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Professor Markowitz has been awarded the Nobel Prize for Economics 1990.
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EFFICIENT DIVERSIFICATION OF INVESTMENTS
By Harry M. Markowitz
Yale UNIVERSITY PRESSCopyright © 1959 Cowles Foundation for Research in Economics at Yale University
All rights reserved.
The Analysis of Portfolios
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 securities. It ends with conclusions concerning portfolios as a whole. The purpose of the analysis is to find portfolios which best meet the objectives of the investor.
Various types of information concerning securities can be used as the raw material of a portfolio analysis. One source of information is the past performance of individual securities. A second source of information is the beliefs of one or more security analysts concerning future performances. When past performances of securities are used as inputs, the outputs of the analysis are portfolios which performed particularly well in the past. When beliefs of security analysts are used as inputs, the outputs of the analysis are the implications of these beliefs for better and worse portfolios.
This introductory chapter discusses broad principles upon which the techniques of portfolio analysis are based. The next chapter discusses the inputs, outputs, and objectives of illustrative portfolio analyses. Subsequent parts of the monograph go more deeply into the techniques by which information concerning securities is transformed into conclusions concerning portfolios.
The Uncertainty of Security Returns
Uncertainty is a salient feature of security investment. Economic forces are not understood well enough for predictions to be beyond doubt or error. Even if the consequences of economic conditions were understood perfectly, non-economic influences can change the course of general prosperity, the level of the market, or the success of a particular security. The health of the President, changes in international tensions, increases or decreases in military spending, an extremely dry summer, the success of an invention, the miscalculation of a business management—all can affect the capital gains or dividends of one or many securities.
We are expecting too much if we require the security analyst to predict with certainty whether a typical security will increase or decrease in value. Even if he could assemble all information, including information available only to the managers of the corporation and information available only to its competitors, the security analyst might still be forced to conclusions such as:
This security may be expected to do well if securities in general do well. It must be expected to do poorly if securities in general do poorly. Even this following of the market is not certain. There are weaknesses which may cause it to do poorly even though securities in general are performing well: The possibility of a labor dispute or of an aggressive competitor cannot be ignored. On the other hand, there are potentialities which may bring success greater than even the corporation management dares hope. The new styling of the product, the (not inexpensive) advertising campaign, and the expansion of production facilities may prove to be a magic combination, fulfilling all expectations for it.
Only the clairvoyant could hope to predict with certainty. Clairvoyant analysts have no need for the techniques of this monograph.
The existence of uncertainty does not mean that careful security analyses are valueless. The security analyst may be expected to arrive at reasonable opinions to the effect that:
The return (including capital gains and dividends) on security A is less uncertain than that on security B; the return on security C is more closely connected to the course of the general market than is that on security D; the growth of security E is more certain but has less potential than that of security F; only if the demand for their industry's product continues to expand (as it is likely, but not certain, to do) will the return on securities G and H be satisfactory.
Carefully and expertly formed judgments concerning the potentialities and weaknesses of securities form the best basis upon which to analyze portfolios.
Correlation among Security Returns
A second salient feature of security investment is the correlation among security returns. Like most economic quantities, the returns on securities tend to move up and down together. This correlation is not perfect: individual securities and entire industries have at times moved against the general flow of prosperity. On the whole, however, economic good and ill tend to spread, causing periods of generally high or generally low economic activity.
If security returns were not correlated, diversification could eliminate risk. It would be like flipping a large number of coins: we cannot predict with confidence the outcome of a single flip; but if a great many coins are flipped we can be virtually sure that heads will appear on approximately one-half of them. Such canceling out of chance events provides stability to the disbursements of insurance companies. Correlations among security returns, however, prevent a similar canceling out of highs and lows within the security market. It is somewhat as if 100 coins, about to be flipped, agreed among themselves to fall, heads or tails, exactly as the first coin falls. In this case there is perfect correlation among outcomes. The average outcome of the 100 flips is no more certain than the outcome of a single flip. If correlation among security returns were "perfect"—if returns on all securities moved up and down together in perfect unison—diversification could do nothing to eliminate risk. The fact that security returns are highly correlated, but not perfectly correlated, implies that diversification can reduce risk but not eliminate it.
The correlation among returns is not the same for all securities. We generally expect the returns on a security to be more correlated with those in the same industry than those of unrelated industries. Business connections among corporations, the fact that they service the same area, a common dependence on military expenditures, building activity, or the weather can increase the tendency of particular returns to move up and down together.
To reduce risk it is necessary to avoid a portfolio whose securities are all highly correlated with each other. One hundred securities whose returns rise and fall in near unison afford little more protection than the uncertain return of a single security.
Objectives of a Portfolio Analysis
It is impossible to derive all possible conclusions concerning portfolios. A portfolio analysis must be based on criteria which serve as a guide to the important and unimportant, the relevant and irrelevant.
The proper choice of criteria depends on the nature of the investor. For some investors, taxes are a prime consideration; for others, such as non-profit corporations, they are irrelevant. Institutional considerations, legal restrictions, relationships between portfolio returns and the cost of living may be important to one investor and not to another. For each type of investor the details of the portfolio analysis must be suitably selected.
Two objectives, however, are common to all investors for which the techniques of this monograph are designed:
1. They want "return" to be high. The appropriate definition of "return" may vary from investor to investor. But, in whatever sense is appropriate, they prefer more of it to less of it.
2. They want this return to be dependable, stable, not subject to uncertainty. No doubt there are security purchasers who prefer uncertainty, like bettors at a horse race who pay to take chances. The techniques in this monograph are not for such speculators. The techniques are for the investor who, other things being equal, prefers certainty to uncertainty.
The portfolio with highest "likely return" is not necessarily the one with least "uncertainty of return." The most reliable portfolio with an extremely high likely return may be subject to an unacceptably high degree of uncertainty. The portfolio with the least uncertainty may have an undesirably small "likely return." Between these extremes would lie portfolios with varying degrees of likely return and uncertainty.
If portfolio A has both a higher likely return and a lower uncertainty of return than portfolio B and meets the other requirements of the investor, it is clearly better than portfolio B. Portfolio B may be eliminated from consideration, since it yields less return with greater uncertainty than does another available portfolio. We refer to portfolio B as "inefficient." After eliminating all such inefficient portfolios—all such portfolios which are clearly inferior to other available portfolios—we are left with portfolios which we shall refer to as "efficient." These consist of: the portfolio with less uncertainty than any other with a 6% likely return, the portfolio with less uncertainty than any other with a 7% likely return, and so on. It cannot be said of two efficient portfolios "the first is clearly better than the second since it has a larger likely return and less uncertainty." All such cases have been eliminated.
The proper choice among efficient portfolios depends on the willingness and ability of the investor to assume risk. If safety is of extreme importance, "likely return" must be sacrificed to decrease uncertainty. If a greater degree of uncertainty can be borne, a greater level of likely return can be obtained. An analysis of the type presented in this monograph:
first, separates efficient from inefficient portfolios;
second, portrays the combinations of likely return and uncertainty of return available from efficient portfolios;
third, has the investor or investment manager carefully select the combination of likely return and uncertainty that best suits his needs; and fourth, determines the portfolio which provides this most suitable combination of risk and return.
ILLUSTRATIVE PORTFOLIO ANALYSES
Inputs to an Illustrative Portfolio Analysis
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 promising securities. Not all these securities would appear in the final desirable portfolio. They enter the analysis as candidates for a place in the desirable portfolio.
The returns on the nine securities, during the years 1937-54, are presented in Table 1 and illustrated in Figure 1. The return during a year is defined to be
(the closing price for the year) minus
(the closing price for the previous year) plus
(the dividends for the year) all divided by
(the closing price of the previous year).
For example, the return in 1948 is
(closing price, 1948) - (closing price, 1947) + (dividends, 1948)/(closing price, 1947)
This is the amount which an investor would have made or lost if he invested 51.00 at the end of 1947, collected the dividends declared in 1948, and sold at the closing price of 1948. A loss is represented by a negative return. For example, if the closing price of 1947 were 50, that of 1948 were 45, and $2 of dividends were declared during 1948, then the return in 1948 would be
45 - 50 + 2/50 = -.06,
or a loss of 6 % per dollar invested.
Our example portfolio analysis will consider performances of portfolios with respect to "return" thus defined. This assumes that a dollar of realized or unrealized capital gains is exactly equivalent to a dollar of dividends, no better and no worse. This assumption is appropriate for certain investors, for example, some types of tax-free institutions. Other ways of handling capital gains and dividends, which are appropriate for other investors, are discussed later.
Our nine securities differed in the amount of return which they yielded on the average. For example, the average of the annual returns on United States Steel Common Stock was 14.6 cents per dollar invested; that on Coca-Cola Common was 5.5 cents per dollar invested. On the average the return on U.S. Steel was higher than that on Coca-Cola.
Securities also differ with respect to their stability of return. For example, the greatest loss incurred on A. T. & T. was 18 cents per dollar invested (in 1941). On the other hand, the greatest loss on Sharon Steel was 43 cents per dollar in vested .(in 1937). In three other years Sharon Steel showed losses exceeding 20 cents per dollar. Clearly, A. T. & T. showed less variability of return than did Sharon Steel.
Portfolio selection should be based on reasonable beliefs about future rather than past performances per se. Choice based on past performances alone assumes, in effect, that average returns of the past are good estimates of the "likely" return in the future; and variability of return in the past is a good measure of the uncertainty of return in the future. Later we shall see how considerations other than past performances can be introduced into a portfolio analysis. For the present it is convenient to discuss an analysis based on past performances alone.
Suppose that a portfolio consisted of 20 cents' worth of Atchison, Topeka & Santa Fe per dollar invested, plus 80 cents' worth of Coca-Cola per dollar invested. The return in 1954 on such a portfolio would be
(.2) times (the return of A. T. & Sfe in 1954) plus
(.8) times (the return of Coca-Cola in 1954)
= (.2)(.469) + (.8)(.077)
Return can be calculated similarly for any combination of securities in any year.
The average return on the portfolio consisting of 80% Coca-Cola and 20% A. T. & Sfe was equal to
(.8) times (the average return on Coca-Cola) plus
(.2) times (the average return on A. T. & Sfe)
= (.8)(.055) + (.2)(.198)
This is higher than the average return on Coca-Cola and lower than the average return on A. T. & Sfe. Inevitably the average return on a portfolio lies somewhere between the highest and the lowest average return on the securities contained in the portfolio.
One might conjecture that the variability of return on a portfolio can, similarly, be no smaller than that of the least variable security in the portfolio. But this is not so. The return on A. T. & Sfe was rather unstable during the period 1937-54 (showing a maximum loss of 45 cents on the dollar). The return on Coca-Cola was more stable, showing a maximum loss of only 25 cents. The return on the 80 %-20 % combination of Coca-Cola and A. T. & Sfe, respectively, was still more stable. Its maximum loss was only 18 cents on the dollar. In Figure 2 we have plotted the annual returns on the portfolio consisting of 80 cents Coca-Cola, 20 cents A. T. & Sfe. For comparison we have also plotted the return on Coca-Cola.
"Largest loss" is not the only possible measure of variability. Another measure, better for our purposes, is discussed later. In terms of this measure also, the variability of A. T. & Sfe is greater than that of Coca-Cola, while that of Coca-Cola is, nevertheless, greater than that of the portfolio. For the present we assume that Figure 2 and the reader's eye confirm the statement that the variability of the particular portfolio was less than that of either of the securities contained in it.
Our 20%-80% portfolio had both a higher average return and a lower variability of return than a portfolio consisting of 100% Coca-Cola. On the whole, the "diversified" portfolio was both more profitable and more stable than Coca-Cola alone. One might wonder whether or not there was some other portfolio—some other combination of our ten securities (nine securities and cash)—which had both greater average return and greater stability than even the 20%-80% mixture. Or perhaps there was a portfolio with greater average return and the same stability; or greater stability and the same average return.
Excerpted from Portfolio Selection by Harry M. Markowitz. Copyright © 1959 Cowles Foundation for Research in Economics at Yale University. Excerpted by permission of Yale UNIVERSITY PRESS.
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Table of Contents
PART I. INTRODUCTION AND ILLUSTRATIONS....................
1. Introduction.................... 3
2. Illustrative Portfolio Analyses.................... 8
PART II. RELATIONSHIPS BETWEEN SECURITIES AND PORTFOLIOS...................
3. Averages and Expected Values.................... 37
4. Standard Deviations and Variances.................... 72
5. Investment in Large Numbers Of Securities.................... 102
6. Return in the Long Run.................... 116
PART III. EFFICIENT PORTFOLIOS....................
7. Geometric Analysis of Efficient Sets.................... 129
8. Derivation of E, V Efficient Portfolios.................... 154
9. The Semi-Variance.................... 188
PART IV. RATIONAL CHOICE UNDER UNCERTAINTY....................
10. The Expected Utility Maxim.................... 205
11. Utility Analysis Over Time.................... 243
12. Probability Beliefs.................... 257
13. Applications to Portfolio Selection.................... 274
A. The Computation of Efficient Sets.................... 316
B. A Simplex Method for Portfolio Selection.................... 337
C. Alternative Axiom Systems for Expected Utility.................... 340
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