Negative Dependence Concept in Copulas and the Marginal Free Herd Behavior Index

TL;DR

This paper introduces a new set of copulas representing negative extreme dependence, explores their properties, and applies them to optimize variance and define a herd behavior index in finance and actuarial science.

Contribution

It presents a novel set of minimal copulas for negative dependence, extending the understanding of dependence structures beyond the bivariate case.

Findings

Proposed copulas coincide with countermonotonic copula in bivariate case.

Identified the minimal copulas in concordance ordering for multivariate cases.

Derived the sharp lower bound of a herd behavior index using the new copulas.

Abstract

We provide a set of copulas that can be interpreted as having the negative extreme dependence. This set of copulas is interesting because it coincides with countermonotonic copula for a bivariate case, and more importantly, is shown to be minimal in concordance ordering in the sense that no copula exists which is strictly smaller than the given copula outside the proposed copula set. Admitting the absence of the minimum copula in multivariate dimensions greater than 2, the study of the set of minimal copulas can be important in the investigation of various optimization problems. To demonstrate the importance of the proposed copula set, we provide the variance minimization problem of the aggregated sum with arbitrarily given uniform marginals. As a financial/actuarial application of these copulas, we define a new herd behavior index using weighted Spearman's rho, and determine the sharp lower bound of the index using the proposed set of copulas.