Value At Risk Using Garch Volatility Models Augmented With Extreme Value Theory

Abstract

Accurate estimation of Value at Risk (VaR) and Expected Shortfall ( ES) is critical in the management of extreme market risks. These risks occur with small probability, but the financial impacts could be large.Both symmetric and non-symmetric GARCH stochastic volatility models are investigated as traditional methods to estimate 99% 10 day VaR and ES. Some important practical problems in GARCH model fitting are highlighted, especially the convergence of these models when the sample period contains extreme return observations. As a solution, Extreme Value Theory (EVT) models that focus especially on extreme market returns are considered. The Peaks Over Threshold (POT) approach is followed and a proposal is provided for the scaling of one day to ten day estimates.As a novel approach, this paper considers the augmentation of the GARCH models with EVT forecasts during periods where the first do not converge. Various combinations are investigated and applied to the JSE Financials Index (J580). Model performance is judged by the actual number of VaR and ES violations compared to the expected number, where the latter is taken as the number of return observations multiplied by 0.01. This augmentation approach provided impressive results for the data under consideration, although it is also clear that no single forecasting model is universally optimal and that the choice will depend on the nature of the data.