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

Abstract

We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure (Formula presented.). 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 (Formula presented.) -arbitrage, and prove that it cannot be excluded—unless (Formula presented.) is as conservative as the worst-case risk measure. After providing a primal characterization of (Formula presented.) -arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterization of (Formula presented.) -arbitrage. We show that the absence of (Formula presented.) -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 (Formula presented.). A special case of our result shows that the market does not admit (Formula presented.) -arbitrage for Expected Shortfall at level (Formula presented.) if and only if there exists an EMM (Formula presented.) such that (Formula presented.).