A Varying-Coefficient Expectile Model for Estimating Value at Risk

Abstract

This article develops a nonparametric varying-coefficient approach for modeling the expectile-based value at risk (EVaR). EVaR has an advantage over the conventional quantile-based VaR (QVaR) of being more sensitive to the magnitude of extreme losses. EVaR can also be used for calculating QVaR and expected shortfall (ES) by exploiting the one-to-one mapping from expectiles to quantiles, and the relationship between VaR and ES. Previous studies on conditional EVaR estimation only considered parametric autoregressive model set-ups, which account for the stochastic dynamics of asset returns but ignore other exogenous economic and investment related factors. Our approach overcomes this drawback and allows expectiles to be modeled directly using covariates that may be exogenous or lagged dependent in a flexible way. Risk factors associated with profits and losses can then be identified via the expectile regression at different levels of prudentiality. We develop a local linear smoothing technique for estimating the coefficient functions within an asymmetric least squares minimization set-up, and establish the consistency and asymptotic normality of the resultant estimator. To save computing time, we propose to use a one-step weighted local least squares procedure to compute the estimates. Our simulation results show that the computing advantage afforded by this one-step procedure over full iteration is not compromised by a deterioration in estimation accuracy. Real data examples are used to illustrate our method. Supplementary materials for this article are available online.

KEY WORDS:

• Asymmetric squared error loss
• α-mixing
• Expected shortfall
• Local linear smoothing
• One-step weighted least squares
• Value at risk

SUPPLEMENTARY MATERIALS

The supplementary file contains proofs of Lemmas A.1 and A.2.

ACKNOWLEDGMENTS

The authors are grateful for the financial support provided by grants from the following bodies: National Natural Science Foundation of China (No. 71203025 (Xie), No. 71271128 (Zhou)), Strategic Research Grant Scheme, City University of Hong Kong (No. 7008134 (Wan)), National Natural Science Funds for Distinguished Young Scholar (No. 70825004 (Zhou)), NCMIS (Zhou), and State Key Program of National Natural Science Foundation of China (No. 71331006 (Xie and Zhou)). The authors thank the Editor, Associate Editor, and several referees for their constructive comments. All remaining errors, if any, are of the authors.