Risk Measures, VaR, and Volatility

A trading desk closes Friday with a portfolio that lost 1.8% on the day. The risk report flags the move as a once-per-month event. Monday it loses 3.2% and the same model now flags it as once-per-decade. Either the world changed over the weekend, or the model's volatility input was three days stale. The job is to read those numbers correctly, knowing exactly which assumption you bought when you accepted them.

Risk managers measure portfolios in two numbers when they can: expected return and standard deviation. The mean-variance framework treats the standard deviation of returns as the risk metric. Two portfolios with the same expected return are ranked by volatility, lower volatility wins.

Mean and variance completely specify a normal return distribution; mean-variance choice is also justified by quadratic utility, although that utility has restrictive assumptions. Real asset returns have fat tails, skewness, and time-varying variance. Mean-variance is still a useful first cut, but tail-aware metrics fill the gap.

KEY: Variance penalizes upside and downside symmetrically. A fund with strong positive skew can have a higher variance than a riskier fund with negative skew.

Common mistakes

• Forgetting the negative sign on VaR. Returns above the mean are gains; returns below are losses. VaR is a positive number representing the loss magnitude. Students compute $\mu - z_c \sigma$, get a negative number, and report it as the answer. Trap: a problem reports VaR as -$5M; choices include$5M and -$5M, and the wrong-sign answer wins points.
• Confusing 1-tail and 2-tail z-scores. VaR is one-sided. Only the loss tail matters. The 99% z-score is 2.326, not 2.576 (which is the two-sided 99% confidence interval). Trap: a question gives the z-table and choice C uses 2.576 for 99% one-sided VaR. The right number is 2.326.
• Using square-root scaling under GARCH. Square-root volatility scaling requires equal finite daily variances and no serial covariance; scaling VaR also needs the distributional assumptions stated above.

Bottom line

• VaR gives a loss quantile at confidence c over horizon h. Parametric VaR assumes normal returns on a P&L basis: $\text{VaR} = -\mu + z_c \sigma$; daily rules drop the mean.
• Three VaR methods: parametric (assumes a distribution), historical simulation (reads the empirical quantile), Monte Carlo (simulates a calibrated process).
• Expected Shortfall averages the worst $1-c$ probability mass; under normality $\text{ES}_c = -\mu + \sigma \phi(z_c)/(1-c)$, where $\mu$ is expected return. ES is at least VaR at the same confidence and is coherent.
• Coherent risk measures satisfy monotonicity, subadditivity, positive homogeneity and translation invariance. VaR can fail subadditivity for non-elliptical loss distributions.

Exam shortcut

When a question asks for parametric VaR, identify three numbers: $\sigma$ (volatility per period), $z_c$ (one-sided quantile at the confidence level), and the position size. Multiply volatility, quantile and position size. If the horizon differs, scale zero-mean VaR by $\sqrt{h}$ only when independent normal returns or another stated scaling assumption justify it. Neglect the mean only when its contribution is immaterial.

The full lesson (about 6,599 words, 44 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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