Checking Stress Direction Against Product Economics

Why Stress Direction Needs a Product-Level Check

A stress result is not plausible merely because its sign matches a general rule such as “higher mortality is adverse” or “higher investment returns are favorable.” The effect depends on what the contract pays, when it pays, and which measure is being reported. Higher mortality can increase the cost of a death benefit but reduce the expected cost of a benefit payable only on survival. A higher lapse rate can remove future premiums and benefits while also triggering surrender payments.

The purpose of a directional check is not to replace a valuation. It is to test whether the result is consistent with the product’s cash-flow structure, the stress definition, and the chosen reporting measure. This reasoning is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A stress-direction check compares the sign of a stressed change in a stated measure with the changes in the product cash flows that drive it. Always identify the measure first: for example, expected claim present value, liability, asset surplus, or profit. A higher liability is generally adverse to an insurer, whereas a higher profit is generally favorable; the same cash-flow change can therefore be described with different signs under different measures.

For a clean comparison, change one assumption at a time and hold the others fixed unless the scenario explicitly combines stresses. State the valuation date, cash-flow timing, probability or rate assumptions, and whether the result is from the insurer’s or policyholder’s perspective. In more complete models, reserves, premiums, options, taxes, reinsurance, and interactions can change the magnitude or even the direction suggested by a simplified calculation.

A One-Period Mortality Direction Check

Suppose a policy has a fixed death benefit $B$, payable at the end of one year if death occurs during the year. Let $q$ be the one-year death probability and $i$ the effective annual interest rate. Assume no expenses, lapses, or other payments. The expected present value of the claim is $qBv$, where $v=1/(1+i)$. If the premium $P$ is fixed at time 0, the expected profit in this simplified model is $P-qBv$.

\text{EPV of claim}=qBv,\qquad \text{expected profit}=P-qBv

The direction follows directly by comparing two mortality assumptions. Increasing $q$ by $\Delta q>0$ increases the expected claim present value by $\Delta q Bv$, so it decreases expected profit by the same amount. For a benefit payable only if the policyholder survives, the one-year expected benefit present value is instead $(1-q)Cv$, where $C$ is the survival benefit. Increasing $q$ reduces that expected benefit present value. This derivation explains why mortality stress does not have a universal direction across products.

\Delta \text{EPV of death claims}=(q_{\text{stress}}-q_{\text{base}})Bv

\Delta \text{EPV of survival benefits} =(q_{\text{base}}-q_{\text{stress}})Cv

Worked Example: The Same Mortality Stress on Two Benefit Designs

Consider two hypothetical one-year contracts, valued at time 0 with an effective annual interest rate of $i=5\%$. Mortality is the only uncertain event; the one-year death probability is $q=0.01$ under the base assumption and $q=0.02$ under stress. The death-benefit contract collects a fixed premium of $12 at time 0 and pays $1,000 at the end of the year on death. The survival-benefit contract pays $1,000 at the end of the year only if the policyholder survives. There are no expenses, lapses, or other payments. Check the direction of each result.

Method. Calculate each expected present value under both mortality assumptions, then compare like with like. The discount factor is $v=1/1.05=0.95238095$.

Death-benefit contract, base assumption. The expected present value of claims is $0.01\times \$1{,}000\times 0.95238095=\$9.5238095$. Expected profit is the time-0 premium less that claim value: $\$12-\$9.5238095=\$2.4761905$.

Death-benefit contract, stressed assumption. The expected present value of claims is $0.02\times \$1{,}000\times 0.95238095=\$19.0476190$. Expected profit is $\$12-\$19.0476190=-\$7.0476190$. The profit change is $-\$7.0476190-\$2.4761905=-\$9.5238095$. Higher mortality increases the death-claim cost and reduces expected profit, as the cash-flow structure predicts.

Survival-benefit contract, base assumption. The survival probability is $1-0.01=0.99$. The expected present value of benefits is $0.99\times \$1{,}000\times 0.95238095=\$942.8571405$.

Survival-benefit contract, stressed assumption. The survival probability is $1-0.02=0.98$. The expected present value of benefits is $0.98\times \$1{,}000\times 0.95238095=\$933.3333310$. The benefit liability decreases by $\$933.3333310-\$942.8571405=-\$9.5238095$. Higher mortality reduces the expected cost of this survival-contingent benefit. Round only the final dollar results to cents: the death-contract profit falls by $9.52, while the survival-benefit present value falls by $9.52. The arithmetic checks: the mortality increase is $0.01$, and $0.01\times \$1{,}000\times v=\$9.5238095$ in either comparison.

Common mistake avoided. Treating higher mortality as adverse for both contracts would assign the same economic direction to opposite benefit triggers. Check whether payment is conditional on death or survival before interpreting the sign.

Lapse Stress: Compare Lapse and Continuance

A lapse stress changes the probability of leaving the contract, but the direction depends on the relative value of the two possible paths. At the lapse date, compare the insurer’s cash flow if the policy lapses with the value, at that same date, of continuing the policy. If lapse triggers a large surrender payment, increased lapses may be costly. If continuing entails a more costly guarantee and the surrender payment is smaller, increased lapses may reduce the insurer’s expected loss.

E[\text{profit at lapse date}] =l\,\Pi_L+(1-l)\,\Pi_C

Here $l$ is the lapse probability, $\Pi_L$ is profit conditional on lapse, and $\Pi_C$ is the present value of profit conditional on continuance, measured from the lapse date. Holding those conditional values fixed, an increase in lapse probability changes expected profit by $(l_{\text{stress}}-l_{\text{base}})(\Pi_L-\Pi_C)$. The expression makes the direction test explicit: it depends on which path has the greater profit.

Worked Example: A Higher Lapse Rate Reduces a Guaranteed Loss

At time 1, a hypothetical policy either lapses or continues. If it lapses, the insurer pays a surrender benefit of $80 immediately. If it continues, the insurer receives a premium of $20 immediately and pays a guaranteed benefit of $120 at time 2. Use an effective annual interest rate of $i=5\%$ to value the continuation cash flows at time 1. Assume no mortality, expenses, or other cash flows. The lapse probability is 10% at baseline and 30% under stress. Calculate the change in expected profit at time 1.

Method. First find the time-1 profit conditional on each path. The lapse-path profit is $\Pi_L=-\$80$. The continuation-path profit is the premium less the discounted future benefit: $\Pi_C=\$20-\$120/1.05=\$20-\$114.2857143=-\$94.2857143$.

Baseline expected profit. With lapse probability $0.10$, expected profit is $0.10(-\$80)+0.90(-\$94.2857143)$ $=-\$8-\$84.8571429=-\$92.8571429$.

Stressed expected profit. With lapse probability $0.30$, expected profit is $0.30(-\$80)+0.70(-\$94.2857143)$ $=-\$24-\$66=-\$90$. The stressed-minus-base change is $-\$90-(-\$92.8571429)=+\$2.8571429$. Round to cents: expected profit improves by $2.86. This is economically plausible because the loss on lapse is smaller than the present-value loss on continuance.

Common mistake and correction. A wrong calculation might treat the extra $0.20$ probability of lapse as an additional surrender loss and report $0.20(-\$80)=-\$16$. That calculation omits the continuation path being replaced. The correct change is $0.20(\Pi_L-\Pi_C)=0.20[-\$80-(-\$94.2857143)]=+\$2.8571429$. The check is to compare the two conditional outcomes, not to consider the lapse payment in isolation.

Expense and Investment Stresses: Trace the Net Cash Flow

A higher expense assumption ordinarily increases the insurer’s outgo, but the reported change depends on which expenses are included, when they are paid, and whether any related charges or allowances also change. Similarly, lower investment returns usually reduce the assets available to meet a fixed guarantee, but a different liability basis or a change in policyholder behavior could affect a reported reserve or profit measure. A direction check should trace the full cash-flow path rather than infer a sign from the name of the stress.

Name the measure and perspective.
State whether the result is a liability, expected profit, or surplus, and whose position it measures.

Write the affected cash flows.
Separate premiums, benefits, expenses, surrender payments, and accumulated assets by timing and condition.

Apply only the stated stress.
Keep other assumptions fixed, unless the scenario specifies joint changes.

Check the sign against the contract.
Ask whether the stress increases outgo, reduces income, changes the chance of a payment, or affects both sides.

Worked Example: Separating a Return Stress from an Expense Stress

A hypothetical one-year contract has $100 of assets available at time 0 to meet a guaranteed payment of $104 at time 1. The assets earn an effective annual return of 6% under the base assumption. A $2 expense is also paid at time 1. Define time-1 surplus as accumulated assets less the guarantee and the expense. Consider (a) a return stress reducing the annual return to 3%, (b) an expense stress increasing the expense to $4, and (c) both stresses together. Assume no mortality, lapses, premiums, or other cash flows.

Method. Accumulate the assets for one year, then subtract the time-1 payment and expense. This is a time-1 surplus comparison, so no further discounting is applied.

Base surplus. Assets accumulate to $\$100(1.06)=\$106$. Therefore, base surplus is $\$106-\$104-\$2=\$0$.

Return stress only. Stressed assets accumulate to $\$100(1.03)=\$103$. With the original $2 expense, surplus is $\$103-\$104-\$2=-\$3$. The return stress reduces surplus by $3.

Expense stress only. Keep the base accumulated assets of $106 and increase the expense to $4. Surplus is $\$106-\$104-\$4=-\$2$. The expense stress reduces surplus by $2.

Both stresses. Use stressed assets of $103 and the stressed expense of $4: $\$103-\$104-\$4=-\$5$. Relative to the base surplus of zero, the combined change is $-\$5$, equal to the sum of the two changes in this simple additive setup. Round to cents: the three stressed surpluses are $-\$3.00$, $-\$2.00$, and $-\$5.00$. The check is direct: the lower return removes $3 of assets and the higher expense removes a further $2.

Common mistake avoided. Reporting only an expense increase or only an asset accumulation change would miss part of the combined scenario. Calculate each cash-flow component at its correct time, then combine them using the stated surplus definition.

When a Plausible Direction Is Not a Universal Rule

A direction check is a diagnostic, not a theorem about every product. A mortality increase can raise death claims while reducing survival benefits; when both types exist, compare their amounts, timing, and probabilities. A lapse increase can remove future premiums as well as future benefits, and may trigger a surrender value. An expense increase can be partly offset if a contractual charge rises with expenses. Lower investment returns can increase a guarantee’s cost, while also affecting the discounting or projected value used in a particular measure.

For a reserve or profit result, keep the measure’s sign convention visible. For example, an increase in the present value of benefits is an increase in a liability; it is not automatically a decrease of the same amount in reported profit if the calculation also changes reserves, premiums, or other cash flows. If a result appears counterintuitive, verify the event being stressed, the conditional payment, timing, discount rate, comparison basis, and perspective before concluding that the model is wrong.

Key takeaway: The economically plausible sign comes from the contract’s complete stressed cash flows and the measure being reported. State both before interpreting the direction; do not rely on a generic rule about mortality, lapse, expenses, or returns.

Check Your Understanding

For each problem, identify the relevant cash-flow comparison before checking the final answer.

A one-year death benefit of $50,000 is paid at year-end. The death probability rises from 0.012 to 0.018, with an effective annual interest rate of 4%. By how much does the expected present value of claims change? (Answer: increases by $288.46.)
A one-year survival benefit of $3,000 is payable at year-end only if the insured survives. The death probability rises from 0.08 to 0.11, and $i=5\%$. What is the change in the expected present value of the benefit? (Answer: decreases by $85.71.)
At a lapse date, lapse produces an immediate $50 payment by the insurer. Continuance produces a $10 premium immediately and a $70 benefit one year later. At $i=5\%$, the lapse probability rises from 0.15 to 0.25. What is the change in expected profit at the lapse date? Assume no other cash flows or decrements. (Answer: increases by $0.67.)
A contract has $250 of assets at time 0 and a $255 guaranteed payment at year-end. The annual return falls from 4% to 2%, and a fixed year-end expense is $3. What is the change in year-end surplus due to the return stress? (Answer: decreases by $5.00.)
A hypothetical block has 100 identical policies, each with a $40 expense payable one year from now. A 10% expense-level stress increases each policy’s expense by 10%; use $i=5\%$. By how much does the present value of total expenses increase? (Answer: increases by $380.95.)

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