case study: calculating capm beta in excel

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.


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):


Next,weplotthescatterplotforthetwodatasetsanddrawalineartrendlinetooutlinethecorrelationbetweenthetwo:

UsingthelinearregressionwizardinNumXL,designatethemonthlyexcessreturnsofMicrosoftasthedependentvariable(Y)andthoseofRussell3000astheindependentvariable(i.e.X).


FromtheOptionstabintheregressiondialogbox,settheintercept/constantvaluetozero.

Note:Youmayleavetheintercept/constantfloating(i.e.unset)andtheregressionwillfinditinsignificant.Tryit.Whenwearefinished,clickOK.Theregressionwizardwillgenerateseveraloutputtables.


Theregressionmodel(i.e.CAPM)isstatisticallysignificant(ANOVAtable)andcapturesabout40%ofMSFTmonthlyexcessreturnvariance.TheBeta(i.e.Russell3000coefficient)hasanaveragevalueof0.98withanerrorof0.10.Thisisgoodsofar,soletsexaminethestandardizedresidualsoftheregression(rightmosttable).Theresidualsexhibitapositiveskewandfattails,andthusitfailsthenormalitytest.Togetabetterideaabouttheresidualsdistribution,wecreatetheQQplotwithaGaussiantheoreticaldistribution:


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.


Now,ClickOK.TheWizardgeneratesthestatisticalstabilitytestoutputtable.

TheBetavalueisstablethroughoutoursampledataset(2001to2013).Letscomputeandplotthebetavaluethroughoutthedataset.Theshadedareaisour95%confidenceinterval.

Q:Aretheregressionsstandardizedresidualsserially(akaauto)correlated?A:Thewhitenoisetestanswersthisspecificquestion,andisavailableintheNumXLstatisticalteststab.


IntheOptionstab,setthemaximumlagorderto12(1year).ClickOK.

Theresidualstimeseriesexhibitsnosignificantserialcorrelation.Sofar,wefoundthefollowing:

ThemonthlyreturnsofMicrosoftstockhaveanaveragesensitivityof0.98withtheoverallmarket.

Theresidual(akaidiosyncratic)risk(i.e. )isaround5.54%.Q:Dowehaveobservation(s)thatsignificantlyaffecttheregressionmorethanothers(i.e.Influentialdata)?


Toanswerthequestionabove,wecomputetheCooksdistanceforeachobservationinthesampledata.Furthermore,weusetheheuristicthresholdof 4

Ntoidentifythoseinfluentialpoints.Nisthe

numberofnonmissingvaluesinthedataset.

Tohandleinfluentialandatapoint,wedecidedtoremoveitbysettingtheMSFTreturnsto#N/A,thusremovingtheobservationfromanyanalysis.Weremoveoneobservation(theonewiththehighestCooksdistance)atatime,thenrecalculatetheCooksdistancefortheremainingdatapointsusingthereduceddataset.Notethatthethresholdslightlyincreasesaswedropobservations.Wecontinuewiththeprocessuntilnoapparentinfluentialdataisinsight.


Notethatthe 4Nthresholdisaheuristic,soweaccepteddatapointswhoseCooksdistanceisslightly

higherthanthethreshold.Recalculatingtheregression(SHIFT+F9),weobservethenewBetavalue(1.21)andregressionerror(5.07%).

PlottingtheCAPMBetavaluethroughoutthesampledata,weobservethattheBetaslightlychangesovertimeandistrendinglowerovertime.OnemayconcludethatMSFTssensitivitytomarketriskisgoingdown,duetoitsmarketcaporthenatureofinvestmentthatthecompanyitselfisundertaking.


Case2:IBMInternationalBusinessMachines(IBM)Corporationprovidesinformationtechnology(IT)productsandservicesworldwide.Thecompanyoperatesinfivesegments:GlobalTechnologyServices,GlobalBusinessServices,Software,SystemsandTechnology,andGlobalFinancing.IBMispubliclytraded,listedonNYSEwithamarketcapof233B.LetsplottheIBMmonthlyexcessreturnsalongwiththeRussell3000(marketproxy)excessreturns.

Next,weplotthescatterplotforthetwodatasetsanddrawalineartrendlinetooutlinethecorrelationbetweenthetwo.


Thetwoseriesdemonstrateastrongcorrelationbetweenthem.Again,usingtheRegressionWizard,designateIBMexcessreturnsasthedependentvariableandtheRussell3000astheindependent,settingtheintercept/constanttozero.

TheoutputtablesshowsimilarresultstowhatwesawwiththeMicrosoftcase.LetsexaminetheresidualsdistributioncloserusingtheQQPlot.


TheQQplotexhibitspositiveskew,withaheavyfattailontheleft(negative)side.BeforewestartusingtheCAPMandourregressionbetatodeterminetheappropriaterequiredreturnofMicrosoft,weoughttoaskourselvesakeyfewquestions:Q:Istheregressionmodelstable?DoestheBetasvaluesignificantlydifferthroughoutthesampledata?Again,welldividethedatasetinto2separatesubsets:dataset1includesallobservationspriorto2008,anddataset2includesallobservationsstartingfromJanuary2008todate.UsingtheNumXLregressionstabilitytest,wespecifytheindependent(X)anddependentvariable(Y)valuesforeachdataset,settheintercepttozero,andclickOK.

Thetestfailed!Wehaveastructuralbreakinthedataset.ThiscanbeinterpretedastheBetavaluechangedsignificantly.


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.


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


usingthereduceddataset.Notethatthresholdslightlyincreasesaswedropobservations.Wecontinuewiththeprocess,untilnoapparentinfluentialdataisinsight.

Recalculatingtheregressionmodel:

Thenonsystematicerrordroppedto3.42%(from4.27%earlier),andalltheresidualsdiagnosistestsarepassed.


PlottingtheCAPMbetavaluethroughoutthesampledata,weobservethattheBetaslightlychangesovertimeandistrendingupwardovertime.OnemayconcludethatMSFTssensitivitytomarketriskisgoingup,duetothenatureofnewinvestmentthatthecompanyisundertaking.


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.

case study: calculating capm beta in excel

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