Investments, 8th editionBodie, Kane and Marcus

Slides by Susan HineSlides by Susan Hine

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

CHAPTER 7CHAPTER 7 Optimal Risky Optimal Risky PortfoliosPortfolios

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Diversification and Portfolio Risk

• Market risk– Systematic or nondiversifiable

• Firm-specific risk– Diversifiable or nonsystematic

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Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

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Figure 7.2 Portfolio Diversification

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Covariance and Correlation

• Portfolio risk depends on the correlation between the returns of the assets in the portfolio

• Covariance and the correlation coefficient provide a measure of the way returns two assets vary

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Two-Security Portfolio: Return

Portfolio Return Bond Weight

Bond Return Equity Weight

Equity Return

p D ED E

P

D

D

E

E

r

rwrwr

w wr r

( ) ( ) ( )p D D E EE r w E r w E r

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= Variance of Security D

= Variance of Security E

= Covariance of returns for Security D and Security E

Two-Security Portfolio: Risk

2 2 2 2 2 2 ( , ) P D D E E D E D Ew w w w Cov r r

2D

2E

( , )D ECov r r

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Two-Security Portfolio: Risk Continued

• Another way to express variance of the portfolio:

2 ( , ) ( , ) 2 ( , )P D D D D E E E E D E D Ew w Cov r r w w Cov r r w w Cov r r

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D,E = Correlation coefficient of returns

Cov(rD,rE) = DEDE

D = Standard deviation of returns for Security DE = Standard deviation of returns for Security E

Covariance

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Range of values for 1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

Correlation Coefficients: Possible Values

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Table 7.1 Descriptive Statistics for Two Mutual Funds

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2p = w1

212 + w2

212

+ 2w1w2 Cov(r1,r2)

+ w323

2

Cov(r1,r3)+ 2w1w3

Cov(r2,r3)+ 2w2w3

Three-Security Portfolio

1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r

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Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

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Table 7.3 Expected Return and Standard Deviation with Various Correlation

Coefficients

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Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

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Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

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Minimum Variance Portfolio as Depicted in Figure 7.4

• Standard deviation is smaller than that of either of the individual component assets

• Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

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Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

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• The relationship depends on the correlation coefficient

• -1.0 < < +1.0• The smaller the correlation, the greater the

risk reduction potential• If = +1.0, no risk reduction is possible

Correlation Effects

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Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two

Feasible CALs

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The Sharpe Ratio

• Maximize the slope of the CAL for any possible portfolio, p

• The objective function is the slope:

( )P fP

P

E r rS

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Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal

CAL and the Optimal Risky Portfolio

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Figure 7.8 Determination of the Optimal Overall Portfolio

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Figure 7.9 The Proportions of the Optimal Overall Portfolio

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Markowitz Portfolio Selection Model

• Security Selection– First step is to determine the risk-return

opportunities available– All portfolios that lie on the minimum-

variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

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Figure 7.10 The Minimum-Variance Frontier of Risky Assets

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Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

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Capital Allocation and the Separation Property

• The separation property tells us that the portfolio choice problem may be separated into two independent tasks– Determination of the optimal risky portfolio

is purely technical– Allocation of the complete portfolio to T-

bills versus the risky portfolio depends on personal preference