A note on the induction of comonotonic additive risk measures from acceptance sets

We present simple general conditions on the acceptance sets under which their induced monetary risk and deviation measures are comonotonic additive. We show that acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables. A generalization of this result applies to risk measures that are additive for random variables with a priori specified dependence structures, e.g., perfectly correlated, uncorrelated, or independent random variables.