IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v62y2014i6p1302-1315.html
   My bibliography  Save this article

Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach

Author

Listed:
  • David Wozabal

    (TUM School of Management, Technische Universität München, 80333 München, Germany)

Abstract

This paper introduces a framework for robustifying convex, law invariant risk measures. The robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure, which represent the investors' beliefs about the distribution of future asset losses. It is shown that under mild conditions, the infinite dimensional optimization problem of finding the worst-case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results, robust versions of several risk measures including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals are derived. The resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems, and a numerical study shows that in most cases they perform significantly better out-of-sample than their nonrobust variants in terms of risk, expected losses, and turnover.

Suggested Citation

  • David Wozabal, 2014. "Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach," Operations Research, INFORMS, vol. 62(6), pages 1302-1315, December.
  • Handle: RePEc:inm:oropre:v:62:y:2014:i:6:p:1302-1315
    DOI: 10.1287/opre.2014.1323
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.2014.1323
    Download Restriction: no

    File URL: https://libkey.io/10.1287/opre.2014.1323?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. Fama, Eugene F & French, Kenneth R, 1992. "The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-465, June.
    2. Jorion, Philippe, 1986. "Bayes-Stein Estimation for Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 279-292, September.
    3. Aharon Ben-Tal & Dimitris Bertsimas & David B. Brown, 2010. "A Soft Robust Model for Optimization Under Ambiguity," Operations Research, INFORMS, vol. 58(4-part-2), pages 1220-1234, August.
    4. Karthik Natarajan & Dessislava Pachamanova & Melvyn Sim, 2009. "Constructing Risk Measures from Uncertainty Sets," Operations Research, INFORMS, vol. 57(5), pages 1129-1141, October.
    5. Pascal J. Maenhout, 2004. "Robust Portfolio Rules and Asset Pricing," The Review of Financial Studies, Society for Financial Studies, vol. 17(4), pages 951-983.
    6. Laurent El Ghaoui & Maksim Oks & Francois Oustry, 2003. "Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach," Operations Research, INFORMS, vol. 51(4), pages 543-556, August.
    7. Zymler, Steve & Rustem, Berç & Kuhn, Daniel, 2011. "Robust portfolio optimization with derivative insurance guarantees," European Journal of Operational Research, Elsevier, vol. 210(2), pages 410-424, April.
    8. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    9. Jun-ya Gotoh & Akiko Takeda, 2011. "On the role of norm constraints in portfolio selection," Computational Management Science, Springer, vol. 8(4), pages 323-353, November.
    10. Jonathan Li & Roy Kwon, 2013. "Portfolio selection under model uncertainty: a penalized moment-based optimization approach," Journal of Global Optimization, Springer, vol. 56(1), pages 131-164, May.
    11. 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.
    12. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation1," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 181-190, November.
    13. Erick Delage & Yinyu Ye, 2010. "Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems," Operations Research, INFORMS, vol. 58(3), pages 595-612, June.
    14. Shushang Zhu & Masao Fukushima, 2009. "Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management," Operations Research, INFORMS, vol. 57(5), pages 1155-1168, October.
    15. David Wozabal, 2012. "A framework for optimization under ambiguity," Annals of Operations Research, Springer, vol. 193(1), pages 21-47, March.
    16. Georg Pflug & David Wozabal, 2007. "Ambiguity in portfolio selection," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 435-442.
    17. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    18. Pflug, Georg Ch. & Pichler, Alois & Wozabal, David, 2012. "The 1/N investment strategy is optimal under high model ambiguity," Journal of Banking & Finance, Elsevier, vol. 36(2), pages 410-417.
    19. Victor DeMiguel & Francisco J. Nogales, 2009. "Portfolio Selection with Robust Estimation," Operations Research, INFORMS, vol. 57(3), pages 560-577, June.
    20. Yitzhaki, Shlomo, 1982. "Stochastic Dominance, Mean Variance, and Gini's Mean Difference," American Economic Review, American Economic Association, vol. 72(1), pages 178-185, March.
    21. D. Goldfarb & G. Iyengar, 2003. "Robust Portfolio Selection Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 1-38, February.
    22. Jun-Ya Gotoh & Keita Shinozaki & Akiko Takeda, 2013. "Robust portfolio techniques for mitigating the fragility of CVaR minimization and generalization to coherent risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 13(10), pages 1621-1635, October.
    23. Georg Ch Pflug & Werner Römisch, 2007. "Modeling, Measuring and Managing Risk," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6478, October.
    24. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    Full references (including those not matched with items on IDEAS)

    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. Alireza Ghahtarani & Ahmed Saif & Alireza Ghasemi, 2022. "Robust portfolio selection problems: a comprehensive review," Operational Research, Springer, vol. 22(4), pages 3203-3264, September.
    2. Alireza Ghahtarani & Ahmed Saif & Alireza Ghasemi, 2021. "Robust Portfolio Selection Problems: A Comprehensive Review," Papers 2103.13806, arXiv.org, revised Jan 2022.
    3. Maria Scutellà & Raffaella Recchia, 2013. "Robust portfolio asset allocation and risk measures," Annals of Operations Research, Springer, vol. 204(1), pages 145-169, April.
    4. Viet Anh Nguyen & Fan Zhang & Shanshan Wang & Jose Blanchet & Erick Delage & Yinyu Ye, 2021. "Robustifying Conditional Portfolio Decisions via Optimal Transport," Papers 2103.16451, arXiv.org, revised Apr 2024.
    5. Panos Xidonas & Ralph Steuer & Christis Hassapis, 2020. "Robust portfolio optimization: a categorized bibliographic review," Annals of Operations Research, Springer, vol. 292(1), pages 533-552, September.
    6. Kolm, Petter N. & Tütüncü, Reha & Fabozzi, Frank J., 2014. "60 Years of portfolio optimization: Practical challenges and current trends," European Journal of Operational Research, Elsevier, vol. 234(2), pages 356-371.
    7. Zhilin Kang & Zhongfei Li, 2018. "An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(2), pages 169-195, April.
    8. Martin Branda & Max Bucher & Michal Červinka & Alexandra Schwartz, 2018. "Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization," Computational Optimization and Applications, Springer, vol. 70(2), pages 503-530, June.
    9. Selim Mankai & Khaled Guesmi, 2014. "Robust Portfolio Protection: A Scenarios-Based Approach," Working Papers hal-04141326, HAL.
    10. Jang Ho Kim & Woo Chang Kim & Frank J. Fabozzi, 2014. "Recent Developments in Robust Portfolios with a Worst-Case Approach," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 103-121, April.
    11. Jose Blanchet & Lin Chen & Xun Yu Zhou, 2022. "Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances," Management Science, INFORMS, vol. 68(9), pages 6382-6410, September.
    12. Pengyu Qian & Zizhuo Wang & Zaiwen Wen, 2015. "A Composite Risk Measure Framework for Decision Making under Uncertainty," Papers 1501.01126, arXiv.org.
    13. Aharon Ben-Tal & Dimitris Bertsimas & David B. Brown, 2010. "A Soft Robust Model for Optimization Under Ambiguity," Operations Research, INFORMS, vol. 58(4-part-2), pages 1220-1234, August.
    14. Wolfram Wiesemann & Daniel Kuhn & Melvyn Sim, 2014. "Distributionally Robust Convex Optimization," Operations Research, INFORMS, vol. 62(6), pages 1358-1376, December.
    15. Liu, Jia & Chen, Zhiping, 2018. "Time consistent multi-period robust risk measures and portfolio selection models with regime-switching," European Journal of Operational Research, Elsevier, vol. 268(1), pages 373-385.
    16. Frank Fabozzi & Dashan Huang & Guofu Zhou, 2010. "Robust portfolios: contributions from operations research and finance," Annals of Operations Research, Springer, vol. 176(1), pages 191-220, April.
    17. Goh, Joel Weiqiang & Lim, Kian Guan & Sim, Melvyn & Zhang, Weina, 2012. "Portfolio value-at-risk optimization for asymmetrically distributed asset returns," European Journal of Operational Research, Elsevier, vol. 221(2), pages 397-406.
    18. Jonathan Li & Roy Kwon, 2013. "Portfolio selection under model uncertainty: a penalized moment-based optimization approach," Journal of Global Optimization, Springer, vol. 56(1), pages 131-164, May.
    19. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    20. Viet Anh Nguyen & Soroosh Shafiee & Damir Filipovi'c & Daniel Kuhn, 2021. "Mean-Covariance Robust Risk Measurement," Papers 2112.09959, arXiv.org, revised Nov 2023.

    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:62:y:2014:i:6:p:1302-1315. 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.