Risk Measures: Robustness, Elicitability, and Backtesting.

Risk measures are used not only for financial institutions' internal risk management but also for external regulation (e.g., in the Basel Accord for calculating the regulatory capital requirements for financial institutions). Though fundamental in risk management, how to select a good risk measure is a controversial issue. We review the literature on risk measures, particularly on issues such as subadditivity, robustness, elicitability, and backtesting. We also aim to clarify some misconceptions and confusions in the literature. In particular, we argue that, despite lacking some mathematical convenience, the median shortfall-that is, the median of the tail loss distribution-is a better option than the expected shortfall for setting the Basel Accords capital requirements due to statistical and economic considerations such as capturing tail risk, robustness, elicitability, backtesting, and surplus invariance. Expected final online publication date for the Annual Review of Statistics, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.