  We revisit mean-risk portfolio selection in a one-period financial market
where risk is quantified by a positively homogeneous risk measure $\rho$. 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 $\rho$-arbitrage, and prove that it cannot be
excluded -- unless $\rho$ is as conservative as the worst-case risk measure.
  After providing a primal characterisation of $\rho$-arbitrage, we focus our
attention on coherent risk measures that admit a dual representation and give a
necessary and sufficient dual characterisation of $\rho$-arbitrage. We show
that the absence of $\rho$-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 $\rho$. A special case of our result shows that the market
does not admit $\rho$-arbitrage for Expected Shortfall at level $\alpha$ if and
only if there exists an EMM $\mathbb{Q} \approx \mathbb{P}$ such that $\Vert
\frac{\text{d}\mathbb{Q}}{\text{d}\mathbb{P}} \Vert_\infty < \frac{1}{\alpha}$.
