Sensitivity to Large Losses and ρ-Arbitrage for Convex Risk Measures

42 Pages Posted: 11 Oct 2021 Last revised: 15 Feb 2022

See all articles by Martin Herdegen

Martin Herdegen

University of Warwick - Department of Statistics

Nazem Khan

University of Warwick - Department of Statistics

Date Written: September 17, 2021

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.

Keywords: portfolio selection, ρ-arbitrage, convex risk measures, dual characterisation, sensitivity to large losses, Expected Shortfall

JEL Classification: G11, D81, C61

Suggested Citation

Herdegen, Martin and Khan, Nazem, Sensitivity to Large Losses and ρ-Arbitrage for Convex Risk Measures (September 17, 2021). Available at SSRN: https://ssrn.com/abstract=3925492 or http://dx.doi.org/10.2139/ssrn.3925492

Martin Herdegen (Contact Author)

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

Nazem Khan

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

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