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 σ wins.
That framework only fully describes risk when returns are normally distributed. The normal distribution is determined by mean and variance alone, so two numbers capture everything. 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.
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 , get a negative number, and report it as the answer. Trap: a problem reports VaR as -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 scaling only works for iid returns. Under GARCH or any mean-reverting volatility process, multi-day variance is less than when current vol is above the long-run level, and more than that when current vol is below.
Bottom line
- VaR answers "what's the worst loss at confidence c over horizon h?" Parametric VaR assumes normal returns on a P&L basis: ; 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 (ES) is the average loss beyond VaR; under normality . ES > VaR at the same confidence; ES is coherent, VaR is not.
- Coherent risk measures satisfy monotonicity, subadditivity, positive homogeneity, translation invariance. VaR fails subadditivity for non-elliptical loss distributions.
Exam shortcut
When a question asks for parametric VaR, identify three numbers: σ (volatility per period), z (one-sided quantile at the confidence level), and the position size. Multiply σ × z × position. If the horizon doesn't match σ, scale by only when iid is assumed. Drop the mean for daily horizons. Memory aid: "VaR reads the threshold; ES averages the tail." Coherent acronym: MSPT: Monotonicity, Subadditivity, Positive homogeneity, Translation invariance.
The full lesson (about 3,279 words, 22 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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