Upload
junjun-santos
View
162
Download
5
Tags:
Embed Size (px)
DESCRIPTION
Citation preview
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