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McGraw-Hill Ryerson © 2003 McGraw–Hill Ryerson Limited

Corporate Finance Ross Westerfield Jaffe Sixth Edition

11Chapter Eleven

An Alternative View of Risk and Return: The APT

Prepared by

Gady JacobyUniversity of Manitoba

and

Sebouh AintablianAmerican University of Beirut

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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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11.1 Factor Models: Announcements, Surprises, and Expected Returns• The return on any security consists of two parts.

1) the expected or normal return: the return that shareholders in the market predict or expect

2) the unexpected or risky return: the portion that comes from information that will be revealed .

Examples of relevant information:

– Statistics Canada figures (e.g., GNP)

– A sudden drop in interest rates

– News that the company’s sales figures are higher than expected

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11.1 Factor Models: Announcements, Surprises, and Expected Returns

• A way to write the return on a stock in the coming month is:

return theofpart unexpected theis

return theofpart expected theis

where

U

R

URR

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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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• 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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Systematic Risk; m

Nonsystematic Risk;

n

Total risk; U

We can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk:

risk icunsystemat theis

risk systematic theis

where

becomes

ε

m

εmRR

URR

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Systematic risk is referred to as market risk.

m influences all assets in the market to some extent.

Is specific to the company and unrelated to the specific risk of most other companies.

0)(,

ji

εεCorr

ε

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11.3 Systematic Risk and Betas

• The beta coefficient, , tells us the response of the stock’s return to a systematic risk.

• In the CAPM, measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio.

• We shall now consider many types of systematic risk.

)(

)(2

,

M

Mii R

RRCov

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11.3 Systematic Risk and Betas• For example, suppose we have identified three

systematic risks on which we want to focus:1. Inflation2. GDP growth3. The dollar-pound spot exchange rate, S($,£)

• Our model is:

risk icunsystemat theis

beta rate exchangespot theis

beta GDP theis

betainflation theis

ε

β

β

β

εFβFβFβRR

εmRR

S

GDP

I

SSGDPGDPII

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Systematic Risk and Betas: Example

• Suppose we have made the following estimates:

1. I = -2.30

2. GDP = 1.50

3. S = 0.50.

• Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return.

εFβFβFβRR SSGDPGDPII

%1ε

%150.050.130.2 SGDPI FFFRR

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We must decide what surprises took place in the systematic factors.

If it was 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%

%150.050.130.2 SGDPI FFFRR

%150.050.1%530.2 SGDP FFRR

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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


= 1% - 4%

= -3%

%150.050.1%530.2 SGDP FFRR

%150.0%)3(50.1%530.2 SFRR

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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


= 0% - 10%

= -10%

%150.0%)3(50.1%530.2 SFRR

%1%)10(50.0%)3(50.1%530.2 RR

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Finally, if it was the case that the expected return on the stock was 8%, then

%150.0%)3(50.1%530.2 SFRR

%12

%1%)10(50.0%)3(50.1%530.2%8

R

R

%8R

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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:

iiii εFβRR

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Relationship Between the Return on the Common Factor & Excess Return

Excess return

The return on the factor F

i

iiii εFβRR

If we assume that there is no

unsystematic risk, then i = 0

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Excess return


If we assume that there is no

unsystematic risk, then i = 0

FβRR iii

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Excess return


Different securities will have different

betas

0.1Bβ

50.0Cβ

5.1Aβ

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Portfolios and Diversification

• We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:

NNiiP RXRXRXRXR 2211

)(

)()( 22221111

NNNN

P

εFβRX

εFβRXεFβRXR

NNNNNN

P

εXFβXRX

εXFβXRXεXFβXRXR

222222111111

iiii εFβRR

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The return on any portfolio is determined by three sets of parameters:

In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

NNP RXRXRXR 2211

1. The weighed average of expected returns.

FβXβXβX NN )( 2211

2. The weighted average of the betas times the factor.

NN εXεXεX 2211

3. The weighted average of the unsystematic risks.

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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.

FβXβXβX

RXRXRXR

NN

NNP

)( 2211

2211

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

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11.5 Betas and Expected Returns

The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.

FβXβXRXRXR NNNNP )( 1111

FβRR PPP

NNP RXRXR 11

that Recall

NNP βXβXβ 11

and

PR Pβ

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Relationship Between & Expected Return

• The relevant risk in large and well-diversified portfolios is all systematic, because unsystematic risk is diversified away.

• If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.

FβRRP

PP

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Relationship Between & Expected Return

Exp

ecte

d re

turn

FR

A B

C

D

SML

)( FPF RRβRR

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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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Multi-factor APT

kFk

FFFβRRβRRβRRRR )(...)( )(

22

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Example: A Canadian study (Otuteye, CIR 1991)with five factors:1. the rate of growth in industrial production2. the changes in the slope of the term structure of

interest rates3. the default risk premium for bonds4. inflation5. The value-weighted return on the market portfolio

(TSE 300)

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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

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11.8 Summary and Conclusions

• The APT assumes that stock returns are generated according to factor models such as:

εFβFβFβRR SSGDPGDPII

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.

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