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MPT an CAPM

An Introduction to Capital AssetPricing Model

• The capital asset pricing model is portrayal ofhow nancial markets price securities andthereby determine expected return on capitalinvestments

• The model provides a methodology for!uantifying risk and translating that risk intoestimates of expected return on e!uity

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MPT an CAPM

Assumptions of CAPM• All investors are Markowit" e#cient investors who want to target

points on the e#cient frontier • Investors can borrow or lend any amount of money at the risk$free

rate of return %&'&(• All investors have homogeneous expectations) that is* they

estimate identical probability distributions for future rates of return• All investors have the same one$period time hori"on such as one$

month* six months* or one year• All investments are in nitely divisible* which means that it is

possible to buy or sell fractional shares of any asset or portfolio• There are no taxes or transaction costs involved in buying or selling

assets• There is no in+ation or any change in interest rates* or in+ation is

fully anticipated• Capital markets are in e!uilibrium

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MPT an CAPM

&isk$'ree Asset• An asset with "ero standard deviation• ,ero correlation with all other risky assets• Provides the risk$free rate of return %&'&(

• -ill lie on the vertical axis of a portfolio graph

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MPT an CAPM

&isk$'ree Asset• Covariance between two sets of returns is

Because the returns for the risk free asset are certain,

Thus R i = E(R i), and Ri - E(Ri) = 0

Conse!uently* the covariance of the risk$free asset with any risky assetor portfolio will always e!ual "ero .imilarly the correlation between anyrisky asset and the risk$free asset would be "ero .

∑==

n

1i j jiiij )]/nE(R -)][R E(R -[R Cov

0RF

=σ

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

• /xpected return• the weighted average of the two returns

This is a linear relationship

))E(R W-(1(RFR)W)E(R iRFRF port +=

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

• The expected variance for a two$asset portfolio is

.ubstituting the risk$free asset for .ecurity 0* and the risky asset for

.ecurity 1* this formula would become

.ince we know that the variance of the risk$free asset is "ero andthe correlation between the risk$free asset and any risky asset i is"ero we can ad2ust the formula

211,221

2

2

2

2

2

1

2

1

2

port r ww2ww)E( σ σ σ σ σ ++=

i RF i RF σ σ σ σ σ iRF,RFRF22

RF22

RF2

port )r w-(1w2)w1(w)E( +−+=

22RF

2 port )w1()E( iσ σ −=

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

• 3iven the variance formula

The standard deviationis

Therefore* the standard deviation of a portfolio that combines therisk$free asset with risky assets is the linear proportion of the

standard deviation of the risky asset portfolio

22RF

2 port )w1()E( iσ σ −=

22RF port )w1()E( iσ σ −=

iσ )w1( RF−=

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

.ince both the expected return and the standarddeviation of return for such a portfolio are linearcombinations* a graph of possible portfolio returnsand risks looks like a straight line between the twoassets

Note to Portfolio Theory - CASE

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

RFR

M

C

AB

.ince both the expected return and the standarddeviation of return for such a portfolio are linearcombinations* a graph of possible portfolio returnsand risks looks like a straight line between the twoassets

)E( portσ

)E(R port

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MPT an CAPM

Combining a &isk$'ree Assetwith a &isky Portfolio

RFR

M

C

AB

.ince both the expected return and the standarddeviation of return for such a portfolio are linearcombinations* a graph of possible portfolio returnsand risks looks like a straight line between the twoassets

To attain a higherexpected returnthan is available atpoint M %in exchangefor accepting higherrisk( either investalong the e#cientfrontier beyondpoint M* such aspoint 4 or* addleverage to theportfolio by

borrowing money atthe risk$free rate)E( portσ

)E(R port

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MPT an CAPM

Portfolio Possibilities Combining the&isk$'ree Asset and &isky Portfolios on

the /#cient 'rontier

RFR

M

C M !

B o r r o "

i n #

! e n d i n #

)E( portσ

)E(R port

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MPT an CAPM

The Market Portfolio• 5ecause portfolio M lies at the point of tangency*

it has the highest portfolio possibility line• /verybody will want to invest in Portfolio M and

borrow or lend to be somewhere on the CM6• Therefore this portfolio must include A66 &I.78

A../T.• 5ecause the market is in e!uilibrium* all assets

are included in this portfolio in proportion to theirmarket value

• 5ecause it contains all risky assets* it is acompletely diversi ed portfolio* which means thatall the uni!ue risk of individual assets%unsystematic risk( is diversi ed away

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MPT an CAPM

.ystematic &isk• 9nly systematic risk remains in the market

portfolio• .ystematic risk is the variability in all risky assets

caused by macroeconomic variables• .ystematic risk can be measured by the standard

deviation of returns of the market portfolio andcan change over time

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MPT an CAPM:umber of .tocks in a Portfolio andthe .tandard 4eviation of Portfolio

&eturn.tandard 4eviation of &eturn

:umber of .tocks in the Portfolio

.tandard 4eviation of theMarket Portfolio %systematicrisk(

.ystematic &isk

Total &isk

;nsystematic%diversi able(&isk

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Competition may bestronger or weaker

than anticipated /xchange rateand Political risk

Pro2ects maydo better or

worse thanexpected

/ntire .ectormay be a

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MPT an CAPM

The CM6 and the .eparation Theorem

• The CM6 leads all investors to invest in the Mportfolio

• Individual investors should diow an investor gets to a point on the CM6 isbased on nancing decisions

• &isk averse investors will lend part of the portfolioat the risk$free rate and invest the remainder inthe market portfolio

MPT CAPM

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MPT an CAPM

The CM6 and the .eparation

Theorem• Investors preferring more risk might borrow fundsat the &'& and invest everything in the marketportfolio

• Investors preferring less risk might lend funds atthe &'& and invest remaining in the marketportfolio

• The decision of both investors is to invest inportfolio M along the CM6 %the investmentdecision(

• The decision to borrow or lend to obtain a pointon the CM6 is a separate decision based on riskpreferences % nancing decision(

• Tobin refers to this separation of the investment

MPT CAPM

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MPT an CAPM

The Capital Asset Pricing Model?/xpected &eturn and &isk

• The existence of a risk$free asset resulted inderiving a capital market line %CM6( that becamethe relevant frontier

• An asset $s covariance with the market portfolio isthe relevant risk measure

• This can be used to determine an appropriateexpected rate of return on a risky asset $ thecapital asset pricing model %CAPM(

MPT CAPM

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MPT an CAPM

The Capital Asset Pricing Model?/xpected &eturn and &isk

• CAPM indicates what should be the expected orre!uired rates of return on risky assets

• This helps to value an asset by providing anappropriate discount rate to use in dividend

valuation models• 8ou can compare an estimated rate of return to

the re!uired rate of return implied by CAPM $over@under valued

MPT CAPM

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MPT an CAPM

The .ecurity Market 6ine%.M6(

• The relevant risk measure for an individual riskyasset is its covariance with the market portfolio%Covi*m(

• This is shown as the risk measure• The return for the market portfolio should be

consistent with its own risk* which is thecovariance of the market with itself $ or itsvariance?

2

mσ

MPT an CAPM

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MPT an CAPM

3raph of .ecurity Market 6ine %.M6(

&'&

.M6

)E(R i

i$Cov2

mσ

$R

MPT an CAPM

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MPT an CAPM

The .ecurity Market 6ine%.M6(

The e!uation for the risk$return line is

RFR)-(R RFR )E(R %i iβ +=

MPT an CAPM

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MPT an CAPM

3raph of .M6 with.ystematic &isk

.M6

:egative5eta

&'&

)E(R i

)&!ta(Cov 2%i$/ σ

0'1

$R

0

MPT an CAPM

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MPT an CAPM

4etermining the /xpected&ate of &eturn for a &isky Asset

• The expected rate of return of a risk asset isdetermined by the &'& plus a risk premium forthe individual asset

• The risk premium is determined by thesystematic risk of the asset %beta( and theprevailing market risk premium %& M$&'&(

RFR)-(R RFR )E(R %i iβ +=

MPT an CAPM

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MPT an CAPM

4etermining the /xpected&ate of &eturn for a &isky Asset

Assume? &'& B D %E E (&M B 01D %E 01(

Implied market risk premium B D

%E E (

E(R A) 0'0 * 0'+0 (0'12-0'0 ) 0'102 10'2,

E(R &) 0'0 * 1'00 (0'12-0'0 ) 0'120 12'0,

E(R C) 0'0 * 1'1 (0'12-0'0 ) 0'12. 12'.,

E(R ) 0'0 * 1' 0 (0'12-0'0 ) 0'1 1 ' ,

E(R E) 0'0 * -0' 0 (0'12-0'0 ) 0'0 2 '2,

Stock Beta

A 0'+0& 1'00C 1'1

1' 0E -0' 0 RFR)-(R RFR )E(R %i iβ +=

MPT an CAPM

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MPT an CAPM

4etermining the /xpected&ate of &eturn for a &isky Asset• In e!uilibrium* all assets and all portfolios

of assets should plot on the .M6• Any security with an estimated return

that plots above the .M6 is underpriced• Any security with an estimated return

that plots below the .M6 is overpriced• A superior investor must derive value

estimates for assets that are consistentlysuperior to the consensus marketevaluation to earn better risk$ad2usted

rates of return than the average investor

MPT an CAPM

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MPT an CAPM

Identifying ;ndervalued and9vervalued Assets

• Compare the re!uired rate of return to theexpected rate of return for a speci c risky assetusing the .M6 over a speci c investmenthori"on to determine if it is an appropriateinvestment

• Independent estimates of return for thesecurities provide price and dividend outlooks

MPT an CAPM

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MPT an CAPM

Plot of /stimated &eturnson .M6 3raph

.M6

1E FE E GE 0 1E 0 FE 0 E 0 GE$ FE$ 1E

11 1E 0G

0 0F 01&m

0E

EG E EF E1

A

5

C

4

/

)E(R i

&!ta0'1

$R

0

MPT an CAPM

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MPT an CAPM

.catter Plot of &ates of&eturn

&M

&i The characteristic line isthe regression line of thebest t through a scatterplot of rates of return

MPT an CAPM

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The Impact of the TimeInterval

• :umber of observations and timeinterval used in regression vary

• Halue 6ine Investment .ervices %H6(

uses weekly rates of return over veyears• Merrill 6ynch* Pierce* 'enner = .mith

%M6( uses monthly return over ve years

• There is no %correct & interval for analysis• -eak relationship between H6 = M6betas due to di

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The /

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-eaknesses of ;sing .=P EEas the Market Proxy

' Includes only ; . stocks ' The theoretical market portfolio should

include ; . and non$; . stocks and bonds*real estate* coins* stamps* art* anti!ues* andany other marketable risky asset from aroundthe world

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&elaxing the Assumptions

• 4i

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&elaxing the Assumptions

• >eterogeneous /xpectations andPlanning Periods ' will have an impact on the CM6 and .M6

• Taxes ' could cause ma2or di

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/mpirical Tests of the CAPM

• .tability of 5eta ' betas for individual stocks are not

stable* but portfolio betas are

reasonably stable 'urther* thelarger the portfolio of stocks andlonger the period* the more stablethe beta of the portfolio

• Comparability of Published/stimates of 5eta

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&elationship 5etween .ystematic&isk and &eturn

• /

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&elationship 5etween .ystematic&isk and &eturn

• /

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The Market Portfolio? Theory versusPractice

• There is a controversy over the marketportfolio >ence* proxies are used

• There is no unanimity about which proxy to use• An incorrect market proxy will a