Copula visualiser
Correlation says how much two risks move together; the copula says where. Four families at the same Kendall's tau, sampled live, so you can see tail dependence appear in the corners rather than memorise it as a formula.
1,500 pairs · ρ = 0.71
Dependence
Lower tail λ_L0.000
Upper tail λ_U0.000
Both in the worst 5%
Joint-lower count26
Joint-upper count31
If independent≈ 4
All four families are shown at the same Kendall’s τ, so overall dependence is held fixed and only its shape changes. Gaussian has no tail dependence at all; the t copula adds it in both tails; Clayton clusters in the joint-lower corner (both risks crash together); Gumbel in the joint-upper. Flip between Gaussian and Clayton at τ = 0.5 and watch the lower-corner count jump while the parameter stays “equivalent” — that gap is why copula choice, not just correlation, drives joint extreme-loss probabilities.
What CS2 wants you to take from this
Sklar’s theorem splits any joint distribution into its margins and a copula, so dependence can be studied on the unit square with the margins stripped away. The families then differ exactly where it matters for insurance: the Gaussian copula has zero tail dependence however high its ρ, the t copula has symmetric tail dependence that grows as ν falls, Clayton concentrates dependence in the lower tail and Gumbel in the upper. The exam asks for the definitions of λ_L and λ_U, the generator functions of the Archimedean families, and the qualitative story this page draws: two portfolios can share a correlation and still have utterly different probabilities of blowing up together.
Make it stick. Extreme joint losses sit alongside ruin in CS2; the ruin theory simulator covers the surplus process, and the distributions cheat sheet the margins. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.
For education only.