Module 03 · Risk Analytics
Risk Dashboard
A ten-year daily history for an illustrative 60/40-with-alternatives portfolio — simulated with a seeded GARCH-style volatility process so the data is identical on every load, and honest about being synthetic. Illustrative simulated data
Annualized where applicable · 2,520 simulated trading days · VaR and ES are one-day, historical (empirical quantile) estimates at 95%.
Cumulative growth of $1. The visible calm-then-storm texture is the volatility clustering built into the generating process.
Drawdown from running peak. Depth gets the attention; duration does the psychological damage.
Rolling 63-day (quarterly) annualized volatility.
Daily returns with VaR95 and ES95 marked; a Gaussian with matching mean and variance overlaid for contrast.
Interpretation
VaR, Expected Shortfall, and the limits of tail arithmetic
Value-at-Risk answers a quantile question: with 95% confidence, tomorrow's loss will not exceed VaR95. Formally it is \( \mathrm{VaR}_\alpha = -\inf\{x : F(x) \ge 1-\alpha\} \) — a single point on the loss distribution. Its virtue is legibility; its vice is that it says nothing about what happens beyond the threshold. Two portfolios can share a VaR while one loses 2% in its bad tail and the other loses 40%. VaR is also not subadditive in general — merging two portfolios can appear to increase risk — which disqualifies it as a coherent risk measure in the Artzner sense.
Expected Shortfall repairs the worst of this by averaging the tail: \( \mathrm{ES}_\alpha = \mathbb{E}[\,L \mid L \ge \mathrm{VaR}_\alpha\,] \). It is coherent, it sees the shape of the tail rather than one point of it, and it is why regulators moved market-risk capital from VaR to ES under FRTB. In the histogram above the gap between the two lines is the tail's verdict on itself.
But neither number escapes the deeper problem: tail estimation is data-starved by construction. A 95% one-day VaR on ten years of data rests on roughly 126 observations of "bad days"; a 99% estimate on about 25. The events that dominate long-run outcomes — the 1987s, 2008s, and March 2020s — appear in a sample zero or one times, and a volatility model fitted to calm regimes will extrapolate calm. This dashboard's own data illustrates the mechanism: because returns are drawn from a clustering, occasionally-jumping process, the empirical tail is fatter than the matched Gaussian — the normal curve visibly underestimates both the peak and the extremes. Historical VaR at least lets the data speak; parametric-normal VaR forces the data through an assumption the tails reject (see the notes for the arithmetic of how badly).