Bayesian Empirical Likelihood For VaR, MS And ES

Measure of market risk is very complicated. The model of VaR allows quantitative calcula-tions of maximum possible loss of a financial portfolio in a holding period. However, in practice, VaR has some shortages. In order to overcome these defects, CVaR (Conditional Value at risk) and ES (Expected Shortfall) are introduced, as well as the MS (Median Shortfall) which appears recently. VaR, MS and ES have more advantages than ever measure methods, which solve more problems and are widely used in practice. On the other hand, Bayesian empirical likelihood effec-tively combines prior information with sample information and inherits the advantages of empirical likelihood. It is a nonparametric method which can use priori information. Under the condition of small samples, make full use of historical information can improve the estimation.This article uses Bayesian empirical likelihood method to estimate VaR, MS and ES. First, we get the posterior distributions of the VaR,MS and ES by combining prior distribution with em-pirical likelihood. Then we use the posterior distributions to estimate the VaR, MS and ES. Finally we prove the consistency of the maximum empirical likelihood estimator and asymptotic normal-ity of the posterior distribution under suitable conditions. Simulations show that, under a fixed confidence level, comparing to empirical likelihood, Bayesian empirical likelihood method has the similar coverage and shorter intervals when prior information is accurate. Under the condition of small sample and accurate prior information, Bayesian empirical likelihood has higher coverage and shorter intervals than empirical likelihood method. If the prior is very far from the truth, the coverage of Bayesian empirical likelihood is very bad.This paper has the following features:1.In part of empirical likelihood, we use Eerson and Owen (2009)’s method to solve the convex hull problem by adding two suitable sample points which can guarantee the sample mean is constant.2.Under suitable conditions, the posterior distributions of VaR, MS and ES have asymptotic normality.3.Bayesian empirical likelihood method can improve the estimation precision by combining prior information with sample information, especially in the case of small sample and accurate prior information. Bayesian empirical likelihood method is a nonparametric method which en-riches the theory of VaR, MS and ES estimation, and provides more tool for practical workers.