Estimating Value at Risk and the Expected Shortfall for Heteroscedastic Financial Log Returns: A Two-Stage Method

Value at Risk and the Expected Shortfall are two measurements of market risks for financial assets. Statistically, they are extreme quantiles of the distribution of financial log returns. Though financial log return data are usually both heteroscedastic and fatter-tailed, most of the existing methods in literature only deal with one of the two properties. Motivated by McNeil and Frey (2000), we propose a two-stage model, which is a combination of a tree-structured GARCH(1,1) and a revised version of the Generalized Pareto Distribution(GPD). In the first-stage model, both the number and the value of the tree nodes are chosen by maximizing some conditional reduction of negative log likelihood or the AIC criterion. In the second stage, the shape parameter of the GPD is defined as a linear function of the log estimated volatilities obtained from the first-stage model. This two-stage model not only considers both of the two data properties, but also allows the model to be different for different extents of market changes. Simulations show that our proposed model has advantages when the underlying model is classical GARCH(1,1) with t innovation, or the underlying model is tree-structured GARCH(1,1). We also applied the proposed method to historical log returns of the NASDAQ index and the MRK stock price.