Modelling asset return using multivariate asymmetric mixture models with applications to estimation of Value-at-Risk

Value-at-Risk (VaR) is a widely used statistical measure in financial risk management for quantifying the level of risk associated with a specific investment portfolio. It is well-known that historical return data exhibit non-normal features, such as heavy tails and skewness. Current analytical (parameteric) calculation of VaR typically assumes the distribution of the portfolio return to be a normal or log-normal distribution, which results in underestimation and overestimation of the VaR at high and low confidence levels, respectively, when a normal distribution is assumed. This study develops a promising approach in modelling asset returns by fitting multivariate mixture models with asymmetric component densities, in particular, members of the skew-symmetric family and other nonelliptically contoured distributions. We focus on component densities with four or more parameters, including the multivariate skew t (MST) distribution, the multivariate normal-inverse-Gaussian (MNIG) distribution, and the multivariate generalized hyperbolic distribution (MGH) distribution. These distributions have proven to be effective in capturing heterogeneous data with asymmetric and heavy tail behaviour, and can flexibly adapt to a variety of distributional shapes. The fitting of these mixture models can be carried out via the Expectation–Maximization (EM) algorithm. This approach has improved the accuracy of VaR estimation, and an example is demonstrated on a portfolio of three Australian stock returns. The asymmetric mixture models were fitted to the monthly returns of the shares Flight Centre Limited (FLT), Westpac Banking Corporation (WBC) and Australia and New Zealand Banking Corporation Group Limited (ANZ) for the period January 2000 to mid 2013. The VaR estimates predicted by the asymmetric mixture models for a range of significance levels of interest compare favourably to traditional methods based on symmetric models, with significant improvements observed in the accuracy of the estimates. Moreover, it is observed that models with relatively restrictive component densities (such as the normal mixtures, skew normal mixtures and shifted asymmetric Laplace mixtures) require a mixture of two components in order to accommodate the skewness and heavy-tails in the data, whereas the more flexible distributions can adequately capture the distributional shape of data with only a single component.