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.

 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

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 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 = 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

We can break down the total risk of holding a stock into two components: systematic risk and unsystematic risk:

2

Total risk
Nonsystematic Risk:

Systematic Risk: m
R

R

U becomes
R

R
 m
 ε where m is the systematic risk
ε is the unsystemat ic risk n
12-5



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.
b  i
Cov ( R i ,
R
M

2
( R
M
)
)
• We shall now consider other types of systematic risk.
12-6



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
F
I
 β
GNP
β
I is the inflation
F
GNP beta
β
GNP is the GNP beta
 β
S
F
S
β
S is the spot exchange rate beta
ε is the unsystemat ic risk
 ε
12-7

Systematic Risk and Betas: Example


R

R
 β
I
F
I
 β
GNP
F
GNP
 β
S
F
S
 ε
Suppose we have made the following estimates:
1.
2.
3.
b
I
= -2.30
b
GNP
= 1.50
b
S
= 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

F
I

1 .
50

F
GNP

0 .
50

F
S

1 %
12-8

Systematic Risk and Betas: Example
R

R

2 .
30

F
I

1 .
50

F
GNP

0 .
50

F
S

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:
F
I
= Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
R

R

2 .
30

5 %

1 .
50

F
GNP

0 .
50

F
S

1 %
12-9

Systematic Risk and Betas: Example
R

R

2 .
30

5 %

1 .
50

F
GNP

0 .
50

F
S

1 %
If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:
F
GNP
= Surprise in the rate of GNP growth
= actual – expected = 1% – 4% = – 3%
R

R

2 .
30

5 %

1 .
50

(

3 %)

0 .
50

F
S

1 %
12-10

Systematic Risk and Betas: Example
R

R

2 .
30

5 %

1 .
50

(

3 %)

0 .
50

F
S

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:
F
S
= 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

 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 i th stock in the list has return:
R i

R i
 β i
F
 ε i
12-13

Relationship Between the Return on the Common Factor & Excess Return
Excess return
 i
R i

R i
 β i
F
 ε i
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
R i

R i
 β i
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 β
B
β
C

1 .
0

0 .
50
Different securities will have different betas.
The return on the factor F
12-16

 We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:
R
P

X
1
R
1

X
2
R
2
  
X i
R i
  
X
N
R
N
R
P

X
1
( R
1
 
X
N
( R
N


β
1
F
 ε
1
)

X
2
( R
2
β
N
F
 ε
N
)

R i

R i
β
2
F

 β i
F
 ε i
ε
2
)

R
P

X
1
R
1

X
1
β
1
F

X
1
ε
1

X
2
R
2

X
2
β
2
F

X
2
ε
2
 
X
N
R
N

X
N
β
N
F

X
N
ε
N 12-17


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.
R
P

X
1
R
1

X
2
R
2
  
X
N


( X
1
X
1
ε
1
β
1


X
X
2
2
ε
2
β
2






X
X
N
N
ε
N
β
N
R
N
) F
In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.
12-18

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.
R

P
(

X
1
X
1
R
1
β
1


X
2
X
2
R
2
β
2
 
  

X
N
X
N
β
N
R
N
) F
In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.
12-19

R
P

X
1
R
1
  
X
N
R
N

( X
1
β
1
  
X
N
β
N
) F
R
P
β
P
Recall that
R
P

X
1
R
1
  
X
N
R
N and
β
P

X
1
β
1
  
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.
R
P

R
P
 β
P
F
12-20

b
 If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.
R
P

R
P
 β
P
F
12-21

b
SML
D
A
B
R
F
C
R

R
F
 β
( R
P

R
F
) b
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





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

 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