Portfolio Optimization

by Michael J. Best

Overview

Eschewing a more theoretical approach, Portfolio Optimization shows how the mathematical tools of linear algebra and optimization can quickly and clearly formulate important ideas on the subject. This practical book extends the concepts of the Markowitz "budget constraint only" model to a linearly constrained model.

Only requiring elementary linear algebra, the text begins with the necessary and sufficient conditions for optimal quadratic minimization that is subject to linear equality constraints. It then develops the key properties of the efficient frontier, extends the results to problems with a risk-free asset, and presents Sharpe ratios and implied risk-free rates. After focusing on quadratic programming, the author discusses a constrained portfolio optimization problem and uses an algorithm to determine the entire (constrained) efficient frontier, its corner portfolios, the piecewise linear expected returns, and the piecewise quadratic variances. The final chapter illustrates infinitely many implied risk returns for certain market portfolios.

Drawing on the author’s experiences in the academic world and as a consultant to many financial institutions, this text provides a hands-on foundation in portfolio optimization. Although the author clearly describes how to implement each technique by hand, he includes several MATLAB^® programs designed to implement the methods and offers these programs on the accompanying downloadable resources.

Product Details

ISBN-13:	9781420085846
Publisher:	Taylor & Francis
Publication date:	03/09/2010
Series:	Chapman and Hall/CRC Financial Mathematics Series
Edition description:	Book and CD
Pages:	222
Product dimensions:	6.10(w) x 9.30(h) x 0.70(d)

About the Author

Michael J. Best is a professor in the Department of Combinatorics and Optimization at the University of Waterloo in Ontario, Canada. He received his Ph.D. from the Department of Industrial Engineering and Operations Research at the University of California, Berkeley. Dr. Best has authored over 37 papers on finance and nonlinear programming and co-authored a textbook on linear programming. He also has been a consultant to Bank of America, Ibbotson Associates, Montgomery Assets Management, Deutsche Bank, Toronto Dominion Bank, and Black Rock-Merrill Lynch.

Table of Contents

Preface ix

Acknowledgments xii

About the Author xii

Chapter 1 Optimization 1

1.1 Quadratic Minimization 1

1.2 Nonlinear Optimization 8

1.3 Extreme Points 12

1.4 Computer Results 15

1.5 Exercises 18

Chapter 2 The Efficient Frontier 21

2.1 The Efficient Frontier 21

2.2 Computer Programs 33

2.3 Exercises 36

Chapter 3 The Capital Asset Pricing Model 41

3.1 The Capital Market Line 41

3.2 The Security Market Line 51

3.3 Computer Programs 54

3.4 Exercises 58

Chapter 4 Sharpe Ratios and Implied Risk Free Returns 59

4.1 Direct Derivation 60

4.2 Optimization Derivation 66

4.3 Free Problem Solutions 73

4.4 Computer Programs 75

4.5 Exercises 78

Chapter 5 Quadratic Programming Geometry 81

5.1 The Geometry of QPs 81

5.2 Geometry of QP Optimality Conditions 86

5.3 The Geometry of Quadratic Functions 92

5.4 Optimality Conditions for QPs 96

5.5 Exercises 103

Chapter 6 A QP Solution Algorithm 107

6.1 QPSolver: A QP Solution Algorithm 108

6.2 Computer Programs 127

6.3 Exercises 136

Chapter 7 Portfolio Optimization with Constraints 139

7.1 Linear Inequality Constraints: An Example 140

7.2 The General Case 151

7.3 Computer Results 159

7.4 Exercises 163

Chapter 8 Determination of the Entire Efficient Frontier 165

8.1 The Entire Efficient Frontier 165

8.2 Computer Results 183

8.3 Exercises 189

Chapter 9 Sharpe Ratios under Constraints, and Kinks 191

9.1 Sharpe Ratios under Constraints 191

9.2 Kinks and Sharpe Ratios 199

9.3 Computer Results 211

9.4 Exercises 213

Appendix 215

References 217

Index 221

What People are Saying About This

From the Publisher

Michael Best’s book is the ideal combination of optimization and portfolio theory. Mike has provided a wealth of practical examples in MATLAB to give students hands-on portfolio optimization experience. The included stand-alone MATLAB code even provides its own quadratic solver, so that students do not need to rely on any external packages.
—David Starer, Stevens Institute of Technology

Overall, this is a nice book that would be ideal as a textbook for one-semester portfolio optimization courses. It can also be good as a supplementary text for courses in operations research and/or financial engineering. The book is self-contained enough to be used as study material for those who want to teach themselves portfolio optimization and related computer programming, be they advanced undergraduate students, graduate students, or financial practitioners.
—Youngna Choi, Mathematical Reviews, Issue 2012a

… an excellent companion text for the course ‘Discrete-Time Models in Finance’ that I have been teaching in the past years. … I think adding your text can make the course more lively. This is what I plan to do in the coming (fall) semester.
—Edward P. Kao, University of Houston, Texas, USA

Editorial Reviews

Michael Best’s book is the ideal combination of optimization and portfolio theory. Mike has provided a wealth of practical examples in MATLAB to give students hands-on portfolio optimization experience. The included stand-alone MATLAB code even provides its own quadratic solver, so that students do not need to rely on any external packages.
—David Starer, Stevens Institute of Technology

Overall, this is a nice book that would be ideal as a textbook for one-semester portfolio optimization courses. It can also be good as a supplementary text for courses in operations research and/or financial engineering. The book is self-contained enough to be used as study material for those who want to teach themselves portfolio optimization and related computer programming, be they advanced undergraduate students, graduate students, or financial practitioners.
—Youngna Choi, Mathematical Reviews, Issue 2012a

… an excellent companion text for the course ‘Discrete-Time Models in Finance’ that I have been teaching in the past years. … I think adding your text can make the course more lively. This is what I plan to do in the coming (fall) semester.
—Edward P. Kao, University of Houston, Texas, USA

From the Publisher