# Portfolio Selection: Efficient Diversification of Investments

Harry M. Markowitz
Pages: 368
Stable URL: http:/stable/j.ctt1bh4c8h

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. PREFACE TO THE SECOND PRINTING
(pp. ix-xii)
H.M.M.
4. PREFACE
(pp. xiii-xiv)
Harry M. Markowitz
5. PART I. INTRODUCTION AND ILLUSTRATIONS
• Chapter I INTRODUCTION
(pp. 3-7)

This monograph is concerned with the analysis of portfolios containing large numbers of securities. Throughout we speak of “portfolio selection” rather than “security selection.” A good portfolio is more than a long list of good stocks and bonds. It is a balanced whole, providing the investor with protections and opportunities with respect to a wide range of contingencies. The investor should build toward an integrated portfolio which best suits his needs. This monograph presents techniques of Portfolio Analysis directed toward determining a most suitable portfolio for the large private or institutional investor.

A portfolio analysis starts with information concerning individual...

• Chapter II ILLUSTRATIVE PORTFOLIO ANALYSES
(pp. 8-34)

The nature and objectives of portfolio analyses may be illustrated by a small example concerned with portfolios made of one or more of nine common stocks and cash. The nine securities, listed in Figures la to li, include a utility, a railroad, a large and a small steel company, and several other manufacturing corporations. Cash is included in the analysis as a tenth “security.” No special significance should be attached to this list of securities other than that it will be used in illustrating principles of portfolio analysis.

An actual portfolio analysis would start from a much longer list of...

6. PART II. RELATIONSHIPS BETWEEN SECURITIES AND PORTFOLIOS
• Chapter III AVERAGES AND EXPECTED VALUES
(pp. 37-71)

The relationships between securities and portfolios, to be discussed, are mathematical in nature. They follow from definitions of terms and properties of numbers. Like the theorems of geometry, they are subject to precise statement and rigorous deduction.

Except for the appendices, this monograph was written to meet the needs of the reader without mathematical training. The writer has attempted to illustrate concepts concretely, to avoid excessively terse proofs, to introduce essential mathematical apparatus in easy stages. Successive chapters build on previously presented concepts, relationships, and apparatus, thus allowing the reader to raise his level of mathematical sophistication gradually.

The non-mathematician...

• Chapter IV STANDARD DEVIATIONS AND VARIANCES
(pp. 72-101)

The definition of thevarianceof a series is illustrated by Table 1. The first column of the table lists returns on a hypothetical security during 5 years. The average of these returns is .01. The second column indicates, for each year, the difference between that year’s return and the average return. Thus, in the first year, return was .09 greater than the five-year average. The third column presents the squares of the numbers in the second column. In the first year, for example, the squared deviation from the average was .09 times .09 = .0081. The average of the...

• Chapter V INVESTMENT IN LARGE NUMBERS OF SECURITIES
(pp. 102-115)

In the last two chapters we presented formulae for computing the expected return and variance of return of a portfolio. These formulae apply to portfolios of any size, to any pattern of correlations among securities. It may be difficult, however, to see broad principles from a cursory inspection of these formulae. The present section derives from these formulae rules of thumb concerning portfolios containing large numbers of securities. These rules of thumb provide insight intodos anddon’ts of large portfolios.

First we discuss the properties of the average of a large number of uncorrelated random variables. We see that...

• Chapter VI RETURN IN THE LONG RUN
(pp. 116-126)

Suppose that the return on a portfolio during eight consecutive years was One dollar invested in the portfolio at the beginning of year 1 became $1.15 by the end of the year. If this$1.15 were reinvested in the same portfolio, it would have become $1.09 = (1.15) ‧ (.95) by the end of the second year. If reinvestment had continued, by the end of the third year the original$1 would have increased to

$1.31= (1.15)‧(.95)‧(1.20) By the end of the eighth year the$1 would have increased to

$1.44= (1.15)‧(.95)‧(1.20)‧ (1.00)‧(.95)‧(1.05).(1.00)‧(1.10).$1.44 is not the same as the...

7. PART III. EFFICIENT PORTFOLIOS
• Chapter VII GEOMETRIC ANALYSIS OF EFFICIENT SETS
(pp. 129-153)

A portfolio isinefficientif it is possible to obtain higher expected (or average) return with no greater variability of return, or obtain greater certainty of return with no less average or expected return. The problem of separating efficient from inefficient portfolios, when standard deviation or variance is used as a measure of uncertainty, is treated in the present chapter, in the following chapter, and in Appendix A. The present chapter presents a geometric analysis of portfolios containing three or four securities. The following chapter presents computing procedures for obtaining efficient portfolios from the means, variances, and covariances of any...

• Chapter VIII DERIVATION OF E, V EFFICIENT PORTFOLIOS
(pp. 154-187)

This chapter presents the ”critical line method” for deriving efficient portfolios. This method, based on principles illustrated in the last chapter, processes the means, variances, and covariances of any number of securities. From this information it obtains the implied efficient portfolios.

Several early sections of this chapter are devoted to matrix algebra. Although it is possible to present the critical line method without reference to matrices, a discussion of this subject is included for three reasons:

1. The required principles of vectors and matrices are not difficult to learn, even for the reader with a meager mathematical background.

2. A much greater...

• Chapter IX THE SEMI-VARIANCE
(pp. 188-202)

In this chapter we consider the semi-variance as a measure of risk. First, we define the semi-variance; next we compare the semi-variance with the variance, noting similarities and differences and pros and cons of each. After this we consider a geometric analysis and computing procedures for the derivation of efficient portfolios based on expected return and semi-variance.

By definition,

${{r}^{-}}=\left\{ \begin{array}{l} r\text{ if }r\text{ is equal to or less than zero,} \\ 0\text{ if }r\text{ is greater than zero}\text{.} \\ \end{array} \right.$

For example.,

S0is defined to be the mean value of (r)2. Ifris a random variable or a future event subject to probability beliefs, then

S0= expt(r)2.

Ifris the past return on a portfolio,S0...

8. PART IV. RATIONAL CHOICE UNDER UNCERTAINTY
• Chapter X THE EXPECTED UTILITY MAXIM
(pp. 205-242)

A portfolio analysis is characterized by

(1) theinformationconcerning securities upon which it is based;

(2) thecriteriafor better and worse portfolios which set the objectives of the analysis; and

(3) thecomputing proceduresby which portfolios meeting the criteria in (2) are derived from the inputs in (1).

The results of a portfolio analysis are no more than the logical consequences of itsinformationconcerning securities. Although they may not be apparent to unaided reason, the results, nevertheless, are but restatements of these inputs.

Broadly conceived,criteriainclude that for distinguishing legitimate from non-legitimate as well as...

• Chapter XI UTILITY ANALYSIS OVER TIME
(pp. 243-256)

We now turn to the problem of rational behavior over time, still assuming that objective probability distributions are known. The problem of behavior when probability distributions are not known is discussed in Chapter XII.

In Part II we used random wheels to represent chance situations. In the present chapter it will be convenient to employ such wheels similarly. Specifically we shall let a series of related wheels describe the opportunities available to an individual over time.

Suppose that the individual must choose one and only one of the bands on a wheel as in Figure 1. Associated with each band...

• Chapter XII PROBABILITY BELIEFS
(pp. 257-273)

The present chapter considers rational choice when objective probabilities are not known for some or all contingencies. Our discussion will draw heavily on the work of Ramsey [31], Savage [32].¹ We shall see that, if the individual follows principles similar to those expressed in the axioms of Chapter X, he will act in the face of such uncertainty as if he attached “personal probabilities” to each contingency. He will maximize expected utility, using these personal probabilities when objective probabilities are not known.

To some, the maxim based on personal probabilities may seem selfevident. Not all persons who have reflected on...

• Chapter XIII APPLICATIONS TO PORTFOLIO SELECTION
(pp. 274-304)

The first sections of this chapter consider portfolio selection when the following three conditions are satisfied:

(1) the investor owns only liquid assets;

(2) he maximizes the expected value ofU(C1,C2, …,CT), whereCt, is the money value of consumption during thetth period (Ctcould, alternatively, represent money expenditure deflated by a cost of living index);

(3) the set of available probability distributions of returns from portfolios remains the same through time (ifCtis deflated consumption, then it is “real return,” taking into account changes in price level whose probability distribution is assumed constant).

Later we...

9. BIBLIOGRAPHY
(pp. 305-307)