Advances on Portfolio Choices

This doctoral dissertation focuses on models, methods, and their applications in portfolio optimization and risk management. The major contribution is developing and analyzing stochastic models on these type of problems. The dissertation consists of five articles, all of which together contribute to the subject by providing novel results and circumventing some shortcomings presented in the literature. Firstly, we analyze the problem of finding optimal mean-variance portfolios under a cardinality constraint. This problem is a mixed integer quadratic programming (MIQP) and is hard to solve. We propose a novel approach to reformulate the problem as a mixed integer linear programming (MILP) problem, which can be solved using any available relevant solver. Computational results show that our MILP approach is significantly more robust and faster than MIQP. Moreover, our MILP approach compares favorably with equivalent problems with coherent risk measures conditional value-at-risk (CVaR) and mean absolute semi-deviation. Secondly, we study the problem of portfolio selection using stochastic dominance (SD) constraints based on second order stochastic dominance (SSD). In particular, we introduce a self-contained theory with some new results. We develop simple arguments to formulate SSD constraints for mathematical programming. We then propose some methods of finding non-dominated optimal solutions and analyze mean-risk models as objectives involving absolute semi-deviation as well as CVaR as risk measures. Moreover, we introduce a relaxation of SSD, named directional SSD (DSSD), and show how it is operationalized for DSSD constrained portfolio optimization. We also presents a thorough comparison of computational performance of alternative approaches to SD constrained optimization. In addition, we consider the out-of-sample study of SSD and DSSD constrained portfolio optimization problems using stock market data of the US. The results suggest that DSSD based approach do well compared to SSD based, given that risk aversion exhibited by the objective function is relatively mild. Lastly, we consider second, third, fourth and fifth order SD (SSD, TSD, FOSD and FISD respectively) as well as decreasing absolute risk aversion (DARA) stochastic dominance (DSD) and stochastic dominance based on exponential utility (ESD). We derive necessary and sufficient efficiency tests under the six types of SD. For well-known SSD and TSD efficiency tests, we provide simple arguments, which are then used to develop new FOSD, FISD, DSD and ESD efficiency tests. Our FOSD and DSD tests circumvent shortcomings in some recent literature. We provide numerical demonstration for each test using stock market data of the US. The results indicate that the market portfolio is inefficient and dominated under those types of SD, for various investment horizons. Interestingly, the results of DSD and ESD are almost identical.