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Value at Risk (VaR) plays a central role in risk management. There are several approaches for the estimation of VaR, such as historical simulation, the variance-covariance (also known as analytical), and the Monte Carlo approaches. Whereas the first approach does not assume any distribution, the last two approaches demand the joint distribution to be known, which in the analytical approach is frequently the normal distribution. The copula theory is a fundamental tool in modeling multivariate distributions. It allows the definition of the joint distribution through the marginal distributions and the dependence between the variables. Recently the copula theory has been extended to the conditional case, allowing the use of copulae to model dynamical structures. Time variation in the first and second conditional moments is widely discussed in the literature, so allowing the time variation in the conditional dependence seems to be natural. This work presents some concepts and properties of copula functions and an application of the copula theory in the estimation of VaR of a portfolio composed by Nasdaq and S&P500 stock indices.

This paper proposes a method for estimating the VaR of a portfolio based on copula and extreme value theory. Each return is modeled by ARMA-GARCH models with the joint distribution of innovations modeled by copula. The marginal distributions are modeled by the generalized Pareto distribution in the left tail (large loss) and empirical distribution otherwise. The copula is estimated by an estimator which gives more weight to observations with large loss. The method is applied to a two-asset portfolio and compared to other traditional methods (JEL: C15, D81,G10).