Stress Testing Banks: formula sheet

Key formulas

VaR versus stress test

VaR = loss not exceeded at confidence level c over horizon h; stress loss = loss under a chosen severe scenario

VaR is probability-based. A stress loss has no stated probability and can exceed VaR by a large margin.

Stress test types

Sensitivity (one factor) | Scenario (multi-factor, historical or hypothetical) | Reverse (start from failure, find the cause)

Reverse stress testing starts from a defined outcome, such as breaching minimum capital, and works back to the scenarios that cause it.

Governance principles (summary)

Board and senior management ownership; written policies; sound data and infrastructure; independent validation; use of results

These are the themes tested. Remember that results must influence decisions.

Scenario criteria

Severe but plausible; covers material risks; forward-looking; tailored to the bank

Scenarios should not be limited to past events and should consider interactions between risks.

Severity vs plausibility

Good scenario = severe but plausible

Severity rising lowers plausibility. Design for the most severe event you can still defend as coherent.

Scenario types

Historical | Hypothetical | Macroeconomic | Reverse

Forward: scenario to loss. Reverse: loss (failure point) to scenario.

Capital impact of a scenario

Post-stress capital ratio = (Capital − stress losses + stress revenue) ÷ Stressed RWA

Compare with the regulatory minimum plus buffers. Stressed RWA can rise as ratings migrate.

Reverse stress test target

Find scenarios where Capital ratio ≤ minimum (or liquidity survival horizon ≈ 0)

The failure point is fixed first. The scenario is the output.

Post-stress capital

Stressed capital = Starting capital + Pre-provision net revenue − Credit losses − Market losses − Operational losses − Other adjustments (taxes, dividends)

Include every risk's impact in the same scenario and same horizon.

Stressed capital ratio

Stressed ratio = Stressed capital ÷ Stressed risk-weighted assets

RWA usually changes under stress (migration, higher risk weights), so do not hold the denominator fixed unless told.

Total integrated impact

Total impact = Σ standalone impacts + interaction (second-round) effects

Interaction effects can be negative for capital, such as fire-sale losses or higher funding costs. Hedges may offset some impacts.

Capital buffer headroom

Headroom = Stressed ratio − Required minimum plus buffers

A negative figure means a shortfall that needs action.

Stress CET1 depletion

Depletion = Actual starting CET1 ratio (start of projection horizon) − Minimum CET1 ratio over the nine quarters of the severely adverse scenario

Peak-to-trough decline, not start to end of the horizon.

Stress capital buffer (US)

SCB = Depletion + (Planned common dividends for quarters 4 through 7 of the planning horizon ÷ RWA), subject to a floor of 2.5%

The dividend add-on uses planned common stock dividends for the fourth through seventh quarters of the planning horizon (four quarters), as a percentage of RWA.

Post-stress requirement

For a firm subject to the SCB: Minimum CET1 requirement = 4.5% + SCB (+ G-SIB surcharge where applicable)

For firms subject to the SCB, it replaces the fixed 2.5% capital conservation buffer. Firms not subject to the SCB keep the 2.5% buffer. The G-SIB surcharge is additive, on top of the SCB.

Projected capital path

Ending CET1 capital = Starting CET1 capital + Pre-provision net revenue − Provisions − Other losses − Distributions

RWA is also projected; the ratio is capital ÷ RWA each quarter.

Program horizon

DFAST/CCAR: nine-quarter projection horizon

Scenarios: baseline, adverse and severely adverse (set by the Federal Reserve).

PPNR

PPNR = Net interest income + Non-interest income − Non-interest expense

Measured before provisions and before taxes. Operational and other losses are usually shown as separate lines.

Pre-tax income

Pre-tax income = PPNR − Provisions − Trading and counterparty losses − Operational losses − Other losses

Losses on securities and fair-valued items may run through income or through other comprehensive income depending on accounting.

Ending capital

Ending CET1 = Starting CET1 + Net income − Dividends − Buybacks ± Other adjustments

Net income = pre-tax income − taxes. A loss reduces capital. Include deduction changes.

Capital ratio

CET1 ratio = CET1 capital ÷ RWA

Compute it each quarter. Compare the minimum with requirement plus buffers.

Expected credit loss

EL = PD × LGD × EAD

Used to project loan losses under each scenario path. Stress raises PD, LGD and often EAD.

Leverage ratio

Leverage ratio = Tier 1 capital ÷ Total leverage exposure

Does not use RWA, so it is unaffected by risk-weight migration.

Stressed net cash outflow

Net outflow = Stressed outflows − Stressed inflows

Apply run-off rates to liabilities and drawdown rates to commitments. Apply haircuts and delays to inflows.

Liquidity surplus or shortfall

Surplus = Liquidity buffer (after haircuts) − Cumulative stressed net outflow

A negative value is a shortfall and should trigger CFP action.

Survival horizon

Survival horizon ≈ Liquidity buffer ÷ Average daily stressed net outflow

A simple approximation. In practice, use the day on which the cumulative net outflow first exceeds the buffer.

Buffer value after haircut

Stressed value = Market value × (1 − haircut)

Use stressed haircuts, not normal ones.

Liquidity coverage ratio

LCR = Stock of HQLA ÷ Total net cash outflows over 30 days ≥ 100%

The Basel regulatory 30-day stress standard. Internal stress tests should be at least as demanding in design.

Scenario loss estimate

Scenario loss = Frequency × Severity (annualised) or total loss = direct loss + fines + remediation + other costs

Expected annual loss from a scenario uses frequency times severity. A single stressed event uses the total of its cost components.

Quick revision

• A stress test examines severe but plausible adverse conditions; it is not a forecast of the most likely outcome.
• Board and senior management own the stress testing programme and must challenge assumptions and act on results.
• Scenarios can be historical, hypothetical, sensitivity-based or reverse; reverse starts from a failure outcome and works back to causes.
• A good scenario is severe, plausible and relevant to the bank's own vulnerabilities.
• Enterprise-wide testing aggregates results across risk types and business lines instead of treating them in silos.
• Ignoring correlations and feedback effects between risks understates stress losses.
• Capital impact comes from losses, changes in pre-provision revenue and changes in risk-weighted assets.
• Supervisory stress tests use common scenarios to compare banks; internal tests should also reflect the bank's own risks.
• Know who runs CCAR, DFAST and the EBA exercise and how results feed capital decisions.
• Liquidity stress tests look at funding outflows and buffer adequacy over defined horizons.
• Operational stress tests use severe loss events and resilience scenarios, where data is scarce and judgement matters.
• Results must lead to action, such as limits, capital planning or contingency plans.

Common mistakes

• Treating stress testing as a replacement for VaR Fix: Remember they complement each other. VaR gives a probability-based threshold; stress tests explore severe events and the tail.
• Saying a stress loss has a stated probability Fix: Scenarios are chosen by plausibility and severity, not by a confidence level.
• Saying historical scenarios are the most forward-looking Fix: Remember they are backward-looking and can miss new risks. Their strength is credibility, not foresight.
• Treating reverse stress testing as a forward scenario with a bigger shock Fix: Reverse starts from a defined failure outcome and searches backward for causes.
• Applying diversification benefits between risk types inside a single stress scenario. Fix: In a scenario-based test the shock is already joint. Add the impacts; do not reduce them by a diversification factor unless the question says so.
• Ignoring second-round effects. Fix: Always ask what the capital loss triggers: funding costs, downgrades, asset sales, credit supply cuts.
• Using the end-of-horizon CET1 ratio instead of the minimum. Fix: Scan every quarter and use the trough.
• Saying DFAST and CCAR are the same test. Fix: DFAST is quantitative; CCAR also assesses capital planning and the distribution plan. Today CCAR's qualitative assessment applies mainly to the largest, most complex (LISCC) firms.
• Treating PPNR as net income after provisions. Fix: PPNR excludes provisions. Subtract provisions and other losses afterwards.
• Dividing stressed capital by starting RWA. Fix: Re-estimate RWA under the scenario, then divide. Say which RWA you used.

Exam tips

• Expect conceptual, case-style questions. Match the described failure to a governance principle such as ownership, challenge or use of results.
• When comparing with VaR, think complementary, not substitute. Options saying one replaces the other are usually wrong.
• Know the three test types: sensitivity, scenario and reverse. Reverse starts from the failure outcome.
• Watch for options that confine responsibility to the technical team. Board and senior management oversight is the favoured answer.
• Read the first clue in the question: past event, new risk, multi-year macro path, or failure outcome. It usually names the type.
• Know one strength and one weakness for each scenario type. Options often swap them.
• The phrase severe but plausible is the standard answer on severity.
• Link scenarios to the bank's own portfolio and business model when the question mentions vulnerabilities.