Risk Measures: Robustness, Elicitability, and Backtesting

Abstract

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.

A comparison of Range Value at Risk (RVaR) forecasting models

“…Thus, ES fulfills the principle of diversification: The portfolio risk is less than or equal to the sum of the risks of the individual assets that make up the portfolio (Subadditivity axiom). Despite ES's conceptual advantage over VaR, its applicability to daily (or even intra-daily) risk management is partially compromised by its lack of robustness, as discussed in Cont et al (2010), Kou et al (2013), and He et al (2022), for instance. Indeed, ES is an example of a more general conflict between coherence and robustness.…”

“…For practical purposes, He et al (2022) emphasize that risk measures should be robust and elicitable . Robustness accommodates model uncertainty, while elicitability allows us to compare the performance of competing econometric models.…”

“…Despite ES's conceptual advantage over VaR, its applicability to daily (or even intra‐daily) risk management is partially compromised by its lack of robustness, as discussed in Cont et al (2010), Kou et al (2013), and He et al (2022), for instance. Indeed, ES is an example of a more general conflict between coherence and robustness.…”

Measuring Tail Risks

“…VaR is more intuitive and more popular. However, it doesn't satisfy the key sub-additivity condition and is thus not a coherent risk measure [1]. This is the main reason regulators are moving towards adopting ES, which is a coherent measure.…”

“…Measuring tail risk is essential in quantitative risk management. There are a few established measures in financial risk management [1,2]. Variance or standard deviation of returns is often used as a risk measure in the context of portfolio optimization and risk attribution.…”