An Alternative View of Risk and Return: The Arbitrage Pricing Theory
Chapter 12
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
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Key Concepts and Skills Discuss the relative importance of systematic
and unsystematic risk in determining a portfolio’s return
Compare and contrast the CAPM and Arbitrage Pricing Theory
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Chapter Outline12.1 Introduction12.2 Systematic Risk and Betas12.3 Portfolios and Factor Models12.4 Betas and Expected Returns12.5 The Capital Asset Pricing Model and the Arbitrage
Pricing Theory12.6 Empirical Approaches to Asset Pricing
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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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Total Risk 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.
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Risk: Systematic and Unsystematic
Systematic Risk: m
Nonsystematic Risk:
n
2
Total risk
We can break down the total risk 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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12.2 Systematic Risk and Betas The beta coefficient, , tells us the response of the stock’s return
to a systematic risk. In the CAPM, measures the responsiveness of a security’s
return to a specific risk factor, the return on the market portfolio.
)(
)(2
,
M
Mii R
RRCov
• We shall now consider other types of systematic risk.
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Systematic Risk and Betas For example, suppose we have identified three
systematic risks: inflation, GNP growth, and the dollar-euro spot exchange rate, S($,€).
Our model is:
risk icunsystemat theis
beta rate exchangespot theis
beta GNP theis
betainflation theis
ε
β
β
β
εFβFβFβRR
εmRR
S
GNP
I
SSGNPGNPII
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Systematic Risk and Betas: Example
Suppose we have made the following estimates:1. I = -2.30
2. GNP = 1.50
3. S = 0.50
Finally, the firm was able to attract a “superstar” CEO, and this unanticipated development contributes 1% to the return.
εFβFβFβRR SSGNPGNPII
%1ε
%150.050.130.2 SGNPI FFFRR
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Systematic Risk and Betas: Example
We must decide what surprises took place in the systematic factors.
If it were the case that the inflation rate was expected to be 3%, but in fact was 8% during the time period, then:
FI = Surprise in the inflation rate = actual – expected
= 8% – 3% = 5%
%150.050.130.2 SGNPI FFFRR
%150.050.1%530.2 SGNP FFRR
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Systematic Risk and Betas: Example
If it were the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then:
FGNP = Surprise in the rate of GNP growth
= actual – expected = 1% – 4% = – 3%
%150.050.1%530.2 SGNP FFRR
%150.0%)3(50.1%530.2 SFRR
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Systematic Risk and Betas: Example
If it were the case that the dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then:
FS = Surprise in the exchange rate
= actual – expected = 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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Systematic Risk and Betas: Example
Finally, if it were the case that the expected return on the stock was 8%, then:
%1%)10(50.0%)3(50.1%530.2 RR
%12
%1%)10(50.0%)3(50.1%530.2%8
R
R
%8R
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12.3 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 has return:
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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Relationship Between the Return on the Common Factor & Excess Return
Excess return
The return on the factor F
If we assume that there is no unsystematic risk, then i = 0.
FβRR iii
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Relationship Between the Return on the Common Factor & Excess Return
Excess return
The return on the factor F
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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Portfolios and DiversificationThe 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 weighted 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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Portfolios and DiversificationSo the return on a diversified portfolio is determined by two sets of parameters:
1. The weighted 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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12.4 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
If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.
FβRR PPP
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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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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.
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12.6 Empirical Approaches to Asset Pricing 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
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Quick Quiz 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?