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Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics

Author

Listed:
  • So Yeon Chun

    (McDonough School of Business, Georgetown University, Washington, DC 20057)

  • Alexander Shapiro

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Stan Uryasev

    (Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida 32611)

Abstract
We discuss linear regression approaches to the estimation of law-invariant conditional risk measures. Two estimation procedures are considered and compared; one is based on residual analysis of the standard least-squares method, and the other is in the spirit of the M -estimation approach used in robust statistics. In particular, value-at-risk and average value-at-risk measures are discussed in detail. Large sample statistical inference of the estimators is derived. Furthermore, finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive Monte Carlo study. Empirical results on the real data (different financial asset classes) are also provided to illustrate the performance of the estimators.

Suggested Citation

  • So Yeon Chun & Alexander Shapiro & Stan Uryasev, 2012. "Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics," Operations Research, INFORMS, vol. 60(4), pages 739-756, August.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:4:p:739-756
    DOI: 10.1287/opre.1120.1072
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    References listed on IDEAS

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    Cited by:

    1. Roman V. Ivanov, 2018. "A Credit-Risk Valuation under the Variance-Gamma Asset Return," Risks, MDPI, vol. 6(2), pages 1-25, May.
    2. Charles-Olivier Amédée-Manesme & Fabrice Barthélémy, 2018. "Ex-ante real estate Value at Risk calculation method," Annals of Operations Research, Springer, vol. 262(2), pages 257-285, March.
    3. Yi-Ting Chen & Edward W. Sun & Yi-Bing Lin, 2019. "Coherent quality management for big data systems: a dynamic approach for stochastic time consistency," Annals of Operations Research, Springer, vol. 277(1), pages 3-32, June.
    4. Roger W. Barnard & Kent Pearce & A. Alexandre Trindade, 2018. "When is tail mean estimation more efficient than tail median? Answers and implications for quantitative risk management," Annals of Operations Research, Springer, vol. 262(1), pages 47-65, March.
    5. Roman V. Ivanov, 2023. "The Semi-Hyperbolic Distribution and Its Applications," Stats, MDPI, vol. 6(4), pages 1-21, October.
    6. Chen Yi-Ting & Sun Edward W. & Yu Min-Teh, 2015. "Improving model performance with the integrated wavelet denoising method," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 19(4), pages 445-467, September.
    7. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498, arXiv.org, revised Feb 2017.
    8. repec:cte:wsrepe:35425 is not listed on IDEAS
    9. Lin, Edward M.H. & Sun, Edward W. & Yu, Min-Teh, 2020. "Behavioral data-driven analysis with Bayesian method for risk management of financial services," International Journal of Production Economics, Elsevier, vol. 228(C).
    10. Roujia Li & Jia Liu, 2022. "Online Portfolio Selection with Long-Short Term Forecasting," SN Operations Research Forum, Springer, vol. 3(4), pages 1-15, December.
    11. Tatiana Labopin-Richard & Fabrice Gamboa & Aur'elien Garivier & Bertrand Iooss, 2014. "Bregman superquantiles. Estimation methods and applications," Papers 1405.6677, arXiv.org, revised Jan 2016.
    12. Denis Chetverikov & Yukun Liu & Aleh Tsyvinski, 2022. "Weighted-average quantile regression," Papers 2203.03032, arXiv.org.
    13. Edward W. Sun & Yu-Jen Wang & Min-Teh Yu, 2018. "Integrated Portfolio Risk Measure: Estimation and Asymptotics of Multivariate Geometric Quantiles," Computational Economics, Springer;Society for Computational Economics, vol. 52(2), pages 627-652, August.
    14. Ivanov Roman V., 2018. "On risk measuring in the variance-gamma model," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 23-33, January.
    15. Rockafellar, R.T. & Royset, J.O. & Miranda, S.I., 2014. "Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk," European Journal of Operational Research, Elsevier, vol. 234(1), pages 140-154.
    16. Marwa Elnahass & Mohamed Marie & Mohammed Elgammal, 2022. "Terrorist attacks and bank financial stability: evidence from MENA economies," Review of Quantitative Finance and Accounting, Springer, vol. 59(1), pages 383-427, July.
    17. Labopin-Richard T. & Gamboa F. & Garivier A. & Iooss B., 2016. "Bregman superquantiles. Estimation methods and applications," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-33, March.
    18. Johanna F. Ziegel, 2013. "Coherence and elicitability," Papers 1303.1690, arXiv.org, revised Mar 2014.
    19. Fulga, Cristinca, 2016. "Portfolio optimization with disutility-based risk measure," European Journal of Operational Research, Elsevier, vol. 251(2), pages 541-553.
    20. R. Tyrrell Rockafellar & Johannes O. Royset, 2018. "Superquantile/CVaR risk measures: second-order theory," Annals of Operations Research, Springer, vol. 262(1), pages 3-28, March.

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    More about this item

    Keywords

    value-at-risk; average value-at-risk; linear regression; least-squares residuals; M-estimators; quantile regression; conditional risk measures; law-invariant risk measures; statistical inference;
    All these keywords.

    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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