Nonparametric Estimation of Expected Shortfall

The expected shortfall is an increasingly popular risk measure in financial risk management and it possesses the desired sub-additivity property, which is lacking for the value at risk (VaR). We consider two nonparametric expected shortfall estimators for dependent financial losses. One is a sample average of excessive losses larger than a VaR. The other is a kernel smoothed version of the first estimator (Scaillet, 2004 Mathematical Finance), hoping that more accurate estimation can be achieved by smoothing. Our analysis reveals that the extra kernel smoothing does not produce more accurate estimation of the shortfall. This is different from the estimation of the VaR where smoothing has been shown to produce reduction in both the variance and the mean square error of estimation. Therefore, the simpler ES estimator based on the sample average of excessive losses is attractive for the shortfall estimation.