Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

Abstract

This paper studies mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal portfolios and the second one is key to ensure the absence of ρ-arbitrage.This leads to a new class of risk measures that are suitable for portfolio selection. We show that ρ belongs to this class if and only if ρ is real-valued and the smallest positively homogeneous risk measure dominating ρ is the worst-case risk measure.We then specialise to the case that ρ is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) ρ-arbitrage as well as the property that ρ is suitable for portfolio selection. Finally, we introduce the new risk measure of Loss Sensitive Expected Shortfall, which is similar to and not more complicated to compute than Expected Shortfall but suitable for portfolio selection -which Expected Shortfall is not.

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.…”