This document summarizes key aspects of arbitrage pricing theory (APT) and multifactor models of risk and return from the textbook "Investments" by Bodie, Kane, and Marcus: 1) APT generalizes the security market line of the CAPM to provide richer insight into the risk-return relationship by allowing for multiple systematic risk factors rather than a single market factor. 2) Multifactor models posit that security returns respond not just to overall market movements but also to specific systematic risk factors like GDP, interest rates, and inflation. 3) The Fama-French three-factor model explains returns based on sensitivities to market, firm size, and book-to-market factors

1. Dr. Muath Asmar
Investment & Portfolio
Management
AN-NAJAH
NATIONAL UNIVERSITY
Faculty of Graduate Studies

2. INVESTMENTS | BODIE, KANE, MARCUS
©2021 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom.
No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
Chapter Ten
Arbitrage Pricing Theory
and Multifactor Models
of Risk and Return

3. INVESTMENTS | BODIE, KANE, MARCUS
• Arbitrage is the exploitation of security
mispricing in such a way that risk-free profits
can be earned
• Most basic principle of capital market theory is
that well-functioning security markets rule out
arbitrage opportunities
• Generalization of the security market line of
the CAPM to gain richer insight into the risk-
return relationship
• Arbitrage pricing theory (APT)
Overview
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4. INVESTMENTS | BODIE, KANE, MARCUS
• Recall stock variability may be decomposed
into the following sources:
• Market (i.e., systemic) risk
• Largely due to macroeconomic events
• Firm-specific (i.e., idiosyncratic) effects
• Risk premiums may depend on correlations
with extra-market risk factors
• E.g., inflation, interest rates, volatility, etc.
Multifactor Models: A Preview
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5. INVESTMENTS | BODIE, KANE, MARCUS
• Single-factor model of excess returns
𝑅𝑖 = 𝐸 𝑅𝑖 + 𝐵𝑖 𝐹 + 𝑒𝑖
𝐸(𝑅𝑖) = expected excess return on stock 𝑖
𝐵𝑖 = sensitivity of firm 𝑖
𝐹 = deviation of the common factor from its expected value
𝑒𝑖 = nonsystematic components of returns
Factor Models of Security Returns
(1 of 3)
©2021 McGraw-Hill Education 10-5

6. INVESTMENTS | BODIE, KANE, MARCUS
• Extra market sources of risk may arise from
several sources
• E.g., uncertainty about interest rates or inflation.
• Multifactor models posit that returns respond
to several systematic risk factors, as well as
firm-specific influences
• Useful in risk management applications
Factor Models of Security Returns
(2 of 3)
©2021 McGraw-Hill Education 10-6

7. INVESTMENTS | BODIE, KANE, MARCUS
𝑅𝑖 = excess return on security i
𝐵 𝐺𝐷𝑃 = sensitivity of returns to GDP
𝐵𝐼𝑅 = sensitivity of returns to interest rates
𝑒𝑖 = nonsystematic components of returns
Factor Models of Security Returns
(3 of 3)
©2021 McGraw-Hill Education 10-7
  iiIRiGDPii eIRGDPRER  

8. INVESTMENTS | BODIE, KANE, MARCUS
• Arbitrage pricing theory (APT) was developed
by Stephen Ross
• Predicts a SML linking expected returns to risk, but
the path is takes to the SML is quite different
• APT relies on three key propositions:
1. Security returns can be described by a factor model
2. There are sufficient securities to diversify away
idiosyncratic risk
3. Well-functioning security markets do not allow for
the persistence of arbitrage opportunities
Arbitrage Pricing Theory
©2021 McGraw-Hill Education 10-8

9. INVESTMENTS | BODIE, KANE, MARCUS
• Arbitrage opportunity exists when an investor
can earn riskless profits without making a net
investment
• E.g., shares of a stock sell for different prices on two
different exchanges
• Law of One Price
• Enforced by arbitrageurs; If they observe a violation,
they will engage in arbitrage activity
• This bids up (down) the price where it is low (high)
until the arbitrage opportunity is eliminated
Arbitrage, Risk Arbitrage, and
Equilibrium
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10. INVESTMENTS | BODIE, KANE, MARCUS
• In a well-diversified portfolio, firm-specific
risk becomes negligible, so that only factor
risk remains
• One effect of diversification is that, when n is
large, nonsystematic variance approaches
zero
• A well-diversified portfolio is on with
each weight small enough that for
practices purposes the nonsystematic
variance is negligible
Well-Diversified Portfolios
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11. INVESTMENTS | BODIE, KANE, MARCUS
Excess Returns as a Function of
the Systematic Factor
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12. INVESTMENTS | BODIE, KANE, MARCUS
Returns as a Function of the Systematic
Factor: An Arbitrage Opportunity
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13. INVESTMENTS | BODIE, KANE, MARCUS
• All well-diversified portfolio with the same beta
must have the same expected return
• Generally, for any well-diversified P, the
expected excess return must be:
• Risk premium on portfolio P is the product of its
beta and the risk premium of the market index
• SML of the CAPM must also apply to well-diversified
portfolios
The SML of the APT
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( ) ( )P P ME R E R

14. INVESTMENTS | BODIE, KANE, MARCUS
An Arbitrage Opportunity
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15. INVESTMENTS | BODIE, KANE, MARCUS
APT
• Built on the foundation
of well-diversified
portfolios
– Cannot rule out a
violation of the expected
return-beta relationship
for any particular asset
• Does not assume
investors are mean-
variance optimizers
• Uses an observable
market index
CAPM
• Model is based on an
inherently unobservable
“market” portfolio
• Provides unequivocal
statement on the
expected return-beta
relationship for all
securities
APT and CAPM
©2021 McGraw-Hill Education 10-15

16. INVESTMENTS | BODIE, KANE, MARCUS
• To this point, we have examined the APT in a
one-factor world
• However, there are several sources of systematic
risk and exposure to these factors will affect a
stock’s appropriate expected return
• APT can be generalized to accommodate these
multiple sources of risk
• Assume a two-factor model:
A Multifactor APT
(1 of 2)
©2021 McGraw-Hill Education 10-16
1 1 2 2( )i i i i iR E R F F e    

17. INVESTMENTS | BODIE, KANE, MARCUS
• Benchmark portfolios in the APT are factor
portfolios
• β=1 for one of the factors and β= 0 for any other
factors
• Factor portfolios track a particular source of
macroeconomic risk, but are uncorrelated with
other sources of risk
• Referred to as a “tracking portfolio”
A Multifactor APT
(2 of 2)
©2021 McGraw-Hill Education 10-17

18. INVESTMENTS | BODIE, KANE, MARCUS
• 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)
Fama-French (FF) Three-Factor
Model
©2021 McGraw-Hill Education 10-18
ittiHMLtiSMBMtiMiit eHMLSMBRR  

19. INVESTMENTS | BODIE, KANE, MARCUS
• Begin by estimating Amazon’s beta on each of
the FF factors
• See Equation 10.10
• Amazon’s expected return in the multifactor
model is lower due to the considerable
hedging value it offers against the size and
value risk factors
• This is ignored by the single-factor model
Estimating and Implementing a
Three-Factor SML
©2021 McGraw-Hill Education 10-19