| Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio that is extremely sensitive to errors in the data. The main goal of this thesis is to formulate robust portfolio selection problems based on VaR when the distribution of the returns is only partially known. The idea is to compute VaR and to optimize a VaR-based portfolio in a way that is robust to uncertainties in both the form of the underlying probability distribution and the estimation errors in the parameters of this distribution, such as mean and covariance matrix.; In our framework, we assume that the distribution of returns is only partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the robust Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problems of computing, and optimizing, the robust VaR, and show that for several useful uncertainty structures these problems can be cast as semidefinite programs (SDP). We consider several extensions and variations of the model. These include incorporating uncertainty in factor models, including support constraints, and relative entropy information. For some of these cases the resulting problem does not seem to be tractable, but we show how to compute and optimize an upper bound on the robust VaR via SDP.; Finally, we examine two related risk measures: worst-case Conditional VaR and worst-case VaR, and show that, for the case when the distribution is only known up to the first two moments, they are equivalent to the robust VaR measure. Numerical examples illustrating the approach are presented. |
