Risk Estimation for GARCH Processes with Heavy-Tailed Innovations

Abstract

The standard measures of risk such as RiskMetrics compute the Value-at-Risk as the maximum probability of loss of an investment over a certain period of time given a chosen confidence level. There are non-parametric and parametric methods to compute VaR. A non-parametric approach simulates the probability of distribution of future returns. The unconditional parametric approach assumes that location and scale components of the returns are constant and reduces the VaR problem to computing the γ-quantile of the returns for a given nominal level γ. A conditional approach assumes non-constant location and scale components. A standard implementation of this approach involves the ARCH/GARCH models. However, the popularity of this approach for risk management is restricted due to risk of misspecification of a GARCH model and of distributions for its conditional innovations. In this paper we focus on investigation of the parametric conditional approach under these both problems. The empirical limitation of the assumption of a Gaussian distributions of portfolio returns has been well documented in the empirical research conducted over the last more than twenty years. It has been shown in several studies that returns exhibit high kurtosis and skewness that are incompatible with the normality assumptions (see, Fama (1965), Blattberg and Gonedes (1974), Marinelli et.al (2006)). A number of studies have showed how dramatically differ the estimates of VaR obtained under wrong assumptions with respect to the underlying return process (see, Beder, 1995). Some recently proposed risk measure frameworks deal with these futures of high-frequency financial data. One natural approach to overcome these inconsistencies is to adopt the model with heavy-tailed innovations (Frey and McNeil 2000, Marinelli et al. 2004, Hang Chan et al. (2007)....). A GARCH model with generalized Pareto distribution for the innovations was considered in Frey and McNeil (2000). They proposed a two-step procedure to compute conditional VaR for GARCH model. Hang Chan et al. (2007) work under different assumption than of Pareto distributed innovations, they suppose the model with heavy-tailed innovations. They compute VaR within the non-parametric framework. However, this paper does not provide a backtesting results. Marinelli et al. (2004) assume that returns are heavy-tailed, e.g., follow a stable law and provide comparison of the approach based on assumption of Paretian stable returns with the approach based on assumption of Gaussian returns and on the Extreme Value Theory. However, this paper does not consider the parametric method for GARCH model with heavy-tailed innovations. Our work is closely related to the last two papers. Our contribution consist of numerical study which compares a backtesting procedure of alternative scheme to compute VaR. We consider the estimates for VaR and Expected Shortfalls (CVaR).