Computational Issues

Proposed portfolio models are computationally attractive as they give rise to linear and mixed integer programming problems. The introduction of real features, when requiring integer and/or binary variables, may increase problem complexity significantly. In this chapter, we address the computational issues related to the solution and comparison of the described models. We briefly discuss the solution of the models through out-of-the shelf software packages and provide, whenever possible, hints to researchers and practitioners facing the solution of LP/MILP portfolio problems. To tackle large size MILP problems, one has to resort to heuristics. General purpose methods with respect to specific algorithms have the advantage that can be applied to large classes of MILP problems. We present a matheuristic, called Kernel Search, that is general and flexible, and requires very limited implementation effort. An important part of the chapter covers the computational issue related to the data needed by a model. The models presented must be fed with data, the most crucial being the rates of return of the assets under the different scenarios. Although several techniques can generate such data, we just present some of them focusing our attention on the one based on historical realizations as the simplest and most frequently used. The type of generation technique adopted affects the number of scenarios that have to be generated, and in turn the time required to solve the portfolio optimization model, where the number of constraints is proportional to the number of scenarios. A part of the chapter is dedicated to the problem of solving large scale LP models. When the number of scenarios, and consequently the number of constraints and slack variables associated with models, grows to the order of several thousands, then solving an LP problem may take a large amount of time. We show how, in such cases, the computational efficiency can be achieved solving the dual problem. Finally, we deal with the comparison and testing of different models. In previous chapters we present different risk/safety performance measures, different ways to include the transaction costs in a model, different constraints that may be added. The question is: Which model is the most appropriate to represent the problem? How can different models be compared? Are there any recognized methodologies? We discuss how models can be validated providing hints based on our personal experience.