Risk Measures Cheat Sheet
Volatility-Based Risk Measures
| Measure | Formula | What It Captures | Limitation |
|---|---|---|---|
| Standard Deviation (σ) | √[Σ(Ri − R̄)² ÷ (n−1)] | Total dispersion of returns around the mean | Treats upside and downside volatility equally |
| Historical Volatility | Annualized σ of log returns | Realized price fluctuation over a period | Backward-looking — may not predict future vol |
| Implied Volatility | Derived from option prices via Black-Scholes | Market’s expectation of future volatility | Can be inflated by supply/demand in options market |
| Downside Deviation | σ of returns below the target (MAR) | Only harmful volatility — the kind investors actually fear | Requires choosing a minimum acceptable return |
| Semi-Variance | Variance using only negative deviations from mean | Focus on below-average returns only | Less intuitive than downside deviation |
Market Risk Measures
| Measure | Formula | What It Captures | Benchmark |
|---|---|---|---|
| Beta (β) | Cov(Ri, Rm) ÷ Var(Rm) | Sensitivity to market movements | β = 1 means moves with market |
| R-Squared (R²) | Correlation² with benchmark | % of returns explained by market | R² > 0.70 = high market dependency |
| Tracking Error | σ of (Rp − Rb) over time | Consistency of deviation from benchmark | Low TE = index-like; high TE = active |
| Systematic Risk | β² × σ²market | Risk that cannot be diversified away | Market-driven risk only |
| Unsystematic Risk | Total Risk − Systematic Risk | Company-specific risk that diversification eliminates | Approaches zero with 25–30 stocks |
Tail Risk & Drawdown Measures
| Measure | Definition | Use Case | Typical Parameters |
|---|---|---|---|
| Value at Risk (VaR) | Maximum loss at a given confidence level over a specific period | Regulatory capital, internal risk limits | 95% or 99% confidence, 1-day or 10-day |
| Conditional VaR (CVaR / ES) | Expected loss given that loss exceeds VaR | Understanding worst-case tail scenarios | Average loss in the worst 5% or 1% of outcomes |
| Max Drawdown | Largest peak-to-trough decline in portfolio value | Worst historical loss experience | Measured from any peak to subsequent trough |
| Drawdown Duration | Time from peak to recovery back to previous peak | How long capital is underwater | Measured in months or years |
| Ulcer Index | RMS of percentage drawdowns over a period | Captures depth and duration of drawdowns | Lower = smoother ride; higher = more painful |
VaR Calculation Methods
| Method | How It Works | Pros | Cons |
|---|---|---|---|
| Historical Simulation | Uses actual past returns to build a distribution | No distributional assumptions; captures fat tails | Assumes past repeats; limited by data history |
| Variance-Covariance (Parametric) | Assumes normal distribution; uses σ and mean | Fast, easy to compute | Underestimates tail risk (returns aren’t normal) |
| Monte Carlo Simulation | Generates thousands of random scenarios | Flexible; can model complex portfolios | Computationally intensive; only as good as the model |
Risk Measure Selection Guide
| Question You’re Asking | Best Measure |
|---|---|
| How volatile is this investment? | Standard deviation |
| How much does it move with the market? | Beta |
| What’s the worst-case loss scenario? | VaR / CVaR |
| What was the worst actual decline? | Max Drawdown |
| Is the manager taking active risk? | Tracking Error |
| How much downside risk exists specifically? | Downside Deviation / Sortino Ratio |
Key Takeaways
- No single risk measure captures all dimensions — use multiple metrics together
- Standard deviation measures total volatility; beta measures market sensitivity
- VaR and CVaR quantify potential losses at specified confidence levels
- Max drawdown shows the worst actual loss — the metric that keeps investors up at night
- Returns are not normally distributed; always account for fat tails and tail risk
Frequently Asked Questions
What is the most important risk measure for retail investors?
Max drawdown is the most intuitive and practically relevant. It tells you the worst peak-to-trough loss you would have experienced. If you can’t stomach a 50% drawdown, you shouldn’t hold 100% equities regardless of other metrics.
What does a VaR of $1 million at 95% confidence mean?
It means there’s a 95% probability that your portfolio will not lose more than $1 million over the specified period (typically one day). Conversely, there’s a 5% chance the loss will exceed $1 million — and VaR doesn’t tell you how much worse it could be.
Why is beta not a complete risk measure?
Beta only captures systematic (market) risk. It ignores company-specific risk, liquidity risk, concentration risk, and tail events. A stock with a beta of 0.5 can still lose 80% of its value if the company faces idiosyncratic problems.
How is CVaR different from VaR?
VaR says “losses won’t exceed X with Y% confidence.” CVaR (Expected Shortfall) says “when losses DO exceed the VaR threshold, the average loss is Z.” CVaR captures the severity of tail losses, making it a more complete risk measure for extreme scenarios.
Can risk measures predict market crashes?
No. Risk measures quantify historical or modeled risk, but they can’t predict when a crash will happen. Rising implied volatility (VIX) can signal increasing fear, but it’s not a reliable timing tool. Risk measures help you prepare for crashes, not predict them.