Measure of financial risk using conditional extreme value copulas with EVT margins

In this paper we propose a method to estimate the value-at-risk (VaR) of a portfolio based on a combination of time series, extreme value theory and copula fitting. Given multivariate financial data, we use a univariate ARMA-GARCH model for each return series. We then fit a generalized Pareto distribution to the tails of the residuals to model the distributions of marginal residuals, followed by a bivariate extreme value copula fitting, which is used to estimate portfolio VaR via simulation. As a first step, this method is applied to two portfolios, each composed of two indexes. As a second step, we extend the method to portfolios based on three indexes. In this case dependence between residuals is modeled by using trivariate nested copulas. The reported results demonstrate that conditional extremevalue copula methods provide a better representation of the dependence structure of multivariate data and produce the most accurate estimates of risk, both for standard and for more extreme VaR quantiles. Comparatively, traditional univariate and multivariate methods result in significantly less accurate risk estimates for most cases. In the context of the international financial crises in the year 2008, the predictive performance of all models decreases significantly. Only copula methods provide acceptable VaR predictions.