COHERENCE AND ELICITABILITY

Johanna F. Ziegel

357 citations0/0 files storedMathematical Finance

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Risk and Portfolio Optimization Financial Risk and Volatility Modeling Probability and Risk Models

Abstract

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile‐based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law‐invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.

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References

1999Coherent Measures of Risk 2007Strictly Proper Scoring Rules, Prediction, and Estimation 2000Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach 2002On the coherence of expected shortfall 1998Evaluating Density Forecasts with Applications to Financial Risk Management 2011Making and Evaluating Point Forecasts 2002Spectral measures of risk: A coherent representation of subjective risk aversion 2001Testing Density Forecasts, With Applications to Risk Management

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2013Generalized quantiles as risk measures 2015What is the best risk measure in practice? A comparison of standard measures 2014An Academic Response to Basel 3.5 2015Risk management with expectiles 2017Elicitability and backtesting: Perspectives for banking regulation 2016A discussion on: Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Ranking, by W. Ehm, T. Gneiting, A. Jordan, and A. Krueger

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