Application of the multivariate skew normal mixture model with the EM Algorithm to Value-at-Risk

Abstract

Since returns of financial assets generally exhibit skewness and kurtosis, modelling returns using a distribution with the ability to capture both of these statistical aspects will increase the accuracy of risk forecasts based on these distributions. The authors propose the use of the multivariate skew normal (MVSN) mixture model to fit asset returns in order to increase the accuracy of Value-at-Risk (VaR) estimates. This paper presents a novel application of the MVSN mixture model to estimate VaR. There is generally no explicit analytical solution for the parameters of the MVSN mixture model via maximum likelihood estimation (MLE), therefore the use of the Expectation Maximization (EM) Algorithm is proposed in order to find the parameter estimates of the model. The example provided in this paper consists of a portfolio of monthly returns of six shares listed on the Australian Securities Exchange (ASX). The shares are BHP Billiton Limited (BHP), Commonwealth Bank of Australia (CBA), Cochlear Limited (COH), News Corporation (NWS), Origin Energy (ORG), and Wesfarmers Limited (WES). Hence, the dimensionality, p, of this portfolio is six. The period of analysis for the data is 01/01/1998-01/04/2011. This paper models the MVSN mixture model with a number of mixtures ranging from one to four. A mixture of multivariate normal densities is modelled for comparison to the MVSN mixture model. We find that for one to three mixtures, the MVSN mixture model provides an improved fit. The improved fit of the MVSN mixture model is translated to the performance of the VaR models, where the results show that for one to three numbers of mixtures, the VaR model using the MVSN mixture model assumption indicates improved risk forecasts when compared to the mixture of multivariate normal densities. Furthermore, for the example examined, we find that the model which incorporates the skewness parameter (MVSN mixture model) requires a fewer number of mixtures when compared to a mixture of normal densities. This is an interesting result as reduced model complexity requires less computational ability, computation time, and will results in decreased computational anomalies.