chapter 12 rwj

8/11/2019 Chapter 12 RWJ

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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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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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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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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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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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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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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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Excess

return


If we assumethat there is no

unsystematic

risk, then i= 0

FRRiii


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Excess

return


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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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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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.

chapter 12 rwj

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