Asymmetric Dependence Implications for Extreme Risk Management

Abstract

Empirical evidence shows that asset returns appear to be more highly correlated on the downside than on the upside, particularly for equities. This is in addition to the well-established property that return variances vary over time. GARCH-type models allow for the latter, but disturbances are typically assumed to be multivariate Gaussian or Student-t, which forces the correlations to be symmetric. This is especially problematical in estimating extreme risk exposures using standard measures like value at risk (VaR) and expected shortfall (ES), for which lower tail dependence causes symmetric models to underestimate downside exposure. Tsafack proves this proposition using a new concept of lowerstochastic ordering. He then explores how much difference it makes to estimate VaR and ES for portfolios of U.S. and Canadian stocks and bonds allowing dependence in the extreme left tail with a Gumbel copula. The data show significant asymmetric tail dependence especially for stocks, and the models that impose symmetry underestimate the risk of extreme tail events. VaR at the 5% level is fairly accurately estimated by standard symmetric models but they do not capture the behavior of the more remote tails, so that VaR at 1% and 0.5% and ES at 5% and beyond are underpredicted.