Nonparametric estimation and comparative analyses of ES in risk measure with applications

Expected shortfall (ES) model developed recently is a powerful mathematical tool to measure and control financial risk. In this paper, two-step kernel smoothed processes are used to develop a two-step nonparametric estimator of ES. Comparisons between the proposed two-step kernel smoothed ES estimator to the existing fully empirical ES estimator and one-step kernel smoothing ES estimator were made by calculating expectation and variance of them. It is of great interest that the proposed two-step kernel smoothed ES estimator has been shown to increases the variance, totally different from the existing result that the kernel smoothed VaR estimator can produces reduction in both the variance and the mean square error. In addition, the simulation results conform to the theoretical analysis. In the related empirical analysis, the close-ended funds in Shanghai and Shenzhen stock markets were explored to compute the empirical ES estimates and kernel smoothing ES estimates for risk analysis. And the RAROC of the funds sample based on weekly return and ES were computed to make the performance evaluation of the funds. The empirical results show that, with weekly return, the method based on ES is higher reliability than those based on VaR.