Gumbel copula
An Archimedean copula with strong dependence in the upper tail and weak dependence elsewhere. It fits risks that only occur together when they are extreme.
Internal model as of December 31, 2025, two European books, windstorm and flood: at equal rank correlation, Gumbel adds 62M EUR at the 99.5th percentile over the Gaussian. No Pearson correlation describes two books weakly correlated in a normal year that collapse together on the day of the catastrophe; a copula can. The Gumbel copula is Archimedean and depends on a parameter theta greater than or equal to one. At theta equal to one the risks are independent, and as theta grows dependence concentrates in the upper tail. Its upper tail dependence coefficient equals two minus two to the power one over theta, strictly positive once theta exceeds one: the probability that one risk is extreme given that the other is does not go to zero. The lower tail is asymptotically independent. That is the profile of an aggregate of natural perils or of lines hit by one macroeconomic shock, which is why internal models reach for it where the Gaussian copula systematically understates coincident extremes.
Aggregation of two European books, windstorm and flood, in an internal model as of December 31, 2025. Observed rank correlation is 0.31, which gives a Gumbel theta of 1.45 and an upper tail dependence coefficient of 0.39. The 99.5th percentile of the aggregate comes out at 610M EUR under Gumbel against 548M EUR under a Gaussian copula with the same rank correlation: 62M EUR of capital added by the choice of copula family and nothing else.
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