Analyzing the use of generalized hyperbolic distributions to value at risk calculations

Resumen

The goal of this paper is to analyze the use of the Generalized Hyperbolic (GH) Distributions to model the US Dollar/Brazilian Real exchange rate in a way to produce more accurate VaR (Value at Risk) measurements. After the GH parameters estimation, several distances were calculated to verify the fitting quality of Normal distribution and GH distribution family to empirical data. The GH Distributions had shown to be more adequate for modeling the US Dollar/Brazilian Real exchange rate, since they produced smaller distances, especially in tails. Additionally, several methodologies for VaR calculation were compared using the Kupiec test: Historical Simulation, RiskMetrics®, unconditional Normal, GH, Normal Inverse Gaussian (NIG) and Hyperbolic, and GARCH models using Normal, GH, Hyperbolic and NIG. The GH Distribution and its subclasses showed better results than unconditional Normal. The use of a GARCH model for volatility forecasting produced satisfactory results, being the main factor of success. Two estimation methods were used: Maximum Log-Likelihood and Minimization of the FOF distance; but both produced similar results. As the Maximum Log-Likelihood showed to be faster we recommend this method. Overall, our recommendation the use of a GH family distribution re-scaled by a GARCH volatility and estimated by Maximum Log-Likelihood.