Chapter Eight
Index Models
INVESTMENTS | BODIE, KANE, MARCUS
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
Chapter Overview
• Advantages of a single-factor model
• Risk decomposition
• Systematic vs. firm-specific
• Single-index model and its estimation
• Optimal risky portfolio in the index model
• Index model vs. Markowitz procedure
8-2
INVESTMENTS | BODIE, KANE, MARCUS
A Single-Factor Market
• Advantages
• Reduces the number of inputs for diversification
• Easier for security analysts to specialize
• Model
ri  E ri   i m  ei
• βi = response of an individual security’s return to
the common factor, m; measure of systematic risk
• m = a common macroeconomic factor
• ei = firm-specific surprises
8-3
INVESTMENTS | BODIE, KANE, MARCUS
Single-Index Model
• Regression equation:
Ri t    i  i RM t   ei t 
• Expected return-beta relationship:
E Ri    i  i E RM 
8-4
INVESTMENTS | BODIE, KANE, MARCUS
Single-Index Model
• Variance = Systematic risk + Firm-specific risk:
      ei 
2
i
2
i
2
M
2
• Covariance = Product of betas × Market index
risk:
Cov  ri , rj   i  j
2
M
Cov(ri , rj )  i  j M2
8-5
INVESTMENTS | BODIE, KANE, MARCUS
Single-Index Model
• Correlation = Product of correlations with the
market index
i  j M2 i M2  j M2
Corr  ri , rj  

 i j
 i M  j M
 Corr  ri , rM   Corr  rj , rM 
8-6
INVESTMENTS | BODIE, KANE, MARCUS
Index Model and Diversification
• Variance of the equally-weighted portfolio of
firm-specific components:
2
1 2
1 2
 e p       ei    e 
n
i 1  n 
n
2
• When n gets large, σ2(ep) becomes negligible
and firm specific risk is diversified away
8-7
INVESTMENTS | BODIE, KANE, MARCUS
Figure 8.1 The Variance of an Equally
Weighted Portfolio with Risk Coefficient βp
8-8
INVESTMENTS | BODIE, KANE, MARCUS
Figure 8.2 Excess Returns on
HP and S&P 500
8-9
INVESTMENTS | BODIE, KANE, MARCUS
Figure 8.3 Scatter Diagram of HP, the S&P
500, and HP’s SCL
RHP t    HP   HP RS &P 500 t   eHP t 
8-10
INVESTMENTS | BODIE, KANE, MARCUS
Table 8.1 Excel Output: Regression Statistics
for the SCL of Hewlett-Packard
8-11
INVESTMENTS | BODIE, KANE, MARCUS
Table 8.1 Interpreting the Output
• 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
8-12
INVESTMENTS | BODIE, KANE, MARCUS
Figure 8.4 Excess Returns on Portfolio Assets
8-13
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• 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).
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.
8-14
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• Single-Index Model Input List
1. Risk premium on the S&P 500 portfolio
2. Estimate of the SD of the S&P 500 portfolio
3. n sets of estimates of
• Beta coefficient
• Stock residual variances
• Alpha values
8-15
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• Optimal risky portfolio in the single-index
model
• Expected return, SD, and Sharpe ratio:
8-16
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• Optimal risky portfolio in the single-index
model is a combination of
• Active portfolio, denoted by A
• Market-index portfolio, the passive
portfolio, denoted by M
8-17
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• Optimal risky portfolio in the single-index
model
• Modification of active portfolio position:
0
*
wA 
when
wA
1  (1   A ) wA
0
 A  1, w  w
*
A
0
A
 A  1, wA  wA
*
8-18
0
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• 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    (eA ) 
2
8-19
2
2
INVESTMENTS | BODIE, KANE, MARCUS
Portfolio Construction and
the Single-Index Model
• 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
8-20
INVESTMENTS | BODIE, KANE, MARCUS
Figure 8.5 Efficient Frontiers with the
Index Model and Full-Covariance Matrix
8-21
INVESTMENTS | BODIE, KANE, MARCUS
Table 8.2 Portfolios from the Single-Index
and Full-Covariance Models
8-22
INVESTMENTS | BODIE, KANE, MARCUS
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 singleindex model
• The single-index model is practical and
decentralizes macro and security analysis
8-23
INVESTMENTS | BODIE, KANE, MARCUS
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
8-24
*
INVESTMENTS | BODIE, KANE, MARCUS
Industry Version of the Index Model
• Adjust beta because
• The average beta over all securities is 1; thus, the
best forecast of the beta would be that it is 1
• Firms may become more “typical” as they age,
causing their betas to approach 1
8-25
INVESTMENTS | BODIE, KANE, MARCUS
Table 8.4 Industry Betas and
Adjustment Factors
8-26
INVESTMENTS | BODIE, KANE, MARCUS