Adjusted empirical likelihood for value at risk and expected shortfall

Value at risk (VaR) and expected shortfall (ES) are widely used risk measures of the risk of loss on a specific portfolio of financial assets. Adjusted empirical likelihood (AEL) is an important non parametric likelihood method which is developed from empirical likelihood (EL). It can overcome the limitation of convex hull problems in EL. In this paper, we use AELmethod to estimate confidence region for VaR and ES. Theoretically, we find that AEL has the same large sample statistical properties as EL, and guarantees solution to the estimating equations in EL. In addition, simulation results indicate that the coverage probabilities of the new confidence regions are higher than that of the original EL with the same level. These results showthat the AEL estimation for VaR and ES deserves to recommend for the real applications.