Value-at-Risk-Estimation in the Mexican Stock Exchange Using Conditional Heteroscedasticity Models and Theory of Extreme Values

This work proposes an approach for estimating value at risk (VaR) of the Mexican stock exchange index (IPC) by using a combination of the autoregressive moving average models (ARMA); three different models of the arch family, one symmetric (GARCH) and two asymmetric (GJR-GARCH and EGARCH); and the extreme value theory (EVT). The ARMA models were initially used to obtain uncorrelated residuals, which were later used for the analysis of extreme values. The GARCH, EGARCH and GJR-GARCH models, by including past volatility, are particularly useful both in instability and calm periods. Moreover, the asymmetric models GJR-GARCH and EGARCH handle differently the impact of positive and negative shocks in the market. The importance of the IPC in the Mexican economy raises the need to study its variations, particularly its downward movement; so, we propose to use VaR to calculate the maximum loss that IPC may have, at a certain level of reliability, in a given period of time, using more efficient models to dynamically quantify volatility. The RiskMetrics approach was parallelly used as a way to compare the methodology proposed. The results indicate that the ARMA-GARCH-EVT methodology showed a better performance than RiskMetrics, because of the simultaneous adjustment of ARMA-GARCH models for returns and variances respectively. Although estimates of the EGARCH models had fewer violations of VaR, the estimates of the three models used for volatility were more accurate than the others, evaluated at the same error and reliability levels through the Kupiec Likelihood Ratio test.