Morbidity Experience and Participating Surplus

Introduction

For a participating disability or long-term care contract, the number of claims is only part of the morbidity experience. The amount paid also depends on how long each claim remains eligible for benefits. More claims than expected increase claim costs, while faster recovery and shorter payable durations can reduce them. The net effect depends on both.

This tutorial measures the morbidity contribution to surplus by comparing claim costs on an actual basis with a specified expected basis. The calculations isolate claim incidence and claim duration; they do not calculate the total surplus available for distribution or determine a policyholder dividend. The treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For a specified group of contracts and observation period, expected claim cost is the claim cost on the morbidity assumptions being used as the comparison basis. Actual claim cost is the cost produced by the observed claims on a consistent benefit basis. Define the morbidity contribution to surplus as expected claim cost minus actual claim cost. A positive amount is favorable morbidity experience; a negative amount is adverse morbidity experience.

The comparison must use the same benefit definition and measurement period on both sides. In the examples below, each claim episode is followed to its end, and costs are undiscounted benefit payments. If claims are still open at the end of an observation period, comparing only cash paid to date with the cost of completed claims would not be consistent; the treatment of remaining claim costs would also have to be made consistent. Expenses, investment experience, and changes in reserves are held outside this morbidity comparison.

Expected Claim Cost and Claim-Weighted Duration

Suppose a contract pays a fixed amount $b$ for each payable month of an eligible claim. For a group with $E$ exposed policy-years, let $q$ be the assumed probability of a claim per policy-year, so the expected claim count is $E q$. If the assumption for payable duration is $D_0$ months per claim, expected undiscounted claim cost is

$$ C_0=E q b D_0. $$

This formula uses a duration assumption that is appropriate to the claims being counted. For example, if the contract has an elimination period, $D_0$ must measure payable months if the formula is estimating benefit payments. It must not instead be a measure of total time disabled unless the elimination period is accounted for when converting duration to payable months.

A careful treatment is also needed when claim count and duration are associated. Let $N$ be the number of claims and $D_j$ the payable duration of claim $j$. Total claim-months are $\sum_{j=1}^{N}D_j$. When $E[N]>0$, define the claim-weighted expected duration by

D_c=\frac{E\left[\sum_{j=1}^{N}D_j\right]}{E[N]}.

This definition gives the identity $E[\sum_{j=1}^{N}D_j]=E[N]D_c$ by construction. If claim durations are independent of the claim count and have a common mean $E[D]$, then $D_c=E[D]$. More generally, that equality requires a condition controlling the relationship between count and duration. Merely saying that claims share the same duration distribution does not justify multiplying the expected count by a duration mean if the count and durations are associated.

Worked Example: Checking the Expected-Cost Formula When Count and Duration Are Associated

A hypothetical small block has either one or three claims, each outcome having probability $0.5$. If there is one claim, its payable duration is 2 months. If there are three claims, each has a payable duration of 4 months. The benefit is $100 per payable month. Calculate expected claim cost, and identify the error in multiplying the expected claim count by the unweighted average of the two possible durations.

Method. Find expected total claim-months directly by conditioning on the two possible outcomes. This avoids assuming that count and duration are independent.

With one claim, total claim-months are $1(2)=2$. With three claims, total claim-months are $3(4)=12$. Therefore,

E\left[\sum_{j=1}^{N}D_j\right] =0.5(2)+0.5(12)=7\text{ claim-months}.

Expected claim cost is $7(100)=\$700$. The expected number of claims is $0.5(1)+0.5(3)=2$. Thus, the claim-weighted expected duration is $7/2=3.5$ months, and $2(3.5)(100)=\$700$, as required.

The unweighted average of the two possible per-claim durations is $(2+4)/2=3$ months. Multiplying this by the expected count would give $2(3)(100)=\$600$, which is wrong: the three-claim outcome contributes more claims, and those claims have the longer duration. The direct calculation checks the result and shows why an unweighted average is not the appropriate duration mean here.

Separating Incidence and Duration Effects

A useful way to interpret an observed difference is to separate the change in claim count from the change in duration. Suppose the expected basis has claim count $N_0$, benefit $b$, and duration $D_0$, giving expected cost $C_0=N_0bD_0$. For actual claims, let the observed count be $N$, and let $T=\sum_{j=1}^{N}D_j$ be the total observed payable claim-months. Actual cost is $C= bT$.

The cost difference can be rearranged exactly:

C-C_0 =b(T-N_0D_0) =b\left[(N-N_0)D_0+(T-ND_0)\right].

The first term, $b(N-N_0)D_0$, is the incidence effect valued at the expected duration. The second, $b(T-ND_0)$, is the duration effect for the actual number of claims. This is an accounting decomposition, not a claim that count and duration are statistically independent. The morbidity contribution to surplus has the opposite sign: it is $C_0-C$.

Worked Example: More Claims but Shorter Durations

A hypothetical disability block has 1,000 exposed policy-years. The assumed claim probability is 4% per policy-year, the monthly benefit is $500, and the assumed average payable duration is 6 months. During the observation, 50 claims are completed with a total of 260 payable claim-months. Calculate expected and actual claim costs, the two effects on cost, and the morbidity contribution to surplus.

Method. Calculate the expected claim count and cost, then compare the actual total claim-months with the cost benchmark. Use the exact incidence-duration decomposition.

Expected claim count is $1{,}000(0.04)=40$. Expected claim cost is

C_0=40(6)(\$500)=\$120{,}000.

Actual mean payable duration is $260/50=5.2$ months. Actual claim cost is

C=260(\$500)=\$130{,}000.

The incidence effect on cost is $(50-40)(6)(\$500)=\$30{,}000$, an increase in cost because there were 10 more claims than expected. The duration effect is $(260-50(6))(\$500)=(-40)(\$500)=-\$20{,}000$. This is a decrease in cost because the actual claims had 40 fewer payable months in total than 50 claims of 6 months each.

The net increase in cost is $\$30{,}000-\$20{,}000=\$10{,}000$. The morbidity contribution to surplus is therefore $\$120{,}000-\$130{,}000=-\$10{,}000$, which is adverse. As a check, the decomposition totals the direct difference between the two costs.

Recovery Experience and Payable Duration

For disability and long-term care claims, recovery experience affects how long benefits are payable. Earlier recovery, when recovery ends benefit eligibility, generally shortens payable duration and reduces claim cost. Slower recovery generally lengthens payable duration and increases claim cost. The relevant comparison is the observed duration of eligible benefit payments against the duration assumption on the same basis.

Recovery should not be inferred from a raw count of claims that happen to close during a period without regard to how long claims have been observed. Claims that remain open at the end of the period have not necessarily ended, and their eventual durations are not yet known. For a completed-claim comparison, use completed episodes; for an in-force period comparison, use a consistent treatment of ongoing claims and their future costs.

Worked Example: Slower Recovery Increases Claim Cost

A hypothetical long-term care block has 30 completed eligible claims. The monthly benefit is $300, and the assumed mean payable duration is 10 months. The total payable duration observed for the 30 claims is 372 months. Calculate the cost difference against the duration assumption and interpret it as a morbidity contribution to surplus.

Method. Hold claim count fixed at the observed 30 and compare actual claim-months with the claim-months expected for 30 claims at the assumed duration. This isolates the duration, or recovery, experience.

At the assumed duration, the benchmark is $30(10)=300$ payable claim-months, for an assumed-basis cost of $300(\$300)=\$90{,}000$. Actual mean payable duration is $372/30=12.4$ months. Actual cost is $372(\$300)=\$111{,}600$.

The increase in cost is $(372-300)(\$300)=72(\$300)=\$21{,}600$. Equivalently, the duration effect is $30(12.4-10)(\$300)=\$21{,}600$. Actual claims lasted longer than assumed, so recovery or termination of eligibility was slower on this measure. The morbidity contribution to surplus is $\$90{,}000-\$111{,}600=-\$21{,}600$, adverse. The direct cost difference and the duration calculation agree.

Elimination Periods and Benefit Measurement

An elimination period creates a common measurement trap. Total time disabled and time for which benefits are payable are not interchangeable. If a contract pays nothing during an initial three-month elimination period, a claim lasting 10 months may produce only 7 payable months, subject to the contract's precise payment rules. When the claim-cost calculation is based on payable months, both the assumed and actual durations must already reflect the elimination period.

Worked Example: Applying an Elimination Period Once

A hypothetical disability contract has a three-month elimination period and pays $400 per month after that period, with no maximum benefit duration. There are 500 exposed policy-years. The assumed claim probability is 6%, and the assumed mean duration is 8 payable months per claim. Actual experience is 36 eligible claims whose total payable duration is 270 months. Calculate the morbidity contribution to surplus and identify the consequence of subtracting the elimination period twice.

Method. Use the specified payable-month durations directly. The 8 assumed months and 270 actual months are already measured after the elimination period.

Expected claim count is $500(0.06)=30$, so expected claim cost is $30(8)(\$400)=\$96{,}000$. Actual claim cost is $270(\$400)=\$108{,}000$. Actual average payable duration is $270/36=7.5$ months.

The incidence effect on cost is $(36-30)(8)(\$400)=\$19{,}200$. The duration effect is $(270-36(8))(\$400)=(-18)(\$400)=-\$7{,}200$. Thus actual cost is $\$19{,}200-\$7{,}200=\$12{,}000$ above the expected basis, and the morbidity contribution to surplus is $-\$12{,}000$, adverse. The direct check is $\$108{,}000-\$96{,}000=\$12{,}000$.

A wrong calculation would subtract the three-month elimination period again from the 7.5 actual payable months, using $36(7.5-3)(\$400)=\$64{,}800$. That understates actual cost because the stated 7.5 months already excludes the unpaid elimination period. Subtracting it again would be appropriate only if the duration data were total time disabled rather than payable duration.

Common Mistakes and Interpretation

Reversing the sign. If actual claim cost exceeds the expected basis, the morbidity contribution to surplus is negative and the experience is adverse.
Using total duration when payable duration is required. Apply elimination periods and other benefit conditions consistently; do not subtract them twice.
Multiplying incompatible averages. Expected claim count times a duration mean is valid when that mean is claim-weighted for the same population of claims. If count and duration are associated, calculate expected aggregate claim-months directly or use the claim-weighted definition.
Calling a lower claim count automatically favorable overall. Fewer claims can be offset by longer durations among those that occur. Compare total claim cost as well as its incidence and duration components.
Treating morbidity surplus as a dividend. This calculation isolates one experience source. It does not by itself determine distributable surplus or the amount allocated to policyholders.

Key takeaway: Measure incidence using claims per consistent unit of exposure, and measure duration on the same payable-benefit basis used for the cost comparison. The exact incidence-duration decomposition helps explain a morbidity gain or loss without requiring claim count and duration to be independent.

Check Your Understanding

Use expected claim cost minus actual claim cost for the morbidity contribution to surplus. Treat all durations stated as payable durations.

A hypothetical block has 800 exposed policy-years, an assumed claim probability of 2.5%, a $250 monthly benefit, and an assumed payable duration of 4 months. Calculate expected claim count and expected claim cost. (Answer: 20 expected claims; $20,000 expected claim cost.)
There are 24 completed claims, each paying $500 per month. Their total payable duration is 240 months, against an assumed duration of 9 months per claim. Calculate the duration effect on cost and the morbidity contribution to surplus, holding count fixed. (Answer: $12,000 adverse duration effect on cost; -$12,000 contribution to surplus.)
A block has 1,200 exposed policy-years, an assumed claim probability of 3%, an assumed payable duration of 7 months, and a $350 monthly benefit. Actual experience is 40 claims with 260 total payable months. Calculate expected cost, actual cost, and the morbidity contribution to surplus. (Answer: $88,200 expected; $91,000 actual; -$2,800 contribution to surplus.)
A claim count is 1 or 3, each with probability 0.5. In the one-claim outcome, duration is 2 months; in the three-claim outcome, each claim lasts 4 months. With a $100 monthly benefit, calculate expected claim cost and the claim-weighted expected duration. (Answer: $700 expected claim cost; 3.5 months claim-weighted expected duration.)

What You'll Learn