Robust portfolio optimisation using risk measures and applications

Portfolio selection is a decision problem that can be formulated as a mathematical optimisation program. Ever since portfolio selection has been first modelled as a mathematical optimisation problem, a number of frameworks have emerged. These different frameworks aim to address the shortfalls and limitations of previous models. However, most of these models rely on the weak assumption, that the input parameters are known exactly. In the existence of uncertainty surrounding the input parameters, the outcome of a deterministic optimisation problem might be overoptimistic with unexpected consequences in certain scenarios. Robust optimisation deals with the uncertainty surrounding the input parameters. This framework approaches the uncertainty as deterministic and the solution provides certain guarantees, given that the realized scenario is within the considered uncertainty set. The consideration of all possible scenarios leads to more sensible decisions. Robust optimisation frameworks are quite popular in engineering, whereas an overoptimistic solution might yield a catastrophic outcome. This thesis aims to investigate the portfolio construction using robust optimisation frameworks. More specifically, we formulate existing deterministic optimisation models as robust optimisation models and show that they remain tractable under several types of uncertainty. In particular, we examine the distributionally robust Omega Ratio maximization (through solving the Omega Ratio as a convex optimisation problem) and we show that it remains tractable under mixture distribution, ellipsoidal and box uncertainty. In order to illustrate this, we first show that the Omega Ratio maximization is a convex optimisation problem. In addition, we show that the robust counterpart of the Equally-weighted Risk Contribution problem can also be formulated as a convex optimisation problem. We finally provide numerical evidence that suggest the existence of a positive premium for the portfolios constructed using robust formulations versus the deterministic models. The numerical evidence is based on real-life data that span the pre-and post- credit crisis periods.