Morbidity Stress in Products with Linked Savings

Why Morbidity Stress Needs a Product-Level View

A morbidity stress can affect more than the expected amount of claims. In a product with linked savings, claim outgo competes with the assets supporting policy values, changes projected profit, and may increase the reserve and capital requirement. The analysis is clearest when the savings account and the morbidity benefit are tracked separately before their results are brought together.

This tutorial uses deliberately simplified one-year projections. Each contract has a linked savings account and a separate morbidity benefit. A qualifying morbidity claim is paid at year-end; it does not cancel or reduce the linked account. Morbidity incidence is the probability that a covered person makes a claim during the year. There are no deaths, lapses, recoveries, expenses beyond those expressly stated, or other benefit payments in the model. In practice, a contract whose benefit continues while a person remains disabled requires a multi-state model that reflects both incidence and duration.

Definition: A morbidity stress is a specified change to morbidity assumptions, such as increasing the probability of a claim, used to measure its effect on projected cash flows and financial measures. In the one-year model here, for a cohort of $N$ policies with claim probability $q$ and claim benefit $B$, expected claim outgo is $NqB$. If it is paid at the end of the year and discounted at effective annual interest rate $i$, its present value is $NqBv$, where $v=1/(1+i)$.

Set the Boundary Between Savings and Morbidity

In each projection, keep a clear ledger. The linked savings balance is the policy value accumulated under the specified account-crediting assumption. The morbidity benefit is an additional claim payment. Assets supporting the contracts receive any stated contributions and investment return, and then pay the morbidity claims at the stated time. The ending assets can be compared with the ending linked-account liability to calculate a simplified projected profit.

The incidence assumption must also be interpreted precisely. Here, $q$ is the probability of a claim during one year for a person in the covered, claim-free group at the start of that year. It is not the probability of being disabled at the end of the year, and it is not a rate of claim payments over multiple years. The baseline and stressed projections use the same benefit, investment return, payment timing, and cohort unless the example says otherwise.

Worked Example: One-Year Reserve Under a Morbidity Stress

A hypothetical product has 80 covered people. Each person may receive a separate $8,000 morbidity benefit if a claim occurs during the year. The baseline annual claim probability is 0.015, and the stressed probability is 0.025. Claims are paid at year-end. The effective annual interest rate is 4%. Find baseline and stressed expected claims and one-year reserves, and their increases.

Method and timing. Calculate expected claim count as $Nq$, multiply by the benefit, and discount the year-end amount by $1.04$. This assumes a single end-of-year payment, not a payment at the time of claim.

Baseline. Expected claims are $80(0.015)=1.2$. Expected claim outgo is $1.2(\$8{,}000)=\$9{,}600$. Thus the one-year reserve is $\$9{,}600/1.04=\$9{,}230.7692\ldots$, or $9,230.77.

Stress. Expected claims are $80(0.025)=2.0$. Expected claim outgo is $2.0(\$8{,}000)=\$16{,}000$. The stressed reserve is $\$16{,}000/1.04=\$15{,}384.6154\ldots$, or $15,384.62.

Increases and check. Expected claim outgo increases by $\$16{,}000-\$9{,}600=\$6{,}400$. The reserve increase, using unrounded values, is $\$6{,}400/1.04=\$6{,}153.8462\ldots$, or $6,153.85. The stressed reserve is larger, as it must be when only the claim probability increases. The common mistake avoided is discounting the claim count but not the benefit, or treating the end-year benefit as an immediate payment.

Justifying the One-Year Reserve and Profit Bridge

The reserve formula follows from the expected present value of the payment. Let $I$ equal 1 if a covered person claims during the year and 0 otherwise. Then the end-year payment is $BI$, and $E[I]=q$. Its expected present value is therefore $E[vBI]=vB E[I]=vqB$. For $N$ people with the same assumption, add the individual expected present values to obtain $NvqB$. This calculation needs the stated one-year incidence and payment timing; it is not a duration model for continuing disability benefits.

V_0=NqBv=\frac{NqB}{1+i}

A simplified profit bridge uses the same cash-flow discipline. Suppose opening assets are $A_0$, net contributions after any stated start-of-year expenses are $C$, and these amounts earn effective annual rate $i$. If expected claims $NqB$ are paid at year-end, projected ending assets are $(A_0+C)(1+i)-NqB$. If the ending linked-account liability is $L_1$, projected profit under this simplified definition is ending assets minus $L_1$. A change in morbidity affects profit through claim outgo, while the linked account itself changes only according to its stated account rules.

Worked Example: Claim Outgo, Asset Share, and Projected Profit

A hypothetical cohort has 100 policies. Each linked account begins at $1,000 and earns 4% for the year, giving an ending account balance of $1,040 per policy. The separate morbidity benefit is $5,000 per claim. At the start of the year, the cohort receives $6,000 of charges and incurs $2,000 of expenses. Treat opening assets, charges, and expenses as invested from the start of the year; claims are paid at year-end. Compare incidence of 0.02 and 0.03. There are no other cash flows.

Method. Accumulate opening assets plus start-of-year net cash flow, subtract expected year-end claims, and compare the result with the ending linked-account liability. The per-policy asset share is the projected ending assets divided by 100; here it is a cohort average based on expected claims.

Assets before claims. Opening assets are $100(\$1{,}000)=\$100{,}000$. Net charges after expenses are $\$6{,}000-\$2{,}000=\$4{,}000$. The amount invested is $\$104{,}000$, and after interest it is $\$104{,}000(1.04)=\$108{,}160$.

Baseline incidence. Expected claims are $100(0.02)=2$, so expected claim outgo is $2(\$5{,}000)=\$10{,}000$. Ending assets are $\$108{,}160-\$10{,}000=\$98{,}160$, or $\$981.60$ per policy. The ending linked-account liability is $100(\$1{,}040)=\$104{,}000$. Projected profit is $\$98{,}160-\$104{,}000=-\$5{,}840$.

Stressed incidence. Expected claims are $100(0.03)=3$, giving claim outgo of $3(\$5{,}000)=\$15{,}000$. Ending assets are $\$108{,}160-\$15{,}000=\$93{,}160$, or $\$931.60$ per policy. With the same $\$104{,}000$ account liability, projected profit is $\$93{,}160-\$104{,}000=-\$10{,}840$.

Check. The stress increases claims by $\$5{,}000$, and profit falls by exactly $\$5{,}000$. The per-policy asset share falls by $\$50$, the additional expected claim cost per policy. The common mistake avoided is applying the morbidity stress to the linked savings balance as though the separate benefit were deducted from each account. In this contract definition, claims are paid from supporting assets, while account balances continue to grow as specified.

From Morbidity Reserve to Capital Need

A reserve is a liability measure under a specified valuation basis. A capital requirement is a separate rule that may use the reserve as an input. To calculate a change in capital, first calculate the reserve on both bases; then apply the stated capital rule to unrounded reserve values. Do not infer a regulatory or company capital formula when none has been supplied.

Worked Example: Mixed Risks and a Stated Capital Rule

A hypothetical cohort has 30 standard-risk members and 20 elevated-risk members. Their baseline annual claim probabilities are 0.02 and 0.04, respectively; under stress they are 0.03 and 0.06. Each claim pays $2,000 at year-end. The effective annual interest rate is 4%. Required capital is defined for this example as 125% of the one-year reserve. Calculate the expected claim increase, reserve increase, and capital increase. Also determine whether $5,000 of available capital meets the stressed requirement.

Method. Calculate expected claims separately by risk group, discount total expected outgo, and apply the capital factor to the reserves. This avoids replacing a mixed cohort with an inappropriate single incidence change.

Baseline expected claims. The expected claim count is $30(0.02)+20(0.04)=0.6+0.8=1.4$. Expected outgo is $1.4(\$2{,}000)=\$2{,}800$. The reserve is $\$2{,}800/1.04=\$2{,}692.3077\ldots$, or $2,692.31.

Stressed expected claims. The expected claim count is $30(0.03)+20(0.06)=0.9+1.2=2.1$. Expected outgo is $2.1(\$2{,}000)=\$4{,}200$. The stressed reserve is $\$4{,}200/1.04=\$4{,}038.4615\ldots$, or $4,038.46.

Reserve and capital increases. Expected outgo increases by $\$4{,}200-\$2{,}800=\$1{,}400$. The reserve increase is $\$1{,}400/1.04=\$1{,}346.1538\ldots$, or $1,346.15. Baseline required capital is $1.25(\$2{,}800/1.04)=\$3{,}365.3846\ldots$, or $3,365.38. Stressed required capital is $1.25(\$4{,}200/1.04)=\$5{,}048.0769\ldots$, or $5,048.08. The capital increase, calculated from unrounded values, is $1.25(\$1{,}400/1.04)=\$1{,}682.6923\ldots$, or $1,682.69.

Available capital and check. The $5,000 available amount is below the stressed requirement by $\$5{,}048.0769\ldots-\$5{,}000=\$48.0769\ldots$, or $48.08. The common mistake avoided is treating all 50 people as if their incidence increased by 0.01: that would produce an incorrect claim increase of $50(0.01)(\$2{,}000)=\$1{,}000$. The actual risk-group increases are $30(0.01)+20(0.02)=0.7$ expected claims, or $\$1{,}400$.

Practical Checks and Common Traps

Before combining morbidity experience with linked-product results, verify the model boundaries. A correct calculation for the wrong payment timing or wrong benefit definition can still give a misleading answer.

Incidence versus prevalence: an annual claim probability is not automatically the proportion disabled at a point in time. Continuing claims require assumptions about duration, recovery, and any recurring benefit.
Per policy versus cohort: multiply expected claim cost per person by the number and mix of covered people. For heterogeneous risks, calculate by group before aggregating.
Timing and discounting: a year-end claim is discounted by $v$; a start-of-year claim is not. Use the specified timing consistently in both assets and reserves.
Linked account versus separate benefit: identify whether a claim terminates, reduces, or leaves the savings account unchanged. Do not assume one treatment for every contract.
Rounding: retain unrounded values through reserve and capital calculations, then round the final dollar answers to cents. Subtracting separately rounded amounts can create a one-cent discrepancy.

Key takeaway: A sound morbidity-stress analysis states the incidence basis, benefit amount, payment timing, and savings-account treatment first. It then traces expected claims through projected assets and profit, and applies only the stated reserve and capital rules.

Check Your Understanding

For each hypothetical case, use the stated assumptions and retain unrounded values until the final dollar amount.

A cohort of 50 has annual claim probability increasing from 0.02 to 0.04. Each year-end claim pays $3,000, and the effective annual rate is 5%. Find the increase in expected claims and the increase in the one-year reserve. (Answer: expected claim outgo increase $3,000; reserve increase $2,857.14)
For 25 covered people, annual incidence increases from 0.04 to 0.06. The year-end benefit is $6,000 and the effective annual rate is 3%. Find the reserve increase. (Answer: $2,912.62)
A group has 20 people whose incidence rises from 0.01 to 0.02 and 30 people whose incidence rises from 0.03 to 0.05. Each year-end claim pays $4,000, and the effective annual rate is 4%. Find the increases in expected claim outgo and reserve. (Answer: expected claim outgo increase $3,200; reserve increase $3,076.92)
A cohort of 40 has claim probability increasing from 0.025 to 0.04. Each claim pays $2,500 at year-end; the effective annual rate is 5%, and required capital is 150% of reserve. Find the increase in required capital. (Answer: $2,142.86)
There are 50 policies with opening assets and account balances of $800 per policy. At the start of the year, charges of $30 and expenses of $10 per policy are added to and deducted from supporting assets, respectively. Assets and linked accounts earn 3%; the separate $2,000 morbidity claim is paid at year-end. Incidence is 0.01 at baseline and 0.02 under stress. Find projected profit under each incidence, where profit is ending assets less ending linked-account liability. (Answer: baseline profit $30; stressed profit −$970)

What You'll Learn