Cost of Capital Formulas - McGraw-Hill资本成本公式

CostofCapitalFormulas

Wehavegatheredtogetherhere(andinsomecasesrepeated)thederivationsofsomeoftheformulasusedinChapter19.Wecoverfivetopics:

DerivingtheWACCformula.

DoesWACCnecessarilydeclinewhenleverageincreases?

Leveringandunlevering.

TheMiles-Ezzellformula.

TheMMformula.

DerivingWACC

Westartwithafirm’sexpectedoperatingcashflowsC1,C2,...,CT.Withall-equityfinancing,theseflowswouldbediscountedattheopportunitycostofcapitalr.ButweassumethefirmisfinancedwithdebtataconstantratioL.Thatis,itfollowsFinancingRule2,sothatD/V=Lineachfutureperiod.

Considerfirmvalueinthenext-to-lastperiod:VT-1=DT-1+ET-1.Wecanwritethetotalcashpayofftodebtandequityinvestorsintwoways:

CashpayoffinT=CT+TCTrDDT-1

=VT-1(1+rDD/V+rEE/V)

TCrDDistheinteresttaxshield.EquatethetwoexpressionsandsolveforVT-1asafunctionofCT.

CT+TCD=VT-1(1+rDD/V+rEE/V)

ThelogicrepeatsforT-2.NotethatthecashpayoffatT-1includesVT-1:

CashpayoffinT-1=CT-1+TCrDDT-2+VT-1

=VT-2(1+rDD/V+rEE/V)

VT-2=

Wecancontinueallthewaybacktot=0:

DoesWACCnecessarilydeclineasleverageincreases?

TheWACCformulaseemstosaythatthecostofcapitaldecreaseswithleveragebecauseofinteresttaxshields.Thatisnotnecessarilyso.Suppose,forexample,thattheextrapersonaltaxondebtinterestexactlyoffsetstheadvantageofthecorporateinteresttaxshield.Inthatcase

WACC=(1-TC)rDD/V+rEE/V=r,

andMM’spropositionIIbecomesrE=r+(r–rD(1-TC))D/E.

Also,thesecuritymarketlinewouldhavetopassthroughtheafter-taxrisk-freerate.

IfyourtaskissimplytoestimateWACCatthefirm’sexistingdebtratio,youdon’tneedtoknowanythingabouttaxespaidbyinvestors.ButifyoutrytocalculateWACCatadifferentdebtratio,youneedtoknowhowrDandrEchangewithleverage.TherelationshipsofrDandrEwithD/Vcandependoninvestortaxes.

Inpracticemostfinancialmanagersdonottrytoadjustforinvestortaxes;theyimplicitlyassumeTp=TpE.WeadoptthisassumptionbelowandinChapter19.

LeveringandUnlevering

Hereisamarket-valuebalancesheetincorporatingthevalueofinteresttaxshields.

Firmvalueif

all-equityfinanced

D

PV(Taxshields)

_____________

E

______

V

V

SupposePV(Taxshields)isexactlyasriskyasthefirm’sassetsandoperations.ThenPV(Taxshields)isvaluedattheopportunitycostofcapitalr.Theoverallrateofreturndemandedbydebtandequityinvestorsequalsr,regardlessofthesizeofPV(Taxshields).

rDD/V+rEE/V=r

ThisgivesStep1ofthethree-steplevering/unleveringprocedure(Section19.3).ForStep2,justsolveforrE:

rE=r+(r–rD)D/E

Inotherwords,taxesdropoutofMM’spropositionIIifinvestorsvalue

PV(Taxshields)attheopportunitycostofcapitalr.

Theseformulas(andourthree-stepprocedure)arenotquiteconsistentwithFinancingRule2.Ifdebtisrebalancedeveryperiod,thennextperiod’sinteresttaxshieldisfixedbythecurrentperiod’sdebtlevelandinterestrate.Thethree-stepprocedureiscorrectonlyifdebtisrebalancedcontinuously.Buttheprocedureisanexcellentapproximationinmostpracticalsettings.

TheMiles-EzzellFormula

Theexactprocedureforleveringandunlevering,givenFinancingRule2,usestheMiles-Ezzellformula,whichcanbederivedasfollows.Startagainatthenext-to-lastperiodT-1.Wewanttheadjustedcostofcapitalrsuchthat:

VT-1=APV =

NotethatDatT-1equalsLVT-1.Then:

wherer=r–LrDTC

YoucaneasilyconfirmthattheMiles-EzzelldiscountrateworksforT–2,T–3,etc.

WiththeMiles-Ezzellformula,youcanleverorunleverWACCinonestep;justplotrasafunctionofL(butrememberthatrDdependsonL).

FinancingRule1andtheMMFormula

TheMMformula:r=r(1-TCL)showstheimplicationsofFinancingRule1(debtfixed).MMassumedacompanyorprojectgeneratingalevel,perpetualstreamofcashflowsfinancedwithfixed,perpetualdebt.Inthatcase:

SinceD=LAPV,

Thereforer*=r(1–TCL).

TheMMformulaisexactonlyforperpetuities,butMyersshowsthattheformuladoesnotleadtolargevaluationerrorsforshorter-livedprojects,SeeFurtherReadingforChapter19.

providedthatdebtisfixedbyFinancingRule1.

It’sinterestingthatWACCandtheMMformulagivethesameanswerforperpetuities,evenunderFinancingRule1.Themarket-valuebalancesheetis:

VA

D

TCD

______

E

______

V

V

whereVAequalsvalueunderall-equityfinancing.Theweighted-averageoftheopportunitycostofcapitalfortheleftsideofthebalancesheetisr(VA/V)+rDTC(D/V).Thisequalstheweightedaveragereturntoinvestors.

rVA/V+rDTCD/V=rDD/V+rEE/V

rVA/V=rD(1-TC)D/V+rEE/V=WACC

NoteVA=V–TCD:

r(V-TCD)/V=WACC

r(1–TCD/V)=r(1–TCL)=