Chapter 12
An Alternative View of Risk and Return: The
Arbitrage Pricing Theory
McGraw-Hill/Irwin
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.
Key Concepts and Skills


Discuss the relative importance of systematic
and unsystematic risk in determining a
portfolio’s return
Compare and contrast the CAPM and
Arbitrage Pricing Theory
12-1
Chapter Outline
12.1 Introduction
12.2 Systematic Risk and Betas
12.3 Portfolios and Factor Models
12.4 Betas and Expected Returns
12.5 The Capital Asset Pricing Model and the Arbitrage
Pricing Theory
12.6 Empirical Approaches to Asset Pricing
12-2
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.
12-3
Total Risk
Total risk = systematic risk + unsystematic risk
 The standard deviation of returns is a measure
of total risk.
 For well-diversified portfolios, unsystematic
risk is very small.
 Consequently, the total risk for a diversified
portfolio is essentially equivalent to the
systematic risk.

12-4
Risk: Systematic and Unsystematic
We can break down the total risk of holding a stock into
two components: systematic risk and unsystematic risk:
2
Total risk
R  R U
becomes
R  Rmε

Nonsystematic Risk: 
where
m is the systematic risk
Systematic Risk: m
ε is the unsystemat ic risk
n
12-5
12.2 Systematic Risk and Betas


The beta coefficient, b, tells us the response
of the stock’s return to a systematic risk.
In the CAPM, b measures the responsiveness
of a security’s return to a specific risk factor,
the return on the market portfolio.
bi 
Cov( Ri , RM )
 ( RM )
2
• We shall now consider other types of systematic risk.
12-6
Systematic Risk and Betas


For example, suppose we have identified three
systematic risks: inflation, GNP growth, and the
dollar-euro spot exchange rate, S($,€).
Our model is:
R  Rmε
R  R  β I FI  βGNP FGNP  βS FS  ε
β I is the inflation beta
βGNP is the GNP beta
βS is the spot exchange rate beta
ε is the unsystemat ic risk
12-7
Systematic Risk and Betas: Example
R  R  βI FI  βGNP FGNP  βS FS  ε

Suppose we have made the following estimates:
1.
2.
3.

bI = -2.30
bGNP = 1.50
bS = 0.50
Finally, the firm was able to attract a “superstar”
CEO, and this unanticipated development
contributes 1% to the return. ε  1%
R  R  2.30  FI  1.50  FGNP  0.50  FS  1%
12-8
Systematic Risk and Betas: Example
R  R  2.30  FI  1.50  FGNP  0.50  FS  1%
We must decide what surprises took place in the
systematic factors.
If it were the case that the inflation rate was expected
to be 3%, but in fact was 8% during the time
period, then:
FI = Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
R  R  2.30  5%  1.50  FGNP  0.50  FS  1%
12-9
Systematic Risk and Betas: Example
R  R  2.30  5%  1.50  FGNP  0.50  FS  1%
If it were the case that the rate of GNP growth
was expected to be 4%, but in fact was 1%,
then:
FGNP = Surprise in the rate of GNP growth
= actual – expected = 1% – 4% = – 3%
R  R  2.30  5%  1.50  (3%)  0.50  FS  1%
12-10
Systematic Risk and Betas: Example
R  R  2.30  5%  1.50  (3%)  0.50  FS  1%
If it were the case that the dollar-euro spot
exchange rate, S($,€), was expected to
increase by 10%, but in fact remained stable
during the time period, then:
FS = Surprise in the exchange rate
= actual – expected = 0% – 10% = – 10%
R  R  2.30  5%  1.50  (3%)  0.50  (10%)  1%
12-11
Systematic Risk and Betas: Example
R  R  2.30  5%  1.50  (3%)  0.50  (10%)  1%
Finally, if it were the case that the expected
return on the stock was 8%, then:
R  8%
R  8%  2.30  5%  1.50  (3%)  0.50  (10%)  1%
R  12%
12-12
12.3 Portfolios and Factor Models



Now let us consider what happens to
portfolios of stocks when each of the stocks
follows a one-factor model.
We will create portfolios from a list of N
stocks and will capture the systematic risk
with a 1-factor model.
The ith stock in the list has return:
Ri  Ri  βi F  εi
12-13
Relationship Between the Return on
the Common Factor & Excess Return
Excess
return
i
Ri  Ri  βi F  εi
If we assume
that there is no
unsystematic
risk, then i = 0.
The return on the factor F
12-14
Relationship Between the Return on
the Common Factor & Excess Return
Excess
return
Ri  Ri  βi F
If we assume
that there is no
unsystematic
risk, then i = 0.
The return on the factor F
12-15
Relationship Between the Return on
the Common Factor & Excess Return
Excess
return
β A  1.5 βB  1.0
Different
securities will
βC  0.50 have different
betas.
The return on the factor F
12-16
Portfolios and Diversification

We know that the portfolio return is the weighted
average of the returns on the individual assets in the
portfolio:
RP  X 1R1  X 2 R2    X i Ri    X N RN
Ri  Ri  βi F  εi
RP  X 1 ( R1  β1 F  ε1 )  X 2 ( R 2  β2 F  ε2 ) 
  X N ( R N  βN F  ε N )
RP  X 1 R1  X 1 β1 F  X 1ε1  X 2 R 2  X 2 β2 F  X 2 ε2 
  X N R N  X N βN F  X N ε N
12-17
Portfolios and Diversification
The return on any portfolio is determined by
three sets of parameters:
1. The weighted average of expected returns.
2. The weighted average of the betas times the factor.
3. The weighted average of the unsystematic risks.
RP  X1 R1  X 2 R 2    X N R N
 ( X 1 β1  X 2 β2    X N βN ) F
 X 1ε1  X 2 ε2    X N ε N
In a large portfolio, the third row of this equation
disappears as the unsystematic risk is diversified away.
12-18
Portfolios and Diversification
So the return on a diversified portfolio is
determined by two sets of parameters:
1.
2.
The weighted average of expected returns.
The weighted average of the betas times the
factor F.
RP  X 1 R1  X 2 R 2    X N R N
 ( X 1 β1  X 2 β2    X N βN ) F
In a large portfolio, the only source of uncertainty is the
portfolio’s sensitivity to the factor.
12-19
12.4 Betas and Expected Returns
RP  X1 R1    X N R N  ( X1 β1    X N βN ) F
βP
RP
Recall that
and
R P  X 1 R1    X N R N
βP  X 1 β1    X N β N
The return on a diversified portfolio is the sum
of the expected return plus the sensitivity of the
portfolio to the factor.
RP  R P  βP F
12-20
Relationship Between b & Expected Return

If shareholders are ignoring unsystematic
risk, only the systematic risk of a stock
can be related to its expected return.
RP  R P  βP F
12-21
Expected return
Relationship Between b & Expected Return
RF
SML
D
A
B
C
b
R  RF  β ( R P  RF )
12-22
12.5 The Capital Asset Pricing Model
and the Arbitrage Pricing Theory




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.
12-23
12.6 Empirical Approaches to Asset Pricing




Both the CAPM and APT are risk-based models.
Empirical methods are based less on theory and
more on looking for some regularities in the
historical record.
Be aware that correlation does not imply causality.
Related to empirical methods is the practice of
classifying portfolios by style, e.g.,


Value portfolio
Growth portfolio
12-24
Quick Quiz



Differentiate systematic risk from
unsystematic risk. Which type is essentially
eliminated with well diversified portfolios?
Define arbitrage.
Explain how the CAPM be considered a
special case of Arbitrage Pricing Theory?
12-25