Star-shaped acceptability indexes

Abstract

We propose the star-shaped acceptability indexes as generalizations of both the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013) in the same vein as star-shaped risk measures generalize both the classes of coherent and convex risk measures. We characterize acceptability indexes through star-shaped risk measures, star-shaped acceptance sets, and as the minimum of some family of quasi-concave acceptability indexes. Further, we introduce concrete examples under our approach linked to Value at Risk, riskadjusted reward on capital, reward-based gain-loss ratio, monotone reward-deviation ratio, and robust acceptability indexes. We also expose an application regarding optimization of performance.

Star-Shaped deviations

“…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.…”