For portfolio management in the real-world, it is required that a portfolio has a manageable number of assets and stable performance. However, much research has pointed out that the Markowitz model, which is a classical model in portfolio theory, forms a portfolio with many different assets that may have unstable performance.
Therefore, in this paper, we focus on developing a portfolio selection model which constructs a sparse and robust optimal portfolio. In order to achieve our research goal, we introduce two kinds of optimization problems. The first one is a $L_2$ -norm regularized cardinality constraint portfolio and the second one is cardinality constrained robust optimization portfolio with ellipsoidal uncertainty set. Moreover, we formulate a convex optimization problem for these proposed models using semi-definite relaxation. The outcomes of our empirical tests show that portfolios obtained by our model have smaller cardinalities and better out-of-sample performances than those of cardinality constrained Markowitz optimal portfolios.
A large part of financial business is now being automated. Our portfolios give the investors new opportunity to obtain the desired properties; sparsity and robustness.