Designing Market Stress Scenarios for Insurers

Why Insurers Need Purposeful Market Stress Scenarios

A market stress scenario is useful only if it tests a vulnerability that matters to the insurer. A large equity decline may be material for an insurer with a substantial equity portfolio, while a rate rise or fall may matter more to an insurer whose assets and long-term liabilities respond differently to interest rates. Scenario design therefore begins with the portfolio and the question being asked, not with a list of dramatic market moves.

A scenario specifies one or more market-factor changes over a stated horizon and applies them to a defined portfolio. The result is an estimate of the change in asset value, liability value, or surplus under the scenario’s assumptions. A scenario is not, by itself, a forecast or a statement of probability. Unless a model explicitly supplies a probability, a stress result should not be described as a loss that is expected to occur with a particular frequency.

Definition: A market stress scenario is a specified set of adverse or challenging changes in market variables, together with a defined horizon and valuation basis, used to assess the effects on an insurer’s financial position. Its shocks should be stated in measurable units, such as basis points, percentage returns, or changes in an exchange rate.

For an insurer, the relevant outcome is often not an asset loss alone. If $A$ is asset value and $L$ is liability value on a consistent valuation basis, surplus is $A-L$. A stress can reduce both assets and liabilities; the change in surplus depends on their relative changes. A well-designed scenario therefore identifies the exposed assets, the exposed liabilities, and any important offsets before drawing a conclusion.

Specify the Scenario Before Calculating

Each scenario should state its factors, shock directions and sizes, horizon, and valuation assumptions. “Rates rise sharply” is not sufficiently specific for a calculation. A statement such as “the relevant yield increases by 75 basis points over an instantaneous valuation shock” identifies a unit, magnitude, and horizon convention. If the scenario instead represents a one-year development, the analyst should explain whether investment income, cash flows, rebalancing, and liability changes during the year are included.

For a small yield change, a bond or other fixed-income value can be approximated with modified duration $D_{\text{mod}}$ and convexity $C$, measured on the same valuation basis:

\frac{\Delta V}{V}\approx -D_{\text{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2

Here, $\Delta y$ is the change in yield in decimal form: 50 basis points is $0.005$, not $50$ or $0.50$. The approximation assumes the position is adequately represented by the stated duration and convexity for the shock being applied. It does not fully revalue options, embedded guarantees, or other nonlinear features; a direct repricing may be needed when those features are material. For spread risk, spread duration can be used similarly as a first-order approximation to the effect of a change in credit spread, provided other relevant market factors are held fixed in that calculation.

The approximation follows from a second-order Taylor expansion of value as a function of yield. By definition, modified duration is approximately the negative percentage slope of value with respect to yield, while convexity describes the curvature. Thus the first-order term is $-D_{\text{mod}}\Delta y$, and the second-order term is $\frac{1}{2}C(\Delta y)^2$. The approximation becomes less reliable as the shock grows or as the position becomes more nonlinear.

Equity shocks are usually expressed as a percentage return applied to the exposed market value. Currency shocks require a clear exchange-rate convention: specify which currency is the reporting currency and whether the quoted exchange rate is reporting-currency units per unit of foreign currency. Under that convention, a foreign currency depreciation reduces the reporting-currency value of an unhedged foreign asset. It also reduces the reporting-currency value of a foreign-currency liability, which can partly offset the asset effect.

Worked Applications

Worked Example: A Rate Rise and Insurer Surplus

A hypothetical insurer has fixed-income assets worth $ \$120$ million and liabilities worth $ \$108$ million. On a consistent market-value basis, assume asset modified duration is $5.0$ and convexity is $34$, while liability modified duration is $7.0$ and convexity is $50$. Design an instantaneous parallel yield-rise stress of 75 basis points and estimate the change in surplus. Assume the stated measures apply to this shock, and exclude cash flows, income, taxes, and management actions.

Method. Apply the duration-convexity approximation separately to assets and liabilities, then subtract the liability change from the asset change. The shock in decimal form is $\Delta y=75/10{,}000=0.0075$.

Assets. The first-order percentage change is $-5.0(0.0075)=-0.0375$. The convexity adjustment is $\frac{1}{2}(34)(0.0075)^2=17(0.00005625)=0.00095625$. Therefore,

\frac{\Delta A}{A}\approx -0.0375+0.00095625=-0.03654375

The asset value change is $\Delta A\approx \$120\text{ million}(-0.03654375)=-\$4.38525\text{ million}$.

Liabilities. The first-order percentage change is $-7.0(0.0075)=-0.0525$. The convexity adjustment is $\frac{1}{2}(50)(0.0075)^2=25(0.00005625)=0.00140625$. Thus $\Delta L/L\approx-0.0525+0.00140625=-0.05109375$, and $\Delta L\approx \$108\text{ million}(-0.05109375)=-\$5.518125\text{ million}$.

The change in surplus is $\Delta(A-L)=\Delta A-\Delta L=-4.38525-(-5.518125)=+\$1.132875$ million. Initial surplus is $120-108=\$12$ million, so stressed surplus is approximately $12+1.132875=\$13.132875$ million. Rounded to the nearest cent, the estimated surplus increase is $ \$1,132,875.00$. The check is that liabilities fall more in dollar terms than assets, so surplus rises in this specified rate-rise scenario.

Common mistake. Treating the liability decline as an additional loss would give the wrong sign. A liability decrease improves surplus: subtracting $\Delta L<0$ increases $\Delta(A-L)$.

Worked Example: Combining Spread and Equity Stresses

A hypothetical insurer holds $ \$40$ million of corporate bonds with spread duration $4.2$, and $ \$25$ million of equities. Construct a six-month stress in which credit spreads widen by 120 basis points and the equity portfolio falls by 18%. For this calculation, hold government yields and all other factors fixed, assume no defaults, and exclude income, hedging changes, taxes, and rebalancing. Estimate the direct asset-value change.

Bond spread effect. Convert the spread shock to $\Delta s=120/10{,}000=0.012$. Using the first-order spread-duration approximation, the bond percentage change is $-4.2(0.012)=-0.0504$, or a 5.04% decline. The dollar change is $ \$40\text{ million}(-0.0504)=-\$2.016$ million.

Equity effect. The specified equity return is $-0.18$, so the dollar change is $ \$25\text{ million}(-0.18)=-\$4.5$ million.

Total direct asset effect. Adding the two specified changes gives $-2.016-4.5=-\$6.516$ million. The estimated stressed value of these holdings is $40+25-6.516=\$58.484$ million, compared with an initial combined value of $ \$65$ million.

This scenario is tailored to the two stated exposures: a spread-sensitive bond portfolio and an equity portfolio. The calculation is a conditional scenario result, not a probability-weighted estimate. It also does not claim that a 120-basis-point spread widening and an 18% equity decline will occur together with any particular likelihood.

Common mistake. Applying the spread shock to the bond’s value as if it were a 120% change confuses basis points with a percentage return. The decimal shock is $0.012$; the estimated bond loss is $4.2\times0.012=5.04\%$, not $504\%$.

Worked Example: Currency Exposure After Matching Foreign Liabilities

A hypothetical insurer reports in U.S. dollars. It has an unhedged euro-denominated asset worth $€30$ million and a euro-denominated liability valued at $€18$ million. The initial exchange rate is $ \$1.10$ per euro. Design a one-year currency stress in which the euro depreciates by 12% against the dollar. Assume the euro amounts remain unchanged, and exclude local investment returns, liability cash flows, and hedging.

Step 1: Find the stressed exchange rate. A 12% depreciation means the dollar value of one euro falls to $1-0.12=0.88$ of its initial value. The stressed rate is $1.10(0.88)=\$0.968$ per euro.

Step 2: Translate the asset. Initially, the asset is $€30\text{ million}\times\$1.10/€=\$33.0$ million. Under stress it is $€30\text{ million}\times\$0.968/€=\$29.04$ million. Its change is $29.04-33.0=-\$3.96$ million.

Step 3: Translate the liability. Initially, the liability is $€18\text{ million}\times\$1.10/€=\$19.8$ million. Under stress it is $€18\text{ million}\times\$0.968/€=\$17.424$ million. Its change is $17.424-19.8=-\$2.376$ million.

Step 4: Calculate the surplus effect. The liability decline offsets part of the asset decline: $\Delta(A-L)=-3.96-(-2.376)=-\$1.584$ million. The check is that net initial euro exposure is $€30-€18=€12$ million; its initial dollar value is $ \$13.2$ million, and a 12% decline is $ \$1.584$ million. The two methods agree.

Common mistake. Applying the currency depreciation only to assets overstates the net loss when the insurer also has foreign-currency liabilities. Both sides must be translated using the same stressed exchange rate and reporting-currency convention.

Choosing Shock Direction, Size, and Horizon

The shock direction should challenge the actual exposure. A yield rise and a yield fall can have different consequences for asset and liability values; equity losses are generally adverse to long equity positions, while an insurer with short positions or guarantees may have different sensitivities. For currency risk, the adverse direction depends on whether the insurer is net long or net short the foreign currency. “Adverse” is therefore a conclusion about the insurer’s financial position, not a fixed label attached to a market movement.

Shock size should be large enough to reveal a material weakness but should remain interpretable in light of the stress purpose. It can be informed by historical market moves, hypothetical severe conditions, internal risk appetite, or a specified decision question. The selected basis should be documented. A number described as “severe” is not self-explanatory, and should not be presented as a probability estimate without an explicit probability model.

The horizon must also fit the exposure and decision. An instantaneous revaluation isolates immediate price sensitivity but excludes the time needed for an insurer to act. A longer horizon may allow cash flows, reinvestment, hedging, and management actions to change the result. The analyst should specify which are included, rather than silently mixing an immediate market-value shock with a year of projected income or actions. If a result is meant to inform a decision, the chosen horizon should allow the relevant decision process to be considered.

Map material exposures.
Identify which assets and liabilities respond to rates, credit spreads, equity prices, or exchange rates. Record the measure, currency, and valuation basis for each exposure.

State the stress question.
Decide whether the scenario is intended to test immediate sensitivity, a planning horizon, or a particular management decision. This determines what is held fixed and what is allowed to change.

Choose and document shocks.
Specify directions, magnitudes, units, and horizon. Explain why the chosen movements challenge the portfolio rather than merely selecting large numbers without reference to exposure.

Revalue consistently and check the result.
Apply suitable approximations or repricing to assets and liabilities, then calculate the net financial effect. Check signs, units, offsets, and excluded effects before interpreting the outcome.

When several factors are shocked in one scenario, the assumptions should make clear how the factors are applied and what the result does and does not represent. The analyst can use a combined scenario to assess a deliberately specified set of conditions without asserting that the movements are statistically related or assigning a likelihood to their joint occurrence. If nonlinear positions are important, duration-based approximations or simple percentage changes may not capture their full response; the scenario should then use an appropriate repricing method and say so.

Common Design and Interpretation Traps

Using the same shock for every insurer. A standard shock can aid comparison, but it may miss the exposures that drive a particular insurer’s risk. Explain why the selected factors matter to the portfolio.
Ignoring liabilities. Asset performance alone does not determine the change in insurer surplus. Apply the scenario consistently to material liabilities as well as assets.
Mixing units. Convert basis points to decimal yield changes before using duration. Apply percentage returns to the correct market value, and state exchange rates in an unambiguous quotation convention.
Leaving the horizon unstated. A market shock with no horizon or revaluation convention cannot be interpreted reliably. State whether cash flows, income, hedges, or management actions are included.
Overstating what a stress means. A scenario result is conditional on the specified shocks and assumptions. It is not automatically a forecast, a capital requirement, or a probability-based risk measure.

Key takeaway: A useful insurer stress scenario connects explicit market shocks to actual asset and liability vulnerabilities. Its direction, size, horizon, valuation method, and exclusions should all be clear enough for another analyst to reproduce and interpret the result.

Check Your Understanding

For each problem, use only the stated assumptions and keep signs consistent with the effect on surplus.

A hypothetical insurer owns $ \$50$ million of bonds with spread duration $3.6$. Estimate the first-order value change under a spread widening of 80 basis points, holding other factors fixed. (Answer: $-\$1.44$ million.)
A hypothetical insurer has $ \$12$ million invested in an unhedged equity portfolio. Under an instantaneous 22% decline, calculate the value change and the stressed portfolio value. (Answer: value change $-\$2.64$ million; stressed value $ \$9.36$ million.)
A U.S.-dollar-reporting insurer has a $€15$ million asset and an $€8$ million liability. The exchange rate falls from $ \$1.25$ to $ \$1.10$ per euro, with euro amounts fixed. Calculate the net surplus change. (Answer: asset change $-\$2.25$ million, liability change $-\$1.20$ million, net surplus change $-\$1.05$ million.)
A hypothetical insurer has assets of $ \$80$ million with modified duration $4$ and convexity $20$, and liabilities of $ \$70$ million with modified duration $6$ and convexity $30$. Estimate the surplus change under a parallel yield decline of 50 basis points using the duration-convexity approximation for both sides. (Answer: asset change $+\$1.62$ million; liability change $+\$2.12625$ million; surplus change $-\$0.50625$ million.)
An insurer wants to test whether its net foreign-currency position could materially reduce surplus over the next year. Name two scenario specifications, in addition to the shock magnitude, that should be fixed before calculating the result. (Answer: specify the exchange-rate quotation and reporting currency, and state the horizon and whether foreign cash flows, hedges, or management actions are included.)

What You'll Learn