Multi-Objective Portfolio Optimization by Mixed Integer Programming

Abstract

In this PhD dissertation the mathematical programming methods of operations research for multi-criteria optimization are presented. The PhD dissertation deals with the problem of selection of methods and numerical tools for solving portfolio optimization problems with different objectives. In particular, the research efforts were concentrated on mixed integer programming formulations. The need for solving multi-objective portfolio optimization models by mixed integer programming can be illustrated for the portfolio models with Value-at-Risk (VaR) as a risk measure, as well as, when the number of assets (investment ventures) is one of the optimality criteria. An alternative, multi-objective portfolio optimization problems is formulated with Conditional Value-at-Risk (CVaR) as a risk measure or with symmetric measure of risk - covariance (variance) of return - as in Markowitz portfolio.

The portfolio models with CVaR and with covariance (variance) of historical return were being solved with the use of mathematical programming with the continuous variables. The proposed multi-objective portfolio models are constructed with the expected return as a performance measure and the expected worst-case return as a risk measure, using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). These measures allow the evaluation of worst-case return and shaping of the resulting return distribution through the selection of the optimal portfolio. The mathematical programming models are constructed and solved using weighting, lexicographic and reference point approach. The presented portfolio models are single-, bi- and triple-objectives and the optimization criteria considered are risk, return and number of stocks.

The main research problem considered in this Ph.D. dissertation is the way for finding the best multi-objective portfolio formulation with risk. The additional research problem is to find the relation between the optimization results with Value-at-Risk solved by mixed integer programming and the results obtained with the use of linear and quadratic programming portfolio models (Conditional Value-at-Risk, Markowitz).

Computational experiments have been conducted for multi-criteria portfolio models of stock exchange investments. The input data for computations consist of historical daily returns of stocks quoted on Warsaw Stock Exchange. The number of selected securities for input data varies from 46 to 240 assets. The historical stocks quotations come from the period from March 10th, 1997 to February 2nd, 2009. This time period includes data from the increase of stock exchange quotations, as well as the economic crisis period. The considered number of data in historical time series is from 500 to 3000 days with assets quoted each day in the whole historical horizon. The portfolios were optimized in an increased time window, which was helpful in evaluating the results of optimization (time-varying optimal portfolio).

The multi-criteria portfolio optimization models with Conditional Value-at-Risk (CVaR) as a risk measure can be used to support on-line stock market investments, since the computational times required to find the optimal solution is relatively short, regardless of the size of the input data. The presented models provide a decision maker with a tool for evaluating the relationship between expected and worst-case returns.

The results obtained from computational experiments proved, that multi-objective portfolio optimization models with Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) could be used to shape the distribution of portfolio returns in a favorable way for a decision maker. The portfolios obtained with both methods (mixed-integer or linear programming) are often similar results, which shows their capability of solving the corresponding problems. It means that a suboptimal portfolio for the integer program with Value-at-Risk (VaR) as optimality criterion can be found by solving the corresponding linear program for the portfolio problem with Conditional Value-at-Risk (CVaR) as an optimality criterion. The proposed scenario-based portfolio optimization problems under uncertainty, formulated as a single- or multi-objective mixed integer program were solved using commercially available software (AMPL/CPLEX) for mixed integer programming.

In addition to the multi-objective approach for portfolio optimization of securities (e.g. stocks) from stock exchanges presented in this dissertation, the selected multi-objective mixed integer programming models are shown for supporting services in medical care institutions, based on an assignment problem.

Keywords: Multi-Criteria Decision Making, Mathematical Programming, Mixed Integer Programming, Linear Programming, Quadratic Programming, Portfolio Optimization, Conditional Value-at-Risk, Value-at-Risk, Weighting Approach, Lexicographic Approach, Reference Point Method

JEL Classification: C02, C6, C61, C63, E44, G11, G17, G31