copulas-and-dependencelisted

Separate the marginals from the dependence - Gaussian, Student t, Clayton and Gumbel copulas, Kendall's tau, tail dependence coefficients, and what fitting the wrong family costs in the joint tail. TRIGGER - copula, Gaussian copula, t copula, Student t copula, Clayton copula, Gumbel copula, Archimedean copula, Sklar's theorem; tail dependence, lower tail dependence, upper tail dependence, lambda_U, "correlation is not dependence", "correlations go to one in a crisis", joint tail probability, joint exceedance; Kendall's tau, Spearman rho, rank correlation, pseudo-observations, inversion of Kendall's tau, copula MLE, copulas python, copulae, statsmodels copula; diversification benefit, "my VaR says the portfolio is safe". SKIP for estimating a covariance matrix and shrinkage (covariance-and-risk-models), for VaR/CVaR methods and their backtests (risk-measures-var-cvar), and for GARCH marginals (volatility-models).
howard-lynn-ye/fin-skills · ★ 1 · AI & Automation · score 77

Install: claude install-skill howard-lynn-ye/fin-skills

# Copulas and dependence **Correlation pins down the middle of a joint distribution and says nothing at all about the corner.** Four copulas can agree on Kendall's tau, on the linear correlation and on both marginals, land within 3% of each other on Spearman's rho, and disagree by a factor of five about how often two assets break their 1% quantile on the same day. Every number marked ✅ Measured is printed by `scripts/copulas.py` (numpy 2.2.6 + scipy 1.13.0, seed 20260909, **about 30 s**). ✅ source-verified means read in the installed scipy source. ## 1. The closed forms | family | Kendall's tau | lower tail dep. | upper tail dep. | |---|---|---|---| | Gaussian(ρ) | `(2/π)·arcsin(ρ)` | **0** for every ρ < 1 | **0** | | t(ρ, ν) | `(2/π)·arcsin(ρ)` | `2·t_{ν+1}(−sqrt((ν+1)(1−ρ)/(1+ρ)))` | same (radially symmetric) | | Clayton(θ) | `θ/(θ+2)` | `2^(−1/θ)` | 0 | | Gumbel(θ) | `1 − 1/θ` | 0 | `2 − 2^(1/θ)` | 🔑 **The tau column is why this skill exists**: `tau` for the t copula does not contain `ν`, so a t copula and a Gaussian copula at the same `ρ` have *identical* Kendall's tau and *identical* linear correlation. ⚠️ Spearman's rho is the one rank measure that does see `ν` — and it barely does; the table below measures it. At **ρ = 0.5**, every family below is set to the same **τ = 1/3**: Gaussian ρ=0.5, t ρ=0.5 ν=4, Clayton θ=1, Gumbel θ=1.5. ✅ Measured — the samplers, against those taus, on 200,000 draws (the Gumbel sampler goes through a positive-stable variable and Clay