Kendall tau and Spearman rho
Dependence measures built on ranks rather than values, invariant under increasing transformation, and the only measures consistent with copula calibration.
Maurice Kendall published his rank correlation measure in 1938 in Biometrika. Pearson correlation measures linear dependence between values, so it changes if you take logarithms, and it can read 0.7 for a pair and 0.4 for the same pair expressed differently. Rank measures avoid that trap. Kendall tau counts the proportion of concordant pairs minus the proportion of discordant pairs; Spearman rho is Pearson correlation applied to ranks. Both are invariant under strictly increasing transformation. They therefore depend only on the copula and not on the marginal laws, and that is exactly what you want when separating dependence structure from each risk's individual behavior. Their practical use is calibration. Every copula family admits a closed relation between its parameter and Kendall tau: for the Gumbel copula with parameter theta, tau equals one minus one over theta. Estimate tau on the data, invert the relation, and you get the parameter without going through an unstable maximum likelihood fit.
Calibration of a Gumbel copula between annual windstorm and flood losses, data from 1995 to 2025, thirty-one observations. Observed Kendall tau 0.31. The relation tau equals one minus one over theta gives theta equal to one over one minus 0.31, that is 1.45. The Pearson correlation of the same series was 0.52, and using it directly as the copula parameter would have been an error of kind, not of degree.
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