The estimation of VaR and ES based on Gaussian mixture model–Take the Shanghai Composite Index as an example

The key to calculating Value at Risk is whether the assumption of return distribution of financial assets is reasonable. In the first place, we introduce the basic theoretical knowledge of Value at Risk and expected shortfall. In the second place, since the distribution of financial assets returns has the characteristics of sharp peaks and heavy tails, we propose a Gaussian mixture model to describe this feature, and uses EM algorithm to estimate the parameters of the model. In addition, the calculation formulas of Value at Risk and expected loss under Gaussian mixture model are given. In this paper, a case study of Shanghai Composite index is carried out. The optimal order of Gaussian mixture model is determined by Kolmogorov-Smirnov test and AIC criterion. In the end, Value at Risk and expected shortfall are estimated respectively under Gaussian distribution hypothesis and Gaussian mixture distribution hypothesis. The results show that the Gaussian mixture distribution can better fit the return distribution of financial assets at the significance level of 1% and 5%.