An Empirical Characteristic Function Approach to VaR Under a Mixture-of-Normal Distribution with Time-Varying Volatility

Abstract

This article considers risk measures constructed under a discrete mixture-of-normal distribution on the innovations of a GARCH model with time-varying volatility. The authors use an approach based on a continuous empirical characteristic function to estimate the parameters of the model using several daily foreign exchange rates’ return data. This approach, compared to the likelihood-based approach, has several important advantages as a method for estimating this type of models; in particular, the characteristic function, unlike its likelihood function counterpart, is always uniformly bounded over the model’s parameter space due to the Fourier transformation. To evaluate VaR and expected shortfall measures obtained from alternative specifications, the authors construct several evaluation criteria, such as the number of violations and the sum square of violations. Based on these criteria, the authors find that both the VaR and expected shortfall measures obtained from the proposed model outperform those obtained from other competing models.

Keywords: Value at Risk, Expected Shortfall, Mixtures of Normal, GARCH, Characteristic Function

JEL Classification: C22, C53, G17