Investments, 8th 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

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• Reduces the number of inputs for diversification

• Easier for security analysts to specialize

Advantages of the Single Index Model

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ßi = index of a securities’ particular return to the factor

m = Unanticipated movement related to security returns

ei = Assumption: a broad market index like the S&P 500 is the common factor.

Single Factor Model

( )i i i ir E r m e

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Single-Index Model

• Regression Equation:

• Expected return-beta relationship:

( ) ( ) ( )t i t M iR t R t e t

( ) ( )i i i ME R E R

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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 ie

2( , )i j i j MCov r r

2 2 2

( , ) ( , ) ( , )i j M i M j Mi j i M j M

i j i M j M

Corr r r Corr r r xCorr r r

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Index Model and Diversification

• Portfolio’s variance:

• Variance of the equally weighted portfolio of firm-specific components:

• When n gets large, becomes negligible

222 2

1

1 1( ) ( ) ( )

n

P ii

e e en n

2 2 2 2 ( )P P M Pe

2 ( )Pe

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Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient

βp in the Single-Factor Economy

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

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Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic

Line (SCL) for HP

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

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Figure 8.4 Excess Returns on Portfolio Assets

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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

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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

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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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Optimal Risky Portfolio of the Single-Index Model

• Maximize the Sharpe ratio

– Expected return, SD, and Sharpe ratio:1 1

1 1

12 21 1 1

2 2 2 2 2 22

1 1

( ) ( ) ( )

( ) ( )

( )

n n

P P M P i i M i ii i

n n

P P M P M i i i ii i

PP

P

E R E R w E R w

e w w e

E RS

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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*

01 (1 )A

AA A

ww

w

* 01,A A Aw w

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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):

22 2

( )A

P MAe

s s

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

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Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance

Models

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

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Table 8.4 Industry Betas and Adjustment Factors