index models.pptffffffffffffffffffffffffffff

ffffffffffffffffff

1. Investments, 8
th
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 8CHAPTER 8
Index ModelsIndex Models

2. 8-2
•Reduces the number of inputs for
diversification
•Easier for security analysts to specialize
Advantages of the Single Index Model

3. 8-3
ß
i = index of a securities’ particular return to the
factor
m = Unanticipated movement related to security
returns
e
i
= Assumption: a broad market index like the
S&P 500 is the common factor.
Single Factor Model
( )
i i i i
r Er m e  

4. 8-4
Single-Index Model
•Regression Equation:
•Expected return-beta relationship:
() () ()
t i t M i
R t R t e t   
( ) ( )
i i i M
ER ER  

5. 8-5
Single-Index Model Continued
•Risk and covariance:
–Total risk = Systematic risk + Firm-specific
risk:
–Covariance = product of betas x market index
risk:
–Correlation = product of correlations with the
market index
2 2 2 2
( )
i i M i
e    
2
( , )
i j i j M
Cov r r
2 2 2
( , ) ( , ) ( , )
i j M i M j M
i j i M j M
i j i M j M
Corr r r Corr r r xCorr r r
     
   
  

6. 8-6
Index Model and Diversification
•Portfolio’s variance:
•Variance of the equally weighted portfolio of
firm-specific components:
•When n gets large, becomes negligible
2
2
2 2
1
1 1
( ) ( ) ( )
n
P i
i
e e e
n n
  

 
 
 
 

2 2 2 2
( )
P P M P
e   
2
( )
P
e

7. 8-7
Figure 8.1 The Variance of an Equally
Weighted Portfolio with Risk Coefficient
β
p in the Single-Factor Economy

8. 8-8
Figure 8.2 Excess Returns on HP and
S&P 500 April 2001 – March 2006

9. 8-9
Figure 8.3 Scatter Diagram of HP, the
S&P 500, and the Security Characteristic
Line (SCL) for HP

10. 8-10
Table 8.1 Excel Output: Regression
Statistics for the SCL of Hewlett-Packard

11. 8-11
Figure 8.4 Excess Returns on Portfolio
Assets

12. 8-12
Alpha and Security Analysis
•Macroeconomic analysis is used to estimate
the risk premium and risk of the market index
•Statistical analysis is used to estimate the
beta coefficients of all securities and their
residual variances, σ
2
( e i )
•Developed from security analysis

13. 8-13
Alpha and Security Analysis Continued
•The market-driven expected return is
conditional on information common to all
securities
•Security-specific expected return forecasts are
derived from various security-valuation models
–The alpha value distills the incremental risk
premium attributable to private information
•Helps determine whether security is a good or
bad buy

14. 8-14
Single-Index Model Input List
•Risk premium on the S&P 500 portfolio
•Estimate of the SD of the S&P 500 portfolio
•n sets of estimates of
–Beta coefficient
–Stock residual variances
–Alpha values

15. 8-15
Optimal Risky Portfolio of the Single-
Index Model
•Maximize the Sharpe ratio
–Expected return, SD, and Sharpe ratio:
1 1
1 1
1
2
21 1 1
2 2 2 2 2 2
2
1 1
( ) ( ) ( )
( ) ( )
( )
n n
P P M P i i M i i
i i
n n
P P M P M i i i i
i i
P
P
P
E R E R w E R w
e w w e
E R
S
   
      

 
 
 
 
   
 
 
        
   

 
 

16. 8-16
Optimal Risky Portfolio of the Single-
Index Model Continued
•Combination of:
–Active portfolio denoted by A
–Market-index portfolio, the (n+1)th asset
which we call the passive portfolio and
denote by M
–Modification of active portfolio position:
–When
0
*
0
1 (1 )
A
A
A A
w
w
w

 
* 0
1,
A A A
w w 

17. 8-17
The Information Ratio
•The Sharpe ratio of an optimally constructed
risky portfolio will exceed that of the index
portfolio (the passive strategy):
2
2 2
( )
A
P M
A
e
s s


 
 
 
 

18. 8-18
Figure 8.5 Efficient Frontiers with the
Index Model and Full-Covariance Matrix

19. 8-19
Table 8.2 Comparison of Portfolios from
the Single-Index and Full-Covariance
Models

20. 8-20
Table 8.3 Merrill Lynch, Pierce, Fenner &
Smith, Inc.: Market Sensitivity Statistics

21. 8-21
Table 8.4 Industry Betas and Adjustment
Factors