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Quantitative Risk Management / Multivariate Models and Copulas

Elliptical Distributions: Theory, Properties, and Factor Models

The multivariate normal and the multivariate tt share a structural property: their density depends on the data only through the Mahalanobis distance (x−μ)⊤Σ−1(x−μ)(\mathbf{x}-\boldsymbol{\mu})^\top\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu}), making their level sets ellipsoids. This common structure defines the elliptical family — a class broad enough to include heavy-tailed and light-tailed distributions while preserving the linear algebra that makes the normal tractable for portfolio analysis. The result that matters most for risk management: for any linear portfolio L=a⊤XL = \mathbf{a}^\top\mathbf{X} with X\mathbf{X} elliptical, LL is also elliptical with the same characteristic generator, so the VaR of any portfolio reduces to a one-dimensional quantile calculation scaled by a⊤Σa\sqrt{\mathbf{a}^\top\boldsymbol{\Sigma}\mathbf{a}}. This justifies the analytic VaR formula for elliptical models and makes the class operational.
Learning objectives. By the end of this note you should be able to:
  • define spherical and elliptical distributions and state their characteristic generators;
  • give the stochastic representation X=RAS\mathbf{X} = R A \mathbf{S} and explain the radial variable;
  • derive that linear combinations of elliptical vectors are elliptical (Proposition 6.27);
  • state why Proposition 6.28 (correlation preserved under extreme conditioning) matters for stress tests;
  • apply the Kendall's τ\tau-based estimator of correlation for elliptical data;
  • describe factor models and PCA as dimension-reduction tools for large portfolios.

In this lesson

    1. Spherical distributions
    1. Elliptical distributions
    1. Properties of elliptical distributions
    1. Estimation of correlation for elliptical data
    1. Factor models and PCA

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