Nothing Special   »   [go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Recent Developments in Robust Portfolios with a Worst-Case Approach

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Robust models have a major role in portfolio optimization for resolving the sensitivity issue of the classical mean–variance model. In this paper, we survey developments of worst-case optimization while focusing on approaches for constructing robust portfolios. In addition to the robust formulations for the Markowitz model, we review work on deriving robust counterparts for value-at-risk and conditional value-at-risk problems as well as methods for combining uncertainty in factor models. Recent findings on properties of robust portfolios are introduced, and we conclude by presenting our thoughts on future research directions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For more complete references on robust optimization, see Bertimas, Brown, and Caramanis [20] and Ben-Tal, El Ghaoui, and Nemirovski [21].

  2. Here the uncertainty of the covariance matrix is represented as a set \(\mathcal{U} _{\varSigma}\), but a factor model approach is used in the original work for defining the uncertainty set. This is further described in Sect. 6.

  3. The structure of uncertainty sets mentioned in this section are described in further detail in Lobo and Boyd [22] and Fabozzi et al. [13, 27].

  4. Stubbs and Vance [28] cover ways to empirically approximate the matrix Σ μ .

  5. We suggest the series of papers by Lu [29, 31, 32] for more detail.

  6. Fabozzi, Huang, and Zhou [34] also provide a comprehensive review on methods for minimizing worst-case VaR and CVaR.

  7. In our definition, we solve for probability at most 1−ε, instead of ε to be consistent with formulations discussed in the following sections.

  8. Natarajan, Pachamanova, and Sim [44] introduce a more general approach for computing the worst-case CVaR when the moments of a distribution are given.

  9. They show that this problem can be formulated as a linear programming problem and Bertsimas, Pachamanova, and Sim [52] provide argument that this uncertainty set is a special case of the norm defined as the D-norm.

  10. To increase the scale of the uncertainty set, they actually test with several point estimates and deviation parameters in addition to a scaling factor.

  11. While their analysis is performed at the portfolio-level, Kim, Kim, and Fabozzi [58] investigate the composition of robust equity portfolios to also confirm that robust portfolios have higher portfolio beta compared to the classical mean–variance model. Kim, Kim, and Fabozzi also conclude that robust optimization forms portfolios that are less diversified but more conservative by allocating less to each stock.

References

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

    Google Scholar 

  2. Markowitz, H.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959)

    Google Scholar 

  3. Black, F., Litterman, R.: Global portfolio optimization. Financ. Anal. J. 48(5), 28–43 (1992)

    Article  Google Scholar 

  4. Michaud, R.O.: The Markowitz optimization enigma: is “optimized” optimal? Financ. Anal. J. 45, 31–42 (1989)

    Article  Google Scholar 

  5. Broadie, M.: Computing efficient frontiers using estimated parameters. Ann. Oper. Res. 45(1), 21–58 (1993)

    Article  MATH  Google Scholar 

  6. Chopra, V.K., Ziemba, W.T.: The effect of errors in means, variances, and covariances on optimal portfolio choice. J. Portf. Manag. 19(2), 6–11 (1993)

    Article  Google Scholar 

  7. Best, M.J., Grauer, R.R.: On the sensitivity of mean–variance-efficient portfolios to changes in asset means: some analytical and computational results. Rev. Financ. Stud. 4(2), 315–342 (1991)

    Article  Google Scholar 

  8. Best, M.J., Grauer, R.R.: Sensitivity analysis for mean–variance portfolio problems. Manag. Sci. 37(8), 980–989 (1991)

    Article  MATH  Google Scholar 

  9. Ceria, S., Stubbs, R.A.: Incorporating estimation errors into portfolio selection: portfolio construction. J. Asset Manag. 7, 109–127 (2006)

    Article  Google Scholar 

  10. Scherer, B.: Can robust portfolio optimization help to build better portfolios? J. Asset Manag. 7(6), 374–387 (2007)

    Article  Google Scholar 

  11. Santos, A.A.P.: The out-of-sample performance of robust portfolio optimization. Braz. Rev. Finance 8(2), 141–166 (2010)

    Google Scholar 

  12. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)

    MATH  Google Scholar 

  13. Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A., Focardi, S.M.: Robust Portfolio Optimization and Management. Wiley, Hoboken (2007)

    Google Scholar 

  14. Soyster, A.L.: Convex programming with set-inclusive constraints and application to inexact linear programming. Oper. Res. 21(5), 1154–1157 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  15. El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. El Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9, 33–52 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goldfarb, D., Iyengar, G.: Robust convex quadratically constrained programming. Math. Program. 97, 495–515 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robustness Optimization. Princeton University Press, Princeton (2009)

    Google Scholar 

  22. Lobo, M.S., Boyd, S.: The worst-case risk of a portfolio. Technical report, Stanford University (2000). http://www.stanford.edu/~boyd/papers/pdf/risk_bnd.pdf

  23. Halldórsson, B.V., Tütüncü, R.H.: An interior-point method for a class of saddle-point problems. J. Optim. Theory Appl. 116(3), 559–590 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28(1), 1–38 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tütüncü, R.H., Koenig, M.: Robust asset allocation. Ann. Oper. Res. 132, 157–187 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Costa, O.L.V., Paiva, A.C.: Robust portfolio selection using linear-matrix inequalities. J. Econ. Dyn. Control 26(6), 889–909 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A., Focardi, S.M.: Robust portfolio optimization. J. Portf. Manag. 33, 40–48 (2007)

    Article  Google Scholar 

  28. Stubbs, R.A., Vance, P.: Computing Return Estimation Error Matrices for Robust Optimization. Axioma, Inc., New York (2005)

    Google Scholar 

  29. Lu, Z.: A new cone programming approach for robust portfolio selection. Technical report, Department of Mathematics, Simon Fraser University, Burnaby, BC (2006)

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Lu, Z.: Robust portfolio selection based on a joint ellipsoidal uncertainty set. Optim. Methods Softw. 26(1), 89–104 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lu, Z.: A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set. Math. Program. 126, 193–201 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. 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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. Natarajan, K., Pachamanova, D., Sim, M.: Incorporating asymmetric distributional information in robust value-at-risk optimization. Manag. Sci. 54(3), 573–585 (2008)

    Article  MATH  Google Scholar 

  36. Chen, X., Sim, M., Sun, P.: A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058–1071 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Huang, D., Fabozzi, F.J., Fukushima, M.: Robust portfolio selection with uncertain exit time using worst-case VaR strategy. Oper. Res. Lett. 35, 627–635 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Rockafeller, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

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

    Article  Google Scholar 

  40. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Huang, D., Zhu, S., Fabozzi, F.J., Fukushima, M.: Portfolio selection with uncertain exit time: a robust CVaR approach. J. Econ. Dyn. Control 32, 594–623 (2007)

    Article  MathSciNet  Google Scholar 

  43. Huang, D., Zhu, S., Fabozzi, F.J., Fukushima, M.: Portfolio selection under distributional uncertainty: a relative robust CVaR approach. Eur. J. Oper. Res. 203, 185–194 (2010)

    Article  MATH  Google Scholar 

  44. Natarajan, K., Pachamanova, D., Sim, M.: Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129–1141 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  45. Ma, X., Zhao, Q., Qu, J.: Robust portfolio optimization with a generalized expected utility model under ambiguity. Ann. Finance 4, 431–444 (2008)

    Article  MATH  Google Scholar 

  46. Garlappi, L., Uppal, R., Wang, T.: Portfolio selection with parameter and model uncertainty: a multi-prior approach. Rev. Financ. Stud. 20(1), 41–81 (2007)

    Article  Google Scholar 

  47. Ruan, K., Fukushima, M.: Robust portfolio selection with a combined WCVaR and factor model. Technical report, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan (2011)

  48. Fama, E.F., French, K.R.: Common risk factors in the returns on stocks and bonds. J. Financ. Econ. 33(1), 3–56 (1993)

    Article  MATH  Google Scholar 

  49. Pflug, G., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  50. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  51. Gregory, C., Darby-Dowman, K., Mitra, G.: Robust optimization and portfolio selection: the cost of robustness. Eur. J. Oper. Res. 212, 417–428 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  52. Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32, 510–516 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  53. Gülpinar, N., Katata, K., Pachamanova, D.: Robust portfolio allocation under discrete asset choice constraints. J. Asset Manag. 12(1), 67–83 (2011)

    Article  Google Scholar 

  54. Kim, W.C., Kim, J.H., Fabozzi, F.J.: Deciphering robust portfolios. Working paper (2012)

  55. Kim, W.C., Kim, J.H., Ahn, S.H., Fabozzi, F.J.: What do robust equity portfolio models really do? Ann. Oper. Res. 205(1), 141–168 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  56. Kim, W.C., Kim, J.H., Mulvey, J.M., Fabozzi, F.J.: Focusing on the worst state for robust investing. Working paper (2013)

  57. Kim, W.C., Kim, M.J., Kim, J.H., Fabozzi, F.J.: Robust portfolios that do not tilt factor exposure. Eur. J. Oper. Res. (2013). doi:10.1016/j.ejor.2013.03.029

    Google Scholar 

  58. Kim, J.H., Kim, W.C., Fabozzi, F.J.: Composition of robust equity portfolios. Finance Res. Lett. (2013). doi:10.1016/j.frl.2013.02.001

    Google Scholar 

  59. Ben-Tal, A., Margalit, T., Nemirovski, A.: Robust modeling of multi-stage portfolio problems. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 303–328. Kluwer Academic, Dordrecht (2000)

    Chapter  Google Scholar 

  60. Dantzig, G.B., Infanger, G.: Multi-stage stochastic linear programs for portfolio optimization. Ann. Oper. Res. 45, 59–76 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  61. Bertsimas, D., Pachamanova, D.: Robust multiperiod portfolio management in the presence of transaction cost. Comput. Oper. Res. 35, 3–17 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful for the suggestions of the anonymous referees and for the support by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2012R1A1A1011157).

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea

    Jang Ho Kim & Woo Chang Kim

  2. EDHEC Business School, Nice, France

    Frank J. Fabozzi

Authors
  1. Jang Ho Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Woo Chang Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frank J. Fabozzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank J. Fabozzi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, J.H., Kim, W.C. & Fabozzi, F.J. Recent Developments in Robust Portfolios with a Worst-Case Approach. J Optim Theory Appl 161, 103–121 (2014). https://doi.org/10.1007/s10957-013-0329-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0329-1

Keywords

Search

Navigation