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Case Study: Calculating CAPM Beta in Excel
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Transcript of Case Study: Calculating CAPM Beta in Excel
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CalculatingCAPMBetaTutorial 1 SpiderFinancialCorp,2013
CalculatingCAPMBetaInthispaper,wewilllookatthecapitalassetpricingmodel(CAPM),asimplebutwidelyusedfactormodelinfinance.CAPMsmainstrengthanditsprimaryweaknessisthatitassumesonesinglesourceofrisk(i.e.marketrisk)andthenbucketseverythingelseasidiosyncratic(i.e.nonsystematic).Thispaperwillpavethewaytomoreadvancedfactormodelingtechniquesincomingissues.Wewillbeginbydiscussingtheunderlyingassumptions,definesystematicandidiosyncraticrisk,andoutlinetheirinfluenceonthecovarianceamongassets.Next,usingasimpleregressionmodel,wewillattempttocomputetheCAPMsensitivityfactor(Beta)fortwodifferenttechstocks:MicrosoftandIBM.OurcoalinapplyingCAPMtothesetechstocksistocomputeeachassetssensitivity(i.e.Beta)tonondiversifiablemarketrisk.Todothat,wewilluseasimplelinearregressionmodel,thenanormalprocesstovalidatethemodelsassumptionsandensureitsstabilityoverthedatasample.Forsampledata,weusedthemonthlyreturnsbetweenJuly2001andMay2013(140observations).Forthemarketrisk,weselectedmonthlyreturnsoftheRussell3000Index,andforriskfree,weoptedforthe4weektreasurybills(TBILL)returns.
BackgroundInfinance,thecapitalassetpricingmodel(CAPM)isusedtodeterminetheappropriaterequiredrateofreturnofanasset(oraportfolio).TheCAPMtakesintoaccounttheassetssensitivitytothenondiversifiablerisk(akasystematicormarketrisk).
[ ] ( [ ] )
[ ][ ]
T T T Ti f i M f
T Ti f
i T TM f
E R R E R R
E R RE R R
Where
[ ]TiE R istheexpectedreturnofanassetIoveraholdingperiodT. TfR istheriskfreereturnovertheperiodT. i isthesensitivityoftheassetsexcessreturnovertheexpectedexcessmarketreturn. [ ]TME R istheexpectedmarketreturnoveraholdingperiodT. [ ]T TM fE R R isthemarketpremium(expectedexcessmarketreturn). [ ]T Ti fE R R isreferredtoastheriskpremium(expectedexcessassetsreturn).Inotherwords,
theassetsriskpremiumequalsthemarketpremiummultipliedbyitsbeta.
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CalculatingCAPMBetaTutorial 2 SpiderFinancialCorp,2013
Theequationabovedescribesasimplelinearregressionmodel(withzerointercept),betweentheassetsexcessreturnsandtheexcessmarketreturn.
2
( )
~ . . ~ (0, )
T T T Ti f i M f i
i
R R R R
i i d N
2 isoftenreferredtoastheidiosyncraticrisk(i.e.riskthatisspecifictotheassetitself,ratherthanthe
overallmarket).
Finally,the i istheslope(sensitivity)andcanbeexpressedasfollows: ( , )
( )
T Ti M
i TM
Cov R RVar R
Furthermore,fortwoassets,thecovariancecanbecomputedusingCAPMasfollows:
2
2
( , ) [ ] [( )( )]
( , ) [( ( ) )( ( ) )]
( , ) [ ( ) ( ) ( ) ]
( , ) [( ) ] ( )
i j i j i f j f
i j i M f i j M f j
i j i j M f i M f j j M f i i j
i j i j M f i j M
Cov R R E R R E R R R RCov R R E R R R R
Cov R R E R R R R R R
Cov R R E R R Var R
BasedontheCAPM,thevariance(orrisk)ofeachassetconsistsoftwocomponents:systematicandidiosyncraticrisk.
2 2( ) ( )T Ti i MVar R Var R Whydowecare?BasedontheCAPMtheory,wecancomputenotonlytheexpectedreturns,butalsoconstructacovariancematrixofthedifferentassets.Notethatthevarianceofeachassetconsistsoftwocomponents.Case1:MicrosoftMicrosoftCorporationdevelops,licenses,andsupportssoftwareproductsandservices,aswellasdesigningandsellinghardwareworldwide.Microsoftisapubliclytradedcompany,listedonNASDAQwithamarketcapitalof290B.LetsplotthemonthlyexcessreturnsofMicrosoftandRussell3000(marketproxy):
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CalculatingCAPMBetaTutorial 3 SpiderFinancialCorp,2013
Next,weplotthescatterplotforthetwodatasetsanddrawalineartrendlinetooutlinethecorrelationbetweenthetwo:
UsingthelinearregressionwizardinNumXL,designatethemonthlyexcessreturnsofMicrosoftasthedependentvariable(Y)andthoseofRussell3000astheindependentvariable(i.e.X).
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CalculatingCAPMBetaTutorial 4 SpiderFinancialCorp,2013
FromtheOptionstabintheregressiondialogbox,settheintercept/constantvaluetozero.
Note:Youmayleavetheintercept/constantfloating(i.e.unset)andtheregressionwillfinditinsignificant.Tryit.Whenwearefinished,clickOK.Theregressionwizardwillgenerateseveraloutputtables.
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CalculatingCAPMBetaTutorial 5 SpiderFinancialCorp,2013
Theregressionmodel(i.e.CAPM)isstatisticallysignificant(ANOVAtable)andcapturesabout40%ofMSFTmonthlyexcessreturnvariance.TheBeta(i.e.Russell3000coefficient)hasanaveragevalueof0.98withanerrorof0.10.Thisisgoodsofar,soletsexaminethestandardizedresidualsoftheregression(rightmosttable).Theresidualsexhibitapositiveskewandfattails,andthusitfailsthenormalitytest.Togetabetterideaabouttheresidualsdistribution,wecreatetheQQplotwithaGaussiantheoreticaldistribution:
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CalculatingCAPMBetaTutorial 6 SpiderFinancialCorp,2013
TheQQPlotshowsasmalldeviationfromnormalityatpositivevalues(i.e.skew)andafatlefttail(negative).BeforewestartusingtheCAPMandourregressionbetatodeterminetheappropriaterequiredreturnofMicrosoft,weshouldaskourselvesakeyfewquestionsfirst:Q:Istheregressionmodelstable?DoestheBetasvaluesignificantlydifferthroughoutthesampledata?A:Toanswerthisquestion,letsdividethesampledataintotwosubsets:dataset1includesallobservationspriorto2008(~70observations)anddataset2coversobservationsstartingfromJanuary2008toMay2013(~70observations).UsingtheRegressionStabilityTestWizardinNumXL,weconductthisimperativetest.Similartowhatwedidwiththeregressionwizard,theRussellsexcessreturnsaretheindependent(X)variable,andtheMSFTreturnsarethedependentvariable(Y).
IntheOptionstab,settheintercept/constanttozero.
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CalculatingCAPMBetaTutorial 7 SpiderFinancialCorp,2013
Now,ClickOK.TheWizardgeneratesthestatisticalstabilitytestoutputtable.
TheBetavalueisstablethroughoutoursampledataset(2001to2013).Letscomputeandplotthebetavaluethroughoutthedataset.Theshadedareaisour95%confidenceinterval.
Q:Aretheregressionsstandardizedresidualsserially(akaauto)correlated?A:Thewhitenoisetestanswersthisspecificquestion,andisavailableintheNumXLstatisticalteststab.
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CalculatingCAPMBetaTutorial 8 SpiderFinancialCorp,2013
IntheOptionstab,setthemaximumlagorderto12(1year).ClickOK.
Theresidualstimeseriesexhibitsnosignificantserialcorrelation.Sofar,wefoundthefollowing:
ThemonthlyreturnsofMicrosoftstockhaveanaveragesensitivityof0.98withtheoverallmarket.
Theresidual(akaidiosyncratic)risk(i.e. )isaround5.54%.Q:Dowehaveobservation(s)thatsignificantlyaffecttheregressionmorethanothers(i.e.Influentialdata)?
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CalculatingCAPMBetaTutorial 9 SpiderFinancialCorp,2013
Toanswerthequestionabove,wecomputetheCooksdistanceforeachobservationinthesampledata.Furthermore,weusetheheuristicthresholdof 4
Ntoidentifythoseinfluentialpoints.Nisthe
numberofnonmissingvaluesinthedataset.
Tohandleinfluentialandatapoint,wedecidedtoremoveitbysettingtheMSFTreturnsto#N/A,thusremovingtheobservationfromanyanalysis.Weremoveoneobservation(theonewiththehighestCooksdistance)atatime,thenrecalculatetheCooksdistancefortheremainingdatapointsusingthereduceddataset.Notethatthethresholdslightlyincreasesaswedropobservations.Wecontinuewiththeprocessuntilnoapparentinfluentialdataisinsight.
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CalculatingCAPMBetaTutorial 10 SpiderFinancialCorp,2013
Notethatthe 4Nthresholdisaheuristic,soweaccepteddatapointswhoseCooksdistanceisslightly
higherthanthethreshold.Recalculatingtheregression(SHIFT+F9),weobservethenewBetavalue(1.21)andregressionerror(5.07%).
PlottingtheCAPMBetavaluethroughoutthesampledata,weobservethattheBetaslightlychangesovertimeandistrendinglowerovertime.OnemayconcludethatMSFTssensitivitytomarketriskisgoingdown,duetoitsmarketcaporthenatureofinvestmentthatthecompanyitselfisundertaking.
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CalculatingCAPMBetaTutorial 11 SpiderFinancialCorp,2013
Case2:IBMInternationalBusinessMachines(IBM)Corporationprovidesinformationtechnology(IT)productsandservicesworldwide.Thecompanyoperatesinfivesegments:GlobalTechnologyServices,GlobalBusinessServices,Software,SystemsandTechnology,andGlobalFinancing.IBMispubliclytraded,listedonNYSEwithamarketcapof233B.LetsplottheIBMmonthlyexcessreturnsalongwiththeRussell3000(marketproxy)excessreturns.
Next,weplotthescatterplotforthetwodatasetsanddrawalineartrendlinetooutlinethecorrelationbetweenthetwo.
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CalculatingCAPMBetaTutorial 12 SpiderFinancialCorp,2013
Thetwoseriesdemonstrateastrongcorrelationbetweenthem.Again,usingtheRegressionWizard,designateIBMexcessreturnsasthedependentvariableandtheRussell3000astheindependent,settingtheintercept/constanttozero.
TheoutputtablesshowsimilarresultstowhatwesawwiththeMicrosoftcase.LetsexaminetheresidualsdistributioncloserusingtheQQPlot.
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CalculatingCAPMBetaTutorial 13 SpiderFinancialCorp,2013
TheQQplotexhibitspositiveskew,withaheavyfattailontheleft(negative)side.BeforewestartusingtheCAPMandourregressionbetatodeterminetheappropriaterequiredreturnofMicrosoft,weoughttoaskourselvesakeyfewquestions:Q:Istheregressionmodelstable?DoestheBetasvaluesignificantlydifferthroughoutthesampledata?Again,welldividethedatasetinto2separatesubsets:dataset1includesallobservationspriorto2008,anddataset2includesallobservationsstartingfromJanuary2008todate.UsingtheNumXLregressionstabilitytest,wespecifytheindependent(X)anddependentvariable(Y)valuesforeachdataset,settheintercepttozero,andclickOK.
Thetestfailed!Wehaveastructuralbreakinthedataset.ThiscanbeinterpretedastheBetavaluechangedsignificantly.
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CalculatingCAPMBetaTutorial 14 SpiderFinancialCorp,2013
Whatcanwedonow?LetsfirstplottheBetavalueovertimeinanattempttoidentifythepoint(s)wherestructuralchangecommenced.
TheIBMstockhasundergoneaBetastartingin2008.Thiscanbeduetointernalcompanypolicychange:typeofinvestment,particularmarketexposure,etc.TheimportantfacthereisthattheidentityoftheIBMstockmorphed(withrespecttoCAPM).Insum,weneedtotossawaytheobservationspriorto2008andusethelaterobservations(i.e.2008toMay2013)toestimatetheCAPMBeta.
Examiningtheregressionoutputs(usingpost2008observations),theBetahasameanvalueof0.66.Furthermore,theresidualdiagnosistestsallpassed.Additionally,thenonsystematicrisk(i.e.regressionstandarderror)isaround4%.Inshort,theIBMstockmorphedfrombeingahighbetavalueabove1toavaluelowerthanone.
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CalculatingCAPMBetaTutorial 15 SpiderFinancialCorp,2013
Q:Aretheregressionsstandardizedresidualsserially(akaauto)correlated?A:Thewhitenoisetestanswersthisspecificquestion,andisavailableinNumXLsstatisticalteststab.
Theresidualstimeseriesexhibitsnosignificantserialcorrelation.Q:Dowehaveobservation(s)thatsignificantlyaffecttheregressionmorethanothers(i.e.influentialdata)?Toanswerthequestionabove,wecomputetheCooksdistanceforeachobservationinthesampledatapost2008.
SimilartowhatwedidintheMicrosoftcase,weremovedinfluentialdatabysettingtheMSFTreturnsto#N/A,thusremovingtheobservationfromanyanalysis.Weremoveoneobservation(onewiththehighestcooksdistance)atatime,thenrecalculatetheCooksdistancefortheremainingdatapoints
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CalculatingCAPMBetaTutorial 16 SpiderFinancialCorp,2013
usingthereduceddataset.Notethatthresholdslightlyincreasesaswedropobservations.Wecontinuewiththeprocess,untilnoapparentinfluentialdataisinsight.
Recalculatingtheregressionmodel:
Thenonsystematicerrordroppedto3.42%(from4.27%earlier),andalltheresidualsdiagnosistestsarepassed.
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CalculatingCAPMBetaTutorial 17 SpiderFinancialCorp,2013
PlottingtheCAPMbetavaluethroughoutthesampledata,weobservethattheBetaslightlychangesovertimeandistrendingupwardovertime.OnemayconcludethatMSFTssensitivitytomarketriskisgoingup,duetothenatureofnewinvestmentthatthecompanyisundertaking.
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CalculatingCAPMBetaTutorial 18 SpiderFinancialCorp,2013
ConclusionInthispaper,wedemonstratedtheprocessforcomputingtheCAPMBetafortwotechstock:IBMandMSFT.Inbothcases,weproposedasimplelinearregressionmodelforthestocksmonthlyexcessreturnsversusthemonthlyexcessreturnsoftheRussell3000Index(marketproxy).TheregressionslopeistheempiricalCAPMBetaandtheregressionstandarderrorisviewedasthestocksnonsystematic(idiosyncratic)error.Afterward,wecarriedonaplainregressionanalysisprocess:ANOVA,coefficientsvaluetest,residualsdiagnosis,regressionstabilitytest,andinfluentialdataanalysis.ThecomputedCAPMBetasignificantlyimprovedaswecarriedourthoroughanalysistotheregressionresults.AlltoolsyouneedtocarryonthisexercisearepartofNumXL1.60Pro.TheCAPMisarelativelysimpleonefactormodel.Inlaterissues,welltacklemultifactors(e.g.FamaFrenchthree(3)factormodel(FFM),etc.),whichmayaddsomenumericalcomplexitywhilethebasicstepsandintuitionremainthesame.