Closed-Form Optimal Portfolios of Distributionally Robust Mean-CVaR Problems with Unknown Mean and Variance

Article
  • 100 Downloads

Abstract

In this paper, we consider both one-period and multi-period distributionally robust mean-CVaR portfolio selection problems. We adopt an uncertainty set which considers the uncertainties in terms of both the distribution and the first two order moments. We use the parametric method and the dynamic programming technique to come up with the closed-form optimal solutions for both the one-period and the multi-period robust portfolio selection problems. Finally, we show that our approaches are efficient when compared with both normal based portfolio selection models, and robust approaches based on known moments.

Keywords

Distributionally robust optimization Robust portfolio selection Nested risk measure Conditional value-at-risk Closed-form solution 

Notes

Acknowledgements

The authors are grateful to the editor and two anonymous referees for their insightful, constructive and detailed comments and suggestions, which have helped us to improve the paper significantly in both content and style. This research was supported by the National Natural Science Foundation of China under grant numbers 71371152 and 11571270, and Programme Cai Yuanpei under grant number 34593YE.

References

  1. 1.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Prog. 99(2), 351–376 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton, NJ (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Chen, Z., Liu, J.: Multi-period robust risk measures and portfolio selection models with regime-switching. Working Paper. Xi’an Jiaotong University (2015)Google Scholar
  4. 4.
    Chen, L., He, S., Zhang, S.: Tight bounds for some risk measures, with applications to robust portfolio selection. Oper. Res. 59(4), 847–865 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, Z., Consigli, G., Liu, J., Li, G., Fu, T., Hu, Q.: Multi-period risk measures and optimal investment policies. In: Consigli, G., Kuhn, D., Brandimarte, P. (eds.) Optimal Financial Decision Making Under Uncertainty, pp. 1–34. Springer, Berlin (2017)Google Scholar
  6. 6.
    Cheng, J., Delage, E., Lisser, A.: Distributionally robust stochastic knapsack problem. SIAM J. Optim. 24, 1485–1506 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Delage, E., Iancu, D.A.: Robust multistage decision making. In: INFORMS Tutorials in Operations Research, Operations Research Revolution, pp. 20–46 (2015)Google Scholar
  8. 8.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58, 595–612 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ermoliev, Y., Gaivoronski, A., Nedeva, C.: Stochastic optimization problems with incomplete information on distribution functions. SIAM J. Control Optim. 23(5), 697–716 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Iancu, D.A., Sharma, M., Sviridenko, M.: Supermodularity and affine policies in dynamic robust optimization. Oper. Res. 61(4), 941–956 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, J.Y.: Closed-form solutions for worst-case law invariant risk measures with application to robust portfolio optimization. Working Paper. SSRN 2838518 (2016)Google Scholar
  13. 13.
    Li, D., Ng, W.L.: Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Math. Finance 10, 387–406 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lobo, M.S., Boyd, S.: The worst-case risk of a portfolio. Working Paper. Stanford University (2000)Google Scholar
  15. 15.
    Lotfi, S., Zenios, S.: Equivalence of robust VaR and CVaR optimization. Working Paper. Wharton Financial Institutions Center WP, 16-03 (2016)Google Scholar
  16. 16.
    Mamani, H., Nassiri, S., Wagner, M.R.: Closed-form solutions for robust inventory management. Manag. Sci. (2016). doi: 10.1287/mnsc.2015.2391
  17. 17.
    Mulvey, J.M., Shetty, B.: Financial planning via multi-stage stochastic optimization. Comput. Oper. Res. 31(1), 1–20 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Natarajan, K., Sim, M., Uichanco, J.: Tractable robust expected utility and risk models for portfolio optimization. Math. Finance 20(4), 695–731 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Paç, A., Pınar, M.: On robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity. TOP 22(3), 875–891 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Probabilistic Constrained Optimization, pp. 272–281. Springer, New York (2000)Google Scholar
  21. 21.
    Pflug, G.C., Analui, B.: On distributionally robust multiperiod stochastic optimization. Comput. Manag. Sci. 11, 197–220 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pflug, G.C., Pichler, A., Wozabal, D.: The \(1/N\) investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)CrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T., Uryasev, S.P.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  25. 25.
    Rockafellar, R.T., Uryasev, S.P.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002)CrossRefGoogle Scholar
  26. 26.
    Scarf, H.: A min-max solution of an inventory problem. In: Arrow, K.J., Karlin, S., Scarf, H.E. (eds.) Studies in the Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Stanford (1958)Google Scholar
  27. 27.
    Shapiro, A.: A dynamic programming approach to adjustable robust optimization. Oper. Res. Lett. 39, 83–87 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)CrossRefMATHGoogle Scholar
  29. 29.
    Sharpe, W.F., Alexander, G.J., Bailey, J.: Investments. Prentice Hall, Englewood Cliffs (1995)Google Scholar
  30. 30.
    Steinbach, M.: Markowitz revisited: mean-variance models in financial portfolio analysis. SIAM Rev. 43(1), 31–85 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wozabal, D.: Robustifying convex risk measures for linear portfolios: a nonparametric approach. Oper. Res. 62, 1302–1315 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Xin, L., Goldberg, D.A., Shapiro, A.: Time (in) consistency of multistage distributionally robust inventory models with moment constraints. arXiv preprint. arXiv:1304.3074 (2013)
  33. 33.
    Yu, Y., Li, Y., Schuurmans, D., Szepesvari, C.: A general projection property for distribution families. Adv. Neural Inf. Process. Syst. 22, 2232–2240 (2009)Google Scholar
  34. 34.
    Žácková, J.: On minimax solutions of stochastic linear programming problems. Časopis pro Pěstování Matematiky 91, 423–430 (1966)MathSciNetMATHGoogle Scholar
  35. 35.
    Zhu, S.S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155–1168 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Jia Liu
    • 1
    • 2
  • Zhiping Chen
    • 1
  • Abdel Lisser
    • 2
  • Zhujia Xu
    • 1
  1. 1.Department of Computing Science, School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Laboratoire de Recherche en Informatique (LRI)Université Paris Sud - XIOrsay CedexFrance

Personalised recommendations