Student t copula
An elliptical copula that keeps the Gaussian copula's correlation matrix but adds a degrees-of-freedom parameter producing symmetric, nonzero tail dependence.
David Li, 2000, Journal of Fixed Income: the Gaussian copula goes into CDO tranche pricing, and the crisis of 2007 and 2008 would show what it left out. For any correlation strictly below one, its tail dependence is zero: it asserts that two highly correlated risks stop coinciding far enough into the tail. The Student t copula fixes that flaw while keeping the convenience of a correlation matrix. It adds a degrees-of-freedom parameter, and its tail dependence is strictly positive, symmetric, and stronger the lower the degrees of freedom; as they tend to infinity you recover the Gaussian. It scales to fifty dimensions without difficulty, its decisive advantage over Archimedean copulas when aggregating a whole balance sheet. Its limitation is that symmetry, which forces the same dependence in the upper and lower tail, while in a real book losses coincide and favorable development much less so.
The role of the Gaussian copula in understating joint losses on CDO tranches backed by US mortgages is the most expensive illustration of the problem, laid bare in 2007 and 2008. On an aggregate of eight risk modules with an average correlation of 0.25, switching from a Gaussian copula to a Student t copula with four degrees of freedom raises aggregate SCR by roughly 9% without changing a single marginal distribution.
Student copula, t copula, tail dependence
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