CHAPTER 8 -...

INVESTMENTS | BODIE, KANE, MARCUS

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

CHAPTER 8

Index Models


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

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Advantages of the Single Index Model

• Reduces the number of inputs for diversification

• Easier for security analysts to specialize

8-3

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Single Factor Model

r𝑖 = 𝐸[𝑟𝑖] + 𝛽𝑖𝑚 + 𝑒𝑖 Where:

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&P500 is often used as a proxy for 𝑚

𝑒𝑖 firm-specific surprises

8-4

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Single-Index Model

Regression Equation:

𝑅𝑖 𝑡 = 𝛼𝑖 + 𝛽𝑖𝑅𝑀 𝑡 + 𝑒𝑖 𝑡

The expectation of the residual term 𝑒𝑖 is zero, so the expected return-beta relationship is:

𝐸 𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖𝐸 𝑅𝑀

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Single-Index Model

Risk and covariance:

• Variance - Systemic risk and firm-specific risk, assume noise is uncorrelated:

• Covariance - product of betas x market index risk:

8-6

2 2 2 2 ( )i i M ie

2( , )i j i j MCov r r

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Single-Index Model - Correlation

Product of correlations with the market index:

8-7

ji

Mji

jiji rrCorr

2

, ,

MjMi

MjMi

22

MjMi rrCorrrrCorr ,,

MM

M

2

jijijijiji rrCovrrCorr /,, ,,

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Questions to test your intuition

• What is the stock’s E[𝑟] if (𝑟𝑀−𝑟𝑓) = 0 ?

• What is the responsiveness of the stock to market movements relative to 𝑟𝑓?

• What is the stock-specific component of return (not driven by the market)?

• What is the variance attributable to uncertainty of the market?

• And that attributable to firm-specific events?

8-8

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Index Model and Diversification

• Consider an Equally weighted portfolio and take the expected return 𝑅𝑃 as the average:

8-9

n

i

iMii

n

i

iP eRn

Rn

R11

11

n

i

in 1

1

n

i

MiRn 1

1

n

i

ien 1

1

PPR MPR Pe

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• The portfolio variance by definition: 𝜎𝑃2 = 𝛽𝑃

2𝜎𝑀𝑃 + 𝜎2(𝑒𝑃)

• where the market component comes from the portfolio’s sensitivity to the market:

• and the non-systemic component 𝜎2(𝑒𝑃) is the contribution of all the stocks in the portfolio.

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n

i

iPn 1

1

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• Variance of the non-systemic component of an equally weighted portfolio is (we assume all the stock-specific components are uncorrelated):

𝜎2(𝑒𝑃) = 1

𝑛

2

𝜎2(𝑒𝑖)

𝑛

1

=1

𝑛𝜎 2(𝑒)

• When n gets large, 𝜎2(𝑒𝑃) becomes negligible and firm specific risk can be diversified away.

8-11

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

8-12

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Fig 8.2 Excess Returns on HP and S&P500

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Fig 8.3 Scatter Diagram of HP, the S&P 500, and HP’s Security Characteristic Line (SCL)

8-14

tetRtR HPSPHPHPHP 500

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

8-15

(monthly)

correlation

explanatory power

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Table 8.1 Interpretation

• Correlation of HP with the S&P500 is 0.7238

• The model explains about 52% of the variation in HP

• HP’s alpha is 0.86% per month (10.32% pa), but it is not statistically significant

Q. What does it mean? Why?

• HP’s beta is 2.0348, but the 95% confidence interval (which is +/- ~2 standard errors) is quite wide (~2 x 0.25 = 0.5)

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Figure 8.4 Excess Returns on Portfolio Assets

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• Study pairs of securities

vs the market to

estimate correlations

• Compute stats to

measure correlations

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Study portfolio stats – 1

8-18

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A closer look at correlations

8-19

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Study portfolio stats – 2

8-20

2

Mji

iM ei

222

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Example: build optimal portfolio

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Alpha and Security Analysis

1. Use Macroeconomic analysis to estimate risk premium and risk of the market index (𝑅𝑀 , 𝜎𝑀)

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. Use numerical methods to establish the expected return of each security independently of security analysis (𝛽)

4. Use security analysis to develop your own forecast of the expected returns for each security (𝛼)

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Single-Index Model considerations

• Techniques for estimating 𝛽 are well known

• Estimating alpha requires a deep knowledge of the company behind the stock:

– Positive 𝛼 means overweight in the

portfolio

– What do you do if 𝛼 is negative?

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Recall the Minimum-Variance Frontier

7-25

Chapter 7

took the

entire

universe of

stocks and

used

brute-force

math to

find the

efficient

frontier

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Single-Index Model – Optimization

• Single-Index model offers a simpler optimization than the model in chapter 7 as the model is simplified

• Include the market as asset n+1 to improve diversification. By definition:

– Beta of market index = 1

– Alpha of market index = 0

– emarket_index = 0

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Single-Index Model Input List

• Risk premium on the S&P500 portfolio (𝑅𝑀)

• Estimate of the SD of the S&P500 portfolio (𝜎𝑀)

• n sets of estimates (one set for each stock) of:

–Beta coefficient

–Stock residual variances

–Alpha values

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Single-Index Model steps

• Use RM, alphas and betas to construct n+1 expected returns

• Use betas and 𝜎𝑀 to construct the covariance matrix

• Set up the optimization problem to minimize portfolio variance, given a return, subject to…

• …constraint that weights add up to one

• You could use excel solver to solve this problem and build your efficient frontier

8-28

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Index Model – Recall 𝛼𝑃 and 𝛽𝑃

Consider a generic portfolio and take the excess return RP as the average:

8-29

n

i

iMiii

n

i

iiP eRwRwR11

n

i

iiw1

n

i

Mii Rw1

n

i

iiew1

PPR MPR Pe

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Optimal Risky Portfolio of the Single-Index Model

Now take the portfolio expected excess return:

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PMPP RERE

n

i

iiM

n

i

ii wREw11

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Standard Deviation and Sharpe Ratio:

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PMPP e2222

n

i

iiM

n

i

ii eww1

222

2

1

PPP RES /

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• No need to use Excel as there is an analytical solution

• Solution is a combination of:

–Active portfolio (A), with weight wA

–Market-index passive portfolio (M)

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Optimal Risky Portfolio - wA

Assume for a moment beta=1

Then the optimal weight wA is proportional to

the ratio 𝜎𝐴/𝜎2(𝑒𝐴) to balance excess return

and residual variance from Active portfolio A:

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2

20

M

M

A

A

A RE

ew

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Next, modify of active portfolio weight wA to

optimize, as beta is not necessarily =1:

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Notice that when

0

0*

11 AA

AA

w

ww

0* then 1 AAA ww

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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):

𝑆𝑃2 = 𝑆𝑀

2 +𝛼𝐴𝜎 𝑒𝐴

2

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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Information

Ratio

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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 decouples macro and security analysis.

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INVESTMENTS | BODIE, KANE, MARCUS 8-39

Beta Book: Industry Version of the Index Model

• Use 60 most recent months of price data

• Use S&P500 as proxy for M

• Compute total returns that ignore dividends

• Estimate index model without excess

returns: *ebrar m

instead of

errrr fmf

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INVESTMENTS | BODIE, KANE, MARCUS 8-40

Beta book – alpha and intercept

• The intercept is different in the two

formulas. Rewrite as:

If* rf is constant you have same 𝛽 and 𝑒.

The intercept a is an estimate for

errrr fmf

errr mf 1

1fr

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Beta Book: Industry Version of the Index Model

• 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.

Adjust beta because:

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13

13

2 sampledadjusted

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Table 8.4 Industry Betas and Adjustment Factors

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