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11-0
CHAPTER
11
An Alternative View of Risk
and Return: The APT
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-1
Chapter Outline
11.1 Factor Models: Announcements, Surprises, and
Expected Returns
11.2 Risk: Systematic and Unsystematic
11.3 Systematic Risk and Betas
11.4 Portfolios and Factor Models
11.5 Betas and Expected Returns
11.6 The Capital Asset Pricing Model and the
Arbitrage Pricing Theory
11.7 Parametric Approaches to Asset Pricing
11.8 Summary and Conclusions
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-2
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.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-3
11.1 Factor Models: Announcements,
Surprises, and Expected Returns
The return on any security consists of two parts.
First the expected returns
Second is the unexpected or risky returns.
A way to write the return on a stock in the coming month
is:
R = R +U
where
R is the expected part of the return
U is the unexpected part of the return
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-4
11.1 Factor Models: Announcements,
Surprises, and Expected Returns
Any announcement can be broken down into two
parts, the anticipated or expected part and the
surprise or innovation:
Announcement = Expected part + Surprise.
The expected part of any announcement is part of
the information the market uses to form the
expectation, R of the return on the stock.
The surprise is the news that influences the
unanticipated return on the stock, U.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-5
11.2 Risk: Systematic and Unsystematic
A systematic risk is any risk that affects a large number
of assets, each to a greater or lesser degree.
An unsystematic risk is a risk that specifically affects a
single asset or small group of assets.
Unsystematic risk can be diversified away.
Examples of systematic risk include uncertainty about
general economic conditions, such as GNP, interest rates
or inflation.
On the other hand, announcements specific to a
company, such as a gold mining company striking gold,
are examples of unsystematic risk.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-6
11.2 Risk: Systematic and Unsystematic
We can break down the risk, U, of holding a stock into two
components: systematic risk and unsystematic risk:

R = R +U
Total risk; U
becomes
R = R +m +ε

Nonsystematic Risk; 
where
m is the systematic risk
Systematic Risk; m
ε is the unsystematic risk
n
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-7
11.3 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 measured the responsiveness of a
security’s return to a specific risk factor, the return
on the market portfolio.
bi =
Cov ( Ri , RM )
2 ( RM )
We shall now consider many types of systematic
risk.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-8
11.3 Systematic Risk and Betas
For example, suppose we have identified three systematic risks
on which we want to focus:
1.
2.
3.
Inflation
GDP growth
The dollar-euro
spot exchange
rate, S($,€)
Our model is:
R = R +m +ε
R = R + βI FI + βGDP FGDP + βS FS + ε
βI is the inflation beta
βGDP is the GDP beta
βS is the spot exchange rate beta
ε is the unsystematic risk
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-9
Systematic Risk and Betas: Example
R = R + βI FI + βGDP FGDP + βS FS + ε
Suppose we have made the following estimates:
1. bI = -2.30
2. bGDP = 1.50
3. bS = 0.50.
Finally, the firm was able to attract a “superstar” CEO
and this unanticipated development contributes 1% to
the return.
ε = 1%
R = R - 2.30  FI +1.50  FGDP + 0.50  FS +1%
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-10
Systematic Risk and Betas: Example
R = R - 2.30  FI +1.50  FGDP + 0.50  FS +1%
We must decide what surprises took place in the systematic
factors.
If it was the case that the inflation rate was expected to be
by 3%, but in fact was 8% during the time period, then
FI = Surprise in the inflation rate
= actual – expected
= 8% – 3%
= 5%
R = R - 2.30 5% +1.50  FGDP + 0.50  FS +1%
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-11
Systematic Risk and Betas: Example
R = R - 2.30 5% +1.50  FGDP + 0.50  FS +1%
If it was the case that the rate of GDP growth was expected
to be 4%, but in fact was 1%, then
FGDP = Surprise in the rate of GDP growth
= actual – expected
= 1% – 4%
= – 3%
R = R - 2.30 5% +1.50 ( -3%) + 0.50  FS +1%
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-12
Systematic Risk and Betas: Example
R = R - 2.30 5% +1.50 ( -3%) + 0.50  FS +1%
If it was the case that dollar-euro spot exchange rate,
S($,€), 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%
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-13
Systematic Risk and Betas: Example
R = R - 2.30 5% +1.50 ( -3%) + 0.50  FS +1%
Finally, if it was 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%
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-14
11.4 Portfolios and Factor Models
Now let us consider what happens to portfolios of
stocks when each of the stocks follows a onefactor model.
We will create portfolios from a list of N stocks
and will capture the systematic risk with a 1factor model.
The ith stock in the list have returns:
Ri = Ri + βi F + εi
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-15
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
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-16
Relationship Between the Return on the
Common Factor & Excess Return
Excess
return
Ri - R i = βi F
If we assume
that there is no
unsystematic
risk, then i = 0
The return on the factor F
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-17
Relationship Between the Return on the
Common Factor & Excess Return
Excess
return
βA =1.5 βB = 1.0
βC = 0.50
Different
securities will
have different
betas
The return on the factor F
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-18
Portfolios and Diversification
We know that the portfolio return is the weighted average of the
returns on the individual assets in the portfolio:
RP = X1 R1 + X 2 R2 + L + Xi Ri + L + X N RN
Ri = Ri + βi F + εi
RP = X1 ( R1 + β1 F + ε1 ) + X2 ( R2 + β2 F + ε2 ) +
L + X N ( RN + βN F + εN )
RP = X1 R1 + X1 β1 F + X1 ε1 + X2 R2 + X 2 β2 F + X 2 ε2 +
L + X N RN + X N βN F + X N εN
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-19
Portfolios and Diversification
The return on any portfolio is determined by three sets of
parameters:
1. The weighed 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 + X2 R2 + L + X N RN
+ ( X1 β1 + X2 β2 + L + X N βN ) F
+ X1 ε1 + X2 ε2 + L + X N εN
In a large portfolio, the third row of this equation disappears as the
unsystematic risk is diversified away.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-20
Portfolios and Diversification
So the return on a diversified portfolio is
determined by two sets of parameters:
1. The weighed average of expected returns.
2. The weighted average of the betas times the factor F.
RP = X1 R1 + X 2 R2 + L + X N RN
+ ( X1 β1 + X 2 β2 + L + X N βN ) F
In a large portfolio, the only source of uncertainty is the portfolio’s
sensitivity to the factor.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-21
11.5 Betas and Expected Returns
RP = X1 R1 + L + X N RN + ( X1 β1 + L + X N βN ) F
RP
Recall that
RP = X1 R1 + L + X N RN
βP
and
βP = X1 β1 + L + X N βN
The return on a diversified portfolio is the sum of the
expected return plus the sensitivity of the portfolio to the
factor.
RP = RP + βP F
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-22
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 = RP + βP F
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-23
Expected return
Relationship Between b & Expected Return
RF
SML
D
A
B
C
b
R = RF + β ( RP - RF )
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-24
11.6 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.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-25
11.7 Empirical Approaches
to Asset Pricing
Both the CAPM and APT are risk-based models. There
are alternatives.
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
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
11-26
11.8 Summary and Conclusions
The APT assumes that stock returns are generated according to
factor models such as:
R = R + βI FI + βGDP FGDP + βS FS + ε
As securities are added to the portfolio, the unsystematic risks of
the individual securities offset each other. A fully diversified
portfolio has no unsystematic risk.
The CAPM can be viewed as a special case of the APT.
Empirical models try to capture the relations between returns and
stock attributes that can be measured directly from the data
without appeal to theory.
McGraw-Hill/Irwin
Corporate Finance, 7/e
© 2005 The McGraw-Hill Companies, Inc. All Rights Reserved.
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