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RWJ Chapter 12
An Alternative view of Risk and
return
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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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Arbitrage Pricing Theory
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:
returntheofpartunexpectedtheis
returntheofpartexpectedtheis
where
U
R
URR
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Arbitrage Pricing Theory
Any announcement can be broken down into twoparts, 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, Rof the return on the stock.
The surprise is the news that influences theunanticipated return on the stock, U.
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Risk: Systematic and Unsystematic
A systematic riskis any risk that affects a large numberof assets, each to a greater or lesser degree.
An unsystematic riskis a risk that specifically affects asingle asset or small group of assets.
Unsystematic risk can be diversified away.
Examples of systematic risk include uncertainty aboutgeneral economic conditions, such as GNP, interestrates or inflation.
On the other hand, announcements specific to acompany, such as a gold mining company striking gold,are examples of unsystematic risk.
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Risk: Systematic and Unsystematic
Systematic Risk; m
Nonsystematic Risk;
n
Total risk; U
We can break down the risk,U
, of holding a stock into twocomponents: systematic risk and unsystematic risk:
riskicunsystemattheis
risksystematictheis
where
becomes
m
mRR
URR
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Systematic Risk and Betas
The beta coefficient, b, tells us the response ofthe stocks return to a systematic risk.
In the CAPM, bmeasured the responsiveness ofa securitys return to a specific risk factor, the
return on the market portfolio.
)(
)(2
,
M
Mi
i
R
RRCov
b
We shall now consider many types of systematic risk.
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Systematic Risk and Betas
For example, suppose we have identified three
systematic risks on which we want to focus:1. Inflation2. GDPgrowth3. The dollar-pound spot exchange rate, S($,)
Our model is:
riskicunsystemattheis
betarateexchangespottheis
betaGDPtheis
betainflationtheis
FFFRR
mRR
S
GDP
I
SSGDPGDPII
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Systematic Risk and Betas: Example
Suppose we have made the following
estimates:
1. bI= -2.30
2. bGDP= 1.50
3. bS= 0.50.
Finally, the firm was able to attract asuperstar CEO and this unanticipated
development contributes 1% to the return.
FFFRR SSGDPGDPII
%1
%150.050.130.2 SGDPI
FFFRR
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Systematic Risk and Betas: Example
We must decide what surprises took place in thesystematic factors.
If it was the case that the inflation rate wasexpected to be by 3%, but in fact was 8%during the time period, then
FI = Surprise in the inflation rate
= actualexpected= 8% - 3%
= 5%
%150.050.130.2 SGDPI FFFRR
%150.050.1%530.2 SGDP
FFRR
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Systematic Risk and Betas: Example
If it was the case that the rate of GDPgrowth
was expected to be 4%, but in fact was
1%, then
FGDP= Surprise in the rate of GDPgrowth
= actualexpected
= 1% - 4%
= -3%
%150.050.1%530.2 SGDP FFRR
%150.0%)3(50.1%530.2 S
FRR
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Systematic Risk and Betas: Example
If it was the case that dollar-pound spotexchange rate, S($,), was expected to
increase by 10%, but in fact remainedstable during the time period, then
FS= Surprise in the exchange rate
= actualexpected= 0% - 10%
= -10%
%150.0%)3(50.1%530.2 S
FRR
%1%)10(50.0%)3(50.1%530.2 RR
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Systematic Risk and Betas: Example
Finally, if it was the case that the expected
return on the stock was 8%, then
%150.0%)3(50.1%530.2 S
FRR
%12%1%)10(50.0%)3(50.1%530.2%8
R
R
%8R
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Portfolios and Factor Models
Now let us consider what happens to portfolios of stockswhen each of the stocks follows a one-factor model.
We will create portfolios from a list of Nstocks and willcapture the systematic risk with a 1-factor model.
The ithstock in the list have returns:
iiii FRR
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Relationship Between the Return on
the Common Factor & Excess Return
Excess
return
The return on the factor F
i
iii
i FRR
If we assumethat 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 assumethat there is no
unsystematic
risk, then i= 0
FRRiii
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Relationship Between the Return on
the Common Factor & Excess Return
Excess
return
The return on the factor F
Differentsecurities will
have different
betas
0.1B
50.0C
5.1A
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Portfolios and Diversification
We know that the portfolio return is theweighted average of the returns on theindividual assets in the portfolio:
NNiiP RXRXRXRXR 2211
)(
)()( 22221111
NNNN
P
FRX
FRXFRXR
NNNNN
N
P
XFXRX
XFXRXXFXRXR
222222111111
iiii FRR
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Portfolios and Diversification
The return on anyportfolio is determined bythree 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.
FXXXNN
)( 2211
2. The weighted average of the betas times the factor.
NNXXX 2211
3. The weighted average of the unsystematic risks.
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Portfolios and Diversification
So the return on a diversifiedportfolio isdetermined by two sets of parameters:
1. The weighed average of expected returns.
2. The weighted average of the betas times thefactor F.
FXXX
RXRXRXR
NN
NNP
)( 2211
2211
In a large portfolio, the only source of uncertainty is the
portfolios sensitivity to the factor.
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Betas and Expected Returns
The return on a diversified portfolio is the sum ofthe expected return plus the sensitivity of theportfolio to the factor.
FXXRXRXR NNNNP )( 1111
FRRP
PP
NN
P RXRXR 11
thatRecall
NNP XX 11
and
PR P
b
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Relationship Between b& Expected
Return
If shareholders are ignoring unsystematic risk, only
the systematic risk of a stock can be related to its
expectedreturn.
FRR PPP
l h b d
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Relationship Between b& Expected
Return
Ex
pectedreturn
FR
AB
C
D
SML
)(F
PF
RRRR
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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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Empirical Approaches to Asset Pricing
Both the CAPM and APT are risk-based models.There are alternatives.
Empirical methods are based less on theory andmore on looking for some regularities in thehistorical record.
Be aware that correlation does not implycausality.
Related to empirical methods is the practice ofclassifying portfolios by style e.g. Value portfolio
Growth portfolio
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Summary and Conclusions
The APT assumes that stock returns aregenerated according to factor models such as:
FFFRRSSGDPGDPII
As securities are added to the portfolio, the unsystematicrisks of the individual securities offset each other. A fullydiversified portfolio has no unsystematic risk.
The CAPMcan be viewed as a special case of theAPT.
Empirical models try to capture the relations betweenreturns and stock attributes that can be measured directlyfrom the data without appeal to theory.
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