Gaussian and t Copulas: Tail Dependence and the 2008 Crisis

The Gaussian and $t$ copulas are the workhorses of implicit multivariate modelling: whenever practitioners fit a multivariate normal or Student-$t$ distribution to return data, they are implicitly choosing one of these copulas. The critical difference between them — zero vs. positive tail dependence — is not visible in correlation or rank correlation estimates. It only becomes visible at extreme quantile levels, which is precisely where the models are used for VaR, stress testing, and CDO tranche pricing. A $t_4$ copula generates joint 0.5%-quantile exceedances 2.79 times as often as the Gaussian copula with the same correlation; in a portfolio of five stocks, the factor grows to 7.68. These are not small rounding differences — they change the risk profile by an order of magnitude in the deep tail.

David Li's (2000) CDO pricing model applied the Gaussian copula to model the dependence of default times: a tractable closed-form approach that could be implemented with a single correlation parameter. The fatal flaw: the Gaussian copula's zero tail dependence means that, in the model, joint defaults in the tail occur almost independently. The tranche prices it produced systematically underpriced the cost of correlated defaults, creating the conditions for the 2007–08 structured credit market collapse.

Learning objectives. By the end of this note you should be able to: