Mean-Variance Analysis in Portfolio Choice and Capital Markets

by Markowitz, Harry M.; Todd, G. Peter; Sharpe, William F.

• ISBN13: 9781883249755
• ISBN10: 1883249759
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-02-15
• Publisher: Wiley

Summary

In 1952, Harry Markowitz published "Portfolio Selection," a paper which revolutionized modern investment theory and practice. The paper proposed that, in selecting investments, the investor should consider both expected return and variability of return on the portfolio as a whole. Portfolios that minimized variance for a given expected return were demonstrated to be the most efficient. Markowitz formulated the full solution of the general mean-variance efficient set problem in 1956 and presented it in the appendix to his 1959 book, Portfolio Selection. Though certain special cases of the general model have become widely known, both in academia and among managers of large institutional portfolios, the characteristics of the general solution were not presented in finance books for students at any level. And although the results of the general solution are used in a few advanced portfolio optimization programs, the solution to the general problem should not be seen merely as a computing procedure. It is a body of propositions and formulas concerning the shapes and properties of mean-variance efficient sets with implications for financial theory and practice beyond those of widely known cases. The purpose of the present book, originally published in 1987, is to present a comprehensive and accessible account of the general mean-variance portfolio analysis, and to illustrate its usefulness in the practice of portfolio management and the theory of capital markets. The portfolio selection program in Part IV of the 1987 edition has been updated and contains exercises and solutions.

Author Biography

Harry Markowitz has applied computer and mathematical techniques to various practical decision making areas. In finance: he presented in an article in 1952 and a book in 1959 &quot;modern portfolio theory,&quot; now a standard topic in college courses and widely used by institutional investors for tactical asset allocation, risk control, and attribution analysis.</BR>

Table of Contents

Foreword	ix
Preface to Revised Reissue	xv
Preface	xvii
Part I The General Portfolio Selection Model
Portfolio Selection Models	3	(20)
The Standard Mean-Variance Portfolio Selection	3	(4)
Standard Analysis with Upper Bounds	7	(1)
The Tobin-Sharpe-Lintner Model	8	(3)
Black's Model	11	(1)
Model Requiring Collateral for Short Positions	11	(2)
Nominal Versus Real Returns	13	(2)
Appendix to chapter 1	15	(5)
Mean and variance of weighted sums	15	(5)
General sample spaces	20	(1)
Exercises	20	(3)
The General Mean-Variance Portfolio Selection Model	23	(19)
Three Forms of the General Model	24	(4)
Nonlinear Examples	28	(8)
Historical Note	36	(4)
Exercises	40	(2)
Capabilities and Assumptions of the General Model	42	(31)
Semidefinite Covariance Matrices	42	(1)
Portfolio Constraints in Theory and Practice	43	(1)
Industry Constraints	44	(1)
Models of Covariance	45	(3)
Exogenous Assets	48	(2)
Tracking an Index	50	(1)
Turnover Constraints	51	(1)
Why Mean and Variance?	52	(4)
Bayesian Inference	56	(1)
Implied Single-period Utility Maximization	57	(2)
Quadratic Approximations	59	(4)
Research on EV Approximations	63	(5)
Related Matters	68	(5)
Part II Preliminary Results
Properties of Feasible Portfolio Sets	73	(34)
Notation	74	(3)
The Limit of a Sequence	77	(3)
Convergence in Rn	80	(1)
Closed Sets	81	(1)
Spheres, Balls, and Open Sets	82	(4)
Compact Sets	86	(3)
Convex Sets	89	(3)
Unbounded Constraint Sets	92	(2)
Disallowed Directions and Bounded Feasible Directions	94	(4)
Conical Sets	98	(2)
Appendix to chapter 4	100	(5)
Exercises	105	(2)
Sets Involving Mean, Variance, and Standard Deviation	107	(18)
Relationship Involving E	107	(2)
Relationships Involving V	109	(4)
Compensating Transformations	113	(1)
V along a Straight Line	114	(2)
σ along a Straight Line	116	(1)
Convex Functions	117	(3)
Minimum Obtainable V and σ	120	(2)
Exercises	122	(3)
Portfolio Selection Models with Affine Constraint Sets	125	(26)
Minimization Subject to Constraints	125	(2)
Efficient Portfolios with Affine Constraint Sets	127	(12)
Postscript	139	(4)
Exercises	143	(8)
Part III Solution to the General Portfolio Selection Model
Efficient Sets for Nondegenerate Models	151	(33)
Kuhn-Tucker Conditions	152	(2)
Critical Lines	154	(3)
Efficient Segments	157	(4)
Adjacent Efficient Segments	161	(5)
The Nonsingularity of M	166	(5)
Nonnegativity of X and η	171	(3)
Finiteness of the Critical Line Algorithm	174	(2)
The Efficient EV Set	176	(2)
Choice of Axes	178	(1)
Exercises	179	(5)
Getting Started	184	(15)
The Simplex Method of Linear Programming	185	(6)
Prices and Profitabilities	191	(2)
Starting the Critical Line Algorithm	193	(2)
Exercises	195	(4)
Degenerate Cases	199	(26)
Simpler ``Good Enough'' Methods	200	(2)
Efficient Sets when E is Bounded	202	(12)
Lexicographical Ordering	214	(2)
Unbounded E	216	(3)
Related Matters	219	(4)
Exercises	223	(2)
All Feasible Mean-Variance Combinations	225	(18)
The Top of the Obtainable EV Set	229	(7)
Comaprison of the Top and Bottom of the EV Set	236	(2)
The Sides of the Feasible EV Set	238	(1)
Exercises	239	(4)
Part IV Special Cases
Canonical Form of the Two-Dimensional Analysis	243	(32)
The Standard Three-security Analysis	244	(4)
Canonical Form when Rank is 2	248	(5)
Efficient Sets in the Canonical Analysis (Rank 2)	253	(4)
Kinks in the Set of Efficient EV Combinations	257	(2)
Linear Segments in the Set of Efficient Eσ Combinations	259	(3)
τ* of Rank 1	262	(3)
The k Dimensional Canonical Analysis	265	(5)
Appendix to chapter 11	270	(2)
Exercises	272	(3)
Conical Constraint Sets and the Efficiency of the Market Portfolio	275	(26)
The Market Portfolio	276	(1)
Conical Constraint Sets	277	(3)
Efficiency of the Market Portfolio	280	(2)
A Simple Market Equilibrium Model	282	(2)
How Inefficient can the Market Portfolio Be?	284	(2)
Expected Returns and Betas	286	(2)
Exercises	288	(13)
Part V A Portfolio Selection Program
Program Description	301	(38)
G. Peter Todd
Notation	302	(1)
Statement of the Problem	303	(1)
Program Inputs	304	(1)
The Main Module	305	(1)
The Simplex Method	306	(6)
The Critical Line Algorithm	312	(6)
Appendix A to Chapter 13: Program Listing	318	(16)
Appendix B to Chapter 13: Integration and Spreadsheet	334	(1)
Appendix C to Chapter 13: Sample Problem	335	(4)
Appendix Elements of Matrix Algebra and Vector Spaces	339	(22)
Mathematical Prerequisites	339	(1)
Uses of Matrix Notation	339	(2)
Matrix Operations	341	(2)
Inverses	343	(1)
Substitution of Variables	344	(2)
n Dimensional Geometry	346	(2)
Orthogonality	348	(1)
Independence and Subspaces	348	(2)
Change of Coordinate Systems	350	(7)
Change of Coordinates in Rn	357	(4)
References	361	(6)
Index	367