Dynamic Semiparametric Models for Expected Shortfall (and Value-at-Risk)

Expected Shortfall (ES) is the average return on a risky asset conditional on the return being below some quantile of its distribution, namely its Value-at-Risk (VaR). The Basel III Accord, which will be implemented in the years leading up to 2019, places new attention on ES, but unlike VaR, there is little existing work on modeling ES. We use recent results from statistical decision theory to overcome the problem of “elicitability” for ES by jointly modelling ES and VaR, and propose new dynamic models for these risk measures. We provide estimation and inference methods for the proposed models, and con…rm via simulation studies that the methods have good …nite-sample properties. We apply these models to daily returns on four international equity indices, and …nd the proposed new ES-VaR models outperform forecasts based on GARCH or rolling window models. Keywords: Risk management, tails, crashes, forecasting, generalized autoregressive score. J.E.L. codes: G17, C22, G32, C58. For helpful comments we thank Tim Bollerslev, Rob Engle, Jia Li, Nour Meddahi, and seminar participants at the Bank of Japan, Duke University, EPFL, Federal Reserve Bank of New York, Hitotsubashi University, New York University, Toulouse School of Economics, the University of Southern California, and the 2015 Oberwolfach Workshop on Quantitative Risk Management where this project started. The …rst author would particularly like to thank the …nance department at NYU Stern, where much of his work on this paper was completed. Contact address: Andrew Patton, Department of Economics, Duke University, 213 Social Sciences Building, Box 90097, Durham NC 27708-0097. Email: [email protected].