Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

Abstract

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.