Risk Measures Based on Multivariate Skew Normal and Skew t‐Mixture Models

Abstract

It is widely recognized that financial stock returns do not always follow the normal distribution. Typically, they exhibit non‐normal features such as skewness, heavy tails and kurtosis. In this chapter, we consider the application of multivariate non‐normal mixture models for modelling the joint distribution of the log returns in a portfolio. Formulas are then derived for some commonly used risk measures including probability of shortfall (PS), Value‐at‐Risk (VaR), expected shortfall (ES) and tail‐conditional expectation (TCE), based on these models. Our focus is on skew normal and skew t ‐component distributions. These families of distributions are generalizations of the normal distribution and t ‐distribution, respectively, with additional parameters to accommodate skewness and/or heavy tails, rendering them suitable for handling the asymmetric distributional shape of financial data. As linear transformations of the quantities under consideration also have mixtures of skew‐normal or skew t ‐distributions, the PS, VaR and TCE, and other risk measures of an asset portfolio, can be expressed explicitly in terms of the parameters of the fitted mixture models. This approach is demonstrated on a real example of a portfolio of Australian stock returns and the performances of these models are compared with the traditional normal mixture model.