Stress Testing: formula sheet

Key formulas

Scenario loss (linear approximation)

Loss ≈ Σ (exposure_i × shock_i)

Use for linear positions. Options need delta-gamma or full revaluation because P&L is nonlinear.

Scenario loss with revaluation

Loss = V(base) − V(stressed)

Reprice the whole portfolio after shocking all risk factors together.

Stress loss versus capital

Post-stress capital = Capital − Stress loss

Compare with the minimum required or risk appetite limit.

VaR versus stress test

VaR: probability-based quantile. Stress test: scenario-based, no probability.

Neither gives the maximum possible loss. ES describes the average beyond VaR but is still statistical.

Governance chain

Board and senior management oversight → scenarios and assumptions challenged → results reviewed → action (risk appetite, capital, limits)

No calculation is required. Remember that oversight must lead to action, not only reporting.

Capital buffer check under stress

Post-stress capital ratio = (Capital − stress losses) ÷ post-stress risk-weighted assets

A simple use test. Compare the result with the minimum plus the management buffer. Losses cut the numerator; RWA can rise under stress.

Reverse stress test

Start from a defined failure outcome → find the scenarios that cause it

Works backward and helps find vulnerabilities that forward scenarios miss.

Sensitivity loss (linear approximation)

ΔP ≈ Exposure × Shock

For a bond, ΔP ≈ −DV01 × shock in bp. Use full revaluation for large shocks or options.

Scenario loss

Total loss = Σ (loss from each position under the joint factor moves)

All factors move together as the scenario specifies.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2

Shows why a rise in ρ in stress increases portfolio risk.

Reverse stress test logic

Fix outcome (e.g. capital ratio < minimum) → solve for factor moves causing it

Start at the failure point, not the shock.

Net income under stress

Net income = PPNR − provisions/loan losses − other losses (trading, counterparty, operational) − taxes

PPNR is revenue minus expenses before provisions. Stress usually lowers PPNR as well as raising losses.

Projected capital

Ending CET1 capital = Beginning CET1 + net income − dividends − buybacks ± other adjustments

Capital actions are what CCAR examines; DFAST uses standardized assumptions.

Projected capital ratio

CET1 ratio = CET1 capital ÷ risk-weighted assets

Use the projected RWA for the same quarter, not the starting RWA.

Capital depletion

Decline in ratio = Starting CET1 ratio − Minimum projected CET1 ratio

The minimum over the horizon is used, not the ending value.

Stress capital buffer (US, simplified)

SCB = peak-to-trough CET1 decline (severely adverse) + planned dividends for four quarters as % of RWA, subject to a floor of 2.5%

Stated as a simplified rule. The SCB replaces the fixed 2.5% capital conservation buffer in the US.

Quick revision

• Stress tests examine severe but plausible events, complementing VaR, which does not describe losses beyond its threshold.
• Historical scenarios replay past events; hypothetical scenarios construct new ones and can capture risks that have not yet occurred.
• A reverse stress test starts from an unacceptable outcome, such as failure, and works back to find the events that cause it.
• Sensitivity analysis moves one risk factor at a time; scenario analysis moves several factors together in a coherent story.
• Scenarios need internal consistency: shocked variables should move in ways that make economic sense together.
• The board and senior management should own the programme, and results should inform risk appetite, limits, capital and contingency planning.
• Independent validation and challenge of stress models and assumptions are part of good governance.
• Models calibrated on normal periods may fail in a crisis because correlations and relationships shift.
• Stress tests should consider liquidity, second-round and feedback effects, not only first-round losses.
• CCAR and DFAST are US supervisory programmes; the EBA runs EU-wide stress tests on banks.
• Supervisory tests use prescribed scenarios so results can be compared across institutions.
• Results are only useful if they lead to action, not just a report.

Common mistakes

• Saying a stress test has a confidence level like VaR. Fix: Remember VaR is a statistical quantile. A stress scenario is chosen and carries no stated probability.
• Treating historical scenarios as always the most severe. Fix: The past may not contain the worst plausible event, and the current portfolio may react differently. Hypothetical scenarios can be harsher.
• Treating stress testing as a purely technical model exercise run by the risk team. Fix: Remember that governance requires board and senior management ownership and challenge.
• Saying stress testing replaces VaR. Fix: Stress tests complement VaR by covering extreme events and exposures VaR misses.
• Treating a reverse stress test as a scenario test with a bigger shock. Fix: A reverse test starts with the failure outcome and finds the causes. A scenario test starts with the cause and finds the loss.
• Saying sensitivity tests capture interactions between risk factors. Fix: Sensitivity tests move one factor at a time and ignore joint moves.
• Treating CCAR and DFAST as the same test. Fix: Link CCAR to capital planning and planned capital actions with a qualitative component. Link DFAST to a quantitative projection with standardized assumptions.
• Using the ending capital ratio instead of the minimum ratio. Fix: Read the quarterly path and take the trough. The buffer uses the peak-to-trough decline.

Exam tips

• Expect conceptual contrasts: stress testing versus VaR, and historical versus hypothetical scenarios. Learn one line for each strength and weakness.
• In calculations, the arithmetic is usually simple. Marks are lost on signs and on forgetting a second risk factor.
• Watch for wording such as severe but plausible. Scenarios should be extreme yet believable to be useful to management.
• If options are in the portfolio and the shock is large, expect the answer to need gamma or full revaluation.
• Remember stress tests complement VaR. An option saying one replaces the other is almost always wrong.
• Expect conceptual questions: match a weakness to its remedy or a decision to the right use of stress results.
• Remember the Basel principles: board and senior management involvement, firm-wide scope, severe scenarios, and use in decisions.
• Be careful with absolute words in options. Stress testing complements other tools and does not replace them.