Mean-ρ portfolio selection and ρ-arbitrage for coherent risk measures

Mathematical Finance

39 Pages Posted: 2 Nov 2020 Last revised: 20 Sep 2021

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 11, 2020

Abstract

We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure ρ. We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation ρ-arbitrage, and prove that it cannot be excluded -- unless ρ is as conservative as the worst-case risk measure.

After providing a primal characterisation of ρ-arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterisation of ρ-arbitrage. We show that the absence of ρ-arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of ρ. A special case of our result shows that the market does not admit ρ-arbitrage for Expected Shortfall at level α if and only if there exists an EMM Q ≈ P such that ll dQ/dP ll∞ < 1/α.

Keywords: portfolio selection, coherent risk measures, dual characterisation, ρ-arbitrage, fundamental theorem of asset pricing

JEL Classification: G11, D81, C61

Suggested Citation

Herdegen, Martin and Khan, Nazem, Mean-ρ portfolio selection and ρ-arbitrage for coherent risk measures (September 11, 2020). Mathematical Finance, Available at SSRN: https://ssrn.com/abstract=3691027 or http://dx.doi.org/10.2139/ssrn.3691027

Martin Herdegen

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

Nazem Khan (Contact Author)

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

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