IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1609.04065.html
   My bibliography  Save this paper

Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization

Author

Listed:
  • Jonathan Yu-Meng Li

Abstract

Worst-case risk measures refer to the calculation of the largest value for risk measures when only partial information of the underlying distribution is available. For the popular risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known for the underlying distribution. These results are remarkable since they not only simplify the use of worst-case risk measures but also provide great insight into the connection between the worst-case risk measures and existing risk measures. We show in this paper that somewhat surprisingly similar closed-form solutions also exist for the general class of law invariant coherent risk measures, which consists of spectral risk measures as special cases that are arguably the most important extensions of CVaR. We shed light on the one-to-one correspondence between a worst-case law invariant risk measure and a worst-case CVaR (and a worst-case VaR), which enables one to carry over the development of worst-case VaR in the context of portfolio optimization to the worst-case law invariant risk measures immediately.

Suggested Citation

  • Jonathan Yu-Meng Li, 2016. "Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization," Papers 1609.04065, arXiv.org.
    • Handle: RePEc:arx:papers:1609.04065
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1609.04065
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Alexander Shapiro, 2013. "On Kusuoka Representation of Law Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 142-152, February.
    2. Acerbi Carlo & Simonetti Prospero, 2002. "Portfolio Optimization with Spectral Measures of Risk," Papers cond-mat/0203607, arXiv.org.
    3. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    4. 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.
    5. David Wozabal, 2014. "Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach," Operations Research, INFORMS, vol. 62(6), pages 1302-1315, December.
    6. Wolfram Wiesemann & Daniel Kuhn & Melvyn Sim, 2014. "Distributionally Robust Convex Optimization," Operations Research, INFORMS, vol. 62(6), pages 1358-1376, December.
    7. 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.
    8. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    9. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    10. Li Chen & Simai He & Shuzhong Zhang, 2011. "Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection," Operations Research, INFORMS, vol. 59(4), pages 847-865, August.
    11. 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.
    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. Cornilly, Dries & Vanduffel, Steven, 2019. "Equivalent distortion risk measures on moment spaces," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 187-192.

    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. Viet Anh Nguyen & Soroosh Shafiee & Damir Filipovi'c & Daniel Kuhn, 2021. "Mean-Covariance Robust Risk Measurement," Papers 2112.09959, arXiv.org, revised Nov 2023.
    2. Pengyu Qian & Zizhuo Wang & Zaiwen Wen, 2015. "A Composite Risk Measure Framework for Decision Making under Uncertainty," Papers 1501.01126, arXiv.org.
    3. Vishal Gupta, 2019. "Near-Optimal Bayesian Ambiguity Sets for Distributionally Robust Optimization," Management Science, INFORMS, vol. 65(9), pages 4242-4260, September.
    4. Wang, Fan & Zhang, Chao & Zhang, Hui & Xu, Liang, 2021. "Short-term physician rescheduling model with feature-driven demand for mental disorders outpatients," Omega, Elsevier, vol. 105(C).
    5. 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.
    6. Postek, K.S. & den Hertog, D. & Melenberg, B., 2015. "Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)," Discussion Paper 2015-047, Tilburg University, Center for Economic Research.
    7. Shao, Hui & Zhang, Zhe George, 2023. "Distortion risk measure under parametric ambiguity," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1159-1172.
    8. Su, Xiaoshan & Li, Yuhan, 2024. "Robust portfolio selection with subjective risk aversion under dependence uncertainty," Economic Modelling, Elsevier, vol. 132(C).
    9. Silvana Pesenti & Qiuqi Wang & Ruodu Wang, 2020. "Optimizing distortion riskmetrics with distributional uncertainty," Papers 2011.04889, arXiv.org, revised Feb 2022.
    10. Wei Liu & Li Yang & Bo Yu, 2022. "Kernel density estimation based distributionally robust mean-CVaR portfolio optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 1053-1077, December.
    11. Wei Liu & Li Yang & Bo Yu, 2021. "KDE distributionally robust portfolio optimization with higher moment coherent risk," Annals of Operations Research, Springer, vol. 307(1), pages 363-397, December.
    12. Postek, K.S. & den Hertog, D. & Melenberg, B., 2015. "Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)," Other publications TiSEM eeb9c898-6943-4199-b747-3, Tilburg University, School of Economics and Management.
    13. 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.
    14. Steffen Rebennack, 2022. "Data-driven stochastic optimization for distributional ambiguity with integrated confidence region," Journal of Global Optimization, Springer, vol. 84(2), pages 255-293, October.
    15. Zhi Chen & Melvyn Sim & Peng Xiong, 2020. "Robust Stochastic Optimization Made Easy with RSOME," Management Science, INFORMS, vol. 66(8), pages 3329-3339, August.
    16. James Ming Chen, 2018. "On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles," Risks, MDPI, vol. 6(2), pages 1-28, June.
    17. 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.
    18. 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.
    19. A. Paç & Mustafa Pınar, 2014. "Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 875-891, October.
    20. 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.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:arx:papers:1609.04065. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.