Excess invariance and shortfall risk measures

This paper introduces an axiom of excess invariance for risk measures, meaning insensitivity to the amount by which a portfolio’s value exceeds a benchmark. The paper also introduces the class of shortfall risk measures, which are excess-invariant as well as normalized, non-negative, and monotone non-increasing. Shortfall risk measures are suitable for regulatory or risk management applications in which risk is associated with shortfall beneath a benchmark, whereas excess above the benchmark is not important. Non-negativity is incompatible with the usual construction of a cash-additive risk measure from an acceptance set. It is shown how to construct shortfall risk measures with the property of cash additivity subject to positivity, thus retaining the interpretation, valuable in capital adequacy applications, that the risk of an unacceptable portfolio is measured as the amount of cash that must be added to make it acceptable. Representation theorems are provided that characterize shortfall risk measures that are convex and positively homogeneous.