On the link between monetary and star-shaped risk measures

Star-Shaped deviations

“…This is a contradiction. [17] show that the Star-Shapedness of the minimum of a family of Convex risk measures is closely related to the behavior of each measure around 0. The same holds for deviation measures.…”

“…However, note that beyond Star-Shapedness and Convexity, which play a key role in the Theorem, we ask nothing but the most basic axioms of deviation measures, namely Non-Negativity and Translation Insensitivity. While in [5], they also demand Normalization, the role of Normalization was explored in [17].…”

“…The key-point in this theory is that a monetary risk measure is Star-Shaped if and only if it is the minimum of a family of Convex risk measures. This class gained some attention in the literature when [13] explores allocations of Star-Shaped risk measures, [17] relate them to the broader class of monetary risk measures, [10] consider portfolio optimization and arbitrage, and [24] explores the interplay with Star-Shaped acceptability indexes.…”