The purpose of this paper is to dispel some common misunderstandings about capital adequacy rules based on Expected Shortfall. We establish that, from a theoretical perspective, Expected Shortfall based regulation can provide a misleading assessment of tail behavior, does not necessarily protect liability holders’ interests much better than Value-at-Risk based regulation, and may also allow for regulatory arbitrage when used as a global solvency measure. We also show that, for a value-maximizing financial institution, the benefits derived from protecting its franchise may not be sufficient to disincentivize excessive risk taking. We further interpret our results in the context of portfolio risk measurement. Our results do not invalidate the possible merits of Expected Shortfall as a risk measure but instead highlight the need for its cautious use in the context of capital adequacy regimes and of portfolio risk control.

The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper is to investigate the class of surplus-invariant acceptance sets. We argue that surplus invariance is a reasonable requirement from a regulatory perspective, since the corresponding capital adequacy tests do not depend on the surplus of a financial institution, which benefits exclusively its shareholders, but only on the default profile, which affects its liability holders. We provide a detailed analysis of surplus-invariant acceptance sets and their associated risk measures and we discuss the link with loss-based and excess-invariant risk measures, recently studied by Cont et al. (2013) and by Staum (2013), respectively.

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp(X) does not exceed a pre-specified threshold g(p) for every probability level p∈[0,1]. Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.