The loss operator requires a distributional model for risk-factor changes. The simplest assumption — changes are iid and normal — underlies the variance-covariance method for VaR. That assumption is wrong, empirically and consistently, across every liquid market and every holding period below a month. The departures are not edge cases; they are the stable, reproducible empirical regularities collected under the label stylised facts.
This note documents the six univariate and four multivariate stylised facts. They are the empirical foundation for GARCH models (Topic 3 lessons 2–4), extreme value theory, and copula-based multivariate models. Every risk model that follows in this unit is, in a sense, a response to the failures documented here.
Learning objectives. By the end of this note you should be able to:
- state the six univariate stylised facts and explain what each implies for return modelling;
- compute sample skewness and kurtosis and apply the Jarque–Bera statistic;
- explain volatility clustering and why it implies the iid assumption is wrong;
- describe aggregational Gaussianity and state the horizon at which it approximately holds;
- state the four multivariate stylised facts and explain what tail dependence means for portfolio risk;
- articulate the "correlations go to one in stress" fallacy and why naive rolling correlations can mislead.
In this lesson
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- Setup and notation
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- Six univariate stylised facts
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- The Jarque–Bera normality test
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- Four multivariate stylised facts
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- Why stylised facts matter for capital calculation
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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