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Kernel density estimation based distributionally robust mean-CVaR portfolio optimization

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Abstract

In this paper, by using weighted kernel density estimation (KDE) to approximate the continuous probability density function (PDF) of the portfolio loss, and to compute the corresponding approximated Conditional Value-at-Risk (CVaR), a KDE-based distributionally robust mean-CVaR portfolio optimization model is investigated. Its distributional uncertainty set (DUS) is defined indirectly by imposing the constraint on the weights in weighted KDE in terms of \(\phi \)-divergence function in order that the corresponding infinite-dimensional space of PDF is converted into the finite-dimensional space on the weights. This makes the corresponding distributionally robust optimization (DRO) problem computationally tractable. We also prove that the optimal value and solution set of the KDE-based DRO problem converge to those of the portfolio optimization problem under the true distribution. Primary empirical test results show that the proposed model is meaningful.

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References

  1. Artzner, P., Delbaen, F., Eber, J., et al.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Bayraksan, G., Love, D.K.: (2015) Data-driven stochastic programming using phi-divergences. In: The Operations Research Revolution. INFORMS TutORials in Operations Research, pp. 1–19

  3. Ben-Tal, A., Den Hertog, D., De Waegenaere, A., et al.: Robust solutions of optimization problems affected by uncertain probabilities. Manage. Sci. 59(2), 341–357 (2013). https://doi.org/10.1287/mnsc.1120.1641

    Article  Google Scholar 

  4. Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Bertsimas, D., Gupta, V., Kallus, N.: Robust sample average approximation. Math. Program. 171(1), 217–282 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Calafiore, G.C.: Ambiguous risk measures and optimal robust portfolios. SIAM J. Optim. 18(3), 853–877 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Chen, S.X.: Empilical likelihood-based kernel density estimation. Australian J. Stat. 39(1), 47–56 (1997)

    Google Scholar 

  8. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)

    MathSciNet  MATH  Google Scholar 

  9. DeMiguel, V., Garlappi, L., Uppal, R.: Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Rev. Finan. Stud. 22(5), 1915–1953 (2007)

    Google Scholar 

  10. El Ghaoui, L., Oks, M., Oustry, F.: Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Oper. Res. 51(4), 543–556 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1–2), 37–61 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Fabozzi, F.J., Huang, D., Zhou, G.: Robust portfolios: contributions from operations research and finance. Ann. Oper. Res. 176(1), 191–220 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Fishburn, P.C.: Mean-risk analysis with risk associated with below-target returns. Am. Econ. Rev. 2(67), 116–126 (1977)

    Google Scholar 

  14. Gao R, Kleywegt AJ (2016) Distributionally robust stochastic optimization with Wasserstein distance. arXiv:1604.02199v2 [math.OC]

  15. Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58((4–part–1)), 902–917 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Grünwald, P.D., Dawid, A.P.: Game theory, maximum entropy, minimum discrepancy and robust bayesian decision theory. Ann. Stat. 32(4), 1367–1433 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Hall, P., Minnotte, M.C.: High order data sharpening for density estimation. J. Roy. Stat. Soc. B 64(1), 141–157 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Hall, P., Presnell, B.: Density estimation under constraints. J. Comput. Graph. Stat. 8(2), 259–277 (1999)

    MathSciNet  Google Scholar 

  19. Hall, P., Presnell, B.: Intentionally biased bootstrap methods. J. Roy. Stat. Soc. B 61(1), 143–158 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Hall, P., Turlach, B.A.: Reducing bias in curve estimation by use of weights. Comput. Stat. Data Anal. 30(1), 67–86 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Hanasusanto, G.A., Kuhn, D.: Robust data-driven dynamic programming. In: Burges, C.J.C., Bottou, L., Welling, M., et al. (eds.) Advances in Neural Information Processing Systems 26, pp. 827–835. Curran Associates, Inc., New York (2013)

    Google Scholar 

  22. Hanasusanto, G.A., Kuhn, D., Wallace, S.W., et al.: Distributionally robust multi-item newsvendor problems with multimodal demand distributions. Math. Program. 152(1–2), 1–32 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Hu Z., Hong L.J.: (2012) Kullback-Leibler divergence constrained distributionally robust optimization. Optimization Online http://www.optimization-online.org/DB-HTML/2012/11/3677.html

  24. Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. 158(1), 291–327 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Jones, M.C., Marron, J.S., Sheather, S.J.: A brief survey of bandwidth selection for density estimation. J. Am. Stat. Assoc. 91(433), 401–407 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Konno, H., Yamazaki, H.: Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market. Manage. Sci. 37(5), 519–531 (1991)

    Google Scholar 

  27. Kuhn, D., Mohajerin Esfahani, P., Nguyen, V.A., et al.: Wasserstein distributionally robust optimization: Theory and applications in machine learning. In: Operations Research & Management Science in the Age of Analytics. INFORMS TutORials in Operations Research pp. 130–166 (2019)

  28. Li, Q., Racine, J.S.: Nonparametric Econometrics: Theory and Practice. Princeton University Press, New Jersey (2007)

    MATH  Google Scholar 

  29. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  30. Markowitz, H.: Portfolio Selection: Efficient Diversification of Investments. John Wiley and Sons, New York (1959)

    Google Scholar 

  31. Homem de Mello, T., Bayraksan, G.: Monte carlo sampling-based methods for stochastic optimization. Surv. Oper. Res. Manag. Sci. 19(1), 56–85 (2014)

    MathSciNet  Google Scholar 

  32. Michaud, R.: The markowitz optimization enigma: Is ‘optimized’ optimal? Financ. Anal. J. 45(1), 31–42 (1989)

    Google Scholar 

  33. Mohajerin Esfahani, P., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. 171(1), 115–166 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Morgan JP, Reuters (1996) \({\rm RiskMetrics\it }^{{\rm TM}}\). Technical Document, 4th ed

  35. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Ogryczak, W., Ruszczynski, A.: On consistency of stochastic dominance and mean-semideviation models. Math. Program. 89(2), 217–232 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Owen, A.B.: Empirical Likelihood. CRC Press, Boca Raton (2001)

    MATH  Google Scholar 

  38. Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman and Hall/CRCs, Boca Raton, FL (2006)

    MATH  Google Scholar 

  39. Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065–1076 (1962)

    MathSciNet  MATH  Google Scholar 

  40. Popescu, I.: Robust mean-covariance solutions for stochastic optimization. Oper. Res. 55(1), 98–112 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Postek, K., Den Hertog, D., Melenberg, B.: Computationally tractable counterparts of distributionally robust constraints on risk measures. SIAM Rev. 58(4), 603–650 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Rahimian H., Mehrotra S.: (2019) Distributionally Robust Optimization: A Review. arXiv:1908.05659 [math.OC]

  43. Rockafellar, R.: Convex Analysis Princeton Landmarks in Mathematics and Physics. Princeton University Press, New Jersey (1997)

    Google Scholar 

  44. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)

    Google Scholar 

  45. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)

    Google Scholar 

  46. Scarf, H.: A min-max solution of an inventory problem. In: Scarf, H., Arrow, K., Karlin, S. (eds.) Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, vol. 10, pp. 201–209. Stanford, CA (1958)

    Google Scholar 

  47. Shapiro, A.: Distributionally robust stochastic programming. SIAM J. Optim. 27(4), 2258–2275 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization, 2nd edn. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  49. Silverman, B.W.: Density Estimation for Statistics and Data Analysis. MPS-SIAM Series on Optimization, 2nd edn. Chapman and Hall, New York, NY (1986)

    MATH  Google Scholar 

  50. Smith, J., Winkler, R.: The optimizer’s curse: Skepticism and postdecision surprise in decision analysis. Manage. Sci. 52(3), 311–322 (2006)

    Google Scholar 

  51. Terrell, G.R., Scott, D.W.: Variable kernel density estimation. Ann. Stat. 20(3), 1236–1265 (1992)

    MathSciNet  MATH  Google Scholar 

  52. Wang, Z., Glynn, P.W., Ye, Y.: Likelihood robust optimization for data-driven problems. CMS 13(2), 241–261 (2016)

    MathSciNet  MATH  Google Scholar 

  53. Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Wozabal, D.: Robustifying convex risk measures for linear portfolios: A nonparametric approach. Oper. Res. 62(6), 1302–1315 (2014)

    MathSciNet  MATH  Google Scholar 

  55. Yao, H., Li, Z., Lai, Y.: Mean-cvar portfolio selection: A nonparametric estimation framework. Comput. Oper. Res. 40(4), 1014–1022 (2013)

    MathSciNet  MATH  Google Scholar 

  56. Zhao, C., Guan, Y.: Data-driven risk-averse stochastic optimization with Wasserstein metric. Oper. Res. Lett. 46(2), 262–267 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Zhu, S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155–1168 (2009)

    MathSciNet  MATH  Google Scholar 

  58. Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137, 167–198 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments, which significantly improved the quality of this article. This research was supported by the National Science Foundation of China (11971092, 11571061) and the Fundamental Research Funds for the Central Universities (DUT15RC(3)037, DUT18RC(4)067).

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Authors and Affiliations

  1. School of Applied Mathematics, Beijing Normal University, Zhuhai, 519087, Guangdong, China

    Wei Liu

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116025, Liaoning, China

    Wei Liu & Bo Yu

  3. School of Mathematical Sciences, Dalian University of Technology, Panjin, 124221, Liaoning, China

    Li Yang

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  1. Wei Liu
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Correspondence to Bo Yu.

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Liu, W., Yang, L. & Yu, B. Kernel density estimation based distributionally robust mean-CVaR portfolio optimization. J Glob Optim 84, 1053–1077 (2022). https://doi.org/10.1007/s10898-022-01177-5

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