Risk, VaR, CVaR and their associated Portfolio Optimizations when Asset Returns have a Multivariate Student T Distribution

TL;DR

This paper presents a method to analytically evaluate VaR and CVaR for portfolios with assets following a multivariate Student T distribution, enabling better risk assessment and optimization considering fat tails.

Contribution

It introduces a semi-analytical approach for portfolio optimization under Student T distributed returns, linking risk measures to moments of the distribution.

Findings

Analytical expressions for VaR and CVaR with Student T distributions.

Enhanced understanding of risk properties with fat-tailed distributions.

Practical tool for asset allocation considering tail risks.

Abstract

We show how to reduce the problem of computing VaR and CVaR with Student T return distributions to evaluation of analytical functions of the moments. This allows an analysis of the risk properties of systems to be carefully attributed between choices of risk function (e.g. VaR vs CVaR); choice of return distribution (power law tail vs Gaussian) and choice of event frequency, for risk assessment. We exploit this to provide a simple method for portfolio optimization when the asset returns follow a standard multivariate T distribution. This may be used as a semi-analytical verification tool for more general optimizers, and for practical assessment of the impact of fat tails on asset allocation for shorter time horizons.