Investments, 8th edition

Bodie, 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

7-2

Diversification and Portfolio Risk

• Market risk– Systematic or nondiversifiable

• Firm-specific risk– Diversifiable or nonsystematic

7-3

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

7-4

Figure 7.2 Portfolio Diversification

7-5

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

7-6

Two-Security Portfolio: Return

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED E

P

D

D

E

E

r

r

w

r

w

r

w wr r

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

7-7

= 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 Cov r r

2D

2E

( , )D ECov r r

wE

7-8

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

7-9

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

7-10

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

7-11

Table 7.1 Descriptive Statistics for Two Mutual Funds

7-12

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

7-13

Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

7-14

Table 7.3 Expected Return and Standard Deviation with Various Correlation

Coefficients

7-15

Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

7-16

Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

7-17

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

7-18

Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

7-19

• 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

7-20

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

Feasible CALs

7-21

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

7-22

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

CAL and the Optimal Risky Portfolio

7-23

Figure 7.8 Determination of the Optimal Overall Portfolio

7-24

Figure 7.9 The Proportions of the Optimal Overall Portfolio

7-25

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

7-26

Figure 7.10 The Minimum-Variance Frontier of Risky Assets

7-27

Markowitz Portfolio Selection Model Continued

• We now search for the CAL with the highest reward-to-variability ratio

7-28

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

7-29

Markowitz Portfolio Selection Model Continued

• Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

2

1 1

( , )n n

P i j i ji j

ww Cov r r

7-30

Figure 7.12 The Efficient Portfolio Set

7-31

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

7-32

Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

7-33

The Power of Diversification

• Remember:

• If we define the average variance and average covariance of the securities as:

• We can then express portfolio variance as:

2

1 1

( , )n n

P i j i ji j

ww Cov r r

2 21 1P

nCov

n n

2 2

1

1 1

1

1( , )

( 1)

n

ii

n n

i jj ij i

n

Cov Cov r rn n

7-34

Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and

Uncorrelated Universes

7-35

Risk Pooling, Risk Sharing and Risk in the Long Run

• Consider the following:

1 − p = .999

p = .001Loss: payout = $100,000

No Loss: payout = 0

7-36

Risk Pooling and the Insurance Principle

• Consider the variance of the portfolio:

• It seems that selling more policies causes risk to fall

• Flaw is similar to the idea that long-term stock investment is less risky

2 21P n

7-37

Risk Pooling and the Insurance Principle Continued

• When we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:

2 2 2

( ) ( )

( ) ( )

( )

E n nE

Var n n Var n

SD n n

7-38

Risk Sharing

• What does explain the insurance business?

– Risk sharing or the distribution of a fixed amount of risk among many investors