Portfolio Optimization: Scenario Generation, Models and Algorithms

In single-period portfolio optimization several facets of the problem may influence the goodness of the portfolios selected. Despite that, some of these facets are frequently ignored when the optimization problem is solved. In this thesis, we aim at investigating the impact of these facets on the optimization problem and on the performances of the portfolios selected. Firstly, we consider the problem of generating scenarios. In the domain of single-period portfolio optimization, scenarios are used to compute the expected value of the portfolio return and the value of a risk (or safety) measure. Historical data observations, taken as equally probable scenarios, are frequently used to this aim. However, several other parametric and non-parametric methods can be alternatively applied. Specifically, when dealing with scenario generation techniques practitioners are mainly concerned on how reliable and effective such methods are when embedded into portfolio selection models. To this aim, we survey different techniques to generate scenarios for the rates of return. We also compare these techniques by providing in-sample and out-of-sample analysis of the portfolios selected using these techniques to generate the rates of return. Evidence on the computational burden required to generate scenarios by the different techniques is also provided. As reference model we use the Conditional Value-at-Risk (CVaR) model with transaction costs. Extensive computational results based on different historical data sets from the London Stock Exchange Market (FTSE) are presented and some interesting financial conclusions are drawn. Secondly, we analyze portfolio optimization when data uncertainty is taken into consideration. Data uncertainty is a common feature in most of real-life optimization problems. In deterministic mathematical optimization, it is assumed that all the input data are known with certainty and equal to some nominal values. Nevertheless, the optimal solution of the nominal problem can reveal itself sub-optimal or even infeasible when some of the data take values different from the nominal ones. An area where data uncertainty is a natural concern is portfolio optimization. As a matter of fact, in portfolio selection every optimization model deals with the estimate of the portfolio rate of return. In the literature several techniques that are immune to data uncertainty have been proposed. These techniques are called robust. We investigate the effectiveness of two well-known robust optimization techniques when applied to a specific portfolio selection problem, and compare the portfolios selected by the corresponding robust counterparts. As reference model we consider the portfolio optimization problem with the CVaR as performance measure. We carried out extensive computational experiments based on real-life data from the London Stock Exchange Market (FTSE) under different market behaviors. Thirdly, we study the optimal portfolio selection problem in a rebalancing framework, considering fixed and proportional transaction costs and evaluating how much they affect a re-investment strategy. Specifically, we modify the single-period portfolio optimization model with transaction costs, based on the CVaR as performance measure, to introduce portfolio rebalancing. The aim is to provide investors and financial institutions with an effective tool to better exploit new information made available by the market. We then suggest a procedure to use the proposed optimization model in a rebalancing framework. Extensive computational results based on different historical data sets from the German Stock Exchange Market (XETRA) are presented. The last part of the thesis is devoted to the problem of replicating the performances of a stock market index, but considering transaction costs and without purchasing all the securities that constitute the index, i.e. the index tracking problem. Furthermore, we also consider the problem of out-performing a market index, i.e. the so-called enhanced index tracking problem. We present mixed-integer linear programming formulations of these two problems. Our formulations include both fixed and proportional transaction costs, a cardinality constraint limiting the number of securities that constitute the portfolio, upper and lower limits on the investment in the securities, and a threshold on the total transaction costs paid. Both models can be used to rebalance a current portfolio composition. We also introduce a heuristic framework, called Enhanced Kernel Search, to solve the problem of tracking an index. We test and analyze the behavior of several implementations of the Enhanced Kernel Search framework. We show the effectiveness and efficiency of the framework comparing the performances of the tested heuristics with those of a general-purpose solver. The computational experiments are carried out using benchmark instances for the index tracking problem.