Dynamic Copula Modelling for Value at Risk

This paper proposes dynamic copula and marginals functions to model the joint distribution of risk factor returns affecting portfolios profit and loss distribution over a specified holding period. By using copulas, we can separate the marginal distributions from the dependence structure and estimate portfolio Value-at-Risk, assuming for the risk factors a multivariate distribution that can be different from the conditional normal one. Moreover, we consider marginal functions able to model higher moments than the second, as in the normal. This enables us to better understand why VaR estimates are too aggressive or too conservative. We apply this methodology to estimate the 95%, 99% VaR by using Monte-Carlo simulation, for portfolios made of the SP500 stock index, the Dax Index and the Nikkei225 Index. We use the initial part of the sample to estimate the models, and the the remaining part to compare the out-of-sample performances of the different approaches, using various back-testing techniques.