Robust portfolio selection based on a joint ellipsoidal uncertainty set

Abstract

‘Separable’ uncertainty sets have been widely used in robust portfolio selection models (e.g. see [E. Erdoğan, 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. Tütüncü 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.Tütüncü 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.