investment and portfolio management 5

8/8/2019 Investment and Portfolio Management 5

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INVESTMENT ANALYSIS ANDPORTFOLIO MANAGEMENT

CAPITAL ASSET PRICING MODEL


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OBJECTIVES OF OUR STUDY

Whataretheassumptionsofthe capitalasset

pricingmodel? Whatisarisk-freeassetand whatareitsrisk-

return characteristics?

Whatisthe covarianceand correlation between

therisk-freeassetandariskyassetorportfolioofriskyassets?

Whatistheexpectedreturn whenyou combinetherisk-freeassetandaportfolioofrisky

assets? Whatisthestandarddeviation whenyou

combinetherisk-freeassetandaportfolioofriskyassets?


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OBJECTIVES OF OUR STUDY Whenyou combinetherisk-freeassetanda

portfolioofriskyassetsonthe Markowitzefficientfrontier, whatdoesthesetofpossibleportfolioslooklike?

Giventheinitialsetofportfoliopossibilities with

arisk-freeasset, whathappens whenyouaddfinancialleverage (thatis, borrow)?

Whatisthemarketportfolio, whatassetsareincludedinthisportfolio,and whatarethe

relative weightsforthealternativeassetsincluded?

Whatisthe capitalmarketline (CML)?

Whatdo wemean by completediversification?


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How do wemeasurediversificationforanindividualportfolio?

Whataresystematic andunsystematic risk?

Giventhe CML, whatistheseparationtheorem?

Giventhe CML, whatistherelevantriskmeasureforanindividualriskyasset?

Whatisthesecuritymarketline (SML),andhowdoesitdifferfromthe CML?

Whatisbeta, and whyisitreferredtoasastandardizedmeasureofsystematic risk?

How canyouusethe SML todeterminetheexpected (required) rateofreturnforarisky

asset?


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Usingthe SML, whatdo wemean byanundervaluedandovervaluedsecurity,andhowdo wedetermine whetheranassetisundervaluedorovervalued?

Whatisanassets characteristic line,andhow doyou computethe characteristic lineforanasset?

Whatistheimpactonthe characteristic linewhenyou computeitusingdifferentreturnintervals (suchas weeklyversusmonthly) and

whenyouemploydifferentproxies (thatis,benchmarks) forthemarketportfolio

Whathappenstothe capitalmarketline (CML)whenyouassumetherearedifferencesintherisk-free borrowingandlendingrates?


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Whatisazero-betaassetandhow doesitsuseimpactthe CML?

Whathappenstothesecurityline (SML) whenyouassumetransactions costs,heterogeneousexpectations,differentplanningperiods,andtaxes?

Whatarethemajorquestions considered whenempiricallytestingthe CAPM?

Whataretheempiricalresultsfromteststhatexaminethestabilityofbeta?

How doalternativepublishedestimatesofbetacompare?


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Whataretheresultsofstudiesthatexaminetherelationship betweensystematic riskandreturn?

Whatothervariables besides betahavehadasignificantimpactonreturns?

Whatisthetheoryregardingthe marketportfolioandhow doesthisdifferfromthe

marketproxyusedforthemarketportfolio?

Assumingthereisa benchmarkproblem,whatvariablesareaffected byit?


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RECALLING PORTFOLIO THEORY

One basic assumptionofportfoliotheoryisthatasaninvestoryou wanttomaximizethereturnsfromyour

investmentsforagivenlevelofrisk.

Investorsare basicallyrisk averse,meaningthat,given

a choice betweentwoassets withequalratesofreturn,

they willselecttheasset withthelowerlevelofrisk.

Thisdoesnotimplythateverybodyisriskaverseorthat

investorsare completelyriskaversiveregardingall

financial commitments.

Certainpeople buylotteryticketsandgambleatracetracksorin casinos, whereitisknownthattheexpected

returnsarenegative, whichmeansthatparticipantsare

willingtopayfortheexcitementoftheriskinvolved.


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Ourbasic assumptionisthatmostinvestors committinglargesumsofmoneytodevelopinganinvestmentportfolioareriskaversive.

Itgivesapositiverelationship betweenexpectedreturnandexpectedrisk.

Historically wealsofindthatthereis

generallyapositiverelationship betweentheratesofreturnonvariousassetsandtheirmeasuresofrisk.


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Riskisthe uncertainty of future outcomes Analternativedefinitionmight bethe probability of an

adverse outcome.

Intheearly 1960s,theinvestment communitytalked

aboutrisk, butthere wasnospecific measurefortheterm.

The basic portfoliomodel wasdeveloped byHarryMarkowitz, whoderivedtheexpectedrateofreturnforaportfolioofassetsandanexpectedriskmeasure.

Markowitzshowedthatthevarianceoftherateofreturnwasameaningfulmeasureofrisk.

Thisportfoliovarianceformulaindicatedtheimportanceofdiversifyingyourinvestmentstoreducethetotalriskofaportfolio butalsoshowedhowtoeffectivelydiversify.


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Althoughtherearenumerouspotential

measuresofrisk,varianceorstandard

deviationofreturnsareused because:

thismeasureissomewhatintuitive;

itisa correctand widelyrecognizedrisk

measure;and

ithas beenusedinmostofthetheoreticalassetpricingmodels.



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Expectedreturnofanindividualstock

ComputationofExpected Returnforan

individualriskystock

Expected Return

E( R )=



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Expectedreturnonportfolioofriskyassets

ComputationofExpected Returnforaportfolio

ofriskyassets:



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MeasurementofRisk

Varianceorthestandarddeviationofexpected

returnsisusedasthemeasureofrisk.

Measurementofriskforanindividual

Investment:

Piis the probability of the possible rate of return,

Ri



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COMPUTATIONOFTHE VARIANCEOFTHE EXPECTED RATE

OFRETURNFORAN INDIVIDUAL RISKY ASSET



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MeasurementofRisk (contd)

Measurement of Risk for a portfolio:

Two basic conceptsinstatistics covariance;and

correlation,

must beunderstood before wediscuss

theformulaforthevarianceoftherateofreturnforaportfolio.



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Covariance isameasureofthedegreeto whichtwovariables movetogetherrelativetotheirindividualmeanvaluesovertime. A positive covariancemeansthattheratesofreturnfortwo

investmentstendtomoveinthesamedirectionrelativetotheir

individualmeansduringthesametimeperiod. anegative covarianceindicatesthattheratesofreturnfortwo

investmentstendtomoveindifferentdirectionsrelativetotheirmeansduringspecifiedtimeintervalsovertime.

Themagnitude ofthe covariancedependsonthevariancesoftheindividualreturnseries,as wellasontherelationship

betweentheseries. Fortwoassets,iandj, the covarianceofratesofreturnis

definedas:


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Standard Deviation of a Portfolio


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Followingthedevelopmentofportfoliotheory by

Markowitz,twomajortheorieshave beenputforththatderiveamodelforthevaluationofriskyassets.

Infirstsession I willintroduceoneofthesetwomodelsthatis,the capitalassetpricingmodel (CAPM).

The backgroundonthe CAPM isimportantatthispoint

becausetheriskmeasureimplied bythismodelisanecessaryinputforoursubsequentdiscussiononthevaluationofriskyassets.

Thepresentation concerns capitalmarkettheoryandthecapitalassetpricingmodelthat wasdevelopedalmostconcurrently bythreeindividuals.

Subsequently,analternativemultifactorassetvaluationmodel wasproposed,thearbitragepricingtheory (APT).

Thishasledtothedevelopmentofnumerousothermultifactormodelsthatarethesubjectof2ndsession.


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Capitalmarkettheoryextendsportfoliotheory

anddevelopsamodelforpricingallriskyassets.Thefinalproduct,thecapital asset pricingmodel (CAPM), willallow youtodeterminetherequiredrateofreturnforanyriskyasset.

We begin withthe backgroundofcapitalmarkettheorythatincludestheunderlyingassumptionsofthetheoryandadiscussionofthefactorsthatledtoitsdevelopment.This includes ananalysis of the effect of assuming the

existence of a risk-free asset. Notably,assumingtheexistenceofarisk-free

ratehassignificantimplicationsforthepotentialreturnandriskandalternativerisk-returncombinations.


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AssumptionsofCapital Market Theory

1. Allinvestorsare Markowitzefficientinvestors who wanttotargetpointsontheefficientfrontier. Theexactlocationontheefficientfrontierand,therefore,thespecific portfolioselected willdependontheindividualinvestorsrisk-returnutilityfunction.

2. Investors can borrow orlendanyamountofmoneyattherisk-freerateofreturn(RFR).

3. Allinvestorshavehomogeneousexpectations;thatis,theyestimateidenticalprobabilitydistributionsforfutureratesofreturn.

4. Allinvestorshavethesameone-periodtimehorizon

5. Allinvestmentsareinfinitelydivisible,

6. Therearenotaxesortransaction costsinvolvedin buyingorsellingassets.

7. Thereisnoinflationorany changeininterestrates,orinflationisfullyanticipated.

8. Capitalmarketsareinequilibrium.


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Theassumptionofarisk-freeassetintheeconomyiscriticaltoassetpricingtheory. Therefore,first we willunderstandthemeaningofarisk-freeassetandshowsitseffectontheriskandreturnmeasures whenthisrisk-free

assetis combined withaportfolio. Astheexpectedreturnonarisk-freeassetisentirely

certain,thestandarddeviationofitsexpectedreturniszero . Therateofreturnearnedonsuchanassetshould betherisk-freerateofreturn(RFR),

When weintroducethisrisk-freeassetintotherisky worldofthe Markowitzportfoliomodel wefindthe covarianceoftherisk-freeasset withanyriskyassetorportfolioofassets willalwaysequalzero.


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Combining a Risk-Free Asset with a RiskyPortfolio

Whathappenstotheaveragerateofreturnand

thestandarddeviationofreturns whenyou

combinearisk-freeasset withaportfolioofriskyassets?

1. ExpectedReturn Liketheexpectedreturnfora

portfoliooftworiskyassets,theexpectedrateof

returnforaportfoliothatincludesarisk-freeassetisthe weightedaverageofthetworeturns:


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Combining a Risk-Free Asset with a Risky Portfolio

2. Standard Deviation theexpectedvarianceforatwo-assetportfoliois:

Substitutingforthevaluesforriskfreeassets:

Weknow thatthevarianceoftherisk-freeassetiszero,thatis, 2RF =0. Becausethe correlation betweentherisk-freeassetandanyriskyasset isalsozero,thefactorrRF,i intheprecedingequationalsoequalszero. Therefore,any componentofthevarianceformulathathaseitheroftheseterms willequalzero.Whenyoumaketheseadjustments,theformula becomes:

Thestandarddeviationis

Therefore,thestandarddeviationofaportfoliothat combinestherisk-freeasset withriskyassetsisthe linear proportion of the standarddeviation of the risky asset portfolio.


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Combining a Risk-Free Asset with a Risky Portfolio


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Combining a Risk-Free Asset with a Risky Portfolio Therefore,both return and risk increase in a

linear fashion along the original Line RFR-M, and

thisextensiondominateseverything below the

lineontheoriginalefficientfrontier. Thus,youhaveanew efficientfrontier:the

straightlinefromtheRFRtangentto Point M.

Thislineisreferredtoasthecapital market line

(CML)


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The Market Portfolio

Because Portfolio M liesatthepointoftangency,ithasthehighestportfoliopossibilityline,andeverybody willwanttoinvestin Portfolio M and borrow orlendto besomewhereonthe CML. Thisportfoliomust,therefore,

includeall risky assets. Thisportfoliothatincludesallriskyassetsisreferredtoas

themarket portfolio.

Becausethemarketportfolio containsallriskyassets,itisacompletely diversified portfolio, whichmeansthatall

theriskuniquetoindividualassetsintheportfolioisdiversifiedaway.

Specifically,a completelydiversifiedportfolio wouldhavea correlation withthemarketportfolioof+1.00. Thisislogical because completediversificationmeanstheeliminationofalltheunsystematic oruniquerisk.


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The Market Portfolio


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Now we canproceedtouseittodetermineanappropriateexpectedrateofreturnonariskyasset. Thissteptakesusintothecapital assetpricing model (CAPM), whichisamodelthatindicates whatshould betheexpectedorrequiredratesofreturnonriskyassets.

Toaccomplishtheforegoing, wedemonstratethecreationofasecuritymarketline (SML) thatvisuallyrepresentstherelationship betweenriskandtheexpectedortherequiredrateofreturnon

anasset. Theequationofthis SML,togetherwithestimates

forthereturnonarisk-freeassetandonthemarketportfolio, cangenerateexpectedorrequiredratesofreturnforanyasset basedonits

systematic risk.


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CAPITAL ASSET PRICING MODEL Weknow thattherelevantriskmeasureforanindividualriskyassetis

its covariance withthemarketportfolio (Covi,M). Therefore, we candraw therisk-returnrelationshipasshown below

withthesystematic covariancevariable (Covi,M) astheriskmeasure.


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Thereturnforthemarketportfolio (RM) should be

consistent withitsownrisk, whichisthe covarianceofthemarket withitself. Ifyourecalltheformulaforcovariance,you willseethatthe covarianceofanyasset withitselfisitsvariance,

Covi,i= i2.

Inturn,the covarianceofthemarket withitselfisthevarianceofthemarketrateofreturn CovM,M = M2.

Therefore,theequationfortherisk-returnlineinthepreviousgraph will be:

Defining CoviM / M2 as beta, (i),thisequation can be

stated:

E(Ri) =RFR+ i(RM RFR)


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Beta can beviewedasastandardizedmeasureofsystematicrisk.

Specifically, wealreadyknow thatthe covarianceofanyassetiwiththemarketportfolio (CoviM) istherelevantriskmeasure.

Betaisastandardizedmeasureofrisk becauseitrelatesthiscovariancetothevarianceofthemarketportfolio.

Asaresult,themarketportfoliohasa betaof1. Therefore,ifthe

iforanassetisabove 1.0,theassethashighernormalized

systematic riskthanthemarket, whichmeansthatitismorevolatilethantheoverallmarketportfolio.

Giventhisstandardizedmeasureofsystematic risk,the SMLgraph can beexpressedasshownonnextpage. Thisisthesamegraphasshownonpreviouspage,exceptthereisadifferentmeasureofrisk. Specifically,thegraph (nextpage)replacesthe covarianceofanassetsreturns withthemarketportfolioastheriskmeasure withthestandardizedmeasureofsystematic risk (beta), whichisthe covarianceofanasset withthemarketportfoliodivided bythevarianceof themarketportfolio.


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Determiningthe Expected RateofReturnfora Risky Asset Todemonstratehow you would computetheexpectedorrequired

ratesofreturn, considerthefollowingexamplestocksassumingyouhavealready computed betas:

Assumethat weexpecttheeconomysRFRto be 6 percent (0.06) andthereturnonthemarketportfolio (RM) to be 12 percent (0.12). Thisimpliesamarketriskpremiumof6 percent (0.06).Withtheseinputs,the SML equation wouldyieldthefollowingexpected (required) rates

ofreturnforthesefivestocks:


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Stock A haslowerriskthantheaggregatemarket,soyoushouldnotexpect (require) itsreturnto beashighasthereturnonthemarketportfolioofriskyassets. Youshouldexpect (require) Stock A toreturn 10.2 percent.

Stock B hassystematic riskequaltothemarkets (beta=

1.00),soitsrequiredrateofreturnshouldlikewise beequaltotheexpectedmarketreturn (12 percent).

Stocks C and D havesystematic riskgreaterthanthemarkets,sotheyshouldprovidereturns consistent withtheirrisk.

Finally, Stock E hasanegative beta (whichisquiterareinpractice),soitsrequiredrateofreturn,ifsuchastockcould befound, would be below theRFR.


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EMPIRICAL TESTSOFTHECAPM Whentestingthe CAPM,therearetwomajorquestions.

First,How stable is the measure of systematic risk(beta)?Because betaisourprincipalriskmeasure,itisimportanttoknow whetherpast betas can beusedasestimatesoffuture betas.Also,how dothealternativepublishedestimatesofbeta compare?

Second,Is there a positive linear relationship ashypothesized between beta and the rate of return onrisky assets?Morespecifically,how welldoreturnsconformtothefollowing SML equation,discussedearlier


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EMPIRICAL TESTSOFTHECAPM Numerousstudieshaveexaminedthestabilityofbeta

andgenerally concludedthattheriskmeasure wasnotstableforindividualstocks butthestabilityofthe betafor

portfolios ofstocksincreaseddramatically. Thelargertheportfolioofstocks (e.g.,over50 stocks)

Thelongertheperiod (over26 weeks),themorestablethe betaoftheportfolio.

Also,the betastendedtoregresstowardthemean.

Specifically,high-betaportfoliostendedtodeclineovertimetowardunity (1.00), whereaslow-betaportfoliostendedtoincreaseovertimetowardunity.


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Anotherfactorthataffectsthestabilityofbetaishowmanymonthsareusedtoestimatethe original betaandthetest beta. Roenfeldt, Griepentrog,and Pflamm (RGP)compared betasderivedfrom48monthsofdatatosubsequent betasfor12, 24,36,and48months.

The48- month betas werenotgoodforestimatingsubsequent 12-month betas but werequitegoodforestimating 24-,36-,and48-month betas.

Tosummarize,individual betas weregenerallyvolatileovertime whereaslargeportfolio betas werestable.

Also,itisimportanttouseatleast36 monthsofdatatoestimate betaand be consciousofthestockstradingvolumeandsize.


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ARBITRAGEPRICING THEORY

Whatisthearbitragepricingtheory (APT)

and whatareitssimilaritiesand

differencesrelativetothe CAPM?


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ARBITRAGEPRICING THEORY Arbitrage Pricing Theory (APT), was

developed by Rossinthemid-1970s withthreemajorassumptions:1. Capitalmarketsareperfectly competitive.

2. Investorsalwaysprefermore wealthtoless wealthwith certainty.

3. Thestochastic processgeneratingassetreturns can

beexpressedasalinearfunctionofasetofKriskfactors (orindexes).


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ARBITRAGEPRICING THEORY

investment and portfolio management 5

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