## Abstract

'Separable' uncertainty sets have been widely used in robust portfolio selection models (e.g. see [E. Erdoan, D. Goldfarb, and G. Iyengar, Robust portfolio management, manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004; D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp.1-38; R.H. Tutuncu and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp.157-187]). For these uncertainty sets, each type of uncertain parameter (e.g. mean and covariance) has its own uncertainty set. As addressed in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009- 0271-z], these 'separable' uncertainty sets typically share two common properties: (1) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set, is unknown, and it can be much higher than the desired one; and (2) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified, as observed in the computational experiments conducted in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z; R.H.Tutuncu and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp.157-187]. To combat these drawbacks, we consider a factor model for random asset returns. For this model, we introduce a 'joint' ellipsoidal uncertainty set for the model parameters and show that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. We further show that the robust maximum risk-adjusted return (RMRAR) problem with this uncertainty set can be reformulated and solved as a cone programming problem. The computational results reported in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z] demonstrate that the robust portfolio determined by the RMRAR model with our 'joint' uncertainty set outperforms that with Goldfarb and Iyengar's 'separable' uncertainty set proposed in the seminal paper [D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp.1-38] in terms of wealth growth rate and transaction cost; moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar's is surprisingly highly non-diversified.

Original language | English (US) |
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Pages (from-to) | 89-104 |

Number of pages | 16 |

Journal | Optimization Methods and Software |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2011 |

Externally published | Yes |

### Bibliographical note

Funding Information:The author gratefully acknowledges comments from Dimitris Bertsimas, Victor DeMiguel, Darinka Dentcheva, Andrzej Ruszczyński and Reha Tütüncü at the 2006 INFORMS annual meeting in Pittsburgh, USA. Also, the author is in debt to two anonymous referees for insightful comments and suggestions, which have greatly improved the article. This author was supported in part by NSERC Discovery Grant.

## Keywords

- cone programming
- linear regression
- maximum risk-adjusted return portfolio selection
- mean-variance portfolio selection
- robust optimization