A Dual Characterisation of Regulatory Arbitrage for Coherent Risk Measures

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