Adjusted Empirical Likelihood For VaR And ES

In the field of finance, risk management is the basic business. With the developing of liberal-ization, globalization, innovation of finance, financial institutions face more and more complicated environment and invisibility of risk. So theory scholars and application workers focus on how to characterize and estimate risk. Value at Risk(VaR),Conditional Value at Risk(CVaR) and Expected Shortfall(ES) can effectively solve the problem that traditional risk measurement method can’t do. So it has been accepted and applied by financial institutions.In this paper, we mainly study the confidence regions for VaR and ES and the confidence interval for ES, based on the research results of predecessors. Baysal and Staum (2008)[empirical likelihood for value at risk and expected shortfall, The Journal of Risk.2008,11(1):3-32.] discuss the empirical likelihood for VaR and ES. But considering the problem that whether there is a solution for the empirical likelihood method, so we try to use the Adjusted Empirical Likelihood method to estimate the confidence regions for VaR and ES and the confidence interval for ES.We firstly discuss the confidence regions using the adjusted empirical likelihood based on the semiparametric model. Secondly, we discuss the confidence regions for VaR and ES and the confidence interval for ES and prove the limits properties of the estimations. In addition, we compare our methods with previous works by simulation. In theory, we find that it has same properties between adjusted empirical likelihood and empirical likelihood. The simulation results indicate that the higher coverage probabilities of the new confidence regions can be obtained. More importantly, the methods and computation of adjusted empirical likelihood are more simple and easy to explain. Besides, it always has solution. These results have important significance for the application workers.In the following, we summary some characteristics in this paper:1. In fact, it is shown to be very effective on solving some application problems associated with the use of adjusted empirical likelihood. It is well known that, we can overcome the short-coming of empirical likelihood that the convex hull{g(xi,θ),i=1,…,n} can contain0as its inner point. For all cases there are solutions to reduce the difficulty of complicated computation.2.Under certain conditions, the higher coverage probabilities of the new confidence regions can be obtained in this paper compared with previous.3.The conclusions of this paper can be riched and improved the theory of estimation of risk measurement tool and provide convenient tool to the practical application workers.