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Recent Developments in Robust Portfolios with a Worst-Case Approach

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.

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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.

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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).

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Correspondence to Frank J. Fabozzi.

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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

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

Keywords

  • Mean-variance model
  • Robust portfolio
  • Worst-case optimization
  • Uncertainty sets