Handbook of Finance: Valuation, Financial Modeling, and Quantitative Tools

Chapter 66. Principles of Optimization for Portfolio Selection

STOYAN V. STOYANOV, PhD

Chief Financial Researcher, FinAnalytica Inc.

SVETLOZAR T. RACHEV, PhD, Dr Sci

Chair-Professor, Chair of Econometrics, Statistics and Mathematical Finance, School of Economics and Business Engineering, University of Karlsruhe and Department of Statistics and Applied Probability University of California, Santa Barbara

FRANK J. FABOZZI, PhD, CFA, CPA

Professor in the Practice of Finance, Yale School of Management

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."

Keywords: optimization, optimal, objective function, set of feasible solutions, set of feasible points, constraint set, saddle point, linear problems, quadratic problems, convex problems, unconstrained optimization, convex functions, function, functional, quasi-convex functions, quasi-concave, convex ...