Clayton copula
Archimedean copula, the mirror image of Gumbel, with dependence concentrated in the lower tail. It suits assets and default risk rather than claims.
Clayton, 1978, Biometrika: David Clayton proposes the model for association in bivariate life tables, in studies of familial disease. The Clayton copula depends on a strictly positive parameter theta. It produces strong lower tail dependence and asymptotically zero upper tail dependence, and its lower tail dependence coefficient equals two to the power minus one over theta. The variables collapse together without rising together, which is how asset returns behave in a crisis and how defaults behave in a credit book. For an insurer it mainly serves the financial side of the balance sheet and the counterparty default risk module, while Gumbel serves the technical side. Applying Clayton to extreme losses amounts to assuming catastrophes do not happen together, an assumption that understates capital exactly where it matters. Like any Archimedean copula, it goes beyond two dimensions only with a hierarchical structure or a single parameter for the whole portfolio, which is rarely credible.
Counterparty default risk module of an insurer as of June 30, 2026, exposure to twelve reinsurers rated A or better. A Clayton copula with theta 0.9 gives a lower tail dependence coefficient of 0.46, against zero under independence. The 99.5th percentile loss moves from 31M to 58M EUR, because the copula allows two simultaneous defaults, which the independence assumption made nearly impossible.
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