Estimating Market Risk Measures

A trading desk inherits a 2-year return history and is asked for a 99% one-day VaR. Plug it into a normal formula and you get one number. Run historical simulation on the same data and you get another, often 30% larger after a stress period. Fit a generalized Pareto to the worst losses and the number can shift again. Every method buys a different assumption. Your job is to know which one you bought.

Every market-risk measure answers the same question: at confidence level c over horizon h, what loss won't you exceed? The methods differ on how they estimate the tail. Three families dominate the FRM:

• Parametric: assume a distribution (normal, lognormal, Student-t), estimate parameters, read the quantile in closed form.
• Non-parametric: let the data speak, sort realized returns, take the empirical quantile.
• Semi-parametric / EVT: non-parametric for the body, parametric for the tail.

Part I covered VaR and Expected Shortfall under the parametric normal lens. Part II layers on the alternatives (lognormal, weighted historical, EVT) that risk managers reach for when the normal...

Common mistakes

• Reading parametric VaR off the wrong z-score. 99% VaR is one-sided: z = 2.326. Candidates use 2.576 (the two-sided 99% confidence interval) and overstate VaR by 11%. Trap: a problem provides a z-table; choice C uses 2.576 for 99% and is wrong.
• Treating historical simulation as the "true" VaR. Empirical quantiles have sampling error. With 250 observations, the 99% VaR is the 2nd or 3rd worst; change one extreme observation and the number jumps 20%. The reported VaR carries a wide confidence interval the framework hides.
• Ignoring the threshold-choice problem in EVT. Setting u too low contaminates the GPD with body observations and biases the tail-shape parameter toward zero. Setting u too high leaves too few exceedances and parameter estimates become unstable.

Bottom line

• Parametric VaR assumes a distribution and reads its quantile: normal gives ${\text{VaR}}_c=-\mu +z_c\sigma$; lognormal gives $S_0[1-\exp (\mu -z_c\sigma )]$, capping loss at 100%.
• Expected Shortfall averages the losses beyond VaR: ${\text{ES}}_c=E[L\mid L>{\text{VaR}}_c]$, estimated as the average of the tail VaRs. Always ${\text{ES}}_c\ge {\text{VaR}}_c$; ES is coherent, VaR is not.
• Historical simulation sorts realized returns and reads the empirical quantile: no distribution assumption, but model-dependent on the window representing the future.
• Weighted historical schemes (age, volatility, correlation, filtered) each fix a specific equally-weighted blind spot: recent shocks, vol regimes, shifting correlations.

Exam shortcut

When a question describes a vol regime shift or correlation breakdown, the right answer is filtered historical simulation or volatility-weighted HS, not equally-weighted historical and not parametric normal. When a question asks for tail risk at very high confidence (99.5% and beyond), POT-EVT is the defensible choice because the empirical quantile is too noisy and parametric normal underweights the tail.

The full lesson (about 4,886 words, 33 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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