Investigating Extreme Dependences: Concepts and Tools

TL;DR

This paper compares six dependence measures for extreme events in financial models, revealing their differing behaviors and emphasizing the multidimensional nature of tail dependence.

Contribution

It provides explicit formulas and numerical estimates for six dependence measures in extreme regimes across various models, highlighting their differences and implications.

Findings

Conditional correlations can differ significantly from unconditional ones.

Dependence measures show opposite behaviors in tail regimes.

Tail dependence exhibits multidimensional characteristics.

Abstract

We investigate the relative information content of six measures of dependence between two random variables $X$ and $Y$ for large or extreme events for several models of interest for financial time series. The six measures of dependence are respectively the linear correlation $\rho^+_v$ and Spearman's rho $\rho_s(v)$ conditioned on signed exceedance of one variable above the threshold $v$, or on both variables ($\rho_u$), the linear correlation $\rho^s_v$ conditioned on absolute value exceedance (or large volatility) of one variable, the so-called asymptotic tail-dependence $\lambda$ and a probability-weighted tail dependence coefficient ${\bar \lambda}$. The models are the bivariate Gaussian distribution, the bivariate Student's distribution, and the factor model for various distributions of the factor. We offer explicit analytical formulas as well as numerical estimations for these six measures of dependence in the limit where $v$ and $u$ go to infinity. This provides a quantitative proof that conditioning on exceedance leads to conditional correlation coefficients that may be very different from the unconditional correlation and gives a straightforward mechanism for fluctuations or changes of correlations, based on fluctuations of volatility or changes of trends. Moreover, these various measures of dependence exhibit different and sometimes opposite behaviors, suggesting that, somewhat similarly to risks whose adequate characterization requires an extension beyond the restricted one-dimensional measure in terms of the variance (volatility) to include all higher order cumulants or more generally the knowledge of the full distribution, tail-dependence has also a multidimensional character.