Introduc corporate finance ch11

11-1

Chapter Eleven

An AlternativeCorporate
View ofFinance
Risk
Ross Westerfield Jaffe
and Return: The APT
•

•

11

Sixth Edition

Sixth Edition

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11-2

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
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11-3

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.

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11-4

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
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11-5

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.

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11-6

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.
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11-7

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:
σ
Total risk; U

ε
Nonsystematic Risk; ε
Systematic Risk; m

R = R +U
becomes
R = R+m+ε
where
m is the systematic risk
ε is the unsystematic risk
n

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11-8

11.3 Systematic Risk and Betas
• The beta coefficient, β, tells us the response of the
stock’s return to a systematic risk.
• In the CAPM, β measured the responsiveness of a
security’s return to a specific risk factor, the return on
the market portfolio.

βi =

Cov ( Ri , RM )

σ ( RM )
2

• We shall now consider many types of systematic risk.
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11-9

11.3 Systematic Risk and Betas
• For example, suppose we have identified three
systematic risks on which we want to focus:
1. Inflation
2. GDP growth
3. The dollar-pound spot exchange rate, S($,£)

• Our model is:

R = R+m+ε
R = R + β I FI + βGDP FGDP + βS FS + ε
β I is the inflation beta
βGDP is the GDP beta
βS is the spot exchange rate beta
ε is the unsystematic risk
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11-10

Systematic Risk and Betas: Example
R = R + β I FI + βGDP FGDP + βS FS + ε

• Suppose we have made the following estimates:
1 . βI = -2.30
2 . βGDP = 1.50
3 . βS = 0.50.

• Finally, the firm was able to attract a “superstar”
CEO and this unanticipated development
ε = 1%
contributes 1% to the return.
R = R − 2.30 × FI + 1.50 × FGDP + 0.50 × FS + 1%
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11-11

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%

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11-12

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%
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11-13

Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1%

If it was the case that dollar-pound 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%
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11-14

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%

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11-15

11.4 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 have returns:

Ri = R i + βi F + εi

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11-16

Relationship Between the Return on the
Common Factor & Excess Return
Excess
return

εi

Ri − R i = βi F + εi
If we assume
that there is no
unsystematic
risk, then εi = 0
The return on the factor F

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11-17

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 εi = 0

The return on the factor F

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11-18

Relationship Between the Return on the
Common Factor & Excess Return
Excess
return

β A = 1.5 β B = 1.0
βC = 0.50

Different
securities will
have different
betas

The return on the factor F

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11-19

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 1 R1 + X 2 R2 +  + X i Ri +  + X N RN
Ri = R i + βi F + εi

RP = X 1 ( R1 + β1 F + ε1 ) + X 2 ( R 2 + β2 F + ε2 ) +
 + X N ( R N + βN F + εN )
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
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11-20

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 = X 1 R1 + X 2 R 2 +  + X N R N
+ ( X 1 β1 + X 2 β2 +  + X N β N ) 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.
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11-21

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 = X 1 R1 + X 2 R 2 +  + X N R N
+ ( X 1 β1 + X 2 β2 +  + X N β N ) F

In a large portfolio, the only source of uncertainty is
the portfolio’s sensitivity to the factor.
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11-22

11.5 Betas and Expected Returns
RP = X 1 R1 +  + X N R N + ( X 1 β1 +  + X N β N ) 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 + β P F
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11-23

Relationship Between β & 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 + β P F

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Relationship Between β & Expected Return
Expected return

11-24

RF

SML
A

D
B
C
β

R = RF + β ( R P − RF )
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11-25

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.

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