Security D and Security E
INVESTMENTS | BODIE, KANE, MARCUS
7-9
Two-Security Portfolio: Risk
• Another way to express variance of the portfolio:
2 P wD wDCov(rD , rD ) wE wE Cov(rE , rE ) 2wD wE Cov(rD , rE )
INVESTMENTS | BODIE, KANE, MARCUS
7-20
Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
INVESTMENTS | BODIE, KANE, MARCUS
7-15
Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions
INVESTMENTS | BODIE, KANE, MARCUS
7-16
Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions
INVESTMENTS | BODIE, KANE, MARCUS
7-5
Figure 7.2 Portfolio Diversification
INVESTMENTS | BODIE, KANE, MARCUS
7-6
Covariance and Correlation
• Portfolio risk depends on the correlation between the returns of the assets in the portfolio
CHAPTER 7
Optimal Risky Portfolios
INVESTMENTS | BODIE, KANE, MARCUS
McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
– Systematic or nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
INVESTMENTS | BODIE, KANE, MARCUS
7-4
Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio
INVESTMENTS | BODIE, KANE, MARCUS
7-17
The Minimum Variance Portfolio
• The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation, the portfolio with least risk. • When correlation is less than +1, the portfolio standard deviation may be smaller than that of either of the individual component assets. • When correlation is -1, the standard deviation of the minimum variance portfolio is zero.
7-14
Three-Asset Portfolio
E (rp ) w1E (r1 ) w2 E (r2 ) w3 E (r3 )
2 2 2 2 2 p w1212 w2 2 w3 3 2w1w21,2 2w1w31,3 2w2 w3 2,3
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
7-25
Markowitz Portfolio Selection Model
• Security Selection – The first step is to determine the riskreturn opportunities available. – All portfolios that lie on the minimumvariance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations
3. Security selection of individual assets within each asset class
INVESTMENTS | BODIE, KANE, MARCUS
7-3
Diversification and Portfolio Risk
• Market risk
7-21
The Sharpe Ratio
• Maximize the slope of the CAL for any possible portfolio, P. • The objective function is the slope:
SP
E (rP ) rf
P
• The slope is also the Sharpe ratio.
INVESTMENTS | BODIE, KANE, MARCUS
7-26
Figure 7.10 The Minimum-Variance Frontier of Risky Assets
INVESTMENTS | BODIE, KANE, MARCUS
7-27
Markowitz Portfolio Selection Model
• We now search for the CAL with the highest reward-to-variability ratio
INVESTMENTS | BODIE, KANE, MARCUS
wE
D E
D
1 wD
INVESTMENTS | BODIE, KANE, MARCUS
7-13
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
INVESTMENTS | BODIE, KANE, MARCUS
7-8
Two-Security Portfolio: Risk
2 2 2 2 2 p wD D wE E 2wD wECovrD , rE
2 D = Variance once of Security E
CovrD , rE = Covariance of returns for
• Covariance and the correlation coefficient provide a measure of the way returns of two assets vary
INVESTMENTS | BODIE, KANE, MARCUS
7-7
Two-Security Portfolio: Return
INVESTMENTS | BODIE, KANE, MARCUS
7-12
Correlation Coefficients
• When ρDE = 1, there is no diversification
P wE E wD D
• When ρDE = -1, a perfect hedge is possible
INVESTMENTS | BODIE, KANE, MARCUS
7-22
Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio
rp rP rD rE
Portfolio Return Bond Return Equity Return
wr
D
D
wE r E
wD Bond Weight wE Equity Weight
E (rp ) wD E (rD ) wE E (rE )
INVESTMENTS | BODIE, KANE, MARCUS
INVESTMENTS | BODIE, KANE, MARCUS
7-11
Correlation Coefficients: Possible Values
Range of values for 1,2
+ 1.0 > > -1.0 If = 1.0, the securities are perfectly positively correlated If = - 1.0, the securities are perfectly negatively correlated
INVESTMENTS | BODIE, KANE, MARCUS
7-18
Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation