Encyclopedia of Financial Models I

Principles of Optimization for Portfolio Selection

STOYAN V. STOYANOV, PhD

Professor of Finance at EDHEC Business School and Head of Research for EDHEC Risk Institute-Asia

SVETLOZAR T. RACHEV, PhD, Dr Sci

Frey Family Foundation Chair-Professor, Department of Applied Mathematics and Statistics, Stony Brook University, and Chief Scientist, FinAnalytica

FRANK J. FABOZZI,, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: The mathematical theory of optimization has a natural application in the field of finance. From a general perspective, the behavior of economic agents in the face of uncertainty involves balancing expected risks and expected rewards. For example, the portfolio choice problem concerns the optimal trade-off between risk and reward. A portfolio is said to be optimal in the sense that it is the best portfolio among many alternative ones. The criterion that measures the quality of a portfolio relative to the others is known as the objective function in optimization theory. The set of portfolios among which we are choosing is called the “set of feasible solutions” or the “set of feasible points.”

In optimization theory there is a distinction between two types of optimization problems depending on whether the set of feasible solutions is constrained or unconstrained. If the optimization problem is a constrained one, then the set of feasible solutions is defined by means of certain linear and/or nonlinear equalities and inequalities. These functions ...