CHAPTER 8
Index Models
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McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
8-2
Advantages of the Single Index
Model
• Reduces the number of inputs for
diversification
• Easier for security analysts to specialize
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Single Factor Model
ri  E (ri )  i m  ei
βi = response of an individual security’s return
to the common factor, m. Beta measures
systematic risk.
m = a common macroeconomic factor that
affects all security returns. The S&P 500 is
often used as a proxy for m.
ei = firm-specific surprises
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Single-Index Model
• Regression Equation:
Ri t    i  i RM t   ei t 
• Expected return-beta relationship:
E Ri    i  i E RM 
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Single-Index Model
• Risk and covariance:
– Variance = Systematic risk and Firmspecific risk:
 i2  i2 M2   2 (ei )
– Covariance = product of betas x market
index risk:
Cov(ri , rj )  i  j M2
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Single-Index Model
• Correlation = product of correlations with
the market index
i  j M2 i M2  j M2
Corr (ri , rj ) 

 Corr (ri , rM ) xCorr (rj , rM )
 i j
 i M  j M
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Index Model and Diversification
• 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, σ2(ep) becomes
negligible and firm specific risk is diversified
away.
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Figure 8.1 The Variance of an Equally Weighted
Portfolio with Risk Coefficient βp
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Figure 8.2 Excess Returns on HP and
S&P 500
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Figure 8.3 Scatter Diagram of HP, the S&P 500, and
HP’s Security Characteristic Line (SCL)
RHP t    HP   HP RS &P 500 t   eHP t 
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Table 8.1 Excel Output: Regression
Statistics for the SCL of Hewlett-Packard
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Table 8.1 Interpretation
• Correlation of HP with the S&P 500 is 0.7238.
• The model explains about 52% of the variation
in HP.
• HP’s alpha is 0.86% per month(10.32%
annually) but it is not statistically significant.
• HP’s beta is 2.0348, but the 95% confidence
interval is 1.43 to 2.53.
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Figure 8.4 Excess Returns on Portfolio
Assets
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Alpha and Security Analysis
1. Use macroeconomic analysis to estimate the
risk premium and risk of the market index.
2. Use statistical analysis to estimate the beta
coefficients of all securities and their residual
variances, σ2 (ei).
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Alpha and Security Analysis
3. Establish the expected return of each
security absent any contribution from
security analysis.
4. Use security analysis to develop private
forecasts of the expected returns for each
security.
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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
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8-17
Optimal Risky Portfolio of the
Single-Index 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
2
n 1
n 1




2 2
2
2
2 2
 P    P M   (eP )    M   wi  i    wi  (ei ) 
 i 1
  i 1

E ( RP )
SP 
1
2
1
2
P
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Optimal Risky Portfolio of the
Single-Index Model
• Combination of:
– Active portfolio denoted by A
– Market-index portfolio, the passive
portfolio denoted by M
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Optimal Risky Portfolio of the
Single-Index Model
Modification of active portfolio position:
0
w
*
A
wA 
1  (1   A ) wA0
When
 A  1, w  w
*
A
0
A
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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
  


s P s M  (eA ) 
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The Information Ratio
• The contribution of the active portfolio
depends on the ratio of its alpha to its
residual standard deviation.
• The information ratio measures the extra
return we can obtain from security
analysis.
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Figure 8.5 Efficient Frontiers with the
Index Model and Full-Covariance Matrix
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Table 8.2 Portfolios from the Single-Index
and Full-Covariance Models
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Is the Index Model Inferior to the
Full-Covariance Model?
• Full Markowitz model may be better in principle,
but
– Using the full-covariance matrix invokes
estimation risk of thousands of terms.
– Cumulative errors may result in a portfolio that
is actually inferior to that derived from the
single-index model.
– The single-index model is practical and
decentralizes macro and security analysis.
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Beta Book: Industry Version of the
Index Model
• Use 60 most recent months of price data
• Use S&P 500 as proxy for M
• Compute total returns that ignore
dividends
• Estimate index model without excess
returns:
*
r  a  brm  e
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Beta Book: Industry Version of the
Index Model
Adjust beta
because:
• The average beta over
all securities is 1. Thus,
our best forecast of the
beta would be that it is 1.
• Also, firms may become
more “typical” as they
age, causing their betas
to approach 1.
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Table 8.4 Industry Betas and Adjustment
Factors
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