This paper extends the Log-robust portfolio management approach to the case with short sales, i.e., the case where the manager can sell shares he does not yet own. We model the continuously compounded rates of return, which have been established in the literature as the true drivers of uncertainty, as uncertain parameters belonging to polyhedral uncertainty sets, and maximize the worst-case portfolio wealth over that set in a one-period setting. The degree of the manager's aversion to ambiguity is incorporated through a single, intuitive parameter, which determines the size of the uncertainty set. The presence of short-selling requires the development of problem-specific techniques, because the optimization problem is not convex. In the case where assets are independent, we show that the robust optimization problem can be solved exactly as a series of linear programming problems, and of convex programming problems with one variable; as a result, the approach remains tractable for large numbers of assets. We also provide insights into the structure of the optimal solution. In the case of correlated assets, we develop and test three heuristics. Numerical results suggest that the manager should select the heuristic where correlation is maintained only between assets invested in. In computational experiments, the proposed approach exhibits performance superior to that of the traditional robust approach.
Technical Report, Lehigh University, Department of Industrial and Systems Engineering, Bethlehem, PA, 2008.