When should I use a Student-t copula instead of a Gaussian copula?

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AcadiFi · Accepted Answer

The Gaussian copula has zero asymptotic tail dependence for any correlation less than one. In plain terms: no matter how correlated two assets appear in normal times...

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The Student-t copula, parameterized by a correlation matrix plus degrees of freedom v, produces symmetric positive tail dependence: lambda = 2 * T_{v+1}( -sqrt((v+1)(1-rho)/(1+rho)) ). For rho = 0.5 and v = 5, lambda is roughly 0.24 — meaningful joint crash probability. As v rises toward infinity, you recover the Gaussian.

At Kestrel, suppose you're pricing a basket default swap on 20 names. A Gaussian copula with average asset correlation 0.3 might give a first-to-default spread of say 180 bps. Switching to a Student-t with v = 4 and the same correlation could widen the senior tranche spread by 40-60% because of clustered defaults.

Practical advice for AcadiFi candidates: pick v by maximum-likelihood on historical joint exceedances. If your estimated v is above 30, Gaussian is a defensible simplification. If v lands near 3-8, you need the Student-t. Never assume Gaussian in regulated risk reporting without testing tail fit.

When should I use a Student-t copula instead of a Gaussian copula?

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