Chap 010

CHAPTER 10 Arbitrage Pricing
Theory and
Multifactor
Models of Risk
and Return
Investments, 8th edition

Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Single Factor Model
• Returns on a security come from two sources

– Common macro-economic factor
– Firm specific events
• Possible common macro-economic factors
– Gross Domestic Product Growth
– Interest Rates
10-2
Single Factor Model Equation
ri  E (ri )  i F  ei
ri = Return for security I

i = Factor sensitivity or factor loading or factor
beta
F = Surprise in macro-economic factor
(F could be positive, negative or zero)
ei = Firm specific events
10-3
Multifactor Models
• Use more than one factor in addition to

market return
– Examples include gross domestic product,
expected inflation, interest rates etc.
– Estimate a beta or factor loading for each
factor using multiple regression.
10-4
Multifactor Model Equation
ri = E(ri) + iGDP GDP + iIR IR + ei
ri = Return for security i

i = Factor sensitivity for GDP
iGDP
IR = Factor sensitivity for Interest Rate
ei = Firm specific events
10-5
Multifactor SML Models
E(r) = rf + iGDPRPGDP + IRi RPIR
GDP
i = Factor sensitivity for GDP
RPGDP = Risk premium for GDP

i
IR = Factor sensitivity for Interest Rate
RPIR = Risk premium for Interest Rate
10-6
Arbitrage Pricing Theory
Arbitrage - arises if an investor can construct a

zero investment portfolio with a sure profit
• Since no investment is required, an investor
can create large positions to secure large
levels of profit
• In efficient markets, profitable arbitrage
opportunities will quickly disappear
10-7
APT & Well-Diversified Portfolios
rP = E (rP) + PF + eP
F = some factor
• For a well-diversified portfolio:
eP approaches zero
Similar to CAPM,
10-8
Figure 10.1 Returns as a Function of the
Systematic Factor
10-9
Figure 10.2 Returns as a Function of the
Systematic Factor: An Arbitrage
Opportunity
10-10
Figure 10.3 An Arbitrage Opportunity
10-11
Figure 10.4 The Security Market Line
10-12
APT and CAPM Compared
• APT applies to well diversified portfolios and

not necessarily to individual stocks
• With APT it is possible for some individual
stocks to be mispriced - not lie on the SML
• APT is more general in that it gets to an
expected return and beta relationship without
the assumption of the market portfolio
• APT can be extended to multifactor models
10-13
Multifactor APT
• Use of more than a single factor

• Requires formation of factor portfolios
• What factors?
– Factors that are important to performance
of the general economy
– Fama-French Three Factor Model
10-14
Two-Factor Model
ri  E (ri )  i1 F1   i 2 F2  ei
• The multifactor APR is similar to the one-
factor case
– But need to think in terms of a factor portfolio
• Well-diversified
• Beta of 1 for one factor
• Beta of 0 for any other
10-15
Example of the Multifactor Approach
• Work of Chen, Roll, and Ross

– Chose a set of factors based on the ability
of the factors to paint a broad picture of the
macro-economy
10-16
Another Example:
Fama-French Three-Factor Model
• The factors chosen are variables that on
past evidence seem to predict average
returns well and may capture the risk
premiums
rit   i  iM RMt  iSMB SMBt  iHML HMLt  eit
• Where:
– SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in
excess of the return on a portfolio of large stocks
– HML = High Minus Low, i.e., the return of a portfolio of stocks with a
high book to-market ratio in excess of the return on a portfolio of stocks
with a low book-to-market ratio
10-17
The Multifactor CAPM and the APM
• A multi-index CAPM will inherit its risk factors

from sources of risk that a broad group of
investors deem important enough to hedge
• The APT is largely silent on where to look for
priced sources of risk
10-18

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CHAPTER 10 Arbitrage Pricing

Investments, 8th edition

Slides by Susan Hine

• Returns on a security come from two sources

ri = Return for security I

• Use more than one factor in addition to

ri = E(ri) + iGDP GDP + iIR IR + ei

ri = Return for security i

RPGDP = Risk premium for GDP

Arbitrage - arises if an investor can construct a

• APT applies to well diversified portfolios and

• Use of more than a single factor

• Work of Chen, Roll, and Ross

• A multi-index CAPM will inherit its risk factors

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