Extreme Negative Risk Dependent Nature

In 2-dimensional space, Frechet-Hoffding upper and lower bounds define comono-tonicity and countermonotonicity, respectively. Similarly, in the multidimensional case, comonotonicity can be defined using the Frechet-Hoffding upper bound. However, since the multidimensional Frechet-Hoffding lower bound is not a distribution function, there is no obvious extension of countermonotonicity in multidimensions. In this paper, we show that several equivalent results among countermonotonicity, mutually exclusive, Σ-countermonotonicity in Frechet spaces, linking them to probabilistic optimization problems. Firstly, we establish several relations and properties among convex order, supermodular order, concordance order and inverse distribution functions. Next, we summarize several equivalent conditions about countermonotonicity, mutually exclusive in the supermodular order, convex order, variance order, joint distribution function, and equally distributed conditions in expected utility theory and distorted expectation theory. Also, we give a new definition Σ’-countermonotonicity, which is closely related to mutually exclusive. The last but not the least, we investigate Σ-countermonotonic properties, which is supported in any Frechet space. It has also many good conclusions and constitute a minimal class of copulas in some limited conditions. In the last, we will give a financial interpretation of Σ-countermonotonicity via the existing herd be-havior index and some simple properties among the herd behavior indices. Throughout this paper, properties of various extreme negative dependence will be emphasized.