# Weak comonotonicity

## TL;DR

This paper introduces weak comonotonicity, a flexible dependence concept that generalizes classical comonotonicity, providing new optimality conditions for risk aggregation and sharing in finance and insurance.

## Contribution

It proposes the notion of weak comonotonicity, extending classical dependence, and derives necessary and sufficient conditions for optimization problems involving risk measures.

## Key findings

- Weak comonotonicity generalizes classical comonotonicity.
- Necessary and sufficient conditions for Value-at-Risk and Expected Shortfall aggregation.
- Application to risk-sharing problems as an intermediate dependence concept.

## Abstract

The classical notion of comonotonicity has played a pivotal role when solving diverse problems in economics, finance, and insurance. In various practical problems, however, this notion of extreme positive dependence structure is overly restrictive and sometimes unrealistic. In the present paper, we put forward a notion of weak comonotonicity, which contains the classical notion of comonotonicity as a special case, and gives rise to necessary and sufficient conditions for a number of optimization problems, such as those arising in portfolio diversification, risk aggregation, and premium calculation. In particular, we show that a combination of weak comonotonicity and weak antimonotonicity with respect to some choices of measures is sufficient for the maximization of Value-at-Risk aggregation, and weak comonotonicity is necessary and sufficient for the Expected Shortfall aggregation. Finally, with the help of weak comonotonicity acting as an intermediate notion of dependence between the extreme cases of no dependence and strong comonotonicity, we give a natural solution to a risk-sharing problem.