On the role of norm constraints in portfolio selection

Several optimization approaches for portfolio selection have been proposed in order to alleviate the estimation error in the optimal portfolio. Among them are the norm-constrained variance minimization and the robust portfolio models. In this paper, we examine the role of the norm constraint in portfolio optimization from several directions. First, it is shown that the norm constraint can be regarded as a robust constraint associated with the return vector. Second, the reformulations of the robust counterparts of the value-at-risk (VaR) and conditional value-at-risk (CVaR) minimizations contain norm terms and are shown to be highly related to the ν-support vector machine (ν-SVM), a powerful statistical learning method. For the norm-constrained VaR and CVaR minimizations, a nonparametric theoretical validation is posed on the basis of the generalization error bound for the ν-SVM. Third, the norm-constrained approaches are applied to the tracking portfolio problem. Computational experiments reveal that the norm-constrained minimization with a parameter tuning strategy improves on the traditional norm-unconstrained models in terms of the out-of-sample tracking error.