application to portfolio theory and the capital asset ...econ446/econ446/march30/appendix c.pdf ·...

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C Application to Portfolio Theory andthe Capital Asset Pricing Model

Appendix B derives the laws of mathematical expectation and variance, whichare summarized in Table B.2. These laws have many important applications. Forexample, they provide the foundation for the methods of statistical inference devel-oped throughout this book, as emphasized especially in the early chapters.

But the laws of mathematical expectation and variance see use in many otherareas of knowledge. One purpose of this appendix is to illustrate this by showinghow they underlie modern financial theory. A second purpose of this appendix is toprovide the theoretical background for the capital asset pricing model, an empiricalapplication studied in the appendix to Chapter 5.

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..C.1 Introduction

Most investments are risky: Their return is uncertain. There are many reasons whythis is so. If the asset is tradeable, often the future price is unknown. Sometimesthere is a risk of default. There may be uncertainty with respect to any income tobe generated by the asset, such as rent from real estate, dividends from stock, orreturns to education. In the case of a capital investment project undertaken by a firm(the construction of a new plant, opening of a mine, acquisition of new equipment,and so on), the returns from the project will depend on a host of contingencies,including future market conditions.

Consider, for example, stock, purchased at the beginning of period t for a priceof Pt−1 per share, that is to be sold at the end of the period for Pt . The net returnover the period is

Rt = Pt − Pt−1

Pt−1= Pt

Pt−1− 1. (C.1)

This return is uncertain because the future price Pt is unknown at the time ofpurchase. If dividends are declared, then the formula must be modified to reflectthis, and the uncertainty associated with dividend payments will contribute to theoverall uncertainty of return.

C

SECTION C.2 Risky Assets C-1

In the special case in which there is no uncertainty with respect to the futureprice in (C.1), the asset offers a risk-free rate of return rft defined by

rft = Pt − Pt−1

Pt−1.

For example, this is the relationship governing the prices and yields of governmenttreasury bills; these are sold at auction, the price Pt−1 being determined as adiscount from the maturity value Pt . The relationship is often rearranged into theform

Pt−1 = Pt

1 + rft, (C.2)

which says that the current or present value of the asset is determined as a discountof the future payment stream—in this case, just the maturity payment Pt . Thedenominator 1 + rft is the discount factor reflecting positive time preference onthe part of economic agents. That is, in return for a promise to receive Pt in thefuture, agents are normally only willing to pay somewhat less than this today, Pt−1,even if the promise is a sure thing. This is, of course, nothing more than sayingthat interest rates are normally positive: rft > 0. In this sense, rft is the “price oftime”—the reward that must be offered to induce agents to postpone consumptioninto the future.

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...C.2 Risky Assets

In the case of risky assets, on the other hand, the asset return R given by (C.1) isa random variable. It is natural to interpret its mean E(R) as its expected returnand its variance �2 = V (R) (or standard deviation �) as a measure of risk. It isconceivable that other features of the distribution, such as skewness or kurtosis,may also be of interest to investors. For the sake of the argument, however, letus assume that investors focus on just expected return and risk in making theirinvestment decisions. This leads to mean-variance portfolio analysis, an importantset of ideas that forms the foundation for much of modern financial theory.

What considerations apply when investors choose among alternative assetson the basis of their assessment of risk and return? If investments are evaluatedsolely on the basis of these two criteria, alternative investment opportunities may beportrayed in a diagram as in Figure C.1, in which expected return is measured on thevertical axis and risk (measured by the standard deviation) is on the horizontal axis.The figure shows three investments: R1 and R2 have the same risks �1 = �2 = 0.04but different expected returns of 10 and 5%, whereas R2 and R3 have the sameexpected returns but different risks.

How would an investor choose among these? It is obvious that no rationalinvestor would choose R2 because another investment is available, R1, that has the

C-2 APPENDIX C Application to Portfolio Theory and the Capital Asset Pricing Model

F IGURE C.1

Three investmentopportunities.

�3 � 0.02�

�1 � �2 � 0.04

R1

E(R)

E(R1) � 0.10

E(R2) � E(R3) � 0.05 R2

R3

same risk but a higher expected return. Alternatively, R3 has the same expectedreturn but a lower risk and would also always be preferred to R2. If these arethe only investment opportunities, then R2 can be dismissed as an obviously fool-ish investment; it is said to be dominated by the others. An investment that isdominated by others is also called inefficient.

Do either of R1 or R3 dominate the other? No. Neither can be dismissedas an obviously foolish investment. Although the high expected return of R1 isappealing, it also has high risk. An investor might rationally prefer R3 because ofits lower risk, even though it also has a lower expected return. The higher expectedreturn of R1 is the reward to the investor for bearing the additional risk of thatasset. Whether this higher expected return will be adequate to induce a particularinvestor to bear the additional risk will depend on that individual’s tastes for riskversus return. Investors who are relatively risk averse (like the proverbial “widowsand orphans”) will not be induced to bear the additional risk of R1 and will holdR3 instead.

As is always true in economics, agents make choices on the basis of theirpreferences. The choice between risk and return is determined by the degree ofrisk aversion of preferences. Indeed, the theory of choice under uncertainty is animportant component of modern microeconomics and is the foundation of mean-variance portfolio theory, although in a way that need not be made explicit here.You may find it useful to consult the treatment of uncertainty in a microeconomicstextbook such as Varian (1996, Chaps. 12–13).

Ultimately, the expected returns on assets are endogenous to the economy.From (C.1), the expected return depends on the expected future asset price E(Pt)

and the price Pt−1 at which the asset currently trades in the market:1

E(Rt ) = E(Pt )

Pt−1− 1. (C.3)

1 This is merely an application of Law B.1 in Appendix B, because the expectation is formed at t − 1when the initial price Pt−1 is a known value.

SECTION C.2 Risky Assets C-3

The current price Pt−1 reflects the equilibration of supply and demand in the market;in turn, these forces of supply and demand reflect the assessment of risk by marketparticipants. Suppose, for example, that a particular asset suddenly comes to be seenas riskier than previously thought. Then demand for that asset will fall and, ceterusparibus, so will its current price Pt−1. Even if the mean of the future distribution,E(Pt), is unchanged, the expected return E(Rt ) is higher as compensation forbearing the newly perceived higher risk. In this way, at any point in time all assetprices and their implied expected returns reflect an equilibrium associated with anassessment of risk by market participants.

Assessments of risk are just one of many factors influencing the demand andsupply of assets and thus determining equilibrium asset prices and returns. Forexample, a decline in the willingness of households to save, as a result of, say, theestablishment of a social security system, would shift the demand for assets to theleft. This would lead, ceterus paribus, to a decline in current asset prices, implyinghigher expected returns on the lower level of saving remaining. An increase in thesupply of assets arising from new debt or equity issuance by firms seeking to exploitnew investment opportunities—associated with, say, the opening up of China orthe former Soviet Union—would similarly necessitate higher expected returns asan inducement for the increased flows of investment funds. It is important to keepin mind that these forces of economic equilibrium ultimately underlie any analysisof the relationship between risk and return.

General economic forces such as these tend to affect the expected returns ofall assets roughly equally—they determine the overall level of expected returns.Risk, on the other hand, is specific to particular assets. Some investments are veryrisky and others less so. Some may be essentially riskless: A bank balance coveredby deposit insurance, for example, or a government treasury bill.

Indeed in comparing different assets, a great deal of evidence establishes thatrisk is the most important determinant of expected return. In the words of Malkiel(1996, pp. 232–233), “One of the best-documented propositions in the field offinance is that, on average, investors have received higher rates of return for bear-ing greater risk.” To see just how true this is, consider the figures in Table C.1.Jagannathan and McGrattan (1995) show that different asset classes offer verydifferent average returns and variabilities.

. . . During the 66-year period from 1926 to 1991, for example, Standard & Poor’s500-stock price index (the S&P 500) earned an average annual return of 11.9 per-cent whereas U.S. Treasury bills (T-bills) earned only 3.6 percent. Since the averageannual inflation rate was 3.1 percent during this period, the average real return on T-bills was hardly different from zero. S&P stocks, therefore, earned a hefty risk pre-mium of 8.3 percent over the nominal risk-free return on T-bills. The performance ofthe stocks of small firms was even more impressive; they earned an average annualreturn of 16.1 percent.

To appreciate the economic importance of these differences in annual average,consider how the value of a dollar invested in each of these types of assets in 1926would have changed over time. . . . by 1991, $1 invested in S&P stocks would be


TABLE C.1 U.S. Financial Asset Returns, 1926–1991a

Stocks U.S. Treasury

S&P 500 Small firms Bonds Bills

Average 11.94 16.05 4.94 3.64

Variability 20.22 31.02 7.62 0.94

Source: Adapted from Jagannathan and McGrattan (1995,Table 1). aAnnual percentage rate of return.

worth about $675, whereas $1 invested in T-bills would be worth only $11. That’snot much considering the fact that a market basket of goods costing $1 in 1926would cost nearly $8 in 1991. . . .

Notice. . . that the assets with higher average returns over 1926–91 also had morevariable returns. This correspondence suggests that the higher average returns werecompensation for some perceived higher risk. For example, small-firm stocks, whichyielded the highest return in this period, had the highest standard deviation too. (Jagan-nathan and McGrattan 1995, pp. 3–4)

Expressing this correspondence formally, for a particular asset i

E(Ri) = f (riski ), (C.4)

where f (·) represents the functional relationship between the two. To put the propo-sition even more strongly, it may be that risk is the only important influence causingexpected returns to differ across assets. To again quote Malkiel (1996, p. 227): “. . .when all is said and done, risk is the only variable worth a damn in the market.”

This appendix develops a model to specify the function f (·) and the appropriatemeasure of risk; the model is called the capital asset pricing model (CAPM).The CAPM was originally developed by William Sharpe (1964), John Lintner(1965), and J. Mossin (1966). Their work was in turn founded on the portfolioanalysis of Harry Markowitz (1952, 1959). Sharpe and Markowitz, jointly witha third financial economist Merton Miller, were awarded the 1990 Nobel Prizein economics for their ideas, which today constitute the foundation of the theoryof finance.

Like any economic model, the CAPM is based on many artificial and unre-alistic assumptions about agent behavior. We have already encountered one: thatinvestors assess risky assets solely on the basis of expected return and variance.As a result—and again like any model—it has implications that may seem tobe grossly inconsistent with observed behavior. For example, we shall find thatthe model implies that all investors hold a common portfolio of risky assets—themarket portfolio.

The CAPM, in fact, refers to a family of related models; different variantsof the model attempt to relax certain of these assumptions. Because the enormousliterature associated with these variants is far beyond our scope to survey, our focusis on the original Sharpe-Lintner version of the CAPM.

SECTION C.3 Portfolios C-5

It is natural to begin with Markowitz’s portfolio theory, which is largely anapplication of the distribution theory in Appendix B. Section C.3 considers thenature of the investment opportunities that become available when two risky assetsare held; Section C.4 generalizes this to a larger number of assets. This analysismakes precise the notion that it is possible to reduce risk through diversification.Section C.5 introduces a risk-free asset into the analysis. A condition under whichdiversification of risk takes place is derived in Section C.6; this in turn motivatesthe adoption of a new concept of risk in Section C.6.1. The optimal constructionof portfolios is considered in Section C.7, which ultimately leads us to the CAPM.

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..C.3 PortfoliosAt any point in time investors typically hold a number of risky assets. One might,for example, invest in several stocks and in other financial assets such as bonds.Firms are often engaged simultaneously in a number of capital investment projects.Many people hold a significant amount of their wealth in the form of home owner-ship, also belong to an employer-sponsored pension plan, and perhaps have personalholdings of financial assets as well. Most university students have very little wealthin financial form, but possess a considerable amount of human capital.

When wealth is held in several forms, this is called a portfolio of assets.Because the returns on the individual assets constituting the portfolio are random,so too is the return on the portfolio as a whole. What is the relationship betweenindividual asset returns and the overall return on a portfolio of those assets? Whatconsiderations apply in attempting to construct an “optimal” portfolio from a groupof assets, and what exactly is meant by optimality in this context? Is it possible toemploy these considerations as an aid in making practical investment decisions?The distribution theory developed in Appendix B provides us with a means ofaddressing these questions.

Let us begin with the simplest case of two uncertain returns R1 and R2. Eachhas some univariate probability distribution (which need not be specified—theycould be discrete or continuous) with means E(R1) and E(R2) and variancesV (R1) = �2

1 and V (R2) = �22, respectively. Suppose as investors we split all our

available wealth between these two risky assets, putting proportion a of our fundsinto R2 and the remainder 1 − a into R1. The resulting portfolio has a return thatmay be denoted Rp and that is determined as2

Rp = (1 − a)R1 + aR2. (C.5)

The proportions (1 − a) and a are called the portfolio weights.

E X A M P L E 1 Suppose you decide to invest $600 in one stock and $400 in another. It subse-quently turns out that the first yields a 10% return, and the second 5%. What is thepercentage return on your total investment?

2 This is derived in the appendix to Chapter 3, Example 4.


S O L U T I O N Because the first yields $600(0.10) = $60 and the second $400(0.05) = $20, thetotal return is $80, or 8% on the investment of $1000. Equivalently, applying (C.5)for a = 0.4 gives 0.6(0.10) + 0.4(0.05) = 0.08.

The notable feature of the expression (C.5) is that the return on a portfoliois a linear combination of the returns on the individual assets. This is importantbecause, as Appendix B demonstrates, special laws of mathematical expectationand variance apply to linear functions.

C.3.1 Expected ReturnFor example, how does the expected return on the portfolio E(Rp) depend on theexpected returns E(R1) and E(R2) of the assets individually? By Law B.3, therelationship is:

E(Rp) = (1 − a)E(R1) + aE(R2). (C.6)

E X A M P L E 2 Suppose one asset has expected return E(R1) = 0.10, the other E(R2) = 0.05, asin Figure C.1. How does the expected return on the portfolio change as we shiftour wealth between the two assets?

S O L U T I O N From (C.6), the expected return on the portfolio is

E(Rp) = (1 − a)(0.10) + a(0.05) = 0.10 − 0.05a. (C.7)

Hence when a = 0 and all our wealth is invested in asset 1, the return on theportfolio is the return on that asset: E(Rp) = 0.10 − 0.05(0) = 0.10 = E(R1).

At the other extreme, when a = 1 and all our wealth is invested in R2, we haveE(Rp) = 0.10 − 0.05(1) = 0.05 = E(R2). As we shift our wealth between the twoassets by varying a between 0 and 1, the expected return on the portfolio variesbetween the two extremes of 10 and 5% associated with the two assets individually.Furthermore, E(Rp) varies in a way that is linear in a.

In general, the expected return on a portfolio ranges between the extremes ofthe expected returns on the lowest and highest yielding assets, depending on howwealth is allocated between them. If one asset has an expected return of 5% andanother of 10%, obviously there is no way to combine these so as to obtain aportfolio having an expected return of 12%.3 Ex post, of course, the realized returnon the portfolio could be 12% or, for that matter, could be negative if the nature of

3 This statement assumes that it is not possible to sell assets short, which is implicit in the specificationthat 0 ≤ a ≤ 1. The availability of short-selling would make possible portfolios with E(Rp) > 0.10,

but at the cost of higher risk. Short-selling is one example of financial leverage; another is encounteredin Section C.5.


the underlying assets is such that a loss is possible. The assets are risky, and actualreturns may be quite different from what is expected. This is nothing more than say-ing that the realization of any random variable may be quite different from its mean.

C.3.2 RiskThat the expected return of a portfolio must be between the expected returns ofthe assets individually contrasts with what may be true of risk. By Law B.4, thevariance of the portfolio, �2

p, is given by

V (Rp) = (1 − a)2�21 + a2�2

2 + 2(1 − a)a�12 (C.8a)

= (1 − a)2�21 + a2�2

2 + 2(1 − a)a�1�2�. (C.8b)

The risk of the portfolio depends, as we expect, on the risks �21 and �2

2 of the assetsindividually and on the proportion a according to which wealth is split betweenthem. Furthermore, for given a, the risk of the portfolio is increasing in each of�2

1 and �22: riskier assets lead to a riskier portfolio.

In addition, however, there is another important determinant of the risk of theportfolio—the covariance �12 between the assets or, equivalently, the correlation� = �12/�1�2. The risk of the portfolio depends not only on the risks of the assetsindividually, but on how their returns covary. When asset returns are positively cor-related, the third term in (C.8) is positive, contributing to the overall risk of the port-folio. The example given in Section B.11 is about two automobile manufacturers.Because both firms are in the same industry, both will tend to be similarly affectedby common industrywide influences. When the economy is doing well and con-sumers are buying new cars, both carmakers will do well; during periods of reces-sion, on the other hand, both will tend to have low returns. Hence, investing in assetshaving positively correlated returns tends to result in a relatively risky portfolio.

By comparison, we would prefer to invest in assets having uncorrelated returnsby, say, selecting firms in unrelated industries because if �12 = 0 the third termin (C.8) disappears, yielding a portfolio having less risk than if the returns werepositively correlated. In fact, it may be possible to do even better because if assetsare available having returns that are inversely related, the third term in (C.8) willbe negative.

In a diagram such as Figure C.1, what risk-return opportunities arise from theability to combine assets into portfolios? It is now apparent that the answer to thisquestion depends on the correlation between the returns. Because any correlationfalls in the range −1 ≤ � ≤ 1, our intuition can be developed by examiningequation (C.8b) in three special cases: � = 1, � = 0, and � = −1. The followingexamples treat each of these in turn.

E X A M P L E 3 � = 1. Consider the returns R1 and R2 in Figure C.1. What risk-return opportunitiesare made available by combining these into portfolios, if � = 1? Of these, whichare efficient?


S O L U T I O N The return on the portfolio Rp is given by equation (C.5); the expected return E(Rp)

by equation (C.6), which, in this example, specializes to (C.7); and the variance�2

p by equation (C.8b). Using the last, when � = 1 the standard deviation of theportfolio �p is

�p =[(1 − a)2�2

1 + a2�22 + 2(1 − a)a�1�2

]1/2

= (1 − a)�1 + a�2.

We have chosen to work in terms of the standard deviation, �p, rather than thevariance because this is what is measured on the horizontal axis in Figure C.1.4

The last line makes use of the observation that the expression in square parenthesesis a perfect square.5 Evaluating �p for the values �1 = �2 = 0.04 of Figure C.1yields �p = (1 − a)(0.04) + a(0.04) = .04. So, not very surprisingly, the risk ofthe portfolio is exactly the same as the common risks of the assets individually,regardless of how wealth is split between the two.

Example 2 shows that the expected return E(Rp) varies between the twoextremes of 5 and 10% as a varies between 0 and 1. Hence, in terms of Figure C.1,the risk-return opportunities available to the investor by combining R1 and R2 intoportfolios take the form of a straight line between them.

�

R1

Efficient pointE(R)

R2

0.10

0.05

0.04

Which of these risk-return opportunities are efficient—that is, are not domi-nated by others? Only one—R1 —because, just as we concluded earlier with respectto R2, any point on the line has the same risk as R1 but a lower expected return.R1 dominates all other investment opportunities.

In Example 3 the ability to create portfolios ends up being of no value toinvestors; they still choose to set a = 0 and invest entirely in R1, just as is

4 Although the diagram could have been expressed in terms of the variance, it turns out to be convenientto use the standard deviation. The reason becomes apparent in Section C.5, where the Tobin frontier isshown to be linear when expressed in terms of the standard deviation.

5 This perfect square has both a positive and negative square root. The positive square root is usedbecause a standard deviation must be positive.


true when the only choice is between investing entirely in either R1 or R2. Thisconclusion is rather anticlimactic: Our intuition suggests that the ability to createportfolios should be beneficial.

In fact, this intuition is correct, in general. The perverse conclusion of Example3 arises from the fact that it involves two artificial features. The first is that, in thisnumerical example, the two assets have the same risk: �1 = �2 = 0.04. Suppose,by contrast, that the second asset had both a lower expected return than R1 and alower risk, like R3 in Figure C.1. It turns out that, for � = 1, the set of risk-returnopportunities associated with portfolios of R1 and R3 would still take the formof a straight line between them. You are asked to confirm this in Exercise C.3.Furthermore, all points on the line would be efficient; all would represent risk-return opportunities that may reasonably be chosen by investors, depending ontheir preferences for risk versus return.

The second artificial feature of Example 3 is that it assumes � = 1. Obviouslywe are unlikely in practice to encounter assets having perfectly correlated returns.It is therefore useful to contrast the findings of Example 3 with the alternativeassumption that � = 0.

E X A M P L E 4 � = 0. Consider the returns R1 and R2 in Figure C.1. What risk-return opportunitiesare made available by combining these into portfolios, if � = 0? Of these, whichare efficient?

S O L U T I O N Returning to equation (C.8b), setting � = 0, and letting �1 = �2 = .04 yields

�p =[(1 − a)2�2

1 + a2�22

]1/2 = [(1 − a) + a]1/2 (0.04).

This expression is nonlinear in a; the easiest way to see what risks become availableas wealth is shifted between the two assets is to evaluate it numerically. The fol-lowing table does this in steps of 0.1 as a varies between 0 and 1; the final columngives the corresponding value of E(Rp) obtained by evaluating equation (C.7).

a �p E(Rp)

0 0.04 0.100.1 0.0362 0.0950.2 0.0272 0.090.3 0.0232 0.0850.4 0.0208 0.080.5 0.0200 0.0750.6 0.0208 0.070.7 0.0232 0.0650.8 0.0272 0.060.9 0.0362 0.0551 0.04 0.05

The lowest attainable risk of �p = 0.02 is achieved by investing 50:50 inthe two assets. Plotting these risk-return values shows that the set of investmentopportunities appears as follows.


�

R1Efficient set

E(R)

R20.05

0.075

0.10

0.040.02

Of these, which are efficient? Only those on the upper portion of the curve,as indicated. Although the lower portion of the curve represents portfolios that arefeasible, no rational individual would invest here because there is always anotherportfolio with the same risk but a higher expected return. In this example, rationalinvestors will always put at least half their wealth into R1. The exact choice of a

in the range 0.5 ≤ a ≤ 1 will, as usual, depend on the individual’s degree of riskaversion.

This example is striking because it shows that, even if an asset is itself inef-ficient (as R2 is), it may still make sense to put some of our wealth into it. Hererational investors might put up to half their wealth into R2, although no more thanthis. The extent to which it is rational to invest in an asset will depend on thecorrelation between that asset and others; in Example 3 we have seen that if � = 1it does not make sense to put any of our wealth into R2.

The phenomenon that, by combining assets into portfolios, it is possible toattain a level of risk lower than that of any of the assets individually, is called riskspreading, risk pooling, or the diversification of risk. Note that this can occurwithout the returns necessarily being uncorrelated; that is, the correlation need notbe as low as zero. Were the analysis of Example 4 repeated based on � = 1/2, forexample, a set of risk-return opportunities similar to the curve in Example 4 wouldbe obtained; it would just not curve so far to the left. You are asked to examine thisin Exercise C.4. Hence, it can be possible to reduce risk below that of either of theassets individually, even when they are positively correlated. A general conditionindicating exactly when risk spreading occurs is derived in Section C.6.

It is now clear that the extent to which risk can be reduced through diversifica-tion depends on the correlation. When � = 1, it is not possible to reduce risk belowthat of the assets individually; in Example 4, where � = 0, the lowest attainablerisk is �p = 0.2, half that of �1 or �2 individually. It is instructive to consider thethird special case of � = −1.

E X A M P L E 5 � = −1. Consider the returns R1 and R2 in Figure C.1. What risk-return opportu-nities are made available by combining these into portfolios, if � = −1? Of these,which are efficient?


S O L U T I O N Returning to equation (C.8b), setting � = −1, and again recognizing a perfectsquare,

�p =[(1 − a)2�2

1 + a2�22 − 2(1 − a)a�1�2

]1/2

= � [(1 − a)�1 − a�2] .

In this case, to ensure a positive value for �p, we must take either the positive ornegative root, depending on the value of a. Evaluating for �1 = �2 = 0.04 yields

�p = �(1 − 2a)(0.04) ={

0.04 − 0.08a, if 0 ≤ a ≤ 1/2;

−0.04 + 0.08a, if 1/2 ≤ a ≤ 1.

The associated risk-return opportunities may be determined by combining theseexpressions with equation (C.7). Inverting these expressions to solve for a yieldsa = 0.5−12.5�p if 0 ≤ a ≤ 1/2, or a = 0.5+12.5�p if 1/2 ≤ a ≤ 1. Substitutingthese into (C.7),

E(Rp) ={

0.075 + 0.625�p, if 0 ≤ a ≤ 1/2;

0.075 − 0.625�p, if 1/2 ≤ a ≤ 1.(C.9)

Hence, when � = −1 the relationship between E(Rp) and �p is linear in the wayshown in the following diagram.

�

R1Efficient set

E(R)

R20.05

0.075

0.10

0.04

Risk is minimized at a = 1/2 where, by equation (C.7), E(Rp) = 0.075.Remarkably, when returns are exactly inversely related it is possible to combinerisky assets in such a way that the returns exactly offset one another and risk iseliminated entirely: �p = 0. As we move from a 50:50 division of wealth betweenthe two assets to putting a higher proportion into R1, we move along the upper linegiven by (C.9). This corresponds to the efficient set. If less than half of our wealthis in R1 the lower line applies, which is inefficient.

These numerical examples illustrate that the diversification of risk occurs incertain circumstances—circumstances largely (but not entirely) determined by thecorrelation between returns. As mentioned, Section C.6 derives a precise conditioncharacterizing the diversification effect. The more immediate task at hand is, first, toextend the analysis to more than two risky assets and, in Section C.5, to incorporatea riskless asset.


E X E R C I S E S

C.1 Consider the bivariate distribution of Example11 of Appendix B. Let X be the random return ona $1 investment in one asset and Y be the returnon a $1 investment in another. You are planningto put proportion a of your wealth into Y and theremainder into X.

(a) Find a numerical expression for E(Rp) as afunction of a.

(b) Find a numerical expression for �2p as a function

of a.

(c) Use calculus to determine the choice of a thatminimizes �2

p. What is this minimum level of risk?

(d) What expected return is associated with thisminimum-risk portfolio?

C.2 Consider the following example given byMalkiel (1996, pp. 236–237).

Let’s suppose we have an island economy with onlytwo businesses. The first is a large resort with beaches,tennis courts, a golf course, and the like. The secondis a manufacturer of umbrellas. Weather affects thefortunes of both. During sunny seasons, the resortdoes a booming business and umbrella sales plum-met. During rainy seasons, the resort owner does verypoorly, while the umbrella manufacturer enjoys highsales and large profits. The following table showssome hypothetical returns for the two businessesduring the different seasons.

Umbrella Resortmanufacturer owner

Rainy season 50% −25%Sunny season −25% 50%

Suppose that, on average, one-half the seasons aresunny and one-half are rainy (i.e., the probability of asunny or rainy season is 1/2). An investor who boughtstock in the umbrella manufacturer would find thathalf the time he earned a 50 percent return and halfthe time he lost 25 percent of his investment. . . .

Suppose, however, that instead of buying onlyone security, an investor. . . diversified and put half hismoney in the umbrella manufacturer’s and half in theresort owner’s business.

(a) Find the expected value and variance of theumbrella manufacturer’s return.

(b) Find the expected value and variance of theresort owner’s return.

(c) Find the correlation between the two returns.

(d) Find the expected value and variance of thediversified investor’s return.

C.3 Consider the assets R1 and R3 in Figure C.1.If proportion a of our wealth is invested in R3, theremainder in R1, and the two are related by � = 1,

(a) find a numerical expression for E(Rp) as afunction of a.

(b) find a numerical expression for �p as a functionof a.

(c) derive an equation showing how E(Rp) is deter-mined as a function of �p.

(d) plot the relationship between E(Rp) and �p for0 ≤ a ≤ 1. Identify the set of efficient portfolios.

C.4 Consider the assets R1 and R2 in Figure C.1.If proportion a of our wealth is invested in R2,the remainder in R1, and the two are related by� = 1/2,

(a) find a numerical expression for E(Rp) as afunction of a.

(b) find a numerical expression for �p as a functionof a.

(c) evaluate your expressions from parts a and bfor 0 ≤ a ≤ 1 using steps of 0.1.

(d) plot the relationship between E(Rp) and �p.

C.5 Consider the assets R2 and R3 in Figure C.1.If proportion a of our wealth is invested in R3, theremainder in R2, and the two are related by � = 0,

(a) find a numerical expression for E(Rp).

(b) find a numerical expression for �2p as a function

of a.

(c) use calculus to determine the choice of a thatminimizes �2

p. What is this minimum level of risk?

(d) plot the relationship between E(Rp) and �p for0 ≤ a ≤ 1. Identify the set of efficient portfolios.

SECTION C.4 The Markowitz Frontier C-13

....

....

....

..C.4 The Markowitz Frontier

The case of two assets is instructive. However, in the real world there are potentiallythousands of investments available to combine into portfolios. Suppose a portfolioconsists of n assets with returns Ri (i = 1, . . . , n). Generalizing equation (C.5),the return on the portfolio is

Rp =n∑

i=1

aiRi, (C.10)

where the portfolio weights must satisfy∑

ai = 1 (wealth is divided completelyamong the assets). As an application of the generalization (B.18) of Law B.3, theexpected return is

E(Rp) =n∑

i=1

aiE(Ri). (C.11)

By the generalization (B.20) of Law B.4, the variance of the portfolio return is

V (Rp) =n∑

i=1

a2i �2

i +n−1∑i=1

n∑j=i+1

aiaj �ij . (C.12)

In particular, the portfolio risk now depends on the n(n−1)/2 covariances relatingthe n returns. How does this affect the analysis?

Let us begin by noting that, even when many assets are available, it will notnormally be the case that any two have returns that are exactly correlated, eitherdirectly or inversely. That is, the cases � = 1 and � = −1, although they wereinteresting special cases to examine, will not normally apply in practice. From ouranalysis of the case � = 0, it is apparent that this is the more typical shape of therisk-return opportunity set when −1 < � < 1.

This typical situation is shown in Figure C.2(a). Now suppose we introduce athird asset R3. It may be combined in portfolios with either R1 or R2; the resultingrisk-return opportunities are shown in Figure C.2(b) and will, in each case, dependon the respective correlations.

This is not a complete characterization of the available risk-return opportuni-ties, however, because thus far only portfolios consisting of two of the three assetshave been considered. What opportunities become available by combining all threeassets?

Returning to Figure C.2(a), it is the case that each of the portfolios on thecurve constitutes an asset in its own right. Consider, for example, portfolios Aand B indicated in Figure C.2(c). These may be combined with R3; the result-ing risk-return opportunities are shown and, again, will depend on the underlyingcorrelations.

Applying this logic to combining R3 with all points on the curve in Figure C.2(a)yields the set of risk-return opportunities depicted in (d). Clearly a fourth asset couldbe introduced and, by the same reasoning, an expanded opportunity set derived.


F IGURE C.2

The risk-returnopportunity set.

� �

R1

R3

R2R2

R1

R3

R2

R1

B

A

R3

R2

R1

(a) two assets (b) two of three assets

E(R)

�

(c) some portfolios of three assets

E(R)

�

(d) all portfolios of three assets

E(R)

E(R)

F IGURE C.3

The Markowitzfrontier is the effi-cient frontier of therisk-expected returnopportunity set.

�

E(R) Markowitz frontier

Taking any opportunity set obtained in this way, which of these risk-return pos-sibilities are efficient? Obviously, only the outer envelope of points associated withexpected returns above that of the minimum-risk portfolio. This efficient frontieris shown in Figure C.3; it is called the Markowitz frontier after Harry Markowitz(1952, 1959), who was the first to recognize the applicability of distribution theoryto investment analysis.

In conclusion, the efficient set in the multiasset case takes essentially the sameform as it did in the two-asset case—for correlation values that are relevant inpractice. Note that most individual assets will not be efficient; their risk-returnvalues are typically inside the efficient set. To be on the Markowitz frontier, it isusually necessary to hold a portfolio of assets. To put all of our wealth into anysingle asset is almost certainly foolish; a higher expected return and/or lower riskis likely available by spreading our wealth across a number of investments.

SECTION C.4 The Markowitz Frontier C-15

FIGURE C.4

The standard devi-ation of portfolioreturn as a func-tion of the num-ber of securitiesin the portfo-lio: 50 randomlyselected stocks.(Source: Figure 7.9

of FOUNDATIONS OF

FINANCE by EUGENE

F. FAMA. Copyrightc© 1976 by Basic

Books, Inc. Reprinted

by permission of Basic

Books, a member of

Perseus Books, L.L.C.)

0.053

0.067

0.082

0.097

0.111

0.03815105 20 25 30 35 40 45 50

How many assets do we need to hold in order to be on the Markowitz fron-tier? Consider selecting stocks randomly from those listed on, say, the New YorkStock Exchange. A single stock chosen at random will, on average, have a rela-tively high standard deviation. If a second stock is chosen randomly and combinedinto an equally weighted portfolio with the first, the resulting portfolio return willhave a somewhat lower risk, due to the diversification effect. If a third stock isadded, the risk will be lower still, and so on. Many studies of this type have beenundertaken; early examples include Fama (1976, pp. 253–254), Fisher and Lorie(1970), Wagner and Lau (1971), and, in an international context, Solnik (1974).Figure C.4, taken from Fama (1976), is a typical finding. The risk of the portfoliodeclines dramatically at first; however, beyond a certain point very little additionalrisk reduction is achieved by including additional stocks.

What level of wealth is required to be able to construct a portfolio that exploitsthese gains to diversification? As a practical matter, for trading costs not to beprohibitive it is necessary to purchase stocks in 100-share blocks. If an averageshare price is, say, $25, a portfolio of 20 stocks would require wealth in the amountof $50,000. Few private individuals have liquid wealth of this magnitude.

An alternative route to investing in a diversified portfolio is to pool resourceswith other investors. This is, of course, exactly the service provided by a number ofimportant financial institutions. Banks pool small deposits, investing in a diversifiedportfolio of loans to businesses and individuals. Pension funds pool retirementsavings and invest in portfolios of financial securities and real estate. Mutual fundsoffer a means by which small investors can purchase part of a much larger portfolioof securities. The vast sums of money managed by these institutions testifies to thevalue of this service to investors.

C.4.1 The Market PortfolioAs we continue to add assets randomly to the portfolio, we come closer and closerto replicating the market portfolio. That is, suppose in the stock market firm Aaccounts for 1% of the total capitalization of the market, firm B for 0.8%, firm C for1.45%, and so on. If we go on choosing stocks randomly from the population—inthis case, all those stocks traded in the market—ultimately we will end up witha portfolio that replicates the composition of that population. The weights of themarket portfolio are the relative market capitalizations of each firm.


Definition C.1 A market portfolio is a portfolio that has the same composition by value as themarket as a whole.

It is evident from Figure C.4 that not all risk can be eliminated through randomdiversification. Instead, risk can be thought of as consisting of two components.First, diversifiable or firm-specific risk is associated with the idiosyncrasies of theindividual security and can be eliminated through diversification. This is also calledidiosyncratic or nonsystematic risk. But even a fully diversified portfolio—as inthe case, for example, of the market portfolio—still involves some risk. After all,in a few days in October 1987 the New York Stock Exchange lost one-third of itsvalue. Thus, second, market risk is associated with the systematic behavior of themarket as a whole and is nondiversifiable.

In concluding that the market portfolio is fully diversified, it follows that itmust be on the envelope of minimum-risk portfolios. As well, it is not an obviouslyfoolish investment. People do in fact invest in the market portfolio, and so it isnot on the underbelly of the minimum-risk envelope; in other words, it must beon the Markowitz frontier. This is an implication that students tend to find eitherdeeply profound or transparently obvious. In any case, it is important enough thatit merits stating formally.

Result C.1 The market portfolio is on the Markowitz frontier.

Being on the Markowitz frontier, the market portfolio has some appeal as aplace to invest. Indeed, it is well documented that the majority of equity mutualfunds underperform the market—see, for example, Malkiel (1999, p. 186), whonotes, “In the twenty-five-year period to 1998, more than two-thirds of the pro-fessionals who manage mutual-fund common-stock portfolios were outperformedby an unmanaged Standard & Poor’s 500-Stock Index Fund.” This fact has ledmany institutional and retail investors to abandon active portfolio managementin favor of a strategy of passive management, often involving the replication ofsome market index. Unavailable 25 years ago, index funds now account for manybillions of dollars of investment capital (see Malkiel 1996, Chap. 15).

The market portfolio is just one of an infinite number of portfolios that con-stitute the Markowitz frontier. The others are also fully diversified portfolios, but,in contrast to the market portfolio, they are not obtained through random diversifi-cation. Instead, they are associated with portfolio weightings different from that ofthe market portfolio, weightings that give rise to more or less risk than the marketportfolio (and a corresponding reward or penalty in terms of expected return). Thedistinguishing feature of the market portfolio is that it is obtained through randomdiversification. Through nonrandom diversification, it is possible to escape fromsystematic risk, but we must pay to do this—the payment being in the form oflower expected return. A trivial example is to invest entirely in a riskless asset, if

0.0000.0010.0020.0030.0040.0050.0060.0070.0080.0090.0100.0110.0120.0130.0140.0150.0160.0170.0180.0190.0200.0210.0220.0230.0240.025

Exp

ecte

d re

turn

Japan

SwedenBelgium

Hong Kong

Variance

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

DenmarkFrance

AustraliaAustria

Canada

Spain

Germany

Italy

World U.K.

Norway

Netherlands

SwitzerlandU.S.

F IGURE C.5 The minimum variance frontier for 17 country equity returns. (Source: Harvey 1991).


F IGURE C.6

Internationaldiversification.(Source: Solnik 1974,

Figure 9. Copyright

1974, Association for

Investment Manage-

ment and Research.

Reproduced and

republished from

Financial Analysts Jour-

nal with permission

from the Association

for Investment Man-

agement and Research.

All Rights Reserved.)

40

2011.7

60

80

100

30 40 5010 201

Ris

k (%

)

Number of Stocks

U.S. StocksInternational Stocks

one is available; all risk would be eliminated, but at the cost of receiving only therisk-free rate of return. In the next section, a risk-free asset is introduced formallyinto the analysis; we also pursue further the idea that expected return is the rewardfor bearing systematic risk.

Although it is an important and useful concept, in practice the market portfoliomust be defined in terms of some universe of assets. An expanded universe isassociated with a lower level of systematic risk. For example, Figure C.4 takesthe market to be the New York Stock Exchange. However, if our concept of “themarket” is expanded to include other securities such as bonds, a lower level ofsystematic risk is attainable through diversification.

This is also true internationally. Figure C.5 shows the minimum variance fron-tier attainable by investing in the equity markets of 17 countries and the individualcountry returns. Figure C.6 contrasts the pattern of diversification available frominvesting solely in U.S. equities versus including international stocks in the port-folio. Clearly a lower level of systematic risk applies when the definition of whatconstitutes the market is broadened.

Indeed, in the final analysis the market portfolio is not observable because itincludes not only securities but also nonfinancial assets such as real estate, preciousmetals, diamonds—even artwork. It also includes physical and human capital.Thus, although the idea of a market portfolio is well defined conceptually in away that is useful for theory, any specification of it is bound to be incompletein practice. As we see in Section C.7, the concept of the market portfolio playsan important role in the CAPM; its unobservability has been emphasized by Roll(1977) as an important problem in the testing of the model.

From our development of the concept of market risk, it is evident that anyindividual asset embodies both market risk and diversifiable risk. As assets arecombined into portfolios, the diversifiable risk is eliminated, leaving only the non-diversifiable component. The decomposition of the total variability of an asset’sreturn, V (Ri), into its systematic and nonsystematic components is formalized inSection C.6.1.

Given that the market portfolio is one point on the Markowitz frontier, cananything further be said by way of characterizing its location? It turns out thatintroducing a riskless asset explicitly into the analysis allows us to do just that.

SECTION C.5 The Tobin Frontier C-19

....

....

....

..C.5 The Tobin FrontierSuppose that a risk-free asset exists. This means that it offers a sure return rf,which is not a random variable. As we have discussed in Section C.2, rf repre-sents the “price of time”—the reward for delaying consumption. What risk-returnopportunities arise when this risk-free asset is combined with risky assets?

C.5.1 A Riskless Asset Combined with One RiskyAsset

Let us begin with the simplest case of a risk-free asset combined with a singlerisky asset i having random return Ri . The portfolio return is

Rp = (1 − a)rf + aRi, (C.13)

with expectation

E(Rp) = (1 − a)rf + aE(Ri) = rf + a(E(Ri) − rf). (C.14)

The quantity E(Ri) − rf is called a risk premium: It is the (expected) rewardearned for holding the risky asset instead of the safe one. The variance of theportfolio is given by

V (Rp) = a2�2i (C.15)

or, in terms of standard deviations,

�p = a�i . (C.16)

Equation (C.14) is an application of Law B.1 in Appendix B, (C.15) an applicationof Law B.2. As a is varied in the range 0 < a < 1, (C.14) and (C.16) tell us howE(Rp) and �p vary. Inverting the latter as a = �p/�i and substituting this into(C.14) shows that the implied relationship between E(Rp) and �p is

E(Rp) = rf + �p(E(Ri) − rf)

�i

. (C.17)

This indicates that, as depicted in Figure C.7, the set of risk-return opportunities islinear. A portfolio with zero risk (�p = 0) requires that we invest entirely in the

F IGURE C.7

Portfolios involvinga risk-free asset.

�p�i

Slope �

E(Rp)

E(Ri)

rf�i

E(Ri) � rf


risk-free asset (so a = 0) and accept the risk-free rate of return: E(Rp) = rf. Foreach additional unit of risk �p taken on by investing in the risky asset, we receivereward (E(Ri) − rf)/�i related to the risk premium, which has the geometricinterpretation as the slope of the line in Figure C.7. At the extreme of a = 1where we are fully invested in the risky asset i, �p = �i and (C.17) reduces toE(Rp) = E(Ri).

E X A M P L E 6 Consider the risky asset R1 in Figure C.1, and suppose a risk-free asset havingreturn rf = 0.03 is also available. What risk-return opportunities become availableby combining the two into portfolios?

S O L U T I O N Setting rf = 0.03, E(Ri) = 0.10, and �p = 0.04 in (C.17) yields the line

E(Rp) = 0.03 + 1.75�p. (C.18)

Note that when wealth is invested entirely in the risk-free asset, �p = 0 and E(Rp)

is just the risk-free rate of 3%; when we are invested entirely in R1, on the otherhand, �p = 0.04 and E(Rp) = 0.03 + 1.75(0.04) = 0.10 = E(R1).

When we invest a portion of wealth in the risk-free asset, we are effectivelylending at the risk-free rate. Consider the possibility of instead borrowing at therisk-free rate to invest in the risky asset. What risk-return opportunities would thisgenerate?

E X A M P L E 7 In Example 6, every dollar of wealth was divided between R1 and lending at therisk-free rate. Suppose instead that for every dollar of wealth, $a (a > 1) is investedin R1, the difference $a−1 being obtained by borrowing at the risk-free rate. Whatrisk-return opportunities become available?

S O L U T I O N The return on the portfolio is the return on the risky investment less the borrowingcosts associated with that portion of the total investment that is borrowed:

Rp = aRi − (a − 1)rf = (1 − a)rf + aRi.

But this is identical to (C.13). It follows that E(Rp) and �p continue to be rep-resented by (C.14) and (C.16) when a > 1 and the trade-off between the two by(C.17). For the numerical values of Example 6, (C.18) still describes how E(Rp)

varies with �p, even when �p > 0.04.

The extension of the risk-return line made available by borrowing is shown inFigure C.7. Borrowing at the risk-free rate is an example of financial leverage: Itmakes possible an expected return greater than that of the risky asset, but at thecost of higher risk.

SECTION C.5 The Tobin Frontier C-21

C.5.2 A Riskless Asset Combined with ManyRisky Assets

Consider introducing a risk-free asset into our analysis of the Markowitz frontier.In principle, the risk-free asset could be combined with any of the points in theopportunity set; examples are R1 or R2 in Figure C.8(a), which depicts the straight-line risk-return opportunities that arise. However Figure C.8(b) makes it apparentthat these are dominated by other opportunities that lie on a line running from rfto a point of tangency with the Markowitz frontier; this point of tangency is calledthe tangent portfolio. The line extends beyond the tangent portfolio, reflecting thepossibility of borrowing at the risk-free rate. This line is called the Tobin frontierafter James Tobin (1958b), who was the first to formulate this analysis of the roleof a riskless asset in risky portfolios.6

The implications of this analysis are profound. First, it is apparent that in thepresence of a risk-free asset most portfolios on the Markowitz frontier are notefficient; they are dominated by the Tobin frontier. The two frontiers have onlya single point in common—the tangent portfolio. This is the only point on theMarkowitz frontier that is efficient.

It follows that efficient investments consist of lending or borrowing at therisk-free rate and investing in the tangent portfolio. Different investors may havedifferent preferences, but this determines only the amount of lending or borrowing.Although different investors may make different choices about the proportion oftheir wealth invested in risky assets, all hold the same portfolio of risky assets—thetangent portfolio. Thus, there is a separation of the two decisions.

Result C.2 (Tobin Separation Theorem) Once the amount of their wealth to be held in theform of risky assets is determined, all investors hold the same risky portfolio. Thisis the tangent portfolio.

F IGURE C.8

A risk-free assetand the Markowitzfrontier.

�

R1

E(R)

R2

rf�

E(R)

rf

(a) introduction of a riskless asset (b) the Tobin frontier

Tangentportfolio

6 Tobin’s goal was to derive an interest-elastic function for money demand (“liquidity preference”).He regarded money as the risk-free asset and showed that its demand would be inversely related to theexpected return on the tangent portfolio—the interest rate on competing assets.


But if all investors hold the same risky portfolio, this must be the marketportfolio! In equilibrium asset prices, and thus their portfolio weights (i.e., theirrelative market capitalizations), reflect investors’ assessments of risk.

Result C.3 The market portfolio corresponds to the tangent portfolio. The market portfolio isthe only point on the Markowitz frontier that is efficient in the presence of a risk-freeasset.

Applying (C.17) to the market portfolio yields the following equation for theTobin frontier:7

E(Rp) = rf + �p(E(Rm) − rf)

�m. (C.19)

The points on this line correspond, as we have seen, to different amounts of lendingand borrowing by economic agents. Indeed, the risk-free rate (and hence the slope ofthe line) would be determined endogenously so as to equilibrate aggregate lendingand borrowing in the economy.

Hence, Figure C.8(b) reflects the equilibration of two sets of markets. One isthe market for borrowing and lending at the risk-free rate. The other is the mar-ket for risky assets, in which the relative market capitalizations of assets (i.e., theweights in the market portfolio) are determined by the equilibration of asset pricesto reflect their risk characteristics. These prices adjust to make the relative capital-izations in the market consistent with what investors are willing to hold. These twosets of markets—the markets through which an economy’s real capital investmentis financed—are called the capital markets. Because the Tobin frontier reflectsequilibrium in these capital markets, it is also called the capital market line.

The capital market line describes the trade-off between risk and return forefficient portfolios. These efficient portfolios consist of either lending or borrowingand investing in the market portfolio. Because the market portfolio incorporatesonly systematic risk, investors choose to bear different degrees of systematic riskby locating at different points on the line. A highly risk-averse investor mightbe unwilling to bear any systematic risk; investing entirely in the riskless assetachieves �p = 0 and E(Rp) = rf. As more risk, measured by �p, is taken on bymoving up the capital market line, the investor is rewarded with a higher expectedreturn in the amount

E(Rp) − rf = �p(E(Rm) − rf)

�m. (C.20)

Hence the expected return on the portfolio, E(Rp) given by (C.19), consists oftwo components. One is the “price of time” rf, which represents compensation

7 The reason for our original decision to express the risk-return diagram in terms of the standarddeviation rather than the variance is now revealed. It leads to a convenient linear representation for theTobin frontier. If the variance were on the horizontal axis, the Tobin frontier would be nonlinear.

SECTION C.6 The Diversification Effect C-23

for postponing consumption into the future. The other is the risk premium (C.20),which represents compensation for bearing risk. The risk premium is the productof the amount of additional risk borne, �p, and the “price of systematic risk”(E(Rm) − rf)/�m; the latter is the slope of the capital market line.

For the efficient portfolios described by the capital market line, therefore, thestandard deviation of the portfolio (or its variance) is an appropriate measure ofthe quantity of risk. However, this is only because these efficient portfolios involveonly systematic risk. In contrast, as we have seen, individual assets incorporateboth market and idiosyncratic risk. This has implications for the appropriate meansof measuring the risk of individual assets and the reward the market will pay forbearing that risk. Pursuing this observation requires that the diversification effectbe studied in somewhat more detail.

....

....

....

..C.6 The Diversification EffectWe have seen that, depending on the correlation structure of returns, it may bepossible to reduce risk through diversification. This section derives a condition thatcharacterizes precisely the circumstances under which the diversification effectoccurs.

For this purpose it is useful to return to the framework of Section C.3 andconsider portfolios of just two assets, so that returns are related by equation (C.5).Reproducing equation (C.8a), the variance of the portfolio return is

V (Rp) = (1 − a)2�21 + a2�2

2 + 2(1 − a)a�12.

Suppose that initially a = 0 so that all our wealth is in R1; we want to knowwhether it is possible to lower our risk by diversifying into the other asset R2.The interesting case is shown in Figure C.9, where V (R2) > V (R1). (After all, ifV (R2) < V (R1) risk will always be reduced by diversifying into R2.) The figuregraphs V (Rp) as wealth is shifted between the two assets. There are only twopossibilities. If no diversification effect exists, then, as wealth is shifted out of R1and into R2, V (Rp) does not fall below V (R1). This case is shown by curve B;

F IGURE C.9

The diversificationeffect.

V(Rp)

V(R2)

V(R1)

E(Rp)E(R1) E(R2)

R1

R2

A

B

a � 1a � 0


it applies, as we know from the analysis of Section C.3, when � = 1, but is notnecessarily limited to this extreme value of the correlation.

In the other case, shown by curve A, the diversification effect exists. By diver-sifying out of R1 into R2, it is possible to construct portfolios having risk lowerthan either of the assets individually. The distinguishing feature of this case is thatthe slope of curve A is negative at the point a = 0. Calculus provides a simplemeans of characterizing this condition mathematically:8

dV (Rp)

da

∣∣∣∣a=0

< 0.

Evaluating this derivative yields

dV (Rp)

da

∣∣∣∣a=0

= −2(1 − a)�21 + 2a�2

2 + (2 − 4a)�12

∣∣∣a=0

= 2(�12 − �21),

which will be negative as long as�12

�21

< 1. (C.21)

This is the condition under which it is possible to reduce risk by diversifyingout of R1 into R2. Notice that, from the perspective of investors who hold R1and are attempting to decide whether to invest in R2 as well, it is this conditionthat is of interest. The investors wish to know how the risk of their portfoliowill be affected by the inclusion of a new investment, and the quantity �12/�2

1gives the answer; the variance of the new investment, in this case �2

2, is of noparticular interest. This is analogous to the fact that, in general in economics, it isthe marginal effect that is relevant in making a decision. Thus, in the context ofa portfolio decision, the quantity �12/�2

1 is the relevant measure of the risk of R2,not its variance.

C.6.1 BetaIn general, then, the relevant measure of risk for asset i when combined withanother asset j is �ij /�2

j . Let us denote this quantity by �ij . By the Tobin separationtheorem, once the borrowing/lending decision has been made all investors hold themarket portfolio. So let this other asset j be the market portfolio; in consideringwhether to diversify out of the market portfolio into asset i (that is, to overweightthe portfolio in favor of i) the relevant measure of risk is

�im = �im

�2m

. (C.22)

This quantity is termed the beta of asset i.

8 That the curves in Figure C.9 extend outside the range 0 ≤ a ≤ 1 implies that short-selling ispermitted. This avoids a discontinuity at the point a = 0.

SECTION C.6 The Diversification Effect C-25

There are three possibilities:

�im < 1, and asset i is said to be defensive.�im = 1, and asset i is said to be neutral.�im > 1, and asset i is said to be aggressive.

That is, an asset is defensive if risk is reduced by overweighting the marketportfolio in favor of it; overweighting in i offers an escape from the systematicrisk of the market. In this sense, the asset has low systematic risk relative to themarket. If the asset is neutral, then overweighting it does not change the risk ofthe portfolio; its systematic risk is the same as that of the market. Overweightingan aggressive asset increases the risk of the portfolio and thus is associated withhigh systematic risk relative to that of the market.

In this respect, then, beta is a measure of systematic risk relative to the market.

Result C.4 From the perspective of an investor holding the market portfolio, �im is theappropriate measure of the risk of asset i relative to the market.

Beta has several interesting features. First, because it is defined in terms of thecovariance �im it can be restated in terms of the correlation:

�im = �im�i

�m. (C.23)

In Section B.10.4 it is shown that � is a dimensionless quantity; somewhat similarly,beta is invariant to variable scalings when the same scaling is applied to bothvariables. For example, if the returns on both asset i and the market double, beta isunaffected. You are asked to show this in Exercise C.6g. As a measure of relativerisk, it makes sense that beta should have this property.

A measure of the total risk of the asset, on the other hand, is one that shoulddepend on any scaling of its returns. Such a measure is

Total systematic risk of asset i = �im�m = �im

�m. (C.24)

In Exercise C.6h, you are asked to show that a doubling of returns doubles thismeasure of total risk.

The reason for the choice of the notation � is that, comparing expression(C.23) with equation (3.8b) from our initial discussion of the regression model inChapter 3, it is apparent that beta may be interpreted as the slope coefficient in theregression model

Ri = �i + �imRm + �. (C.25)

This in turn means that, given observations over time t on the returns of asseti and the market, Rit and Rmt , beta may be estimated as an application of sim-ple regression analysis. This regression model is known as the market model(see, for example, Fama 1976, Chap. 4). At this point, the market model is just adescriptive statistical relationship between the two returns, obtained by observing


the equivalence between the expressions (C.23) and (3.8b). It turns out, however,that it is related to the CAPM, as we see in the next section.

Our study of the diversification phenomenon has led to the understanding thatthe total variability of the return on any individual asset i involves two components:systematic and idiosyncratic risk. Applying Corollaries B.1.2 and B.4.1 in AppendixB to the market model yields

V (Ri) = �2imV (Rm) + V (�) = (�im�m)2 + V (�). (C.26)

This indicates that, indeed, the total variability V (Ri) decomposes into the squareof total systematic risk (C.24) and idiosyncratic risk represented by V (�). It alsosuggests another interpretation of beta. An aggressive asset is one for which a unitincrease in the volatility of the market gives rise to a greater-than-unit increase inthe volatility of the asset; a defensive asset is affected with a less-than-unit increasein its volatility.

Finally, beta has the interesting property that it aggregates linearly. Supposewe have a portfolio of n assets having returns Ri (i = 1, . . . , n). The portfolioreturn, expected return, and variance are given by equations (C.10), (C.11), and(C.12), respectively. In particular, the last expression indicates that, if the varianceis used as the measure of risk, then the relationship between the portfolio risk �2

p

and the risks �2i of the assets individually is the rather complex one given by (C.12)

that depends on the n(n − 1)/2 covariances �ij .We have concluded, however, that the variance is not the appropriate measure

of risk. Instead beta is. Not only do the individual assets have their betas definedas �im = �im/�2

m, but so too the portfolio has its beta �pm = �pm/�2m. What is

the relationship between the portfolio beta �pm and the individual asset betas �im?In contrast to (C.12), the relationship is a linear one:

�pm =n∑

i=1

ai�im.

Exercise C.7 asks you to establish this result in the case of n = 2 assets.In Section C.4 it is concluded that individual assets embody both market and

idiosyncratic risk. Idiosyncratic risk can easily be diversified away by combiningassets into portfolios, and so investors should not be rewarded for bearing thiscomponent of risk. The bearing of systematic risk, on the other hand, should berewarded; after all, in the case of efficient portfolios the capital market line offerssuch a reward. In the case of an individual asset, the amount of this reward shoulddepend on the amount of systematic risk it embodies. We now have a measure ofthe amount of this systematic risk—beta. The next section derives the role of betain the risk premium of an individual asset.

E X E R C I S E S

C.6 Suppose the returns on asset i and the marketare scaled by a factor v, and so are now vRi

and vRm. (For example, if returns double, then

v = 2.) Use the results in Appendix B to answerthe following.

SECTION C.7 Portfolio Optimization C-27

(a) How is the return on a portfolio of the twoaffected?

(b) How is the expected return on each affected?

(c) How is the expected return on a portfolioaffected?

(d) How is the variance of each affected? Thestandard deviation?

(e) How is the covariance �im affected? The cor-relation �im?

(f) How is the variance of a portfolio affected? Thestandard deviation?

(g) Use (C.23) to prove that beta is invariant to thisscaling.

(h) Use (C.24) to prove that this measure of totalrisk is scaled by the factor v.

C.7 Consider two assets with random returns R1

and R2, expected returns E(R1) and E(R2), andstandard deviations �1 and �2. These returns arerelated by the covariance

�12 = cov(R1, R2)

= E[(R1 − E(R1))(R2 − E(R2))].

(a) Consider a portfolio consisting of proportion a

invested in R2 and 1 − a invested in R1.i. Write the expression for the return on the port-folio, Rp.

ii. Write the expression for the expected return onthe portfolio, E(Rp).

iii. Write the expression for the variance of thereturn on the portfolio, V (Rp).

(b) Suppose we adopt as a measure of risk for thesetwo assets their betas with the market portfolio:

�1 = �1m

�2m

, �2 = �2m

�2m

.

i. Consider the portfolio in part a and the covari-ance of its return Rp with the market return Rm:

�pm = cov(Rp, Rm).

Derive the relationship between �pm and the indi-vidual asset covariances �1m and �2m.

ii. Consider the beta of the portfolio with themarket, �p = �pm/�2

m. Prove that it is a linearfunction of the individual asset betas, �1 and �2.What are the coefficients in this linear combination?

....

....

....

...C.7 Portfolio Optimization

A portfolio (C.10) is on the Markowitz frontier if it offers the highest expectedreturn E(Rp) for any given variance �2

p. In this section, we ask what more can besaid by way of characterizing these optimal portfolios. Specifically, the implicationsof choosing the portfolio weights ai so as to construct a portfolio on the Markowitzfrontier are investigated.

Given a universe of n risky assets, the problem is one of choosing the ai

weights to maximize the portfolio expected return (C.11)n∑

i=1

aiE(Ri) (C.27)

for any level of variance

�2p =

n∑i=1

n∑j=1

aiaj �ij . (C.28)


This expression uses the notation �ii = �2i when i = j and is equivalent to (C.12).

The only restriction on the choice of these weights is that they must sum to 1:n∑

i=1

ai = 1. (C.29)

The method of Lagrange multipliers is the most convenient way of maximiz-ing (C.27) subject to the constraints (C.28) and (C.29). The expression for theLagrangian is

L =n∑

i=1

aiE(Ri) − 1

(n∑

i=1

ai − 1

)− 1

22

n∑

i=1

n∑j=1

aiaj �ij − �2p

,

where 1 and 12 2 are defined as the Lagrange multipliers associated with each of

the constraints.9 The first n first-order conditions are of the form10

∂L

∂ai

= E(Ri) − 1 − 2

n∑j=1

aj �ij = 0 (i = 1, . . . , n). (C.30)

How are these first-order conditions characterizing the optimal choice of port-folio weights to be interpreted? Rearranging to isolate E(Ri), and using the lawsof summation from Appendix A, yields the following derivation.

E(Ri) = 1 + 2

n∑j=1

aj �ij

= 1 + 2

n∑j=1

ajE[(Ri − E(Ri))(Rj − E(Rj ))]

= 1 + 2E

(Ri − E(Ri))

n∑j=1

aj (Rj − E(Rj ))

= 1 + 2E

(Ri − E(Ri))(

n∑j=1

ajRj − E(

n∑j=1

ajRj ))

= 1 + 2E[(Ri − E(Ri))(Rp − E(Rp))]

= 1 + 2�ip. (C.31)

This establishes the following result.

9 The introduction of the constant 1/2 is a convenience, the utility of which becomes apparent whenthe derivative is taken at the next step.

10 There are two additional first-order conditions associated with the derivatives with respect to 1 and2, but these simply reproduce the constraints (C.29) and (C.28).


Result C.5 If a portfolio is on the Markowitz frontier, a linear relationship exists between theexpected return on each asset in the portfolio and the covariance of the return onthat asset with that of the portfolio.

C.7.1 The Capital Asset Pricing ModelThe coefficients in this linear relationship are the Lagrange multipliers 1 and 2.What interpretations can be attached to these? Note first that, if asset i has zerocovariance with the portfolio, then �ip = 0 and E(Ri) = 1. One example11 ofsuch an asset is the risk-free asset; because its return rf is certain, the covarianceof this with any other random return is zero.12 Thus,

1 = rf. (C.32)

Turning to 2, Result C.5 applies to any point on the Markowitz frontier. Onesuch point is the market portfolio. Applied to the market portfolio, and using theresult (C.32), equation (C.31) becomes

E(Ri) = rf + 2�im (i = 1, . . . , n). (C.33)

This is depicted in Figure C.10, which shows that the relationship between theexpected returns of individual assets and the covariance of those returns with themarket portfolio is a linear one.13 This linear relationship applies to the expectedreturns on individual securities (in contrast to the capital market line, which appliesonly to efficient portfolios) and must characterize equilibrium in the market forthose securities; it is therefore called the security market line.

F IGURE C.10

The security marketline.

E(Ri) � rf 2�i m

�2m

Ri

rf

E(Rm)

cov(Ri, Rm)

Slope � 2

�2m

E(Rm) � rf�

11 In general, any zero-beta asset qualifies. This observation is the basis for the zero-beta version ofthe CAPM derived by Black (1972), which does not require the existence of a riskless asset.

12 Recall the result in Section B.10.3 that the covariance of any random variable with a constant iszero.

13 This figure is a slightly different diagrammatic interpretation of the SML equation (C.34) from thatin Figure 5.7. The only difference is that �im = cov(Ri , Rm) is plotted on the horizontal axis insteadof �i = cov(Ri , Rm)/V (Rm), with the slope of the line redefined accordingly.


As shown in the figure, one point on the security market line pertains to themarket portfolio itself; the expected return on the market, E(Rm), is linearly relatedto the covariance of the return on the market portfolio with itself, cov(Rm, Rm) =V (Rm) = �2

m:

E(Rm) = rf + 2�2m.

Solving for 2 yields

2 = E(Rm) − rf

�2m

.

Substituting this back into the general relationship (C.33) and recalling thedefinition of beta from (C.22),

E(Ri) = rf + �im

�2m

(E(Rm) − rf) = rf + �im(E(Rm) − rf). (C.34)

It is this equation that is known as the capital asset pricing model.14 What does it tellus? Note first that the commonsense meaning of the model is very simple: It is justa precise statement of the relationship between an asset’s expected return and riskthat is expressed earlier as equation (C.4). It is more precise in two respects. First,the appropriate measure of the relative risk of asset i is �im. Second, the nature ofthe functional relationship between the expected return of the asset E(Ri) and itssystematic risk �im has been established; most notably, it is linear.

One way of interpreting this linear relationship is by rewriting (C.34) as

E(Ri) − rf = �im

�m· E(Rm) − rf

�m. (C.35)

In analogy with the intuition we have given to the capital market line, the CAPMdescribes the expected return on an asset, E(Ri) in (C.34), as consisting of twocomponents: (1) the “price of time” rf and (2) the risk premium (C.35), which iscompensation for bearing systematic risk. The risk premium is the product of twofactors. The first factor is the total systematic risk of asset i given by (C.24); thesecond is the price of systematic risk derived in Section C.5.2 in connection withthe capital market line (C.20). Hence, the CAPM states that

Risk premium of asset i = (Systematic risk of asset i)

×(Price of systematic risk). (C.36)

Thus we see that investors are indeed rewarded for bearing systematic risk. Thebearing of idiosyncratic risk is not rewarded because it can be diversified away.An asset with no systematic risk (�im = 0) would earn only the risk-free rateof return, even though its return may be uncertain, because this random return isentirely idiosyncratic.

Suppose an asset exists having an expected return greater than that suggestedby the CAPM. Then it would be a very attractive one to investors, who have an

14 More precisely, the Sharpe-Lintner version of that model.


incentive to seek out such assets. Their desire to purchase it will drive up its price(that is, Pt−1 in (C.1)) until its expected return is reduced to a level consistent withthe CAPM (C.34). It is in this sense that the CAPM explains the determination ofreturns in terms of the equilibration of the supply and demand for assets based ontheir risk characteristics.

The CAPM equation (C.34) explains the determination of a risky asset’sexpected return. In what sense is it an “asset pricing model”? Rearranging equation(C.3) into a present value form yields

Pt−1 = E(Pt)

1 + E(Rit ),

which is analogous to the earlier present value expression (C.2). Substituting in(C.34),

Pt−1 = E(Pt)

1 + rf + �im(E(Rm) − rf)

Thus, the current price of a risky asset is determined as a discount from its expectedfuture price, analogous to the present value formula (C.2), which applies to ariskless asset. The discount factor includes not only the “price of time” rf but alsothe risk premium (C.35). Other things being equal, a risky asset sells at largerdiscount than a riskless asset does.

C.7.2 Empirical ImplementationLike any economic model, the CAPM is a description of ex ante behavior; inparticular, it is a prediction about relationships between expected returns. It statesthat rational behavior on the part of investors should result in assets being priced insuch a way that, on average, investors receive higher expected returns for bearingadditional systematic risk. Ex post, of course, realized returns may be very differentfrom what was expected, and it is not necessarily the case that (C.34) holds forobserved historical data.15

Nevertheless, we expect that, for historical data over time t on the returnsof a representative security Rit , the market Rmt , and the risk-free rate rft , theestimation of a regression model analogous to (C.34) will yield a useful estimateof �im. (Useful in the sense that securities having large estimated betas should alsoexperience large realized returns on average, as predicted by the CAPM.)

More formally, an empirical counterpart to (C.34) may be derived as follows.First, recall the market model (C.25):

Rit = �i + �imRmt + �t .

Taking the expected value,

E(Rit ) = �i + �imE(Rmt ).

15 Malkiel (1996, Chap. 10) provides some amusing examples of the inconsistencies that can arisebetween the ex ante prediction of the CAPM and the historical experience of certain securities or timeperiods.


Substituting this back into the market model so as to eliminate �i yields

Rit = E(Rit ) + �im(Rmt − E(Rmt )) + �t .

Finally, substituting the CAPM (C.34) into the right-hand side and rearrangingleads to

Rit − rft = �im(Rmt − rft ) + �t , (C.37)

which is the sought-after empirical counterpart to (C.34).Note that if the risk-free rate were constant over time—as might be true of,

say, rates paid on bank deposits during some periods— then this regression modelcould be rewritten as

Rit = (1 − �im)rf + �imRmt + �t .

This is just the market model (C.25) with �i = (1 − �im)rf.What does the CAPM give us that the market model does not? For the more

typical situation in which the risk-free rate varies over time, the estimation of(C.37) implies the testable restriction of a zero intercept.16 After all, if inclusionof an intercept in the estimation of (C.37) yields an estimate that is significantlydifferent from zero, this suggests that investors are earning higher or lower expectedreturns than are justified by the systematic risk they are assuming, contradicting(C.36). The empirical exercises in the appendix to Chapter 5 provide you with anopportunity to estimate the CAPM and test this hypothesis.

E X E R C I S E S

C.8 In our development of the model, we beganby considering portfolio optimization based on n

risky assets and subsequently introduced the risk-free asset into the analysis. Suppose instead that webegin by including the risk-free asset as an (n+1)stasset with portfolio weight an+1.

(a) What is the appropriate expression for thereturn on the portfolio, analogous to (C.10).

(b) What is the appropriate expression for theexpected return on the portfolio, analogous to(C.11).

(c) What is the appropriate expression for therestriction on the portfolio weights, analogous to(C.29).

(d) What is the appropriate expression for the

variance of the return on the portfolio, analogous to(C.28). (Hint: Use Corollary B.2.2 in Appendix B.)

(e) Set up the appropriate Lagrangian function forthis optimization problem.

(f) Do the derivatives of the Lagrangian withrespect to the ai (i = 1, . . . , n) differ from (C.30)?

(g) Consider the derivative of the Lagrangianwith respect to an+1. What interpretation of 1

is revealed?

C.9 If rft is nonrandom, show that the decom-position (C.26) of total risk into its systematicand idiosyncratic components follows from (C.37).Which law from Appendix B is the key to yourconclusion?

16 There is no inconsistency between a rate of return varying over time and yet being nonrandom.The rate on 30-day treasury bills varies with each issue but, once purchased, the yield-to-maturity isessentially riskless.

SECTION C.8 Further Reading C-33

....

....

....

...C.8 Further Reading

The literature on the CAPM, and on asset pricing models generally, is vast. In part,this is because of the alternative versions of the model that arise from attempts torelax some of the strong assumptions underlying it. In addition, there has been muchcontroversy over its empirical implementation, in particular the question of whatconstitutes an appropriate test of the model. An excellent starting point is Malkiel(1996, Chaps. 9–10), who offers a realistic assessment of the CAPM accessibleto the popular reader interested in practical investment matters. A more systematictreatment, less analytical than that presented here, is Houthakker and Williamson(1996, Chap. 6).

There are many textbook developments of portfolio theory and the CAPM.Copeland and Weston (1988) and Bodie, Kane, and Marcus (1989) are examplesof MBA-level finance textbooks that include the material. Levy and Sarnat (1984)and Elton and Gruber (1987) are particularly exhaustive in their treatment. All offermore coverage of applications of the CAPM than is possible here.

Presentations of the material oriented toward undergraduates include Sharpe(1985), Berndt (1991, Chap. 2), and Varian (1996, Chap. 13); the last provides abrief intuitive development of the CAPM explicitly in the context of the theory ofchoice under uncertainty. Berndt provides a data set, which has since come to bewidely used for pedagogical purposes, and extensive empirical exercises.

Recent advanced treatments are Dumas and Allaz (1996) and Campbell, Lo, andMacKinlay (1997); the latter focus on contemporary econometric testing method-ology. Jagannathan and McGrattan (1995) assess the state of the empirical testingliterature. For a survey of asset pricing theory and tests and a collection of classicpapers, see Grauer (2003).

Credit Figure C-5, page C-17. (Harvey, C.R. (1991) “The World Price of Covariance Risk,” Journal of Finance 46, 111—158, Blackwell Publishing. Reprinted with the permission of Blackwell Publishing.)

application to portfolio theory and the capital asset ...econ446/econ446/march30/appendix c.pdf ·...

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