Why a Stress Result Can Look Backward
A stress result can be mathematically correct and still appear inconsistent with the product’s economics. A mortality stress, for example, does not necessarily increase every benefit: it may change how many policyholders reach a state that triggers a different, more costly benefit. Likewise, an increase in a future claim amount does not by itself determine the change in current solvency surplus. Discounting, asset movements, and reserve changes also matter.
The reliable approach is to trace the reported measure back to its components. Confirm what event triggers each payment, which assumptions apply in each period, how transitions change the exposed population, and whether the output is a cash-flow measure, a reserve, a profit, or a surplus. The methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
That last relationship is a sign convention, not a claim that a higher reserve is always bad in every business interpretation. A reserve may rise because the expected cost of benefits has increased, while a reported profit or capital measure may use a different sign or include other components. Always identify the reported quantity before interpreting its direction.
A Diagnostic Sequence
When an output moves in an unexpected direction, do not immediately reverse the stress or alter the model. First break the result into stages. Each stage tests a different source of error or genuine nonlinear effect.
Write down whether the reported result is a claim cost, asset value, reserve, profit, loss, or surplus. State explicitly whether an increase is favorable or adverse for that measure.
For each payment, specify the event or state that makes it payable. A benefit payable on entering disability is not payable merely because a policy remains in force or dies.
Confirm that transition probabilities are mutually consistent and that each benefit is multiplied by the probability of its own trigger. Do not apply a benefit to all policies if only a subset reaches the relevant event.
Apply each annual assumption for the number of years it actually operates. Check whether inflation affects the benefit amount, the expense, or both, and whether the changed discount rate applies at the valuation date.
Compare stressed and baseline components separately. For surplus, subtract the reserve change from the asset change; for profit, use the stated reserve-release convention consistently.
Why Surplus and Profit Signs Differ
The surplus relationship follows directly from its definition. If the stress changes assets from \(A\) to \(A+\Delta A\) and the reserve from \(R\) to \(R+\Delta R\), then the stressed surplus is \(A+\Delta A-(R+\Delta R)\). Subtracting the baseline surplus \(A-R\) gives the change.
A one-period profit calculation often uses the opposite-looking reserve movement because a release of reserve adds to profit. Under a simple convention, with premiums and investment income as inflows, and claims and expenses as outflows, profit is
The reserve term is \(R_{\text{start}}-R_{\text{end}}\), not \(R_{\text{end}}-R_{\text{start}}\). This follows because the profit calculation begins with the resources previously held for the liability and ends by setting aside the closing reserve. An increase in the closing reserve therefore reduces the period’s profit, all else equal. This simplified equation assumes the listed items capture the period’s relevant cash flows and that the reserve basis is consistent at the start and end dates.
Worked Example: A Mortality Stress Reduces Total Claims
A hypothetical portfolio has 1,000 active policies at the start of a year. During the year, each policy makes exactly one of three mutually exclusive transitions: to disability, to death, or to remain active. Under the baseline, the probabilities are 0.12, 0.08, and 0.80, respectively. Under a stress, they are 0.05, 0.15, and 0.80. A payment of $10,000 is made only when a policy transitions to disability, and a payment of $2,000 is made only on death. There are no other payments. Compare expected claims under the two sets of probabilities.
Method. Calculate the expected number of each benefit-triggering transition, then multiply by its own payment. The transition probabilities in each scenario sum to 1.00, so each scenario is internally consistent.
Baseline. Expected disability transitions are \(1{,}000(0.12)=120\), and expected deaths are \(1{,}000(0.08)=80\). Expected claims are
Stress. Expected disability transitions are \(1{,}000(0.05)=50\), and expected deaths are \(1{,}000(0.15)=150\). Expected claims are
The mortality probability rises by 0.07, but expected claims fall by \(\$1{,}360{,}000-\$800{,}000=\$560{,}000\). In this specified stress, probability shifts away from disability and toward death, and disability has the larger payment. This result does not show that mortality generally reduces claim costs; it reflects the particular transition changes and benefit triggers given here.
Check. Baseline transition probabilities sum to \(0.12+0.08+0.80=1.00\), and stressed probabilities sum to \(0.05+0.15+0.80=1.00\). The expected transition counts also sum to 1,000 in each scenario. The common mistake is to multiply the disability benefit by all policies who do not die, or to assume that a mortality increase must increase every claim measure. Only the specified disability transition triggers the disability payment.
Timing and Discounting Must Match the Cash Flow
A stress may affect a payment’s amount and its present value in different ways. Inflation changes a future nominal amount; a discount rate converts that future amount to a value at an earlier date. A trigger probability determines whether the payment is expected at all. Each factor must be applied once, for the correct duration.
For a single payment \(B_n\) at the end of year \(n\), with a fixed annual effective discount rate \(i\), its present value is \(B_n/(1+i)^n\). If an annual inflation assumption applies to the payment for all \(n\) years, the inflated amount is \(B_0(1+g)^n\), where \(g\) is the annual inflation rate. For a payment made only if an event occurs with probability \(p\), expected present value is \(pB_n/(1+i)^n\), provided the payment amount and discounting are conditional on that event as specified.
Worked Example: Inflation and a Discount-Rate Shock
A hypothetical contract pays $1,000 at the end of year 2 if a specified event occurs. The probability of that event is 0.25 in both scenarios. Under the baseline, the benefit is not inflated and the annual effective discount rate is 5%. Under stress, benefit inflation is 10% per year for both years and the annual effective discount rate is 4%. Assume no other payments or changes in the event probability. Find the baseline and stressed expected present values.
Method. First determine the payment amount at year 2, then multiply by the trigger probability, and finally discount two years.
Baseline. The year-2 benefit is $1,000, so expected present value is
Stress. The benefit is inflated for two years: \(\$1{,}000(1.10)^2=\$1{,}210\). The expected payment is \(0.25(\$1{,}210)=\$302.50\). Discounting at 4% for two years gives
The stressed expected present value is higher by \(\$279.68-\$226.76=\$52.92\). The trigger probability remains 0.25; it is not appropriate to value the payment as though all contracts receive it.
Check. The stressed nominal benefit is greater than $1,000, and the stressed discount rate is lower than the baseline rate, so both changes increase the present value. A common timing error is to apply inflation for only one year, using $1,100 instead of $1,210. That would give \(0.25(\$1{,}100)/1.0816=\$254.25\), understating the stressed value by $25.43. Values are rounded to the nearest cent only at the final step.
Reserve Movements Can Reverse the Apparent Story
A stress can increase claims and still have a reported effect that differs from the claims change alone because reserve movements and other items also enter the result. To diagnose the output, calculate the baseline and stressed profit independently, then reconcile the difference component by component. This avoids relying on an informal statement such as “higher claims mean lower profit” when the metric includes more than claims.
Worked Example: Finding the Reserve Sign in a Profit Result
For one hypothetical policy over a year, use the profit convention defined above. Baseline values are a premium of $150, investment income of $24, claims of $110, expenses of $18, a starting reserve of $700, and an ending reserve of $730. Under stress, the premium and starting reserve are unchanged; investment income is $20, claims are $125, expenses are $20, and the ending reserve is $760. Calculate each profit and the stress change.
Method. Substitute each scenario into the complete profit equation, including the reserve release \(R_{\text{start}}-R_{\text{end}}\).
Baseline. The reserve contribution is \(\$700-\$730=-\$30\). Therefore,
Stress. The reserve contribution is \(\$700-\$760=-\$60\). Therefore,
The profit change is \(-\$35-\$16=-\$51\). The component reconciliation gives the same answer: investment income falls by $4, claims rise by $15, expenses rise by $2, and the reserve contribution falls from \(-\$30\) to \(-\$60\), a further decrease of $30. Thus the total change is \(-\$4-\$15-\$2-\$30=-\$51\).
Check. The ending reserve is larger under stress, so \(R_{\text{start}}-R_{\text{end}}\) is more negative, not more positive. The common mistake is to add the reserve increase of $30 to stressed profit. Under this convention it reduces profit by $30 relative to the baseline reserve contribution.
Common Diagnostic Traps
The calculations above illustrate checks that are useful whenever a stress output is surprising. Keep the following distinctions explicit:
- State versus transition: A payment triggered by entering a state is not automatically payable to everyone occupying that state at a later date, or to everyone who avoids another transition.
- Probability versus severity: Expected cost depends on both how often a payment is triggered and its amount. A higher transition probability can coincide with a lower total expected cost if probability moves away from a more expensive trigger.
- Amount versus present value: A higher nominal future benefit may have a different present-value effect when the discount rate or payment timing also changes.
- Reserve level versus reserve movement: A reserve is a liability at a date; profit includes a movement between dates under a stated convention. Do not treat the closing reserve as a profit inflow.
- Stress direction versus output direction: “Adverse” describes the assumption change, not necessarily the direction of every output. Explain the output using its own definition and components.
A useful final check is to calculate totals in two ways when possible: once from the scenario’s full components, and once as a baseline result plus the individual stress changes. If the two calculations do not agree, revisit the trigger, timing, units, probability totals, discount period, or sign convention before interpreting the result.
Check Your Understanding
For each problem, use the stated triggers and sign conventions rather than inferring them from the stress label.
- A hypothetical cohort of 500 active policies has baseline one-year transition probabilities of 0.06 to disability, 0.04 to death, 0.10 to lapse, and 0.80 to remain active. Under stress these probabilities are 0.04, 0.08, 0.08, and 0.80. Disability pays $3,000 on transition and death pays $1,000 on transition; lapse and remaining active pay nothing. Find baseline and stressed expected claims. (Answer: baseline $110,000; stressed $100,000, a decrease of $10,000.)
- A hypothetical benefit of $1,000 is paid at the end of year 2 if an event occurs with probability 0.20. Under stress, benefit inflation is 5% per year for two years, the event probability is unchanged, and the discount rate is 5% effective annually. What is the stressed expected present value? (Answer: \(0.20(1000)(1.05)^2/(1.05)^2=\)$200.00.)
- Using \(\text{Profit}=P+I-C-E+R_{\text{start}}-R_{\text{end}}\), calculate profit if premium is $150, investment income $20, claims $125, expenses $20, starting reserve $700, and ending reserve $760. (Answer: \(-\$35\).)
- Baseline assets are $1,250 and the reserve is $1,100. Under stress, assets are $1,220 and the reserve is $1,085. Calculate the change in surplus and state its direction. (Answer: baseline surplus $150; stressed surplus $135; change \(-\$15\), an adverse decrease.)