A mixed-integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem

In the last years, some scholars developed new models for the optimal portfolio problem, taking into account the return non-normality. The simplest models are straightforward extensions of the Markowitz model. It is commonly accepted that efficiency is measured by the portfolio return expectation, but variance is now replaced by some lower tail return distribution function statistics. All these streams of research did not consider the risk measure that most prominently imposed itself within the financial community in the last ten years, that is, the Value-at-Risk (VaR). The mathematical properties of VaR are not that appealing (at least from a mathematical point of view), since it is not linear nor convex and can have many local minima and maxima. As such, it can prevent portfolio diversification. It is somewhat ironic that it was proposed by the Basel committee (a Committee of Internation Bankers that suggests and dictates new rules for financial managent) and J.P. Morgan helped its diffusion with the implementation of specific software This diffusion justifies the efforts of some authors to develop optimization models where, in the stream of the Markovitz model extensions, variance is replaced by VaR. In this paper, we show that the optimal Mean/Value-at-Risk portfolio problem is NP-hard even when future returns are described by discrete uniform distributions. Furthermore, we propose a mixed integer linear programming formulation, which allows us to solve medium size yet practical instances using Cplex. We also report on our experimental evaluation of this solution approach: for some input data the answer is obtained in a few seconds, while for other data the computational times sharply increase. Luckily, the financial instances are usually tractable. We show that in an application to Italian data the model overperfomed the index, suggesting that it provides useful information to the decision maker. Finally, we propose a polynomial time algorithm whenever the number of assets K is fixed.