Archimedean copula that creates strong dependence in the upper tail and weak dependence elsewhere, the structure that fits risks which only occur together when they are extreme.
Two books can show mediocre linear correlation in a normal year and collapse together on the day of the catastrophe. No Pearson correlation describes that; a copula can. The Gumbel copula belongs to the Archimedean family 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, and it is strictly positive as soon as theta exceeds one, meaning the probability that one risk is extreme given that the other is does not go to zero. Its lower tail, by contrast, is asymptotically independent. That is exactly the profile of an aggregate of natural perils or of lines hit by one macroeconomic shock, and it 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.
copule de Gumbel-Hougaard, Gumbel-Hougaard copula, copule à dépendance de queue supérieure