portfolio selection - wiley

JWPR026-Fabozzi c01 June 22, 2008 6:54

CHAPTER 1

Portfolio SelectionFRANK J. FABOZZI, PhD, CFA, CPAProfessor in the Practice of Finance, Yale School of Management

HARRY M. MARKOWITZ, PhDConsultant

FRANCIS GUPTA, PhDDirector, Research, Dow Jones Indexes

Some Basic Concepts 4Utility Function and Indifference Curves 4The Set of Efficient Portfolios

and the Optimal Portfolio 4Risky Assets versus Risk-Free Assets 4

Measuring a Portfolio’s Expected Return 5Measuring Single-Period Portfolio Return 5The Expected Return of a Portfolio of Risky

Assets 5Measuring Portfolio Risk 6

Variance and Standard Deviationas a Measure of Risk 6

Measuring the Risk of a Portfolio Comprisedof More than Two Assets 8

Portfolio Diversification 8Portfolio Risk and Correlation 9The Effect of the Correlation of Asset Returns on

Portfolio Risk 9Choosing a Portfolio of Risky Assets 9

Constructing Efficient Portfolios 10Feasible and Efficient Portfolios 10Choosing the Optimal Portfolio in the Efficient Set 11

Index Model’s Approximations to the CovarianceStructure 12Single-Index Market Model 12Multi-Index Market Models 13

Summary 13References 13

Abstract: The goal of portfolio selection is the construction of portfolios that maximizeexpected returns consistent with individually acceptable levels of risk. Using bothhistorical data and investor expectations of future returns, portfolio selection usesmodeling techniques to quantify “expected portfolio returns” and “acceptable levelsof portfolio risk,” and provides methods to select an optimal portfolio. It would notbe an overstatement to say that modern portfolio theory has revolutionized the worldof investment management. Allowing managers to quantify the investment risk andexpected return of a portfolio has provided the scientific and objective complement tothe subjective art of investment management. More importantly, whereas at one timethe focus of portfolio management used to be the risk of individual assets, the theoryof portfolio selection has shifted the focus to the risk of the entire portfolio. This theoryshows that it is possible to combine risky assets and produce a portfolio whose expectedreturn reflects its components, but with considerably lower risk. In other words, it ispossible to construct a portfolio whose risk is smaller than the sum of all its individualparts.

Keywords: portfolio selection, modern portfolio theory, mean-variance analysis, utilityfunction, efficient portfolio, optimal portfolio, covariance, correlation,portfolio diversification, beta, portfolio variance, feasible portfolio

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COPYRIG

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ATERIAL


4 Portfolio Selection

In this chapter, we present the theory of portfolio selectionas formulated by Markowitz (1952). This theory is alsoreferred to as mean-variance portfolio analysis or simplymean-variance analysis.

SOME BASIC CONCEPTSPortfolio theory draws on concepts from two fields: finan-cial economic theory and probability and statistical theory.This section presents the concepts from financial economictheory used in portfolio theory. While many of the con-cepts presented here have a more technical or rigorousdefinition, the purpose is to keep the explanations simpleand intuitive so the reader can appreciate the importanceand contribution of these concepts to the development ofmodern portfolio theory.

Utility Function and Indifference CurvesIn life there are many situations where entities (that is,individuals and firms) face two or more choices. The eco-nomic “theory of choice” uses the concept of a utility func-tion developed by von Neuman and Morgenstern (1944),to describe the way entities make decisions when facedwith a set of choices. A utility function assigns a (numeric)value to all possible choices faced by the entity. The higherthe value of a particular choice, the greater the utility de-rived from that choice. The choice that is selected is the onethat results in the maximum utility given a set of (budget)constraints faced by the entity.

In portfolio theory too, entities are faced with a set ofchoices. Different portfolios have different levels of ex-pected return and risk. Also, the higher the level of ex-pected return, the larger the risk. Entities are faced withthe decision of choosing a portfolio from the set of allpossible risk/return combinations: where return is a de-sirable which increases the level of utility, and risk is anundesirable which decreases the level of utility. There-fore, entities obtain different levels of utility from differentrisk/return combinations. The utility obtained from anypossible risk/return combination is expressed by the util-ity function. Put simply, the utility function expresses thepreferences of entities over perceived risk and expectedreturn combinations.

A utility function can be expressed in graphical form bya set of indifference curves. Figure 1.1 shows indifferencecurves labeled u1, u2, and u3. By convention, the horizontalaxis measures risk and the vertical axis measures expectedreturn. Each curve represents a set of portfolios with dif-ferent combinations of risk and return. All the points on agiven indifference curve indicate combinations of risk andexpected return that will give the same level of utility to agiven investor. For example, on utility curve u1, there aretwo points u and u′, with u having a higher expected re-turn than u′, but also having a higher risk. Because the twopoints lie on the same indifference curve, the investor hasan equal preference for (or is indifferent to) the two points,or, for that matter, any point on the curve. The (positive)slope of an indifference curve reflects the fact that, to

Exp

ecte

d re

turn

Risk

Utilityincreases

u3

u3

u2

u2

u1

u1

u

′

Figure 1.1 Indifference Curves

obtain the same level of utility, the investor requires ahigher expected return in order to accept higher risk.

For the three indifference curves shown in Figure 1.1,the utility the investor receives is greater the further theindifference curve is from the horizontal axis, because thatcurve represents a higher level of return at every level ofrisk. Thus, for the three indifference curves shown in theexhibit, u3 has the highest utility and u1 the lowest.

The Set of Efficient Portfoliosand the Optimal PortfolioPortfolios that provide the largest possible expected re-turn for given levels of risk are called efficient portfolios.To construct an efficient portfolio, it is necessary to makesome assumption about how investors behave when mak-ing investment decisions. One reasonable assumption isthat investors are risk averse. A risk-averse investor is aninvestor who, when faced with choosing between two in-vestments with the same expected return but two differentrisks, prefers the one with the lower risk.

In selecting portfolios, an investor seeks to maximizethe expected portfolio return given his tolerance for risk.Alternatively stated, an investor seeks to minimize therisk that he is exposed to given some target expected re-turn. Given a choice from the set of efficient portfolios,an optimal portfolio is the one that is most preferred by theinvestor.

Risky Assets versus Risk-Free AssetsA risky asset is one for which the return that will be re-alized in the future is uncertain. For example, an investor


INVESTMENT MANAGEMENT 5

who purchases the stock of Pfizer Corporation today withthe intention of holding it for some finite time does notknow what return will be realized at the end of the hold-ing period. The return will depend on the price of Pfizer’sstock at the time of sale and on the dividends that the com-pany pays during the holding period. Thus, Pfizer stock,and indeed the stock of all companies, is a risky asset.

Securities issued by the U.S. government are also risky.For example, an investor who purchases a U.S. govern-ment bond that matures in 30 years does not know thereturn that will be realized if this bond is held for onlyone year. This is because changes in interest rates in thatyear will affect the price of the bond one year from nowand that will impact the return on the bond over thatyear.

There are assets, however, for which the return that willbe realized in the future is known with certainty today.Such assets are referred to as risk-free or riskless assets.The risk-free asset is commonly defined as a short-termobligation of the U.S. government. For example, if aninvestor buys a U.S. government security that maturesin one year and plans to hold that security for one year,then there is no uncertainty about the return that will berealized. (Note: Here “return” refers to the nominal return.The “real” return, which adjusts for inflation, is uncertain.)The investor knows that in one year, the maturity date ofthe security, the government will pay a specific amount toretire the debt. Notice how this situation differs for the U.S.government security that matures in 30 years. While the1-year and the 30-year securities are obligations of the U.S.government, the former matures in one year so that thereis no uncertainty about the return that will be realized. Incontrast, while the investor knows what the governmentwill pay at the end of 30 years for the 30-year bond, hedoes not know what the price of the bond will be one yearfrom now.

MEASURING A PORTFOLIO’SEXPECTED RETURNWe are now ready to define the actual and expected returnof a risky asset and a portfolio of risky assets.

Measuring Single-Period Portfolio ReturnThe actual return on a portfolio of assets over some spe-cific time period is straightforward to calculate using thefollowing:

Rp = w1 R1 + w2 R2 + . . . + wG RG (1.1)

whereRp = rate of return on the portfolio over the periodRg = rate of return on asset g over the periodwg = weight of asset g in the portfolio (that is, market

value of asset g as a proportion of the market valueof the total portfolio) at the beginning of the period

G = number of assets in the portfolio

In shorthand notation, equation (1.1) can be expressedas follows:

Rp =G∑

g=1

wg Rg (1.2)

Equation (1.2) states that the return on a portfolio (Rp)of G assets is equal to the sum over all individual assets’weights in the portfolio times their respective return. Theportfolio return Rp is sometimes called the holding periodreturn or the ex post return.

For example, consider the following portfolio consistingof three assets:

Market Value atthe Beginning of Rate of Return Over

Asset Holding Period Holding Period

1 $6 million 12%2 8 million 10%3 11 million 5%

The portfolio’s total market value at the beginning of theholding period is $25 million. Therefore,

w1 = $6 million/$25 million = 0.24, or 24% and R1 = 12%w2 = $8 million/$25 million = 0.32, or 32% and R2 = 10%w3 = $11 million/$25 million = 0.44, or 44% and R3 = 5%

Notice that the sum of the weights is equal to 1. Substi- tut-ing into equation (1.1), we get the holding period portfolioreturn,

Rp = 0.24(12%) + 0.32(10%) + 0.44(5%) = 8.28%

Note that since the holding period portfolio return is8.28%, the growth in the portfolio’s value over the holdingperiod is given by ($25 million) × 0.0828 = $2.07 million.

The Expected Return of a Portfolioof Risky AssetsEquation (1.1) shows how to calculate the actual return of aportfolio over some specific time period. In portfolio man-agement, the investor also wants to know the expected (oranticipated) return from a portfolio of risky assets. Theexpected portfolio return is the weighted average of theexpected return of each asset in the portfolio. The weightassigned to the expected return of each asset is the per-centage of the market value of the asset to the total marketvalue of the portfolio. That is,

E(Rp) = w1 E(R1) + w2 E(R2) + . . . + wG E(RG) (1.3)

The E( ) signifies expectations, and E(RP) is sometimescalled the ex ante return, or the expected portfolio returnover some specific time period.

The expected return, E(Ri), on a risky asset i is calculatedas follows. First, a probability distribution for the possiblerates of return that can be realized must be specified. Aprobability distribution is a function that assigns a proba-bility of occurrence to all possible outcomes for a randomvariable. Given the probability distribution, the expected



Table 1.1 Probability Distribution for the Rate of Return forStock XYZ

n Rate of Return Probability of Occurrence

1 12% 0.182 10 0.243 8 0.294 4 0.165 −4 0.13

Total 1.00

value of a random variable is simply the weighted av-erage of the possible outcomes, where the weight is theprobability associated with the possible outcome.

In our case, the random variable is the uncertain returnof asset i. Having specified a probability distribution forthe possible rates of return, the expected value of the rateof return for asset i is the weighted average of the possi-ble outcomes. Finally, rather than use the term “expectedvalue of the return of an asset,” we simply use the term“expected return.” Mathematically, the expected return ofasset i is expressed as:

E(Ri ) = p1 R1 + p2 R2 + · · · + pN RN (1.4)

where,Rn = the nth possible rate of return for asset ipn = the probability of attaining the rate of return n for

asset iN = the number of possible outcomes for the rate of

return

How do we specify the probability distribution of re-turns for an asset? We shall see later on in this chapterthat in most cases the probability distribution of returns isbased on historical returns. Probabilities assigned to dif-ferent return outcomes that are based on the past perfor-mance of an uncertain investment act as a good estimateof the probability distribution. However, for purpose ofillustration, assume that an investor is considering an in-vestment, stock XYZ, which has a probability distributionfor the rate of return for some time period as given inTable 1.1. The stock has five possible rates of return andthe probability distribution specifies the likelihood of oc-currence (in a probabilistic sense) for each of the possibleoutcomes.

Substituting into equation (1.4) we get

E(RXYZ) = 0.18(12%) + 0.24(10%) + 0.29(8%)+ 0.16(4%) + 0.13(−4%)

= 7%

Thus, 7% is the expected return or mean of the probabilitydistribution for the rate of return on stock XYZ.

MEASURING PORTFOLIO RISKThe dictionary defines risk as “hazard, peril, exposure toloss or injury.” With respect to investments, investors have

used a variety of definitions to describe risk. Markowitz(1952, 1959) quantified the concept of risk using the well-known statistical measures of variances and covariances.He defined the risk of a portfolio as the sum of thevariances of the investments and covariances among theinvestments. The notion of introducing the covariancesamong returns of the investments in the portfolio to mea-sure the risk of a portfolio forever changed how the in-vestment community thought about the concept of risk.

Variance and Standard Deviationas a Measure of RiskThe variance of a random variable is a measure of thedispersion or variability of the possible outcomes aroundthe expected value (mean). In the case of an asset’s return,the variance is a measure of the dispersion of the possiblerate of return outcomes around the expected return.

The equation for the variance of the expected return forasset i, denoted var(Ri), is

var(Ri ) = p1[r1 − E(Ri )]2 + p2[r2 − E(Ri )]2

+ · · · + pN[rN − E(Ri )]2

or,

var(Ri ) =N∑

i=1

pn[rn − E(Ri )]2 (1.5)

Using the probability distribution of the return for stockXYZ, we can illustrate the calculation of the variance:

var(RXYZ) = 0.18(12% − 7%)2 + 0.24(10% − 7%)2

+ 0.29(8% − 7%)2 + 0.16(4% − 7%)2

+ 0.13(−4% − 7%)2

= 24.1%

The variance associated with a distribution of returnsmeasures the compactness with which the distribution isclustered around the mean or expected return. Markowitzargued that this compactness or variance is equivalent tothe uncertainty or riskiness of the investment. If an assetis riskless, it has an expected return dispersion of zero. Inother words, the return (which is also the expected returnin this case) is certain, or guaranteed.

Standard DeviationSince the variance is squared units, it is common to see thevariance converted to the standard deviation by taking thepositive square root of the variance:

SD(Ri ) =√

var(Ri )

For stock XYZ, then, the standard deviation is

SD(RXYZ) =√

24.1% = 4.9%

The variance and standard deviation are conceptuallyequivalent; that is, the larger the variance or standard de-viation, the greater the investment risk.



There are two criticisms of the use of the variance as ameasure of risk. The first criticism is that since the variancemeasures the dispersion of an asset’s return around itsexpected return, it treats both the returns above and belowthe expected return identically. There has been research inthe area of behavioral finance to suggest that investors donot view return outcomes above the expected return inthe same way as they view returns below the expectedreturn. Whereas returns above the expected return areconsidered favorable, outcomes below the expected returnare disliked. Because of this, some researchers have arguedthat measures of risk should not consider the possiblereturn outcomes above the expected return.

Markowitz recognized this limitation and, in fact, sug-gested a measure of downside risk—the risk of realizingan outcome below the expected return—called the semi-variance. The semivariance is similar to the variance ex-cept that in the calculation no consideration is given toreturns above the expected return. However, because ofthe computational problems with using the semivarianceand the limited resources available to him at the time, heused the variance in developing portfolio theory.

Today, practitioners use various measures of downsiderisk. However, regardless of the measure used, the basicprinciples of portfolio theory developed by Markowitzand set forth in this chapter are still applicable. That is,the choice of the measure of risk may affect the calculationbut doesn’t invalidate the theory.

The second criticism is that the variance is only onemeasure of how the returns vary around the expectedreturn. When a probability distribution is not symmetricalaround its expected return, then a statistical measure of theskewness of a distribution should be used in addition tothe variance (see Ortobelli et al., 2005).

Because expected return and variance are the only twoparameters that investors are assumed to consider inmaking investment decisions, the Markowitz formula-tion of portfolio theory is often referred to as a “two-parameter model.” It is also referred to as “mean-varianceanalysis.”

Measuring the Portfolio Risk of a Two-Asset PortfolioEquation (1.5) gives the variance for an individual asset’sreturn. The variance of a portfolio consisting of two assetsis a little more difficult to calculate. It depends not only onthe variance of the two assets, but also upon how closelythe returns of one asset track those of the other asset. The

formula is:

var(Rp) = var(Ri ) + var(Rj ) + 2wiw j cov(Ri , Rj ) (1.6)

where

cov(Ri , Rj ) = covariance between the return for assetsiand j

In words, equation (1.6) states that the variance of theportfolio return is the sum of the squared weighted vari-ances of the two assets plus two times the weighted covari-ance between the two assets. We will see that this equationcan be generalized to the case where there are more thantwo assets in the portfolio.

CovarianceLike the variance, the covariance has a precise mathemati-cal translation. Its practical meaning is the degree to whichthe returns on two assets co-vary or change together. Infact, the covariance is just a generalized concept of thevariance applied to multiple assets. A positive covariancebetween two assets means that the returns on two assetstend to move or change in the same direction, while anegative covariance means the returns tend to move inopposite directions. The covariance between any two as-sets i and j is computed using the following formula:

cov(Ri , Rj ) = p1[ri1 − E(Ri )][r j1 − E(Rj )]

+p2[ri2 − E(Ri )][r j2 − E(Rj )]

+ · · · + pN[ri N − E(Ri )][r j N − E(Rj )]

(1.7)

whererin = the nth possible rate of return for asset irjn = the nth possible rate of return for asset jpn = the probability of attaining the rate of return n for

assets i and jN = the number of possible outcomes for the rate of

return.

The covariance between asset i and i is just the varianceof asset i.

To illustrate the calculation of the covariance betweentwo assets, we use the two stocks in Table 1.2. The first isstock XYZ from Table 1.1 that we used earlier to illustratethe calculation of the expected return and the standard de-viation. The other hypothetical stock is ABC whose data

Table 1.2 Probability Distribution for the Rate of Return for Asset XYZ and Asset ABC

n Rate of Return for Asset XYZ Rate of Return for Asset ABC Probability of Occurrence

1 12% 21% 0.182 10 14 0.243 8 9 0.294 4 4 0.165 −4 −3 0.13

Total 1.00Expected return 7.0% 10.0%Variance 24.1% 53.6%Standard deviation 4.9% 7.3%



are shown in Table 1.2. Substituting the data for the twostocks from Table 1.2 in equation (1.7), the covariance be-tween stocks XYZ and ABC is calculated as follows:

cov(RXYZ, RABC ) = 0.18(12% − 7%)(21% − 10%)

+ 0.24(10% − 7%)(14% − 10%)

+ 0.29(8% − 7%)(9% − 10%)

+ 0.16(4% − 7%)(4% − 10%)

+ 0.13(−4% − 7%)(−3% − 10%)

= 34

Relationship between Covariance and CorrelationThe correlation is analogous to the covariance between theexpected returns for two assets. Specifically, the correla-tion between the returns for assets i and j is defined asthe covariance of the two assets divided by the product oftheir standard deviations:

cor(Ri , Rj ) = cov(Ri , Rj )/[SD(Ri )SD(Rj )] (1.8)

The correlation and the covariance are conceptuallyequivalent terms. Dividing the covariance between thereturns of two assets by the product of their standard de-viations results in the correlation between the returns ofthe two assets. Because the correlation is a standardizednumber (that is, it has been corrected for differences in thestandard deviation of the returns), the correlation is com-parable across different assets. The correlation betweenthe returns for stock XYZ and stock ABC is

cor(RXYZ, RABC) = 34/(4.9 × 7.3) = 0.94

The correlation coefficient can have values ranging from+1.0, denoting perfect co-movement in the same direction,to –1.0, denoting perfect co-movement in the opposite di-rection. Also note that because the standard deviationsare always positive, the correlation can be negative only ifthe covariance is a negative number. A correlation of zeroimplies that the returns are uncorrelated. Finally, thoughcausality implies correlation, correlation does not implycausality.

Measuring the Risk of a PortfolioComprised of More than Two AssetsSo far, we have defined the risk of a portfolio consistingof two assets. The extension to three assets—i, j, and k—isas follows:

var(Rp) = w2i var(Ri ) + w2

j var(Rj ) + w2k var(Rk)

+ 2wiw j cov(Ri , Rj ) + 2wiwk cov(Ri , Rk)

+ 2w jwk cov(Rj , Rk) (1.9)

In words, equation (1.9) states that the variance of theportfolio return is the sum of the squared weighted vari-ances of the individual assets plus two times the sum ofthe weighted pair-wise covariances of the assets. In gen-eral, for a portfolio with G assets, the portfolio variance isgiven by,

var(Rp) =G∑

g=1

G∑

h=1

wgwh cov(Rg, Rh) (1.10)

In (1.10), the terms for which h = g results in the variancesof the G assets, and the terms for which h �= g results inall possible pair-wise covariances amongst the G assets.Therefore, (1.10) is a shorthand notation for the sum of allG variances and the possible covariances amongst the Gassets.

PORTFOLIO DIVERSIFICATIONOften, one hears investors talking about diversifying theirportfolio. By this an investor means constructing a portfo-lio in such a way as to reduce portfolio risk without sacri-ficing return. This is certainly a goal that investors shouldseek. However, the question is how to do this in practice.

Some investors would say that including assets acrossall asset classes could diversify a portfolio. For example, ainvestor might argue that a portfolio should be diversifiedby investing in stocks, bonds, and real estate. While thatmight be reasonable, two questions must be addressedin order to construct a diversified portfolio. First, howmuch should be invested in each asset class? Should 40%of the portfolio be in stocks, 50% in bonds, and 10% inreal estate, or is some other allocation more appropriate?Second, given the allocation, which specific stocks, bonds,and real estate should the investor select?

Some investors who focus only on one asset class suchas common stock argue that such portfolios should also bediversified. By this they mean that an investor should notplace all funds in the stock of one corporation, but rathershould include stocks of many corporations. Here, too,several questions must be answered in order to constructa diversified portfolio. First, which corporations shouldbe represented in the portfolio? Second, how much ofthe portfolio should be allocated to the stocks of eachcorporation?

Prior to the development of portfolio theory, while in-vestors often talked about diversification in these generalterms, they did not possess the analytical tools by which toanswer the questions posed above. For example, in 1945,D. H. Leavens (1945) wrote:

An examination of some fifty books and articles on in-vestment that have appeared during the last quarter ofa century shows that most of them refer to the desirabil-ity of diversification. The majority, however, discuss itin general terms and do not clearly indicate why it isdesirable.

Leavens illustrated the benefits of diversification on theassumption that risks are independent. However, in thelast paragraph of his article, he cautioned:

The assumption, mentioned earlier, that each securityis acted upon by independent causes, is important, al-though it cannot always be fully met in practice. Di-versification among companies in one industry cannotprotect against unfavorable factors that may affect thewhole industry; additional diversification among in-dustries is needed for that purpose. Nor can diversifi-cation among industries protect against cyclical factorsthat may depress all industries at the same time.

A major contribution of the theory of portfolio selectionis that using the concepts discussed above, a quantitativemeasure of the diversification of a portfolio is possible,



and it is this measure that can be used to achieve themaximum diversification benefits.

The Markowitz diversification strategy is primarilyconcerned with the degree of covariance between as-set returns in a portfolio. Indeed a key contribution ofMarkowitz diversification is the formulation of an asset’srisk in terms of a portfolio of assets, rather than in isola-tion. Markowitz diversification seeks to combine assetsin a portfolio with returns that are less than perfectlypositively correlated, in an effort to lower portfolio risk(variance) without sacrificing return. It is the concern formaintaining return while lowering risk through an analy-sis of the covariance between asset returns, that separatesMarkowitz diversification from a naive approach to diver-sification and makes it more effective.

Markowitz diversification and the importance of assetcorrelations can be illustrated with a simple two-assetportfolio example. To do this, we will first show the gen-eral relationship between the risk of a two-asset portfolioand the correlation of returns of the component assets.Then we will look at the effects on portfolio risk of com-bining assets with different correlations.

Portfolio Risk and CorrelationIn our two-asset portfolio, assume that asset C and as-set D are available with expected returns and standarddeviations as shown:

Asset E(R) SD(R)

Asset C 12% 30%Asset D 18% 40%

If an equal 50% weighting is assigned to both stocks Cand D, the expected portfolio return can be calculated asshown:

E(Rp) = 0.50(12%) + 0.50(18%) = 15%

The variance of the return on the two-stock portfolio fromequation (1.6) is:

var(Rp) = w2C var(RC ) + w2

DwD 2 var(RD)+2wCwD cov(RC , RD)

= (0.5)2(30%)2 + (0.5)2(40%)2

+ 2(0.5)(0.5) cov(RC , RD)

From equation (1.8),

cor(RC , RD) = cov(RC , RD)/[SD(RC )SD(RD)]so

cov(RC , RD) = SD (RC ) SD(RD) cor(RC , RD)

Since SD(RC) = 30% and SD(RD) = 40%, then

cov(RC , RD) = (30%)(40%) cor(RC , RD)

Substituting into the expression for var(Rp), we get

var(Rp) = (0.5)2(30%)2 + (0.5)2(40%)2

+ 2(0.5)(0.5)(30%)(40%) cor(RC , RD)

Taking the square root of the variance gives

SD(Rp)=

√(0.5)2(30%)2 + (0.5)2(40%)2 + 2(0.5)(0.5)(30%)(40%) cor(RC , RD)

= √625 + (600) cor(RC , RD)

(1.11)

The Effect of the Correlation of AssetReturns on Portfolio RiskHow would the risk change for our two-asset portfoliowith different correlations between the returns of the com-ponent stocks? Let’s consider the following three cases forcor(RC,RD): +1.0, 0, and –1.0. Substituting into equation(1.11) for these three cases of cor(RC,RD), we get

cor E SD

+1.0 15% 35%0.0 15 25

−1.0 15 5

As the correlation between the expected returns onstocks C and D decreases from +1.0 to 0.0 to –1.0, thestandard deviation of the expected portfolio return alsodecreases from 35% to 5%. However, the expected portfo-lio return remains 15% for each case.

This example clearly illustrates the effect of Markowitzdiversification. The principle of Markowitz diversificationstates that as the correlation (covariance) between the re-turns for assets that are combined in a portfolio decreases,so does the variance (hence the standard deviation) of thereturn for the portfolio. This is due to the degree of corre-lation between the expected asset returns.

The good news is that investors can maintain expectedportfolio return and lower portfolio risk by combiningassets with lower (and preferably negative) correlations.However, the bad news is that very few assets have smallto negative correlations with other assets! The problem,then, becomes one of searching among large numbers ofassets in an effort to discover the portfolio with the min-imum risk at a given level of expected return or, equiva-lently, the highest expected return at a given level of risk.

The stage is now set for a discussion of efficient portfo-lios and their construction.

CHOOSING A PORTFOLIOOF RISKY ASSETSDiversification in the manner suggested by ProfessorMarkowitz leads to the construction of portfolios that havethe highest expected return at a given level of risk. Suchportfolios are called efficient portfolios. In order to con-struct efficient portfolios, the theory makes some basic as-sumptions about asset selection behavior by the entities.The assumptions are as follows:

Assumption 1: The only two parameters that affect an in-vestor’s decision are the expected return and the vari-ance. (That is, investors make decisions using the two-parameter model formulated by Markowitz.)

Assumption 2: Investors are risk averse. (That is, whenfaced with two investments with the same expectedreturn but two different risks, investors will prefer theone with the lower risk.)

Assumption 3: All investors seek to achieve the highestexpected return at a given level of risk.



Table 1.3 Portfolio Expected Returns and Standard Deviationsfor Five Mixes of Assets C and D

Proportion ProportionPortfolio of Asset C of Asset D E(Rp) SD(Rp)

1 100% 0% 12.0% 30.0%2 75 25 13.5% 19.5%3 50 50 15.0% 18.0%4 25 75 16.5% 27.0%5 0 100 18.0% 40.0%

Asset C: E(RC) = 12%, SD(RC) = 30%Asset D: E(RD) = 18%, and SD(RD) = 40%Correlation between Asset C. and D = cor(RC,RD) = −0.5

Assumption 4: All investors have the same expectations re-garding expected return, variance, and covariances forall risky assets. (This is referred to as the homogeneousexpectations assumption.)

Assumption 5: All investors have a common one-periodinvestment horizon.

Constructing Efficient PortfoliosThe technique of constructing efficient portfolios fromlarge groups of stocks requires a massive number of calcu-lations. In a portfolio of G securities, there are (G2 − G)/2unique covariances to calculate. Hence, for a portfolio ofjust 50 securities, there are 1,224 covariances that mustbe calculated. For 100 securities, there are 4,950. Further-more, in order to solve for the portfolio that minimizes riskfor each level of return, a mathematical technique calledquadratic programming must be used. A discussion of thistechnique is beyond the scope of this chapter. However,it is possible to illustrate the general idea of the construc-tion of efficient portfolios by referring again to the simpletwo-asset portfolio consisting of assets C and D.

Recall that for two assets, C and D, E(RC) = 12%,SD(RC) = 30%, E(RD) = 18%, and SD(RD) = 40%. We nowfurther assume that cor(RC,RD) = –0.5. Table 1.3 presentsthe expected portfolio return and standard deviation forfive different portfolios made up of varying proportionsof C and D.

Feasible and Efficient PortfoliosA feasible portfolio is any portfolio that an investor can con-struct given the assets available. The five portfolios pre-sented in Table 1.3 are all feasible portfolios. The collectionof all feasible portfolios is called the feasible set of portfo-lios. With only two assets, the feasible set of portfolios isgraphed as a curve that represents those combinations ofrisk and expected return that are attainable by construct-ing portfolios from all possible combinations of the twoassets.

Figure 1.2 presents the feasible set of portfolios for allcombinations of assets C and D. As mentioned earlier, theportfolio mixes listed in Table 1.3 belong to this set andare shown by the points 1 through 5, respectively. Startingfrom 1 and proceeding to 5, asset C goes from 100% to0%, while asset D goes from 0% to 100%—therefore, all

20%

15%

10%

5%

0%

ER

(Rp)

0% 10% 20% 30% 40% 50%SD(Rp)

Feasible set represented by curve 1–5Markowitz efficient set: portion of curve 3–5

12

3

4 5

Figure 1.2 Feasible and Efficient Portfolios for Assets Cand D

possible combinations of C and D lie between portfolios 1and 5, or on the curve labeled 1–5. In the case of two assets,any risk/return combination not lying on this curve is notattainable, since there is no mix of assets C and D that willresult in that risk/return combination. Consequently, thecurve 1–5 can also be thought of as the feasible set.

In contrast to a feasible portfolio, an efficient portfolio isone that gives the highest expected return of all feasibleportfolios with the same risk. An efficient portfolio is alsosaid to be a mean-variance efficient portfolio. Thus, foreach level of risk there is an efficient portfolio. The collec-tion of all efficient portfolios is called the efficient set.

The efficient set for the feasible set presented in Figure1.2 is differentiated by the bold curve section 3–5. Efficientportfolios are the combinations of assets C and D that re-sult in the risk/return combinations on the bold section ofthe curve. These portfolios offer the highest expected re-turn at a given level of risk. Notice that two of our five port-folio mixes—portfolio 1 with E(Rp) = 12% and SD(Rp) =20% and portfolio 2 with E(Rp) = 13.5% and SD(Rp) =19.5%—are not included in the efficient set. This is be-cause there is at least one portfolio in the efficient set (forexample, portfolio 3) that has a higher expected returnand lower risk than both of them. We can also see thatportfolio 4 has a higher expected return and lower riskthan portfolio 1. In fact, the whole curve section 1–3 is notefficient. For any given risk/return combination on thiscurve section, there is a combination (on the curve section3–5) that has the same risk and a higher return, or the samereturn and a lower risk, or both. In other words, for anyportfolio that results in the return/risk combination onthe curve section 1–3 (excluding portfolio 3), there exists aportfolio that dominates it by having the same return andlower risk, or the same risk and a higher return, or a lowerrisk and a higher return. For example, portfolio 4 domi-nates portfolio 1, and portfolio 3 dominates both portfolios1 and 2.

Figure 1.3 shows the feasible and efficient sets whenthere are more than two assets. In this case, the feasibleset is not a line, but an area. This is because, unlike thetwo-asset case, it is possible to create asset portfolios thatresult in risk/return combinations that not only result in



ER

(Rp)

Risk [SD(Rp)]

Feasible set: All portfolios on and bounded by curve I–II–IIIMarkowitz efficient set: All portfolios on curve II–III

I

II

III

Figure 1.3 Feasible and Efficient Portfolios with MoreThan Two Assets*

* The picture is for illustrative purposes only. The actual shape ofthe feasible region depends on the returns and risks of the assetschosen and the correlation among them.

combinations that lie on the curve I–II–III, but all combi-nations that lie in the shaded area. However, the efficientset is given by the curve II–III. It is easily seen that all theportfolios on the efficient set dominate the portfolios inthe shaded area.

The efficient set of portfolios is sometimes called theefficient frontier, because graphically all the efficient port-folios lie on the boundary of the set of feasible portfoliosthat have the maximum return for a given level of risk.Any risk/return combination above the efficient frontiercannot be achieved, while risk/return combinations ofthe portfolios that make up the efficient frontier dominatethose that lie below the efficient frontier.

Choosing the Optimal Portfolioin the Efficient SetNow that we have constructed the efficient set of portfo-lios, the next step is to determine the optimal portfolio.

Since all portfolios on the efficient frontier provide thegreatest possible return at their level of risk, an investoror entity will want to hold one of the portfolios on theefficient frontier. Notice that the portfolios on the efficientfrontier represent trade-offs in terms of risk and return.Moving from left to right on the efficient frontier, the riskincreases, but so does the expected return. The question iswhich one of those portfolios should an investor hold? Thebest portfolio to hold of all those on the efficient frontieris the optimal portfolio.

Intuitively, the optimal portfolio should depend on theinvestor’s preference over different risk/return trade-offs.As explained earlier, this preference can be expressed interms of a utility function.

ER

(Rp)

SD(Rp)

Markowitzefficientfrontier

u3

u2

u1

u3

u2

u1

PMEF*

u1, u2, u3 = indifference curves with u1 < u2 < u3

PMEF = Optimal portfolio on Markowitz efficient frontier*

PMEF*

Figure 1.4 Selection of the Optimal Portfolio

In Figure 1.4, three indifference curves representing autility function and the efficient frontier are drawn on thesame diagram. An indifference curve indicates the com-binations of risk and expected return that give the samelevel of utility. Moreover, the farther the indifference curvefrom the horizontal axis, the higher the utility.

From Figure 1.4, it is possible to determine the opti-mal portfolio for the investor with the indifference curvesshown. Remember that the investor wants to get to thehighest indifference curve achievable given the efficientfrontier. Given that requirement, the optimal portfoliois represented by the point where an indifference curveis tangent to the efficient frontier. In Figure 1.4, that isthe portfolio P∗

MEF. For example, suppose that P∗MEF cor-

responds to portfolio 4 in Figure 1.2. We know fromTable 1.3 that this portfolio is made up of 25% of assetC and 75% of asset D, with an E(Rp) = 16.5% and SD(Rp) =27.0%.

Consequently, for the investor’s preferences over riskand return as determined by the shape of the indifferencecurves represented in Figure 1.4, and expectations for as-set C and D inputs (returns and variance-covariance) rep-resented in Table 1.2, portfolio 4 is the optimal portfoliobecause it maximizes the investor’s utility. If this investorhad a different preference for expected risk and return,there would have been a different optimal portfolio. Forexample, Figure 1.5 shows the same efficient frontier butthree other indifference curves. In this case, the optimalportfolio is P∗∗

MEF, which has a lower expected return andrisk than P∗

MEF in Figure 1.4. Similarly, if the investor hada different set of input expectations, the optimal portfoliowould be different.

At this point in our discussion, a natural question is howto estimate an investor’s utility function so that the indif-ference curves can be determined. Unfortunately, thereis little guidance about how to construct one. In general,



u2 u1u3

u3

u2

u1

ER

(Rp)

SD(Rp)

Markowitzefficientfrontier

PMEF**

u1, u2, u3 = indifference curves with u1 < u2 < u3

PMEF = Optimal portfolio on Markowitz efficient frontier**

Figure 1.5 Selection of Optimal Portfolio with DifferentIndifference Curves (Utility Function)

economists have not been successful in estimating utilityfunctions.

The inability to estimate utility functions does not meanthat the theory is flawed. What it does mean is that oncean investor constructs the efficient frontier, the investorwill subjectively determine which efficient portfolio is ap-propriate given his or her tolerance to risk.

INDEX MODEL’SAPPROXIMATIONS TO THECOVARIANCE STRUCTUREThe inputs to mean-variance analysis include expectedreturns, variance of returns, and either covariance or cor-relation of returns between each pair of securities. Forexample, an analysis that allows 200 securities as possiblecandidates for portfolio selection requires 200 expectedreturns, 200 variances of return, and 19,900 correlations orcovariances. An investment team tracking 200 securitiesmay reasonably be expected to summarize their analysesin terms of 200 means and variances, but it is clearly unrea-sonable for them to produce 19,900 carefully consideredcorrelations or covariances.

It was clear to Markowitz that some kind of model ofcovariance structure was needed for the practical appli-cation of normative analysis to large portfolios. He didlittle more than point out the problem and suggest somepossible models of covariance for research.

One model Markowitz proposed to explain the correla-tion structure among security returns assumed that the re-turn on the ith security depends on an “underlying factor,the general prosperity of the market as expressed by someindex.” Mathematically, the relationship is expressed as

follows (Markowitz, 1959, pp. 96–101):

ri = αi + βi F + ui (1.12)

whereri = the return on security iF = value of some indexui = error term

(Note that Markowitz (1959) used the notation I in propos-ing the model given by equation (1.12).) The expectedvalue of ui is zero and ui is uncorrelated with F and everyother uj.

The parameters αi and β i are parameters to be estimated.When measured using regression analysis, β i is the ratioof the covariance of asset i’s return and F to the varianceof F.

Markowitz further suggested that the relationship neednot be linear and that there could be several underlyingfactors.

Single-Index Market ModelSharpe (1963) tested equation (1.12) as an explanation ofhow security returns tend to go up and down togetherwith general market index, F. For the index in the mar-ket model he used a market index for F. Specifically,Sharpe estimated using regression analysis the followingmodel:

rit = αi + βi rmt + uit (1.13)

whererit = return on asset i over the period t

rmt = return on the market index over the period tαi = a term that represents the nonmarket component

of the return on asset iβ i = the ratio of the covariance of the return of asset i

and the return of the market index to the varianceof the return of the market index

uit = a zero mean random error term

The model given by equation (1.13) is called the singleindex market model or simply the market model. It is im-portant to note that when Markowitz discussed the pos-sibility of using equation (1.12) to estimate the covariancestructure, the index he suggested was not required to be amarket index.

Suppose that the Dow Jones Wilshire 5000 is used torepresent the market index, then for a portfolio of G assetsregression analysis is used to estimate the values of the βsand αs. The beta of the portfolio (βp), is simply a weightedaverage of the computed betas of the individual assets(β i), where the weights are the percentage of the marketvalue of the individual assets relative to the total marketvalue of the portfolio. That is,

βp =G∑

i=1

wiβi



So, for example, the beta for a portfolio comprised of thefollowing:

Company Weight β

General Electric 20% 1.24McGraw-Hill 25% 0.86IBM 15% 1.22General Motors 10% 1.11Xerox 30% 1.27

would have the following beta:

Portfolio beta = 20%(1.24) + 25%(0.86) + 15%(1.22)+ 10%(1.11) + 30%(1.27)

= 1.14

Multi-Index Market ModelsSharpe concluded that equation (1.12) was as complex acovariance as seemed to be needed. This conclusion wassupported by research of Cohen and Pogue (1967). King(1966) found strong evidence for industry factors in ad-dition to the marketwide factor. Rosenberg (1974) foundother sources of risk beyond a marketwide factor and in-dustry factor.

One alternative approach to full mean-variance analysisis the use of these multi-index or factor models to obtainthe covariance structure.

SUMMARYIn this chapter we have introduced portfolio theory. De-veloped by Markowitz, this theory explains how investorsshould construct efficient portfolios and select the bestor optimal portfolio from among all efficient portfolios.The theory differs from previous approaches to portfolioselection in that he demonstrated how the key parame-ters should be measured. These parameters include therisk and the expected return for an individual asset and aportfolio of assets. Moreover, the concept of diversifyinga portfolio, the goal of which is to reduce a portfolio’s riskwithout sacrificing expected return, can be cast in termsof these key parameters plus the covariance or correlationbetween assets. All these parameters are estimated fromhistorical data and other sources of information and drawfrom concepts in probability and statistical theory.

A portfolio’s expected return is simply a weighted av-erage of the expected return of each asset in the portfolio.The weight assigned to each asset is the market value ofthe asset in the portfolio relative to the total market value

of the portfolio. The variance or the standard deviationof an asset’s returns measures its risk. Unlike the port-folio’s expected return, a portfolio’s risk is not a simpleweighting of the standard deviation of the individual as-sets in the portfolio. Rather, the covariance or correlationbetween the assets that comprise the portfolio affects theportfolio risk. The lower the correlation, the smaller therisk of the portfolio.

Markowitz has set forth the theory for the constructionof an efficient portfolio, which has come to be called a ef-ficient portfolio—a portfolio that has the highest expectedreturn of all feasible portfolios with the same level of risk.The collection of all efficient portfolios is called the effi-cient set of portfolios or the efficient frontier.

The optimal portfolio is the one that maximizes an in-vestor’s preferences with respect to return and risk. Aninvestor’s preference is described by a utility function,which can be represented graphically by a set of indif-ference curves. The utility function shows how much aninvestor is willing to trade off between expected returnand risk. The optimal portfolio is the one where an indif-ference curve is tangent to the efficient frontier.

REFERENCESRosenberg, B. (1974). Extra-market components of covari-

ance in security returns. Journal of Financial and Quanti-tative Analysis 19, 2: 23–274.

Cohen, K. J., and Pogue, G. A. (1967). An empirical eval-uation of alternative portfolio selection models. Journalof Business 40, 2: 166–193.

King, B. F. (1966). Market and industry factors in stockprice behavior. Journal of Business 39, 1 (Part 2: Supple-ment on Security Prices): 139–190.

Leavens, D. H. (1945). Diversification of investments.Trusts and Estates 80: 469–473.

Markowitz, H. M. (1952). Portfolio selection. Journal ofFinance 7, 1: 77–91

Markowitz, H. M. (1959). Portfolio Selection: Efficient Diver-sification of Investments. Cowles Foundation Monograph16, New York: John Wiley & Sons.

Ortobelli, S., Rachev, S. T., Stoyanov, S., Fabozzi, F. J., andBiglova, A. (2005). The proper use of risk measures inportfolio theory. International Journal of Theoretical andApplied Finance 8, 8: 1–27.

Sharpe, W. F. (1963). A simplified model for portfolio anal-ysis. Management Science 9, 2: 277–293.

von Neumann, J., and Morgenstern, O. (1944). Theory ofGames and Economic Behavior. Princeton, NJ: PrincetonUniversity Press.


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