified assumption basis. A negative reserve is possible when the value of future net premiums exceeds the value of future outgo; no minimum reserve floor is imposed in the calculations below.

For a one-year duration, let $V_t$ be the opening reserve before a premium and expense are paid, $P_t$ the premium paid at the start of the year, $E_t$ the start-of-year expense, $B_t$ the benefit paid at year-end, and $V_{t+1}$ the closing reserve after the state transition. With effective annual interest $i$, the profit at year-end is

\Pi_t=(V_t+P_t-E_t)(1+i)-B_t-V_{t+1}.

For a cohort, $B_t$ and $V_{t+1}$ in this equation are expected aggregate amounts. The closing reserve is the sum of the reserves for each destination state, weighted by the expected number in that state. This equation follows from the fund balance: the opening reserve plus the net start-of-year cash flow accumulates with interest, then benefits are paid and the closing reserve is set aside. The remainder is profit. It also explains why comparing benefit outgo alone cannot measure the full profit effect of a morbidity shock.

Result: Under a fixed reserve basis, a morbidity shock changes expected profit through current benefit outgo and the expected closing reserve; it can also change later-duration profits through the changed state counts. If reserves are recalculated on a changed future morbidity basis, the closing reserve reflects that basis change as well. Always state separately the actual transition assumption and the assumption used to value future liabilities.

Worked Reserve and Profit Projection

The following example uses a two-year hypothetical contract to trace a permanent morbidity change through state-specific reserves and first-duration profit. Premiums and expenses are paid at the start of each year for active policies only. A $600 benefit is paid at year-end for every policy disabled during that year, including those already disabled at the start of the year. Disabled policies remain disabled. There are no other benefits, expenses, decrements, or terminal reserves after year 2. The effective annual interest rate is 5%. The premium is $400 and the expense is $20 per active policy per year, so the net start-of-year cash inflow for an active policy is $380.

Worked Example: Permanent Increase in Disability Incidence

The baseline incidence probability is $q=0.05$ in each year. At issue, a permanent shock changes it to $q=0.10$ in both years. Calculate the active and disabled reserves at the start of year 2 under both bases, the issue reserve under each basis, and the first-year expected profit for 100 initially active policies when the opening reserve is the baseline issue reserve but the closing reserve is recalculated on the shocked basis.

Method and reserve at the start of year 2. There is one year remaining. For an active policy, the net premium and expense are paid immediately; a disability benefit is paid at year-end with probability $q$. For a disabled policy, the benefit is certain at year-end and there is no premium. Hence

V_{A,1}(q)=-$380+\frac{$600q}{1.05},\qquad V_{D,1}=\frac{$600}{1.05}.

The disabled reserve is $600/1.05=$571.428571\ldots$ under either basis. Under the baseline, $q=0.05$, so

V_{A,1}(0.05)=-$380+\frac{$30}{1.05} =-$380+$28.571428\ldots =-$351.428571\ldots.

Under the permanent shock, $q=0.10$, giving

V_{A,1}(0.10)=-$380+\frac{$60}{1.05} =-$380+$57.142857\ldots =-$322.857142\ldots.

Issue reserves. At issue, an active policy pays the net $380 immediately. At year-end it may incur the disability benefit and will be in the active or disabled state for the remaining year's reserve. Therefore,

V_{A,0}(q)=-$380+ \frac{$600q+(1-q)V_{A,1}(q)+qV_{D,1}}{1.05}.

For the baseline, the expected closing reserve at the end of year 1 is

0.95\left(-\frac{$2,460}{7}\right)+0.05\left(\frac{$4,000}{7}\right) =-\frac{$2,137}{7} =-$305.285714\ldots.

The expected year 1 benefit is $0.05($600)=$30$. Substituting both amounts gives

V_{A,0}(0.05) =-$380+\frac{$30-$305.285714\ldots}{1.05} =-$380-\frac{$1,927}{7}\frac{20}{21} =-$642.176870\ldots.

For the shocked basis, the expected closing reserve is

0.90\left(-\frac{$2,260}{7}\right)+0.10\left(\frac{$4,000}{7}\right) =-\frac{$1,634}{7} =-$233.428571\ldots.

The expected year 1 benefit is $0.10($600)=$60$. Thus,

V_{A,0}(0.10) =-$380+\frac{$60-$233.428571\ldots}{1.05} =-$380-\frac{$1,214}{7}\frac{20}{21} =-$545.170068\ldots.

The issue reserve increases by $$-545.170068\ldots-(-$642.176870\ldots)=$97.006802\ldots$, or $97.01 per active policy to the nearest cent. The direction is consistent with the assumptions: higher future incidence increases the expected disability outgo, making the active reserve less negative.

First-year profit with the basis change. The opening reserve is the baseline issue reserve, but the year 1 incidence and closing reserves use $q=0.10$. Per initially active policy, the accumulated opening reserve plus net premium is

(V_{A,0}(0.05)+$380)(1.05) =(-$642.176870\ldots+$380)(1.05) =-$275.285714\ldots.

The expected year 1 benefit is $60 and the expected aggregate closing reserve per initial policy is $-$1,634/7=-$233.428571\ldots$. Applying the profit balance gives

\Pi_1=-$275.285714\ldots-$60-(-$233.428571\ldots) =-$101.857142\ldots.

For 100 policies, expected first-year profit is $100(-$101.857142\ldots)$, or -$10,185.71 to the nearest cent. The calculation includes both the higher current benefit and the higher closing reserve relative to the baseline closing reserve. Rounding each reserve before subtracting can change a cent-level result; the difference above is calculated from unrounded values and rounded only at the end.

Temporary Shocks, Duration Profit, and an Exam Trap

A temporary experience shock and a permanent assumption change are different projections. If only year 1 incidence is shocked, the year 2 reserve should generally use the stated year 2 valuation assumption, not automatically carry forward the year 1 shock. The distinction matters even when year 1 actual incidence is identical in both scenarios. The next example isolates this reserve-basis issue.

Worked Example: A One-Year Incidence Shock

Use the two-year contract and 100 initially active policies from the reserve projection above. Actual year 1 incidence is 0.10, but the year 2 incidence assumption remains at the baseline 0.05. The opening reserve is the baseline issue reserve. Calculate year 1 expected profit using the year 2 reserves on the baseline basis, and compare it with the permanent-shock result.

Method. The year 1 expected benefit remains $100(0.10)($600)$. At the end of year 1, 90 policies are expected to be active and 10 disabled. Because the future incidence assumption is 0.05, the appropriate expected closing reserve per initial policy is

0.90V_{A,1}(0.05)+0.10V_{D,1} =0.90\left(-\frac{$2,460}{7}\right)+0.10\left(\frac{$4,000}{7}\right) =-\frac{$1,814}{7} =-$259.142857\ldots.

The accumulated opening reserve plus net premium is still $-$1,927/7=-$275.285714\ldots$ per policy. Therefore the expected profit per initial policy is

\Pi_1=-\frac{$1,927}{7}-$60+\frac{$1,814}{7} =-\frac{$533}{7} =-$76.142857\ldots.

For 100 policies, this is -$7,614.29 to the nearest cent. The permanent-shock calculation gave -$10,185.71. The temporary-shock result is higher by $100($1,814/7-$1,634/7)=$2,571.43$ to the nearest cent, because the temporary scenario has a less adverse future incidence assumption and a correspondingly lower expected closing reserve liability.

Check by duration. In year 2, the baseline reserve at the start of the year is prospective on $q=0.05$. For an active policy, $(V_{A,1}(0.05)+$380)(1.05)=$30$, exactly the expected year 2 benefit $0.05($600)=$30$. For a disabled policy, $V_{D,1}(1.05)=$600$, exactly its year 2 benefit. There is no terminal reserve. Thus expected year 2 profit is zero in either state under the stated assumptions. The year 1 shock changes the number in each state, but it does not create a year 2 profit when the year 2 reserve basis and experience agree.

A common exam error is to use the permanent-shock reserve in the temporary-shock calculation. That wrong calculation uses $-$1,634/7$ for expected closing reserve and gives

-\frac{$1,927}{7}-$60+\frac{$1,634}{7} =-\frac{$713}{7} =-$101.857142\ldots

per policy, instead of the correct $-$76.142857\ldots$. The correction is to value future liabilities using the future incidence assumption actually specified. A separate but related error is applying a later year's incidence rate to the original cohort rather than the remaining active count. Incidence is conditional on active exposure, so the cohort count must be projected before calculating later new disabilities.

Practical Checks for a Morbidity Profit Test

Before interpreting a duration-specific profit result, verify that the projection distinguishes the following items:

Incidence basis: Is the changed probability applied to active policies only, and is the shock temporary or permanent?
Benefit trigger: Is the benefit paid on new disability incidence, on each disabled policy, or under another stated schedule?
State counts: Do active and disabled counts reconcile after each transition? With no other decrements, their sum should remain equal to the original cohort.
Reserve state: Is the closing reserve weighted by the expected count in each destination state, using the correct future valuation assumptions?
Timing and sign: Are start-of-year premiums and expenses accumulated to year-end, and is the closing reserve subtracted from accumulated funds in the profit equation?
Precision: Are intermediate amounts retained unrounded, with money rounded only at the final reporting step?

A higher incidence assumption will often increase current disability benefits and increase the liability for future disabled benefits. At the same time, it can reduce future active exposure, premiums, expenses, and future incidence claims. The net impact on profit is therefore not determined by the direction of the morbidity change alone. It depends on benefit design, state transitions, premium eligibility, reserve assumptions, and the duration being examined.

Key takeaway: Trace a morbidity shock through the transition first, then through cash flows and state-specific reserves, and finally through the profit balance. Do not substitute a changed incidence rate for a changed claim count: the rate acts on the exposure that remains eligible at that duration.

Check Your Understanding

For each problem, state the relevant incidence exposure before calculating. Use unrounded values in intermediate steps.

A hypothetical group has 180 active policies. There is no recovery, mortality, or lapse. Incidence is 0.06 in year 1 and 0.05 in year 2. How many expected new disabilities occur in each year, and how many active policies remain at the start of year 3? (Answer: 10.8 in year 1; 8.46 in year 2; 160.74 active at the start of year 3.)
A block has 120 active policies at issue. The year 1 incidence probability rises from 0.03 to 0.08. A $4,000 lump sum is paid at the end of year 1 for each new disability. What is the increase in expected year 1 benefit outgo? (Answer: $120(0.08-0.03)($4,000)=$24,000$.)
At the start of a year, a hypothetical contract has 70 active and 20 disabled policies. Incidence among active policies is 0.10. A $500 benefit is paid at year-end for every policy disabled by then, including previously disabled policies. What is the expected year-end benefit outgo? (Answer: $(20+70(0.10))($500)=$13,500$.)
For an active policy, the opening reserve is -$200, the start-of-year premium is $500, the expense is $20, the effective annual interest rate is 4%, the expected year-end benefit is $30, and the expected closing reserve is $260. Calculate expected profit. (Answer: $(-200+500-20)(1.04)-30-260=$-10.80$.)
A two-year contract has 100 active policies at issue. Year 1 incidence is 0.10, year 2 incidence is 0.04, and disability is absorbing. How many new disabilities are expected in year 2? (Answer: $100(1-0.10)(0.04)=3.6$.)

What You'll Learn