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Open AccessOpen Access JUSTC Mathematics/Management 25 June 2022

Worst-case conditional value-at-risk and conditional expected shortfall based on covariance information

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2022-0023
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  • Author Bio:

    Tiantian Mao is an Associate Professor with the University of Science and Technology of China (USTC). In 2012, she received her Ph.D. degree in Science from USTC. In May of the same year, she joined the Department of Statistics and Finance, School of Management, USTC, for postdoctoral work. Her research fields include risk measurement, risk management, random dominance and extreme value theory

    Qinyu Wu is currently a Ph.D. candidate at the School of Management, University of Science and Technology of China. His research interests focus on risk management and mathematical finance

  • Corresponding author: E-mail: wu051555@mail.ustc.edu.cn
  • Received Date: 25 January 2022
  • Accepted Date: 30 March 2022
    • Available Online: 25 June 2022
    Abstract

  • Abstract

    In this paper, we study the worst-case conditional value-at-risk (CoVaR) and conditional expected shortfall (CoES) in a situation where only partial information on the underlying probability distribution is available. In the case of the first two marginal moments are known, the closed-form solution and the value of the worst-case CoVaR and CoES are derived. The worst-case CoVaR and CoES under mean and covariance information are also investigated.

    Graphical abstract

    We construct the above new ambiguity set, then propose the optimization problem of CoVaRand CoES based on this ambiguity set, and give the theoretical results.

    Abstract

    In this paper, we study the worst-case conditional value-at-risk (CoVaR) and conditional expected shortfall (CoES) in a situation where only partial information on the underlying probability distribution is available. In the case of the first two marginal moments are known, the closed-form solution and the value of the worst-case CoVaR and CoES are derived. The worst-case CoVaR and CoES under mean and covariance information are also investigated.

    • Public Summary

    • The relationship between CoVaR, CoES and dependence structure are investigated.
    • In case where the first two marginal moments are known, the closed-form solution and the valueof the worst-case CoVaR and CoES are derived.
    • The worst-case CoVaR and CoES under mean and covariance information are investigated.

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    • References(18)
  • [1]
    Basel Committee on Banking Supervision. Minimum capital requirements for market risk. Basel, Switzerland: Bank for International Settlements, 2019.
    [2]
    Adrian T, Brunnermeier M K. CoVaR. American Economic Review, 2016, 106 (7): 1705–1741. doi: 10.1257/aer.20120555
    [3]
    Brownlees C, Engle R F. SRISK: A conditional capital shortfall measure of systemic risk. The Review of Financial Studies, 2017, 30 (1): 48–79. doi: 10.1093/rfs/hhw060
    [4]
    Acharya V V, Pedersen L H, Philippon T, et al. Measuring systemic risk. The Review of Financial Studies, 2017, 30 (1): 2–47. doi: 10.1093/rfs/hhw088
    [5]
    Girardi G, Ergün A T. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 2013, 37 (8): 3169–3180. doi: 10.1016/j.jbankfin.2013.02.027
    [6]
    Huang W Q, Uryasev S. The CoCVaR approach: Systemic risk contribution measurement. Journal of Risk, 2018, 20 (4): 75–93. doi: 10.21314/JOR.2018.383
    [7]
    López-Espinosa G, Moreno A, Rubia A, et al. Short-term wholesale funding and systemic risk: A global CoVaR approach. Journal of Banking & Finance, 2012, 36 (12): 3150–3162. doi: 10.1016/j.jbankfin.2012.04.020
    [8]
    Karimalis E N, Nomikos N K. Measuring systemic risk in the European banking sector: A Copula CoVaR approach. The European Journal of Finance, 2018, 24 (11): 944–975. doi: 10.1080/1351847X.2017.1366350
    [9]
    Popescu I. Robust mean-covariance solutions for stochastic optimization. Operations Research, 2007, 51 (1): 98–112. doi: 10.1287/opre.1060.0353
    [10]
    Bertsimas D, Doan X V, Natarajan K, et al. Models for minimax stochastic linear optimization problems with risk aversion. Mathematics of Operations Research, 2010, 35 (3): 580–602. doi: 10.1287/moor.1100.0445
    [11]
    Delage E, Ye Y. Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 2010, 58 (3): 595–612. doi: 10.1287/opre.1090.0741
    [12]
    Natarajan K, Sim M, Uichanco J. Tractable robust expected utility and risk models for portfolio optimization. Mathematical Finance, 2010, 20 (4): 695–731. doi: 10.1111/j.1467-9965.2010.00417.x
    [13]
    Wiesemann W, Kuhn D, Sim M. Distributionally robust convex optimization. Operations Research, 2014, 62 (6): 1358–1376. doi: 10.1287/opre.2014.1314
    [14]
    Chen L, He S, Zhang S. Tight bounds for some risk measures, with applications to robust portfolio selection. Operations Research, 2011, 59 (4): 847–865. doi: 10.1287/opre.1110.0950
    [15]
    Nelsen R B. An Introduction to Copulas. 2nd edition. New York: Springer, 2006.
    [16]
    Mainik G, Schaanning E. On dependence consistency of CoVaR and some other systemic risk measures. Statistics & Risk Modeling, 2014, 31 (1): 49–77. doi: 10.1515/strm-2013-1164
    [17]
    Ghaoui L E, Oks M, Oustry F. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 2003, 51 (4): 543–556. doi: 10.1287/opre.51.4.543.16101
    [18]
    Bernard C, Pesenti S M, Vanduffel S. Robust distortion risk measures. https://ssrn.com/abstract=3677078 .
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    Figure  1.  CoVaR and CoES with bivariate Gaussian distributions.

    Figure  2.  CoVaR and CoES with bivariate t-distributions.

    Figure  3.  CoVaR and CoES with Pareto marginal and different copulas.

    [1]
    Basel Committee on Banking Supervision. Minimum capital requirements for market risk. Basel, Switzerland: Bank for International Settlements, 2019.
    [2]
    Adrian T, Brunnermeier M K. CoVaR. American Economic Review, 2016, 106 (7): 1705–1741. doi: 10.1257/aer.20120555
    [3]
    Brownlees C, Engle R F. SRISK: A conditional capital shortfall measure of systemic risk. The Review of Financial Studies, 2017, 30 (1): 48–79. doi: 10.1093/rfs/hhw060
    [4]
    Acharya V V, Pedersen L H, Philippon T, et al. Measuring systemic risk. The Review of Financial Studies, 2017, 30 (1): 2–47. doi: 10.1093/rfs/hhw088
    [5]
    Girardi G, Ergün A T. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 2013, 37 (8): 3169–3180. doi: 10.1016/j.jbankfin.2013.02.027
    [6]
    Huang W Q, Uryasev S. The CoCVaR approach: Systemic risk contribution measurement. Journal of Risk, 2018, 20 (4): 75–93. doi: 10.21314/JOR.2018.383
    [7]
    López-Espinosa G, Moreno A, Rubia A, et al. Short-term wholesale funding and systemic risk: A global CoVaR approach. Journal of Banking & Finance, 2012, 36 (12): 3150–3162. doi: 10.1016/j.jbankfin.2012.04.020
    [8]
    Karimalis E N, Nomikos N K. Measuring systemic risk in the European banking sector: A Copula CoVaR approach. The European Journal of Finance, 2018, 24 (11): 944–975. doi: 10.1080/1351847X.2017.1366350
    [9]
    Popescu I. Robust mean-covariance solutions for stochastic optimization. Operations Research, 2007, 51 (1): 98–112. doi: 10.1287/opre.1060.0353
    [10]
    Bertsimas D, Doan X V, Natarajan K, et al. Models for minimax stochastic linear optimization problems with risk aversion. Mathematics of Operations Research, 2010, 35 (3): 580–602. doi: 10.1287/moor.1100.0445
    [11]
    Delage E, Ye Y. Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 2010, 58 (3): 595–612. doi: 10.1287/opre.1090.0741
    [12]
    Natarajan K, Sim M, Uichanco J. Tractable robust expected utility and risk models for portfolio optimization. Mathematical Finance, 2010, 20 (4): 695–731. doi: 10.1111/j.1467-9965.2010.00417.x
    [13]
    Wiesemann W, Kuhn D, Sim M. Distributionally robust convex optimization. Operations Research, 2014, 62 (6): 1358–1376. doi: 10.1287/opre.2014.1314
    [14]
    Chen L, He S, Zhang S. Tight bounds for some risk measures, with applications to robust portfolio selection. Operations Research, 2011, 59 (4): 847–865. doi: 10.1287/opre.1110.0950
    [15]
    Nelsen R B. An Introduction to Copulas. 2nd edition. New York: Springer, 2006.
    [16]
    Mainik G, Schaanning E. On dependence consistency of CoVaR and some other systemic risk measures. Statistics & Risk Modeling, 2014, 31 (1): 49–77. doi: 10.1515/strm-2013-1164
    [17]
    Ghaoui L E, Oks M, Oustry F. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 2003, 51 (4): 543–556. doi: 10.1287/opre.51.4.543.16101
    [18]
    Bernard C, Pesenti S M, Vanduffel S. Robust distortion risk measures. https://ssrn.com/abstract=3677078 .
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