A Detailed Comparison of Value at Risk in International Stock Exchanges

Resumen

This work investigates the performance of different models of Value at Risk (VaR). We include a wider range of methods (Parametric, Historical simulation, Monte Carlo simulation, and Extreme value theory models) and several models to compute the conditional variance (exponential moving averages, GARCH and asymmetric GARCH models) under Normal and Student�s t-distribution of returns. We analyse four European indexes (IBEX-35, CAC40, DAX and FTSE100), the American Dow Jones and S&P 500 indexes, the Japanese Nikkei 225 index and the Hong Kong Hang Seng index. We examine two periods: a stable period and a volatile one. To choose the best model, we employ a two-stage selection approach. First, we test the accuracy of different models of VaR. We use the unconditional and conditional coverage test, the Back-Testing criterion and the dynamic quantile test. A model survived if all tests indicated the model is accurate. With regard to the first stage, the best models are Parametric and Extreme value theory methods, when they use asymmetric and nonasymmetric GARCH models under Student�s t-distribution of returns. Second, we evaluate the loss function of these models. We use several non-parametric tests to test the superiority of a VaR model in terms of the loss function. The result of the second stage indicates that the best model is a Parametric model with conditional variance estimated by asymmetric GARCH model under Student�s t-distribution of returns. Nowadays the Parametric models are not as popular because some authors argue that the most conventional parametric specifications have failed in capturing some rare events. However, this paper shows that these models can obtain successful VaR measures if conditional variance is estimated with a GARCH model to capture the characteristic of the returns. This model is usually an exponential GARCH under Student�s t-distribution of returns.