The Multivariate Normal Distribution and Its Inadequacy for Risk

Every portfolio VaR model must make assumptions about joint risk-factor movements. The variance-covariance method — the simplest and most widely used analytical VaR approach — assumes that risk factors are jointly normally distributed. This assumption is the load-bearing wall under the method: it makes the loss distribution of any linear portfolio analytically tractable, it allows covariance estimation from sample data, and it has zero implementation cost. But the Dow Jones data say otherwise: Mardia's tests reject multivariate normality at daily, weekly, and monthly horizons with effectively zero p-value. The multivariate normal is not wrong in the way a poor approximation is wrong — it fails on the dimension that matters most for risk management. This note develops the multivariate normal's properties precisely, so the failure mode is concrete rather than vague: the joint distribution assigns far too little probability to joint tail events and imposes zero tail dependence.

Learning objectives. By the end of this note you should be able to: