Linear vs. quadratic portfolio selection models with hard real-world constraints

Several risk–return portfolio models take into account practical limitations on the number of assets to be included in the portfolio and on their weights. We present here a comparative study, both from the efficiency and from the performance viewpoint, of the Limited Asset Markowitz (LAM), the Limited Asset mean semi-absolute deviation (LAMSAD), and the Limited Asset conditional value-at-risk (LACVaR) models, where the assets are limited with the introduction of quantity and of cardinality constraints.The mixed integer linear LAMSAD and LACVaR models are solved with a state of the art commercial code, while the mixed integer quadratic LAM model is solved both with a commercial code and with a more efficient new method, recently proposed by the authors. Rather unexpectedly, for medium to large sizes it is easier to solve the quadratic LAM model with the new method, than to solve the linear LACVaR and LAMSAD models with the commercial solver. Furthermore, the new method has the advantage of finding all the extreme points of a more general tri-objective problem at no additional computational cost.We compare the out-of-sample performances of the three models and of the equally weighted portfolio. We show that there is no apparent dominance relation among the different approaches and, in contrast with previous studies, we find that the equally weighted portfolio does not seem to have any advantage over the three proposed models. Our empirical results are based on some new and old publicly available data sets often used in the literature.