We revisit mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure $\rho$. We make three contributions. First, we introduce the new axiom of sensitivity to large expected losses and show that it is key to ensure the existence of optimal portfolios. Second, we give primal and dual characterisations of (strong) $\rho$-arbitrage. Finally, we use our conditions for the absence of (strong) $\rho$-arbitrage to explicitly derive the (strong) $\rho$-consistent price interval for an external financial contract. This is joint work with Martin Herdegen.