Market Risk Measurement - VaR and Beyond › Stress Testing and Scenario Analysis
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VaR and ES are statistical measures based on historical patterns or distributional assumptions. Stress testing complements these by explicitly examining portfolio performance under extreme but plausible scenarios that may not be well-represented in historical data.
Historical Stress Tests
Historical stress tests revalue the portfolio using returns from actual crisis periods. This ensures the scenarios are internally consistent: correlations, volatility spikes, and return magnitudes all reflect observed market dislocations.
Common historical stress scenarios include:
- Black Monday (October 19, 1987): S&P 500 fell 22.6% in a single day
- LTCM Crisis (August 1998): Flight to quality crushed credit spreads while equity volatility spiked
- Global Financial Crisis (September-October 2008): Broad market declines with correlation breakdown
- COVID Crash (March 2020): Rapid 34% decline in equities with liquidity dislocations
# Define historical stress scenarios
# Returns are approximate single-day or short-period moves
stress_scenarios = {
"Black Monday 1987": {
"equities": -0.226,
"bonds": 0.03,
"commodities": -0.05,
},
"LTCM Crisis 1998": {
"equities": -0.06,
"bonds": 0.02,
"commodities": -0.08,
},
"Lehman Bankruptcy 2008": {
"equities": -0.09,
"bonds": -0.02,
"commodities": -0.12,
},
"COVID Crash 2020": {
"equities": -0.12,
"bonds": 0.01,
"commodities": -0.15,
},
"Flash Crash 2010": {
"equities": -0.09,
"bonds": 0.01,
"commodities": -0.03,
},
}
# Current portfolio (using same weights as before)
# Assume Asset 1 = equities, Asset 2 = commodities, Asset 3 = bonds
asset_mapping = ["equities", "commodities", "bonds"]
portfolio_value = 10_000_000
weights = np.array([0.4, 0.35, 0.25])
stress_results = {}
for scenario_name, scenario_returns in stress_scenarios.items():
# Map scenario returns to our 3-asset portfolio
returns_vector = np.array(
[
scenario_returns["equities"],
scenario_returns["commodities"],
scenario_returns["bonds"],
]
)
# Calculate portfolio P&L
portfolio_return = weights @ returns_vector
portfolio_loss = -portfolio_value * portfolio_return
stress_results[scenario_name] = {
"portfolio_return": portfolio_return,
"portfolio_loss": portfolio_loss,
}Historical Stress Test Results ============================================================ Portfolio Value: $10,000,000 Weights: Equities 40%, Commodities 35%, Bonds 25% ------------------------------------------------------------ Black Monday 1987 Return: -10.04% Loss: $1,004,000 LTCM Crisis 1998 Return: -4.70% Loss: $470,000 Lehman Bankruptcy 2008 Return: -8.30% Loss: $830,000 COVID Crash 2020 Return: -9.80% Loss: $980,000 Flash Crash 2010 Return: -4.40% Loss: $440,000 ------------------------------------------------------------ For comparison: 99% VaR (Monte Carlo): $384,912 99% ES (Monte Carlo): $435,761
The stress test results reveal that all historical crisis scenarios produce losses substantially exceeding the 99% VaR. This is expected: VaR estimates typical tail behavior, while stress tests examine market dislocations that may occur once per decade or less frequently.
Hypothetical Stress Scenarios
Beyond historical events, risk managers construct hypothetical scenarios to test vulnerabilities specific to their portfolios. These might include interest rate shocks, currency devaluations, sector-specific collapses, or combinations that stress multiple risk factors simultaneously.
# Hypothetical stress scenarios
hypothetical_scenarios = {
"Interest Rate Shock +200bp": {
"equities": -0.05,
"bonds": -0.08,
"commodities": -0.03,
},
"Dollar Collapse": {"equities": -0.03, "bonds": -0.02, "commodities": 0.15},
"Stagflation": {"equities": -0.15, "bonds": -0.05, "commodities": 0.10},
"Deflationary Spiral": {
"equities": -0.20,
"bonds": 0.08,
"commodities": -0.25,
},
"Correlation Breakdown": {
"equities": -0.10,
"bonds": -0.05,
"commodities": -0.10,
},
}
hypothetical_results = {}
portfolio_value = 10_000_000
weights = np.array([0.4, 0.35, 0.25])
for scenario_name, scenario_returns in hypothetical_scenarios.items():
returns_vector = np.array(
[
scenario_returns["equities"],
scenario_returns["commodities"],
scenario_returns["bonds"],
]
)
portfolio_return = weights @ returns_vector
portfolio_loss = -portfolio_value * portfolio_return
hypothetical_results[scenario_name] = {
"portfolio_return": portfolio_return,
"portfolio_loss": portfolio_loss,
}Hypothetical Stress Test Results ============================================================ Interest Rate Shock +200bp Return: -5.05% Loss: $505,000 Dollar Collapse Return: 3.55% Loss: $355,000 gain Stagflation Return: -3.75% Loss: $375,000 Deflationary Spiral Return: -14.75% Loss: $1,475,000 Correlation Breakdown Return: -8.75% Loss: $875,000
The hypothetical scenarios highlight that events like a deflationary spiral or correlation breakdown could generate losses exceeding $2 million, significantly outpacing the standard VaR estimates.
Reverse Stress Testing
Traditional stress tests ask "what happens if scenario X occurs?" Reverse stress testing inverts the question: "what scenarios would cause losses exceeding threshold Y?" This helps identify hidden vulnerabilities and concentration risks.
For a portfolio heavily weighted in technology stocks, reverse stress testing might reveal that a 15% tech sector decline combined with rising interest rates would breach the firm's risk limits, even if such a scenario seems unlikely based on recent history.
Visualizing Stress Test Results
