VaR competition: Measuring the degree of adjustment of Value at Risk methodologies

The objective of this paper is the evaluation of different Value at Risk (VaR) methodologies. In particular, four VaR methodologies (Normal, GARCH, Historical Simulation and Extreme Values (EV)), are compared for 36 indexes covering stock-exchanges worldwide. This paper proposes for the EV approach an automatic procedure to obtain the threshold that divides the distribution between extreme values and normal ones, using this threshold to estimate the tail index of the Pareto distribution. This tail index is usually estimated by plotting the Hill Estimator and choosing the value of the threshold in the region where this estimator becomes stable. This procedure is discretional since a decision maker is required in order to fix the threshold. In the present article we propose an automatic procedure based on the computation of successive normality tests over the whole distribution. We establish multicriteria rankings for better hedging the market risk through three concrete measures: the proportion of returns that fell out of VaR value, mean VaR, and finally, the total amount of losses over the VaR. It is shown that, for the lower significance levels, EV methodology with a Pareto distribution for the tails, as built in this paper, is the best suited approach.