### Ch. 8

```CHAPTER 8
Index Models
Investments, 8th edition
Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin
Advantages of the Single Index Model
• Reduces the number of inputs for
diversification
• Easier for security analysts to specialize
8-2
Single Factor Model
ri  E (ri )  i m  ei
factor
m = Unanticipated movement related to security
returns
ei = Assumption: a broad market index like the
S&P 500 is the common factor.
8-3
Single-Index Model
• Regression Equation:
Rt (t )  i   t RM (t )  ei (t )
• Expected return-beta relationship:
E ( Ri )  i  i E ( RM )
8-4
Single-Index Model Continued
• Risk and covariance:
– Total risk = Systematic risk + Firm-specific
risk:  i2  i2 M2   2 (ei )
– Covariance = product of betas x market index
risk:
2
Cov(ri , rj )  i  j M
– Correlation = product of correlations with the
market index
Corr (ri , rj ) 
i  j M2
 i j

i M2  j M2
 i M  j M
 Corr (ri , rM ) xCorr (rj , rM )
8-5
Index Model and Diversification
• Portfolio’s variance:
  
2
P
2
P
  (eP )
2
M
2
• Variance of the equally weighted portfolio
of firm-specific components:
2
1 2
1 2
 (eP )      (ei )   (e)
n
i 1  n 
n
2
• When n gets large,
negligible
 (eP )
2
becomes
8-6
Figure 8.1 The Variance of an Equally
Weighted Portfolio with Risk Coefficient
βp in the Single-Factor Economy
8-7
Figure 8.2 Excess Returns on HP and
S&P 500 April 2001 – March 2006
8-8
Figure 8.3 Scatter Diagram of HP, the
S&P 500, and the Security Characteristic
Line (SCL) for HP
8-9
Table 8.1 Excel Output: Regression
Statistics for the SCL of Hewlett-Packard
8-10
Figure 8.4 Excess Returns on Portfolio
Assets
8-11
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, σ2 ( e i )
• Developed from security analysis
8-12
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
• Helps determine whether security is a good
8-13
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
8-14
Optimal Risky Portfolio of the SingleIndex Model
• Maximize the Sharpe ratio
– Expected return, SD, and Sharpe ratio:
n 1
n 1
i 1
i 1
E ( RP )   P  E ( RM )  P   wi i  E ( RM ) wi i
1
2
n 1
n 1

2


2
2
2
2
  (eP )  2   M   wi  i    wi  (ei ) 
i 1
 i 1



1
 P    P2 M2
SP 
E ( RP )
P
8-15
Optimal Risky Portfolio of the SingleIndex 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:
0
w 
*
A
– When
wA
1  (1   A ) w
0
A
 A  1, w  w
*
A
0
A
8-16
The Information Ratio
• The Sharpe ratio of an optimally constructed
risky portfolio will exceed that of the index
portfolio (the passive strategy):
 A 


s P s M   (e ) 
A 

2
2
2
8-17
Figure 8.5 Efficient Frontiers with the
Index Model and Full-Covariance Matrix
8-18
Table 8.2 Comparison of Portfolios from
the Single-Index and Full-Covariance
Models
8-19
Table 8.3 Merrill Lynch, Pierce, Fenner &
Smith, Inc.: Market Sensitivity Statistics
8-20
Table 8.4 Industry Betas and Adjustment
Factors
8-21
```