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Quantitative Risk Management / Multivariate Models and Copulas

Dependence Measures: Rank Correlation and Tail Dependence

Pearson correlation is the dependence measure that practitioners reach for first, but it breaks badly in precisely the settings that matter most for risk — heavy-tailed losses, non-elliptical copulas, and stress scenarios. Three failures are worth internalising: correlation cannot distinguish between wildly different joint distributions (Fallacy 1), it cannot always attain the full [−1,1][-1, 1] range for given marginals (Fallacy 2), and maximising correlation does not maximise portfolio tail risk (Fallacy 3). The copula-based alternatives — Kendall's τ\tau, Spearman's ρ\rho, and the tail dependence coefficient λU\lambda_U — avoid all three failures. They are also the quantities used to calibrate copulas to data, making them operational, not merely theoretical.
The punchline: the Gaussian copula has λU=0\lambda_U = 0 regardless of ρ\rho. The tt-copula has λU>0\lambda_U > 0 for any ρ>−1\rho > -1. That single contrast, formalised in this lesson, is the precise statement of what the 2008 CDO market got wrong.
Learning objectives. By the end of this note you should be able to:
  • define comonotonicity and state the additivity of quantiles (Proposition 7.20);
  • name and correct the three correlation fallacies with concrete examples;
  • define Kendall's τ\tau and Spearman's ρ\rho and express them as copula functionals;
  • define the upper and lower tail dependence coefficients λU,λL\lambda_U, \lambda_L;
  • compute λU\lambda_U for the Gumbel copula and λL\lambda_L for the Clayton copula;
  • state the result λ=0\lambda = 0 for the Gaussian copula and λ>0\lambda > 0 for the tt-copula.

In this lesson

    1. Comonotonicity and countermonotonicity
    1. The three correlation fallacies
    1. Kendall's τ\tau and Spearman's ρ\rho
    1. Tail dependence coefficients
    1. Gaussian copula: zero tail dependence

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