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Discussion Issues and Derivations 1. A Derivation of the Capital Asset Pricing Model I. Establish the Objects of Choice: Mean versus Variance Theme : Investors are risk averse. They measure reward using expected return and risk using variance. Underlying assumptions : The meanvariance assumption can hold only if (a) all investors have quadratic utility function or (b) returns are normally distributed. Implication : Portfolio A with higher expected return and the same variance as portfolio B will be preferred to B II. Benefits of Diversification For any desired level of risk (s) there exists a portfolio of several assets which yields a higher expected return than any individual security E(R p )= w i E(R i ) 2 p = w i w j Cov ij Efficient portfolios : maximize returns for any level of risk. Implications : (a) Everybody should diversify (b) Investors should try to identify and hold efficient portfolios (c) This method has very heavy computational requirements. III. The Single Index Model: The Logical Limit of Diversification Assumptions : (a) Riskfree lending and borrowing (b) Markets which are frictionless there are no transactions costs (c)Homogeneous expectations Implications : (1) The risky portfolio than when combined with the riskless asset maximizes returns is the market portfolio. (2) Everybody holds some combination of the market portfolio and the risky asset. How much of each is held will be a function of the investor's risk aversion. (3) Since all investors hold the same market portfolio it must contain all assets in the economy in proportion to their value. IV. The Risk of an Individual Asset Step 1: Individuals diversify and hold portfolios Step 2: The risk of a security is the risk it adds to the portfolio Step 3: Everybody holds the market portfolio Step 4: The risk of a security is the risk that it adds to the market portfolio. Step 5: The covariance between an asset "i" and the market portfolio (Cov im ) is a measure of this added risk. The higher the covariance the higher the risk. Step 6: This measure can be standardized by dividing by the market variance. b = Cov im / 2 m . Variants of the Capital Asset Pricing Model I. No Riskless Asset Basis : If no riskless asset exists investors can use a portfolio of risky assets which is uncorrelated with the market portfolio instead as the riskless asset. This portfolio is called the zerobeta portfolio. Properties of the Zerobeta portfolio (1) Of all the the zerobeta portfolios this has the minimum variance (2) The separation principle applies here with the two portfolios, the market portfolio and the zero beta portfolio, i.e. all investors hold combinations of the two. (3) The expected return on any security can be expressed as a linear function of its beta. E(R i ) = E(R z )+ (E(R m ) E(R z )) where E(R z ) is the expected return on a zero beta portfolio II. Riskless Lending but no Riskless Borrowing Basis : (a) There is a piecewise linear relationship between expected return and beta for efficient portfolios. (b) Efficient portfolios with the riskfree asset lie along the segment RfT and those containing only risky assets lies along the segment TMC.

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  • 5/17/2015 Chapter3Derivations

    http://pages.stern.nyu.edu/~adamodar/New_Home_Page/AppldCF/derivn/ch3deriv.html#ch3.1 1/3

    DiscussionIssuesandDerivations

    1. ADerivationoftheCapitalAssetPricingModelI.EstablishtheObjectsofChoice:MeanversusVarianceTheme:Investorsareriskaverse.Theymeasurerewardusingexpectedreturnandriskusingvariance.Underlyingassumptions:Themeanvarianceassumptioncanholdonlyif(a)allinvestorshavequadraticutilityfunctionor(b)returnsarenormallydistributed.Implication:PortfolioAwithhigherexpectedreturnandthesamevarianceasportfolioBwillbepreferredtoBII.BenefitsofDiversificationForanydesiredlevelofrisk(s)thereexistsaportfolioofseveralassetswhichyieldsahigherexpectedreturnthananyindividualsecurityE(Rp)=wiE(Ri)2p=wiwjCovijEfficientportfolios:maximizereturnsforanylevelofrisk.Implications:(a)Everybodyshoulddiversify(b)Investorsshouldtrytoidentifyandholdefficientportfolios(c)Thismethodhasveryheavycomputationalrequirements.III.TheSingleIndexModel:TheLogicalLimitofDiversificationAssumptions:(a)Riskfreelendingandborrowing(b)Marketswhicharefrictionlesstherearenotransactionscosts(c)HomogeneousexpectationsImplications:(1)Theriskyportfoliothanwhencombinedwiththerisklessassetmaximizesreturnsisthemarketportfolio.(2)Everybodyholdssomecombinationofthemarketportfolioandtheriskyasset.Howmuchofeachisheldwillbeafunctionoftheinvestor'sriskaversion.(3)Sinceallinvestorsholdthesamemarketportfolioitmustcontainallassetsintheeconomyinproportiontotheirvalue.IV.TheRiskofanIndividualAssetStep1:IndividualsdiversifyandholdportfoliosStep2:TheriskofasecurityistheriskitaddstotheportfolioStep3:EverybodyholdsthemarketportfolioStep4:Theriskofasecurityistheriskthatitaddstothemarketportfolio.Step5:Thecovariancebetweenanasset"i"andthemarketportfolio(Covim)isameasureofthisaddedrisk.Thehigherthecovariancethehighertherisk.Step6:Thismeasurecanbestandardizedbydividingbythemarketvariance.b=Covim/2m.

    VariantsoftheCapitalAssetPricingModelI.NoRisklessAssetBasis:Ifnorisklessassetexistsinvestorscanuseaportfolioofriskyassetswhichisuncorrelatedwiththemarketportfolioinsteadastherisklessasset.Thisportfolioiscalledthezerobetaportfolio.PropertiesoftheZerobetaportfolio(1)Ofallthethezerobetaportfoliosthishastheminimumvariance(2)Theseparationprincipleappliesherewiththetwoportfolios,themarketportfolioandthezerobetaportfolio,i.e.allinvestorsholdcombinationsofthetwo.(3)Theexpectedreturnonanysecuritycanbeexpressedasalinearfunctionofitsbeta.E(Ri)=E(Rz)+(E(Rm)E(Rz))whereE(Rz)istheexpectedreturnonazerobetaportfolioII.RisklessLendingbutnoRisklessBorrowingBasis:(a)Thereisapiecewiselinearrelationshipbetweenexpectedreturnandbetaforefficientportfolios.(b)EfficientportfolioswiththeriskfreeassetliealongthesegmentRfTandthosecontainingonlyriskyassetsliesalongthesegmentTMC.

  • 5/17/2015 Chapter3Derivations

    http://pages.stern.nyu.edu/~adamodar/New_Home_Page/AppldCF/derivn/ch3deriv.html#ch3.1 2/3

    III.ExistenceofNonMarketableAssets(suchasHumanCapital)Theseparationprinciplestillholdsbut,(a)Investorsholddifferentportfoliosofriskyassetsdependingupontheportfoliosofnonmarketableassetsthattheypossess.(b)ThemarketpriceofriskincludesthevarianceofthemarketandthecovariancebetweenthemarketportfolioandtheportfolioofnonmarketableassetsIV.ExistenceofTaxesModel:Themodelconsidersdifferentialtaxesondividendsandcapitalgainsinaoneperiodcontextwhereinvestorsmaximizetheironeperiodreturns.ThefinalmodelforexpectedreturnhasadividendcomponentE(Ri)=a+i(E(Rm)Rf)+c(diRf)wheredi=DividendyieldonassetiRf=AftertaxriskfreerateV.ExistenceofHeterogeneousExpectationsandInformationModel:Togetstrongconclusionswehavetoassumethatallinvestorshaveacertainclassofutilityfunctions(ConstantAbsoluteriskaversion)andcompletemarkets(Atleastasmanyindependentsecuritiesasstates).

    TestingtheCAPM:IssuesandDiscussionIssue1:TheCAPMcanneverbetestedbecausethemarketportfoliocanneverbeobservedCentraltotheCAPMistheconceptofamarketportfoliowhichincludeseveryassetintheeconomy.TotesttheCAPMthereforeonehastoobserveandbeabletomeasurethisefficientmarketportfolio.IfonecannotdosoonecannottesttheCAPM.OnecannotuseofaninefficientportfolioliketheS&P500ortheNYSE2000oreveneverystockintheeconomytoestimatebetasandtestforlinearity(likeallthestudieshavedone)because(a)ThebetasmeasuredagainstaninefficientportfolioaremeaninglessmeasuresandcannotbeusedtoacceptorrejecttheCAPMwhichisreallyatheoryaboutbetasmeasuredagainsttheefficientmarketportfolio(b)Foreveryinefficientportfoliothereexistsasetofbetaswhichwillsatisfythelinearitycondition.Issue2:TheCAPMisdifficulttotestonindividualassetsThenoisinessinbetaestimatesandthefactthattheCAPMyieldsexpectedreturnsforindividualassetsoverthelongtermmakesitdifficulttotesttheCAPMbytryingtorelateexpectedreturnsonindividualassets(suchasstocks)totheirbetas.WhatmosttestsoftheCAPMdoinsteadistolookatportfoliosofstocks,baseduponbetas,andthencomparethesebetastoexpectedreturnsinthenexttimeperiod.

    MoreonFactorAnalysisandtheArbitragePricingModelCentraltoapplyingthearbitragepricingmodelistheuseofafactoranalysis.Inatypicalfactoranalysis,webeginwithpricingdataonalargenumberofassetsoververylongtimeperiods.Inthefactoranalysis,welookforfactorsthatseemtomovepricesonlargenumbersofassetsinunison.Topreventfactorsfrombeingdoublecounted,weensurethatthefactorsthatemergeareindependentofeachother.Whileallofthisoccursbehindthescreenofthefactoranalysis,whatemergesasoutputfromtheanalysisincludes:(a)thenumberofcommonfactorsthatappearedtoaffectassetpricesovertheperiodforwhichthedataisavailable(b)thebetasofeachassetrelativetoeachfactor,againusingthesamedata(c)the"riskpremiums"associatedwitheachfactorThesefactorbetasandfactorpremiumsarethenused,inconjunctionwithariskfreeratetogetanexpectedreturnforanasset.

    EstimatingtheMacroEconomicFactorsinaMultiFactorModelOncethenumberoffactorshavebeenidentifiedinanarbitragepricingmodel,thetimeseriesbehaviorofeachfactorcanbederivedfromthefactoranalysis.Thesearchthenbeginsformacroeconomicfactorsthatexhibitthesametimeseriesbehavior.Oncemacroeconomicfactorshavebeenmatchedupwiththeunnamedfactorsinthefactoranalysis,thebetasofeachassetarereestimatedagainsttheidentifiedmacroeconomicfactors.Thebetaestimationmaybedonebyrunningamultiple

  • 5/17/2015 Chapter3Derivations

    http://pages.stern.nyu.edu/~adamodar/New_Home_Page/AppldCF/derivn/ch3deriv.html#ch3.1 3/3

    regressionofstockreturns(foreachstock)againstchangesinmacroeconomicvariables(suchasinterestrates,inflationratesandGNPgrowth)overtime.Thecoefficientsontheseregressionsyieldthebetas,andriskpremiumscanbeestimatedalsofromthehistoricaldata.

    BuildingaRegressionModelGenerally,regressionmodelsbeginwiththecrosssectionaldifferencesinreturnsacrossstocksatanypointintime,andtrytoexplainthesedifferencesusingdifferencesonmeasurablefinancialcharacteristicsofthefirmsissuingtheseassets.Asanexample,FamaandFrench,intheirmuchquotedstudy,useddifferencesinmarketcapitalizationandpricetobookratiostoexplaindifferencesinreturnsacrossstocks.Themoredifficultquestionisdecidingwhichfinancialvariablestouseinexplainingreturns.ThebestplacetostartistolookattheempiricalevidencethathasbeenaccumulatedovertimeonmarketefficiencyandtheCAPM.ThisevidencesuggeststhatLowmarketcapitalizationstocksseemtoearnhigherreturns,onaverage,thanhighmarketcapitalizationstocksLowPE,PBVandPSratiostocksseemtoearnhigherreturns,onaverage,thanhighPE,PBVandPSratiostocksHighdividendyieldstocksseemtoearnhigherreturns,onaverage,thanlowdividendyieldstocksWhiletheinitialregressionmayincludeallofthesevariables,manyofthesevariablestendtobecorrelatedwitheachother.Thus,lowPEstockstendtoalsobelowPBVratiostockswhichpayhighdividends.Intheinterestsofefficiency(andtopreventproblemsintheregressionfromindependentvariablesbeingcorrelatedwitheachother),itmakessensetousethemeasurethatismosthighlycorrelatedwithreturnsanddroptheothers.Thus,theuseofpricetobookvalueratiosbyFamaandFrench.

    Whynotusebondbetastoarriveatthecostofdebt?Giventhatweusestockbetastoarriveatexpectedreturnsforstocks,thequestionmayariseastowhywedonotusebondbetastogetexpectedreturnsforbonds.Thereasonliesintheabsenceorpresenceofsymmetryinreturnsforeachoftheseassetclasses.Stocks,whichhavepotentiallyunlimitedupsidepotentialaswellassignificantdownsidepotential,havemuchmoresymmetricreturnsthanbonds.Thus,theytendtofitinmuchmorecleanlyintothemeanvarianceframeworkthandobonds.Corporatebondshavesomeupsidepotential,butitislimitedbythefactthatbondscanatbestbecomedefaultfree.Thus,theupsidepotentialforaAAratedbondisfairlylimited.Consequently,theriskmeasurethatwehavetousehastobeadownsideriskmeasure,whichiswhatdefaultriskandratingsmeasure.Clearly,thelowertheratingofabond,thegreatertheupsidepotential,andthus,thegreaterthelikelihoodthatwecanestimatebondbetasandexpectedreturnsonthem.Forajunkbond,forinstance,itmaybepossibletoestimateabetalikeastockbetaandgetanexpectedreturnfromit.

    CreditScoresasAlternativestoBondRatingsBondratingsareatoolthatweusetomeasuredefaultriskandarriveatacostofdebt.Lenders(suchasbanks)havehistoricallyusedcreditscoresasameasureofdefaultrisk,especiallywhenlendingtoindividualsandprivatebusinessess.Acreditscoreisderivedbymeasuringhowaborrowerscoresonavarietyofmeasures,whichovertimehavebeencorrelatedwithdefaultrisk.


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