COHERENCE AND ELICITABILITY

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.

[1]  Steven Kou,et al.  External Risk Measures and Basel Accords , 2013, Math. Oper. Res..

[2]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[3]  R. Rockafellar,et al.  The fundamental risk quadrangle in risk management, optimization and statistical estimation , 2013 .

[4]  Volker Krätschmer,et al.  Quasi-Hadamard differentiability of general risk functionals and its application , 2014, 1401.3167.

[5]  Yoav Shoham,et al.  Eliciting properties of probability distributions , 2008, EC '08.

[6]  Xianhua Peng,et al.  The Fundamental Review of the Trading Book , 2013 .

[7]  Volker Krätschmer,et al.  Qualitative and infinitesimal robustness of tail-dependent statistical functionals , 2012, J. Multivar. Anal..

[8]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[9]  R. Dana,et al.  A REPRESENTATION RESULT FOR CONCAVE SCHUR CONCAVE FUNCTIONS , 2005 .

[10]  T. Gneiting,et al.  Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules , 2011 .

[11]  Volker Krätschmer,et al.  Comparative and qualitative robustness for law-invariant risk measures , 2012, Finance Stochastics.

[12]  A. S. Cherny,et al.  Weighted V@R and its Properties , 2006, Finance Stochastics.

[13]  C. Goodhart,et al.  An academic response to Basel II , 2001 .

[14]  Alexander Shapiro,et al.  Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics , 2012, Oper. Res..

[15]  A. Müller,et al.  Generalized Quantiles as Risk Measures , 2013 .

[16]  S. Weber,et al.  DISTRIBUTION‐INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY , 2006 .

[17]  William Thomson,et al.  Eliciting production possibilities from a well-informed manager , 1979 .

[18]  R. Koenker.  Quantile Regression: Name Index , 2005 .

[19]  Marco Saerens,et al.  Building cost functions minimizing to some summary statistics , 2000, IEEE Trans. Neural Networks Learn. Syst..

[20]  W. Newey,et al.  Asymmetric Least Squares Estimation and Testing , 1987 .

[21]  Abaxbank,et al.  Spectral Measures of Risk : a Coherent Representation of Subjective Risk Aversion , 2002 .

[22]  Chung-Ming Kuan,et al.  Assessing Value at Risk With CARE, the Conditional Autoregressive Expectile Models , 2008 .

[23]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[24]  Alexander Shapiro,et al.  Uniqueness of Kusuoka Representations , 2013 .

[25]  E. Jouini,et al.  Law Invariant Risk Measures Have the Fatou Property , 2005 .

[26]  Jeremy Berkowitz.  Testing Density Forecasts, With Applications to Risk Management , 2001 .

[27]  Rama Cont,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2010 .

[28]  A. McNeil,et al.  Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach , 2000 .

[29]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

[30]  Stan Uryasev,et al.  Risk Tuning With Generalized Linear Regression , 2007, Math. Oper. Res..

[31]  Romain Deguest,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2008 .

[32]  D. Tasche,et al.  On the coherence of expected shortfall , 2001, cond-mat/0104295.

[33]  Paul Embrechts,et al.  Quantitative Risk Management , 2011, International Encyclopedia of Statistical Science.

[34]  T. Gneiting.  Making and Evaluating Point Forecasts , 2009, 0912.0902.

[35]  Peter Christoffersen,et al.  Elements of Financial Risk Management , 2003 .

[36]  Fabio Bellini,et al.  Elicitable Risk Measures , 2014 .

[37]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[38]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[39]  Johannes O. Royset,et al.  Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk , 2014, Eur. J. Oper. Res..

[40]  Guillaume Carlier,et al.  Core of convex distortions of a probability , 2003, J. Econ. Theory.

[41]  Kent Osband,et al.  Providing incentives for better cost forecasting , 1985 .