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Journal of Financial Econometrics Advance Access published online on November 28, 2007
Journal of Financial Econometrics, doi:10.1093/jjfinec/nbm019
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© 2007 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonparametric Estimation of Expected Shortfall

Song Xi Chen
     Iowa State University

Address for correspondence: Department of Statistics, Iowa State University, Ames, IA 50011-1210, email: songchen{at}iastate.edu


  Abstract

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.

KEYWORDS: expected shortfall, kernel estimator, risk measures, value at risk, weakly dependent

The author thanks the Editor-in-Chief Professor Eric Renault, an Associate Editor and two referees for valuable comments and suggestions which have improved the presentation of the paper. The author also thanks Cheng Yong Tang for valuable computational support and acknowledges support of a National Science Foundation Grant (DMS-0604563).

Received July 12, 2007; revised April 16, 2007; accepted September 24, 2007


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