Multi-asset return risk measures

We revisit the recently introduced concept of return risk measures (RRMs). We extend it by allowing risk management via multiple so-called eligible assets. The resulting new risk measures are called multi-asset return risk measures (MARRMs). We analyze properties of these risk measures. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex. Furthermore, we state conditions to avoid inconsistent risk evaluations. Then, we point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). This is used to obtain various dual representations of MARRMs. Moreover, we compare RRMs, MARMs, and MARRMs in numerous case studies. First, using typical continuous-time financial markets and different notions of acceptability of losses, we compare MARRMs and MARMs and draw conclusions about the cost of risk mitigation. Second, in a real-world example, we compare the relative difference between RRMs and MARRMs in times of crisis.