In this paper, we study extensions of the classical Markowitz mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint which imposes that the expected return of the constructed portfolio must exceed a prescribed return threshold with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the deterministic equivalents are second-order cone programs, and give closed-form formulations. Second, we account for real-world trading constraints (such as the need to diversify the investments in a number of industrial sectors, the non-profitability of holding small positions and the constraint of buying stocks by lots) modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a non-linear branch-and-bound algorithm which features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and is called portfolio risk branching. The two branching rules are implemented and tested using the open-source Bonmin framework. The comparison of the computational results obtained with state-of-the-art MINLP solvers (MINLP BB and CPLEX) and with our approach shows the effectiveness of this latter which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time. The practicality of the approach is illustrated through its use for the construction of four fund-of-funds now available on the major trading markets.

## Citation

Forthcoming in Operations Research.