IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v58y2010i5p1505-1521.html
   My bibliography  Save this article

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Author

Listed:
  • Jörn Dunkel

    (Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom)

  • Stefan Weber

    (Institut für Mathematische Stochastik, Insurance and Financial Mathematics, Gottfried Wilhelm Leibniz Universität Hannover, 30167 Hannover, Germany)

Abstract

Reliable risk measurement is a key problem for financial institutions and regulatory authorities. The current industry standard Value-at-Risk has several deficiencies. Improved risk measures have been suggested and analyzed in the recent literature, but their computational implementation has largely been neglected so far. We propose and investigate stochastic approximation algorithms for the convex risk measure Utility-Based Shortfall Risk. Our approach combines stochastic root-finding schemes with importance sampling. We prove that the resulting Shortfall Risk estimators are consistent and asymptotically normal, and provide formulas for confidence intervals. The performance of the proposed algorithms is tested numerically. We finally apply our techniques to the Normal Copula Model, which is also known as the industry model CreditMetrics. This provides guidance for future implementations in practice.

Suggested Citation

  • Jörn Dunkel & Stefan Weber, 2010. "Stochastic Root Finding and Efficient Estimation of Convex Risk Measures," Operations Research, INFORMS, vol. 58(5), pages 1505-1521, October.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:5:p:1505-1521
    DOI: 10.1287/opre.1090.0784
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1090.0784
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1090.0784?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Jeff Linderoth & Alexander Shapiro & Stephen Wright, 2006. "The empirical behavior of sampling methods for stochastic programming," Annals of Operations Research, Springer, vol. 142(1), pages 215-241, February.
    2. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    3. Stefan Weber, 2006. "Distribution‐Invariant Risk Measures, Information, And Dynamic Consistency," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 419-441, April.
    4. Paul Glasserman & Wanmo Kang & Perwez Shahabuddin, 2008. "Fast Simulation of Multifactor Portfolio Credit Risk," Operations Research, INFORMS, vol. 56(5), pages 1200-1217, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kim, Sojung & Weber, Stefan, 2022. "Simulation methods for robust risk assessment and the distorted mix approach," European Journal of Operational Research, Elsevier, vol. 298(1), pages 380-398.
    2. Zhaolin Hu & Dali Zhang, 2018. "Utility‐based shortfall risk: Efficient computations via Monte Carlo," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(5), pages 378-392, August.
    3. Sojung Kim & Stefan Weber, 2020. "Simulation Methods for Robust Risk Assessment and the Distorted Mix Approach," Papers 2009.03653, arXiv.org, revised Jan 2022.
    4. Vishwajit Hegde & Arvind S. Menon & L. A. Prashanth & Krishna Jagannathan, 2021. "Online Estimation and Optimization of Utility-Based Shortfall Risk," Papers 2111.08805, arXiv.org, revised Nov 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Parrini, Alessandro, 2013. "Importance Sampling for Portfolio Credit Risk in Factor Copula Models," MPRA Paper 103745, University Library of Munich, Germany.
    2. Qinyu Wu & Fan Yang & Ping Zhang, 2023. "Conditional generalized quantiles based on expected utility model and equivalent characterization of properties," Papers 2301.12420, arXiv.org.
    3. Freddy Delbaen, 2021. "Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions," Finance and Stochastics, Springer, vol. 25(3), pages 597-614, July.
    4. Steven Kou & Xianhua Peng, 2016. "On the Measurement of Economic Tail Risk," Operations Research, INFORMS, vol. 64(5), pages 1056-1072, October.
    5. William B. Haskell & Wenjie Huang & Huifu Xu, 2018. "Preference Elicitation and Robust Optimization with Multi-Attribute Quasi-Concave Choice Functions," Papers 1805.06632, arXiv.org.
    6. Ruodu Wang & Yunran Wei, 2020. "Risk functionals with convex level sets," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1337-1367, October.
    7. Fissler Tobias & Ziegel Johanna F., 2021. "On the elicitability of range value at risk," Statistics & Risk Modeling, De Gruyter, vol. 38(1-2), pages 25-46, January.
    8. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
    9. Masato Wada & Felipe Delgado & Bernardo K. Pagnoncelli, 2017. "A risk averse approach to the capacity allocation problem in the airline cargo industry," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(6), pages 643-651, June.
    10. Tim J. Brereton & Dirk P. Kroese & Joshua C. Chan, 2012. "Monte Carlo Methods for Portfolio Credit Risk," ANU Working Papers in Economics and Econometrics 2012-579, Australian National University, College of Business and Economics, School of Economics.
    11. Schur, Rouven & Gönsch, Jochen & Hassler, Michael, 2019. "Time-consistent, risk-averse dynamic pricing," European Journal of Operational Research, Elsevier, vol. 277(2), pages 587-603.
    12. Samuel Drapeau & Michael Kupper, 2013. "Risk Preferences and Their Robust Representation," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 28-62, February.
    13. Mohamed A. Ayadi & Hatem Ben-Ameur & Nabil Channouf & Quang Khoi Tran, 2019. "NORTA for portfolio credit risk," Annals of Operations Research, Springer, vol. 281(1), pages 99-119, October.
    14. Anthony Coache & Sebastian Jaimungal & 'Alvaro Cartea, 2022. "Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning," Papers 2206.14666, arXiv.org, revised May 2023.
    15. Tadese, Mekonnen & Drapeau, Samuel, 2020. "Relative bound and asymptotic comparison of expectile with respect to expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 387-399.
    16. Nicole EL KAROUI & Claudia RAVANELLI, 2008. "Cash Sub-additive Risk Measures and Interest Rate Ambiguity," Swiss Finance Institute Research Paper Series 08-09, Swiss Finance Institute.
    17. Tobias Fissler & Jana Hlavinová & Birgit Rudloff, 2021. "Elicitability and identifiability of set-valued measures of systemic risk," Finance and Stochastics, Springer, vol. 25(1), pages 133-165, January.
    18. Philpott, A.B. & de Matos, V.L., 2012. "Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion," European Journal of Operational Research, Elsevier, vol. 218(2), pages 470-483.
    19. Gundel, Anne & Weber, Stefan, 2008. "Utility maximization under a shortfall risk constraint," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1126-1151, December.
    20. Federico Gatta & Fabrizio Lillo & Piero Mazzarisi, 2024. "CAESar: Conditional Autoregressive Expected Shortfall," Papers 2407.06619, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:58:y:2010:i:5:p:1505-1521. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.