Value-at-Risk via mixture distributions reconsidered

Value-at-Risk (VaR) has evolved as one of the most prominent measures of downside risk in financial markets. Zhang and Cheng M.-H. Zhang, Q.-S. Cheng, An Approach to VaR for capital markets with Gaussian mixture, Applied Mathematics and Computation 168 (2005) 1079–1085] proposed an approach to VaR for daily returns based on Gaussian mixtures, which have become rather popular in empirical economics and finance since the seminal paper of Hamilton J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (2) (1989) 357–384]. However, they do not conduct tests to assess the accuracy of the mixture-implied VaR measures. Recently, Guidolin and Timmermann M. Guidolin, A. Timmermann, Term structure of risk under alternative econometric specifications, Journal of Econometrics, 131 (2006) 285–308] showed that Markov mixture models do well in measuring VaR at a monthly frequency, but the results may not hold for daily returns due to their more pronounced non-Gaussian features. This paper provides an extensive application of various Markov mixture models to VaR for daily returns of major European stock markets, including out-of-sample backtesting. To accommodate the properties of daily returns, we consider both Gaussian and Student’s t mixtures, and we compare the performance of both uni- and multivariate models under different parameter updating schemes. We find that a univariate mixture of two Student’s t distributions performs best overall. However, by the example of the recent turmoil in financial markets, we also highlight a weak point of the approach.