This study contributes to the optimization literature with an approach that would help investors understand how the risk-aversion profile hyperparameter affects excess returns, risk, and Sharpe ratio curves in portfolio optimization problems with short selling constraints. These curves were characterized by studying the original optimization problem and reducing it to a one-dimensional optimization problem. The problem variable was the excess return, and the minimum level of risk is expressed as a function of it. An approach to the functional form of the minimum risk level curve was also proposed, which allows us to determine an analytical expression for the aforementioned curves. The study provides significant results for the financial literature, such as (i) an upper and lower bound for the risk aversion profile hyperparameter; (ii) the optimal value for the risk aversion profile hyperparameter; (iii) a reduced version of the optimization problem that is easier to solve, and of course (iv) an analytical expression for the excess return, risk and Sharpe ratio curves as functions of the aforementioned hyperparameters. All of these results are reported using the Mean Squared Variance (MSV) portfolio optimization problem as the baseline model, representing the two objectives of the problem minimization function (excess return and risk) in the same unit.

This research presents a fast heuristic method for solving large-scale real-life Stock Rebalancing problems with minimum transfer constraints on the arcs, as well as a maximum supply and demand limitation on the nodes, which can be considered as a variation of the multi-commodity network design (MCND) problems. The proposed Rank-based Greedy Heuristic with Swapping (RGHS) ranks all feasible flow combinations according to a profit criteria. Then, the algorithm greedily considers the combinations until demand and supply constraints are met, followed by a flow swapping mechanism to further improve the solution. Furthermore, the RGHS is extended to a Reduced Set Hybrid Model (RSHM) that combines the heuristic approach with a commercial solver on the reduced solution space. The proposed methods are evaluated against the published Modified Greedy (MG) algorithm that showed good results on benchmark instances with a significantly improved computation time compared to other state-of-the-art methods. This study contributes by proposing a fast algorithm tailored for the real-life instances on considerably larger instances compared to existing literature, and introduces the concept of minimum transfer restrictions in contrast to the more common maximum capacities. This paper reports the results on various large-scale real-life instances and larger simulated instances and shows the scalability and solution quality compared to existing methods. • Presents the Stock Rebalancing problem as variant to Multi-Commodity Network Design. • Proposes a Hybrid framework for solving the novel Stock Rebalancing problem. • Introduces a heuristic approach for solving large-scale real-life instances. • Demonstrates very high scalability using extensive computational experiments. • Outperforms existing fast hybrid methods for significantly larger instances.