Archimedean copula, mirror image of Gumbel, concentrating dependence in the lower tail, a structure relevant for assets and default risk rather than for claims.
The Clayton copula depends on a strictly positive parameter theta and produces strong lower tail dependence and asymptotically zero upper tail dependence. Its lower tail dependence coefficient equals two to the power minus one over theta. It therefore describes variables that collapse together but do not rise together, which is the documented behavior of asset returns in a crisis and of defaults in a credit book. For an insurer it mainly serves the financial side of the balance sheet and the counterparty risk module, while Gumbel serves the technical side. The choice between the two is not cosmetic: applying Clayton to extreme losses amounts to assuming catastrophes do not happen together, an assumption that understates capital exactly where it matters. As with any Archimedean copula, going beyond two dimensions requires either 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.
copule de Cook-Johnson, Clayton copula, copule à dépendance de queue inférieure