Chapter 12
AN ALTERNATIVE VIEW OF RISK AND RETURN:
THE ARBITRAGE PRICING THEORY
SLIDES
CHAPTER ORGANIZATION
12.1 Introduction
12.2 Systematic Risk and Betas
12.3 Portfolios and Factor Models
Portfolios and Diversification
12.4 Betas, Arbitrage, and Expected Returns
The Linear Relationship
The Market Portfolio and the Single Factor
12.1 Key Concepts and Skills
12.2 Chapter Outline
12.3 Arbitrage Pricing Theory
12.4 Total Risk
12.5 Risk: Systematic and Unsystematic
12.6 Systematic Risk and Betas
12.7 Systematic Risk and Betas
12.8 Systematic Risk and Betas: Example
12.9 Systematic Risk and Betas: Example
12.10 Systematic Risk and Betas: Example
12.11 Systematic Risk and Betas: Example
12.12 Systematic Risk and Betas: Example
12.13 Portfolios and Factor Models
12.14 Relationship Between the Return on the Common Factor & Excess Returns
12.15 Relationship Between the Return on the Common Factor & Excess Returns
12.16 Relationship Between the Return on the Common Factor & Excess Returns
12.17 Portfolios and Diversification
12.18 Portfolios and Diversification
12.19 Portfolios and Diversification
12.20 Betas, Arbitrage, and Expected Returns
12.21 Relationship Between β & Expected Return
12.22 Relationship Between β & Expected Return
12.23 The Capital Asset Pricing Model and the Arbitrage Pricing Theory
12.24 Empirical Approaches to Asset Pricing
12.25 Quick Quiz
12.5 The Capital Asset Pricing Model and the Arbitrage Pricing Theory
Differences in Pedagogy
Differences in Application
12.6 Empirical Approaches to Asset Pricing
Empirical Models
Style Portfolios
ANNOTATED CHAPTER OUTLINE
Slide 12.0 Chapter 12 Title Slide
Slide 12.1 Key Concepts and Skills
Slide 12.2 Chapter Outline
Slide 12.3 Arbitrage Pricing Theory
Arbitrage, by definition, is a situation in which an investor can
create a zero investment portfolio that generates a positive profit.
This chapter introduces the Arbitrage Pricing Theory (APT) as an
alternative asset-pricing model to the CAPM. The APT relies on
fewer assumptions than the CAPM and allows for multiple
systematic influences on security returns. The APT is derived using
portfolios, not individual securities.
12.1. Introduction
Slide 12.4 Total Risk
Slide 12.5 Risk: Systematic and Unsystematic
Recall that in the prior chapter we explored the two components of
total risk: systematic risk and unsystematic risk. In portfolios, most
unsystematic (or company-specific) risk can be eliminated,
implying that only systematic risk matters. This type of risk is best
measured by beta.
We also learned that the CAPM uses a stock’s beta to estimate the
expected return of that stock.
12.2. Systematic Risk and Betas
Slide 12.6 –
Slide 12.7 Systematic Risk and Betas
The beta () coefficient measures the response of an asset’s return
to a particular systematic risk factor.
The APT does not prescribe the number of factors that impact the
systematic risk and return of securities. If there are k factors, then
the actual return on a security is:
Ri =
_
R
+ i1F1 + i2F2 + … + iKFk + i
Each factor (F) is a systematic surprise. For example, if one of the
factors is inflation, then FINFLATION = Actual Inflation – Expected
Inflation. Under the APT, the return on a security has three
components:
1. Expected Return Component:
This is the portion of total return that remains after the unexpected
systematic and unsystematic returns have been removed. This
component of return is common to all assets.
2. Systematic Return Component:
A systematic risk is a risk that affects a large number of assets. The
total systematic return of a security depends upon the k systematic
factors and the security’s sensitivities to these factors. Surprises in
the systematic factors (F) affect all securities and the magnitude of
impact depends on each security’s sensitivity to these factors.
Possible candidates for the systematic factors include changes in
interest rates, GNP, oil prices, or productivity.
3. Unsystematic Return Component:
An unsystematic risk, is specific to a single asset. It does not
affect any other asset. It depends upon company-specific
information such as changes in management.
Another way to present the return-risk relationship for a security is
to use actual returns on the factors. For example if there are 3
factors (changes in inflation, GNP, and interest rates), the return on
a security can be presented as:
Ri =
_
R
+ i,INFLATIONFINFLATION + i,GNPFGNP + i,RATEFRATE + i
Slide 12.8 –
Slide 12.12 Systematic Risk and Betas: Example
12.3. Portfolios and Factor Models
Slide 12.13 Portfolios and Factor Models
A special case of the k-Factor Model is the One-Factor Model. The
return on the stock market (S&P 500 Index) is commonly chosen
as the factor:
Ri =
_
R
+ iFM + i
Lecture Tip: As in Chapter 11, the unsystematic risk will be
eliminated in well-diversified portfolios because the impact of
covariances far outweighs the impact of variances. The
unsystematic risk disappears in the APT paradigm by definition,
and straightly speaking, the result only applies to well-diversified
portfolios. This is an important point, but a fine point. In an
introductory course we simply point out that the systematic risk in
CAPM cannot be diversified away because securities are positively
correlated and in APT systematic risk exists because the factors
are common to all securities. When there is only one factor, we
obtain the following result under the APT:
Total risk i2) = Systematic risk M2) + Unsystematic risk i2)
This result is identical to the one obtained under the CAPM.
.A Portfolios and Diversification
Slide 12.14 –
Slide 12.16 Relationship Between the Return on the Common Factor &
Excess Return
As securities are added to a portfolio, the return and risk
characteristics of both the individual securities and the portfolio
change in important ways. For a portfolio with N securities, its
expected return, systematic risk, and unsystematic risk are:
N
1i iip RxR
N
1i ikipk x
N
1i iip x
.B The Market Portfolio and the Single Factor
Consider a one-factor model with factor M and an equally
weighted portfolio with N securities (x = 1/N). The return of this
portfolio is simply the average of the N securities. The risk of this
portfolio is:
N. largefor
N
N
1
2
M
2
P
2
i
2
M
2
P
2
i
N
1i 2
2
M
2
P
2
P
2
M
2
P
2
P
Slide 12.17 –
Slide 12.19 Portfolios and Diversification
Consequently, as securities are added to a portfolio, unsystematic
variance as a proportion of total portfolio variance falls rapidly.
12.4. Betas, Arbitrage, and Expected Returns
.A The Linear Relationship
Slide 12.20 Betas and Expected Returns
Slide 12.21 –
Slide 12.22 Relationship Between & Expected Return
Since the unsystematic risk is essentially eliminated in large, well
diversified portfolios, only the expected returns and the systematic
risk affect the actual return. These pieces are just weighted
averages of the individual assets in the portfolio, which provides
the simple result:
R =
_
R
+ PF
.B The Market Portfolio and the Single Factor
Considering the market portfolio as our factor (with the zero beta
arbitrage assumption) produces the following, which is equivalent
to the CAPM:
_
R
P = RF + (
_
R
M-RF)
12.5. The Capital Asset Pricing Model and the Arbitrage Pricing Theory
Slide 12.23 The Capital Asset Pricing Model and the Arbitrage
Pricing Theory
.A Differences in Pedagogy
The CAPM is generally more intuitive for students; however, the
APT is better suited for illustrating the reduction in unsystematic
risk as assets are added to a portfolio.
B. Differences in Application
The APT is a more general model and allows for more than one
systematic factor. On the other hand, the APT does not tell us what
these factors are, nor the number of factors. The CAPM specifies
that there is only one systematic factor, the Market. To apply either
model, the user must relate the systematic factor(s) to available
data. The S&P 500 Index is often used as the Market factor in the
CAPM. Since the APT does not specify the systematic factors,
users are left to find these factors on their own.
12.6. Empirical Approaches to Asset Pricing
Slide 12.24 Empirical Approaches to Asset Pricing
.A Empirical Models
The CAPM and APT are risk-based models. Empirical methods
tend to be based less on theory and more on looking for some
regularities in the historical record. An example is the Fama-
French three-factor model.
.B Style Portfolios
These empirical approaches tend to concentrate on styles such as
growth or value, which is apparent in the book-to-market factor
employed in the three-factor model.
Slide 12.25 Quick Quiz