The comonotonicity coefficient: A new measure of positive dependence in a multivariate setting

In financial and actuarial sciences, knowledge about the dependence structure is of a great importance. Unfortunately this kind of information is often scarce. Many research has already been done in this field e.g. through the theory of comonotonicity. It turned out that a comonotonic dependence structure provides a very useful tool when approximating an unknown but (preferably strongly) positive dependence structure. As a consequence of this evolution, there is a need for a measure which reflects how close a given dependence structure approaches the comonotonic one. In this contribution, we design a measure of (positive) association between n variables (X1,X2, · · · ,Xn) which is useful in this context. The proposed measure, the comonotonicity coefficient _(X) takes values in the range [0, 1]. As we want to quantify the degree of comonotonicity, _(X) is defined in such a way that it equals 1 in case (X1,X2, · · · ,Xn) is comonotonic and 0 in case (X1,X2, · · · ,Xn) is independent. It should be mentioned that both the marginal distributions and the dependence structure of the vector (X1,X2, · · · ,Xn) will have an effect on the resulting value of this comonotonicity coefficient. In a first part, we show how _(X) can be designed analytically, by making use of copulas for modeling the dependence structure. In the particular case where n = 2, we compare our measure with the classic dependence measures and find some remarkable relations between our measure and the Pearson and Spearman correlation coefficients. In a second part, we focus on the case of a discounting Gaussian process and we investigate the performance of our comonotonicity coefficient in such an environment. This provides us insight in the reason why the comonotonic structure is a good approximation for the dependence structure.