D D M

where is the correlation coefficient between RD and RM . Its square measures the ratio of ex-

plained variance to total variance, that is, the proportion of total variance that can be attributed

to market fluctuations. But if beta is negative, so is the correlation coefficient, an indication

that the explanatory and dependent variables are expected to move in opposite directions.

At the extreme, when the correlation coefficient is either 1.0 or 1.0, the security return is

fully explained by the market return, that is, there are no firm-specific effects. All the points

of the scatter diagram will lie exactly on the line. This is called perfect correlation (either pos-

itive or negative); the return on the security is perfectly predictable from the market return. A

large correlation coefficient (in absolute value terms) means systematic variance dominates

the total variance; that is, firm-specific variance is relatively unimportant. When the corre-

lation coefficient is small (in absolute value terms), the market factor plays a relatively un-

important part in explaining the variance of the asset, and firm-specific factors predominate.

5

Note that the average beta of all securities will be 1.0 only when we compute a weighted average of betas (using mar-

ket values as weights), since the stock market index is value weighted. We know from Chapter 5 that the distribution

of securities by market value is not symmetric: There are relatively few large corporations and many more smaller

ones. Thus, if you were to take a randomly selected sample of stocks, you should expect smaller companies to domi-

nate. As a result, the simple average of the betas of individual securities, when computed against a value-weighted

index such as the S&P 500, will be greater than 1.0, pushed up by the tendency for stocks of low-capitalization com-

panies to have betas greater than 1.0.

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Fifth Edition

197

6 Efficient Diversification

<

6. Interpret the eight scatter diagrams of Figure 6.12 in terms of systematic risk,

Concept

diversifiable risk, and the intercept.

CHECK

R1 R3 R4

R2

*

*

**

* *

*

**

* *

* *

** **

*

*

*

* **

* RM * RM RM

RM

*

* * * * *

*

R5 R7 R8

R6

*

* *

*

*

*

* *

*

*

*

*

** *

*

RM RM RM

RM

* *

* * * *

*

* * *

* *

* *

*

*

F I G U R E 6.12

Various scatter diagrams

The direct way to calculate the slope and intercept of the characteristic lines for ABC and XYZ

is from the variances and covariances. Here, we use the Data Analysis menu of Excel to obtain

EXAMPLE 6.4

the covariance matrix in the following spreadsheet.

The slope coefficient for ABC is given by the formula Estimating

Cov(RABC, RMarket) the Index Model

773.31

1.156 Using Historical

ABC

Var(RMarket) 684.01

Data

The intercept for ABC is

Average(RABC) Average(RMarket)

ABC ABC

15.20 1.156 9.40 4.33

Therefore, the security characteristic line of ABC is given by

RABC 1.156 RMarket

4.33

This result also can be obtained by using the “Regression” command from Excel™s Data

Analysis menu, as we show at the bottom of the spreadsheet. The minor differences between

the direct regression output and our calculations above are due to rounding error.

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Fifth Edition

198 Part TWO Portfolio Theory

WEBMA STER

Risk Comparison

Go to www.morningstar.com and select the tab entitled Funds. In the dialog box for

selecting a particular fund, type Fidelity Select and hit the Go button. This will list all of

the Fidelity Select funds. Select the Fidelity Select Multimedia Fund. Find the fund™s top

25 individual holdings from the displayed information. The top holdings are found in

the Style section. Identify the top five holdings using the ticker symbol.

Once you have obtained this information, go to www.financialanalyses.com. From

the Site menu, select the Forecast and Analysis tab and then select the fund™s

Scorecard tab. You will find a dialog box that allows you to search for funds or

individual stocks. You can enter the name or ticker for each of the individual stocks and

the fund. Compare the risk ranking of the individual securities with the risk ranking of

the fund.

1. What factors are likely causing to the differences in the individual rankings and

the annual fund ranking?

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6 Efficient Diversification

Diversification in a Single-Factor Security Market

Imagine a portfolio that is divided equally among securities whose returns are given by the

single-index model in Equation 6.6. What are the systematic and nonsystematic (firm-

specific) variances of this portfolio?

The beta of the portfolio is the simple average of the individual security betas, which we

denote . Hence, the systematic variance equals 2 2 . This is the level of market risk in Fig-

PM

ure 6.1B. The market variance ( 2 ) and the market sensitivity of the portfolio ( P) determine

M

the market risk of the portfolio.

The systematic component of each security return, i RM , is fully determined by the mar-

ket factor and therefore is perfectly correlated with the systematic part of any other security™s

return. Hence, there are no diversification effects on systematic risk no matter how many se-

curities are involved. As far as market risk goes, a single-security portfolio with a small beta

will result in a low market-risk portfolio. The number of securities makes no difference.

It is quite different with firm-specific or unique risk. If you choose securities with small

residual variances for a portfolio, it, too, will have low unique risk. But you can do even better

simply by holding more securities, even if each has a large residual variance. Because the

firm-specific effects are independent of each other, their risk effects are offsetting. This is the

insurance principle applied to the firm-specific component of risk. The portfolio ends up with

a negligible level of nonsystematic risk.

In sum, when we control the systematic risk of the portfolio by manipulating the average

beta of the component securities, the number of securities is of no consequence. But in

the case of nonsystematic risk, the number of securities involved is more important than the

firm-specific variance of the securities. Sufficient diversification can virtually eliminate firm-

specific risk. Understanding this distinction is essential to understanding the role of diversifi-

cation in portfolio construction.

We have just seen that when forming highly diversified portfolios, firm-specific risk be-

comes irrelevant. Only systematic risk remains. We conclude that in measuring security risk

for diversified investors, we should focus our attention on the security™s systematic risk. This

means that for diversified investors, the relevant risk measure for a security will be the secu-

rity™s beta, , since firms with higher have greater sensitivity to broad market disturbances.

As Equation 6.7 makes clear, systematic risk will be determined both by market volatility, 2 , M

and the firm™s sensitivity to the market, .

<

7. a. What is the characteristic line of XYZ in Example 6.4? Concept

b. Does ABC or XYZ have greater systematic risk?

CHECK

c. What percent of the variance of XYZ is firm-specific risk?

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SUMMARY

• The expected rate of return of a portfolio is the weighted average of the component asset

expected returns with the investment proportions as weights.

• The variance of a portfolio is a sum of the contributions of the component-security

variances plus terms involving the correlation among assets.

• Even if correlations are positive, the portfolio standard deviation will be less than the

weighted average of the component standard deviations, as long as the assets are not

perfectly positively correlated. Thus, portfolio diversification is of value as long as assets

are less than perfectly correlated.

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Fifth Edition

200 Part TWO Portfolio Theory

• The contribution of an asset to portfolio variance depends on its correlation with the other

assets in the portfolio, as well as on its own variance. An asset that is perfectly negatively

correlated with a portfolio can be used to reduce the portfolio variance to zero. Thus, it

can serve as a perfect hedge.

• The efficient frontier of risky assets is the graphical representation of the set of portfolios

that maximizes portfolio expected return for a given level of portfolio standard deviation.

Rational investors will choose a portfolio on the efficient frontier.

• A portfolio manager identifies the efficient frontier by first establishing estimates for the

expected returns and standard deviations and determining the correlations among them.

The input data are then fed into an optimization program that produces the investment

proportions, expected returns, and standard deviations of the portfolios on the efficient

frontier.

• In general, portfolio managers will identify different efficient portfolios because of

differences in the methods and quality of security analysis. Managers compete on the

quality of their security analysis relative to their management fees.

• If a risk-free asset is available and input data are identical, all investors will choose the

same portfolio on the efficient frontier, the one that is tangent to the CAL. All investors

with identical input data will hold the identical risky portfolio, differing only in how

much each allocates to this optimal portfolio and to the risk-free asset. This result is

characterized as the separation principle of portfolio selection.

• The single-index representation of a single-factor security market expresses the excess rate

of return on a security as a function of the market excess return: Ri i RM ei .

i

This equation also can be interpreted as a regression of the security excess return on the

market-index excess return. The regression line has intercept i and slope i and is called

the security characteristic line.

• In a single-index model, the variance of the rate of return on a security or portfolio can be

decomposed into systematic and firm-specific risk. The systematic component of variance

equals 2 times the variance of the market excess return. The firm-specific component is

the variance of the residual term in the index model equation.

• The beta of a portfolio is the weighted average of the betas of the component securities.

A security with negative beta reduces the portfolio beta, thereby reducing exposure to

market volatility. The unique risk of a portfolio approaches zero as the portfolio

becomes more highly diversified.

KEY beta, 193 investment opportunity security characteristic

TERMS diversifiable risk, 171 set, 178 line, 195

efficient frontier, 189 market risk, 170 separation property, 192

excess return, 193 nondiversifiable risk, 170 systematic risk, 170

factor model, 192 nonsystematic risk, 171 unique risk, 171

firm-specific risk, 171 optimal risky

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