资本结构的计算方法

CAPITALSTRUCTUREAIMSTheaimofthissectionofthemoduleistoprovidestudentswithanintroductiontothetheoryandpracticeoffirms’capitalstructuredecisions.INTENDEDLEARNINGOUTCOMESOnthesuccessfulcompletionofthissectionofthemodulestudentswillbeableto:1. Explainthescopeandsignificanceofcapital structuretheory.2. Explainandcriticallyevaluatecompeting theoriesofcapitalstructure.3. Evaluatetheevidenceon,andimplicationsof capitalstructuretheory.TOPICOUTLINEi. Introductionii. Modigliani-Miller Hypothesisi. Scope&Significanceii. MeasuringGearingiii. ImpactofDebtFinancingiv. Assumptionsv. MeasuresofCostofCapitali. Assumptionsii. Propositionsiii. Rationaleiv. ArbitrageProofv. MarketImperfectionsvi. Extension-CapitalStructure inaCAPMcontextiii. StaticTrade OffTheoryiv. OtherTheoriesv. Evidence& Implicationsi. ImpactofTaxationii. ImpactofBankruptcyCostsiii. OptionPricing&CapitalStructureiv. AgencyCostsi. TaxExhaustionii. DebtCapacityiii. Controliv. ManagerialPreferencesv. Industryvi. PeckingOrderTheoryi. EmpiricalEvidenceii. ImplicationsforInvestmentAppraisalCAPITALSTRUCTURESCOPECapitalstructureconcernedwiththelevelsofdebtandequityfinancingemployedbyfirmstofinancetheiractivities.Twoquestions:i. Whateffectdoescapitalstructurehaveonthe valueofthefirmtoitsowners?ii. Whateffectdoescapitalstructurehaveonthe costofcapitaltothefirm?SIGNIFICANCEOPTIMUMCAPITALSTRUCTUREIfcapitalstructureaffectsfirmvaluetheremaybesomeoptimalcapitalstructureforthefirm.INVESTMENTAPPRAISALIfcapitalstructureaffectsthecostofcapitalthenwemayhavetoconsiderhowaprojectisfinancedwhenevaluatingit.GEARINGOPERATINGGEARINGReferstotheextenttowhichthefirm’soperatingcostsarefixed.FINANCIALGEARINGMeasurestherelationshipbetweendebtandequityinthefirm’scapitalstructure.Maybemeasuredas:i. IncomeGearingii. CapitalGearingIMPACTOFDEBTFINANCINGDebtappearscheaperthanequityasasourceoffinanceforfirms:i. Lowerriskforinvestorsii. Taxadvantagesiii. LowertransactionscostsButdebtisisariskiersourceoffinanceforfirms:i. Increasesriskoffinancialdistressii. IncreasesvolatilityofreturnstoshareholdersFulthorplcistobesetupwithatotalcapitalof£10million.Expectedresultsforthecompanydependontradingconditionsshownbelow:TradingConditions Poor Normal GoodEBIT(£000) 600 1,500 2,400ROCE 6% 15% 24%Threepossiblefinancingstructuresarebeingconsidered:i. Gearing0%(Equity10million£1shares)ii. Gearing20%(Equity8million£1shares,10%Debt£2 million)iii. Gearing60%(Equity4million£1shares,10%Debt£6 million)i. Gearing0%

EBIT 600 1,500 2,400ShareholderEarnings 600 1,500 2,400EPS(pence) 6 15 24ReturnonEquity 6% 15% 24%ii. Gearing20%EBIT 600 1,500 2,400DebtInterest 200 200 200ShareholderEarnings 400 1,300 2,200EPS(pence) 5 16.25 27.5ReturnonEquity 5% 16.25% 27.5%iii.Gearing60%EBIT 600 1,500 2,400DebtInterest 600 600 600ShareholderEarnings 0 900 1,800EPS(pence) 0 22.5 45ReturnonEquity 0% 22.5% 45%ReturnonEquity% 45

60%gearing

42 39 36 33 30 27

20%gearing

24

0%gearing

21 18 15 12 9 6 3 0 0 2 4 6 8 10 12 14 16 18 20 22 24ROCE%ASSUMPTIONS1. Thecapitalstructureofthefirmisalteredby substitutingdebtforequityandviceversa.2. Thefirmpaysoutitsentirenetincome (earningsafterinterestandtaxes)as dividends.3. Thenetoperatingincomeofthefirmisnot expectedtogrow.4. Thecapitalstructureofthefirmcomprises equityandperpetualdebtonly.MEASURESOFCOSTOFCAPITALCostofequity(ke):

E/VeCostofdebt(kd):

I/VdOverallCostofCapital(ko): ke(Ve/Vo)+kd(Vd/Vo) = X/VoWhere: E = NetIncome

= X-I

I = DebtInterest X = NetOperatingIncome

= E+I

Ve = ValueofEquity Vd = ValueofDebt

Vo = TotalValueofFirm

= Ve+VdTHECAPITALSTRUCTUREDEBATETwobasicviewsoncapitalstructure:1. Capitalstructurehasnoimpactontheoverallcostofcapitaltothefirmoritstotalvalue.(Modigiani-MillerHypothesis)2. Thereisanoptimumcapitalstructureforthefirmatwhichitstotalvalueismaximised.(TraditionalTheory,StaticTrade-offTheory)MODIGLIANI-MILLERHYPOTHESISASSUMPTIONS1. PerfectCapitalMarkets2. Firmscanbecategorisedintoequivalentrisk classes3. Investorshavehomogeneousexpectations4. NoTaxesPROPOSITIONS1. Thetotalvalueofthefirmisindependentofits capitalstructure.2. Thecostofequityincreasestoexactlyoffset anybenefitsfromincreaseduseofcheaper debt.3. Thecut-offrateforinvestmentappraisalis independentofthefirm’scapitalstructureand thereforeofthewayinwhichtheprojectis financed.

keCostofCapital

ko

kd

Leverage(Vd/Ve)CostofCapitalunderModigliani-MillerHypothesis

Thecostofequityincreasestoexactlyoffsetanybenefitsfromincreaseduseofcheaperdebt.Thecostofequityforagearedfirmisgivenby:

Keg = Kou+(kou-kd)(Vd/Ve)Where

Kou = Costofequityofungearedfirmof samerisk Kd = Costofdebtingearedfirm Vd = Valueofdebtingearedfirm Ve = Valueofequityingearedfirm

ForexampleassumeUrplcandGorplcarefirmsinthesameindustrywiththesamerisk.UrisallequitywhereasGorisfinanced60%byequityand40%bydebt.ThecostofequitytoUris12%,thecostofdebttoGorplcis5%.ThecostofequitytoGorplcwillbe:

Ke = .12+(.12-.05)(40/60) = 16.67%TheoverallcostofcapitaltoGorplcis:

Ko = .60(.1667)+.40(.05) = 12%

VoMarketValue

Vd

Ve

Leverage(Vd/Ve)ValueofFirmunderModigliani-MillerHypothesis RATIONALEi. Thevalueofthefirmisdeterminedsolelyby theamountofitsnetoperatingincome(NOI) andthebusinessriskattachedtothatNOI.ii. Capitalstructureaffectsneitherofthose factors-allitaffectsisthedistributionofthe riskoftheNOIbetweendifferentclassesof investorinthefirm. iii. Thereforecapitalstructurecannotaffectthe valueofthefirmoritsoverallcostofcapital.ARBITRAGEPROOFMMarguethattwocompaniesidenticalintermsoftheamountandbusinessriskoftheirnetoperatingincomemusthaveidenticaltotalvalues.Ifthevaluesdifferinvestorscouldgainbymovingoutoftherelativelyovervaluedfirmintotherelativelyundervaluedone.Thisprocesswoulddrivethevaluesofthetwofirmstogether.ThefollowinginformationrelatestoUrplc&Gorplc,companieswhichoperateinthesameindustry.

Urplc

GorplcShareCapital(£1shares)

20,000,000

20,000,0005%IrredeemableDebt

0

40,000,000NetOperatingIncome

12,000,000

12,000,000DebtInterest

0

2,000,000NetIncome

12,000,000

10,000,000Thenetoperatingincomeofbothcompaniesisexpectedtoremainatcurrentlevels.Bothcompaniespayoutallnetincomeasdividendsandareexpectedtocontinuedoingso.1. ValueofGor>ValueofUr

UrGorSharePrice

£5

£4

ValueofEquity

100,000,000

80,000,000ValueofDebt

0

40,000,000

ValueofFirm

100,000,000

120,000,000Bodowns20,000shares(i.e0.1%)inGorplcIncome: 0.001x10,000,000 = 10,000Cost: 20,000x4 = 80,000

BodshouldsellsharesinGorplc,buy0.1%(20,000)ofthesharesinUrplcandborrowanamountequalto0.1%ofGorplc’sdebti.e£40,000.NewIncomeIncomefromUrplc:0.001x12,000,000 = 12,000Lessdebtinterest:0.05x40,000 = -2,000NetIncome: = 10,000CostofInvestmentCostofShares20,000x£5 = 100,000Lessamountborrowed = 40,000NetInvestment = 60,000Bodreceivesthesameincomeforalowernetoutlay.2. ValueofGor<ValueofUr

Ur

GorSharePrice

£6

£3

ValueofEquity

120,000,000

60,000,000ValueofDebt

0

40,000,000

ValueofFirm

120,000,000

100,000,000Bodowns20,000shares(i.e0.1%)inUrplcIncome: 0.001x12,000,000 = 12,000Cost: 20,000x6 = 120,000

BodshouldsellsharesinUrplc,buy0.1%(20,000)ofthesharesinGorplcandbuy0.1%(£40,000)ofGorplc’sdebt.NewIncomeIncomefromsharesinGorplc:0.001x10,000,000 = 10,000Plusdebtinterest:0.05x40,000 = 2,000TotalIncome: = 12,000CostofInvestmentCostofShares20,000x£3 = 60,000CostofDebt = 40,000NetInvestment = 100,000Bodreceivesthesameincomeforalowernetoutlay.Inperfectcapitalmarketsarbitragewouldensurethatthevaluesoftwocompaniesidenticalinallrespectsexceptcapitalstructurewereequal.Butrealworldmarketsarenotperfect:i. Transactioncostshinderthearbitrage process.ii. Personal&corporateleveragearenotperfect substitutes.CAPM&CAPITALSTRUCTURECAPMsupportsModigliani-Millerviewthatcapitalstructurehasnoimpactonthefirm’scostofcapitalortotalvalue.Theoverallcostofcapitalofthefirmdependsonthesystematicriskofitsoperationsi.e.itsassetbeta.Thesystematicriskofasharecanbesplitinto:BusinessSystematicRiskFinancialSystematicRiskTheequitybetaofagearedcompanycanbeestimatedfrom:

βe = βa+(βa–βd)(Vd/Ve)Where:

βe = Equitybeta

βa = AssetBeta

βd = DebtBeta Vd = ValueofDebt Ve = ValueofEquityForexampleUr&Goroperateinthesameindustrywiththesameassetbeta(1.4).

Ur

GorAssetBeta

1.4

1.4ValueofEquity(£million)

100

60Valueof5%Debt(£million)

0

40AssumingadebtbetaofzeroforGorplc’sdebt,theequitybetasforthetwocompanieswillbe

Ur βe = βa = 1.4Gor βe = 1.4+(1.4–0)(40/60) = 2.333Assumingthereturnonthemarketportfoliois10%andtheriskfreerateis5%thecostofequityandoverallcostofcapitaltothecompaniesis:Ur Ke = Ko = .05+1.4(.10-.05) = 12%Gor Ko = .05+1.4(.10-.05) = 12%

Ke = .05+2.333(.10-.05) = 16.67%IMPACTOFTAXATIONDebtcapitalismorefavourablytreatedfortaxpurposesthanequity.Agearedcompanywillhaveahigheraftertaxnetoperatingincomethananungearedcompanywithanidenticalpre-taxnetoperatingincome.Inaworldwithtax,therefore,MMstate:i. Thegearedfirmwillhaveahighervalueand lowerposttaxcostofcapitalthantheungeared firm.ii. Optimumgearingleveliseffectively100%.Forexample,UrisallequityfinancedandGorisfinancedbyamixtureofequityanddebt.ThevalueofUr’sequityis£100million.Gorplc’sdebtcomprises£40million5%loanstockquotedatpar.Thepretaxnetoperatingincomeforeachcompanyis£12million.Thetaxrateis30%

Urplc

GorplcPretaxoperatingincome(EBIT)

12,000

12,000LessDebtinterest

0

2,000Pretaxnetincome

12,000

10,000Taxat30%

3,600

3,000Posttaxnetincome

8,400

7,000Adddebtinterest

0

2,000Posttaxoperatingincome

8,400

9,000Themarketvalueofthegearedcompanyisgivenby:

Vg = Vu+tcVdwhere: Vg = valueofgearedfirm Vu = valueofidenticalungearedfirm Vd = valueofgearedfirmdebt tc = corporatetaxrateThevalueofGorplcistherefore:

Vg = 100,000,000+0.3(40,000,000) = 112,000,000IMPLICATIONSi. Thehigherthegearingthegreaterthevalueofthefirm.ii. Thehigherthetaxratethegreaterthevalueofthefirm.iii. Thetaxsavingsarevaluedatthecostofdebt. Thepost-taxcostofequitytothegearedfirmisgivenby:

Keg = Keu+(Keu–Kd)(1–tc)Vd/Vegwhere

Keg = costofequityforgearedfirm Keu = costofequityforungearedfirm Kd = costofdebtforgearedfirm Veg = valueofequityofgearedfirmTheposttaxcostofequitytoGoristherefore:

Keu = 8,400/100,000 = 8.4% Keg = .084+(.084-.05)(1-.3)(40/72) = 9.72%Thepost-taxoverallcostofcapitaltothegearedfirmisgivenby:

Kog = Keu[1–(tcVd/Vg)]where

Kog = costofcapitalforgearedfirm Keu = costofequityforungearedfirm

TheoverallcostofcapitaltoGoristherefore:

Kog = .084[1–(.3)(40/112)] = 7.5%

CostofCapital

keg

kog

kd

Leverage(Vd/Ve)CostofCapital:Modigliani-Millerwithtax

Vg(tax)Valueof Firm taxshield

Vu

Vg(notax)

Leverage(Vd/Ve)ValueofFirm:Modigliani-Millerwithtax PERSONALTAXESPersonaltaxesmayaffectthevalueofthetaxshield.Takingaccountofthisthevalueofthetaxshieldbecomes:

{1-[(1-tc)(1-te)/(1-td)]}xVdwhere

tc = corporatetaxrate te = personaltaxrateonequityincome td = personaltaxrateondebtincomeForexampleusingthepreviousdataandassumingpersonaltaxratesof0%onequityand20%ondebt,thevalueofGor’staxshieldis:{1-[(1-.3)(1-0)/(1-.2)]}x40million = 5millionThiscomparesto£12millionignoringpersonaltaxes.ThevalueofGorwouldbe£105millioninsteadof£112million.IMPACTOFBANKRUPTCYCOSTSHighgearingincreasestheriskoffinancialdistressandbankruptcy.Therearehighcoststobankruptcy/financialdistress:Directcosts: Administrationcosts SaleofassetsatdistresspricesIndirectcosts:

Lossofreputation

Costof ko(bankruptcy

costs)Capital

ko(nobankruptcycosts)

Leverage(Vd/Ve)CostofCapitalwithbankruptcycosts

Costof Capital

Vo(nobankruptcycosts)

bankruptcycosts

ko(bankruptcy

costs)

Leverage(Vd/Ve)Valueoffirmwithbankruptcycosts STATICTRADEOFFTHEORYCombiningtax&bankruptcycostsleadstoatrade-offtheoryofcapitalstructure.Valueofgearedfirmbecomes:Vg = Vu+tcVd-Bcosts

Where:Bcosts = costofbankruptcy/financial distress.Companyshouldgearupuntilthemarginaltaxbenefitsequalthemarginalbankruptcycosts.STATICTRADE-OFFTHEORY

CostofCapital

BankruptcyCosts

Tax+Bankruptcy

Tax

Leverage(Vd/Ve)CostofCapital:TaxandBankruptcyCosts STATICTRADE-OFFTHEORY

Vo(tax)

Vo(tax&bankruptcycosts)

Costof TaxShield

Capital Vo(notax/bankruptcycosts)

BankruptcyCosts

Vo(bankruptcycosts)

Leverage(Vd/Ve)ValueoffirmwithTax&BankruptcyCosts OPTIONPRICING&CAPITALSTRUCTUREEquitysharesinagearedcompanycanbeviewedasacalloptionontheunderlyingassetsofthefirm.OptionValue: ValueofequityCurrentSharePrice: Valueoffirm’sassetsExercisePrice: RedemptionpriceofdebtVolatilityofShare: Volatilityoffirm’sassets

Expirydate: RedemptiondateofdebtThevalueoftheshareswhenthedebtmaturesisgivenby:

Ve = Max(Vf-D),0where

Vf = Valueoffirm’sassets D = RedemptionpriceofdebtTheequityanddebtcanbevaluedbeforethematuritydateusingtheBlack-Scholesmodel.Forexample,assumeGoskaplcisfinancedbyamixtureofequityandzerocoupondebtrepayableatparin4years.CurrentValueofAssets(£million) 120FaceValueofDebt(£million) 40TimetoMaturity (years) 4Riskfreerate 5%VolatilityofAssets 30%d1 = {ln(120/40)+[.05+.5(.3)2]4}/[.3(4)0.5] = 2.4644d2 = 2.4644-[.3(4)0.5] = 1.8644Therefore:

N(d1)=0.9931;N(d2)=0.9689.

Thevalueoftheequityistherefore:Ve = 120(0.9931)-(40/e0.2)(0.9689) = 87.441millionThevalueofthedebtisthereforeVd = 120-87.441 = 32.559millionIfGoskaplcincreasesthevolatilityofitsassetsfrom30%to70%:d1 = {ln(120/40)+[.05+.5(.7)2]4}/[.7(4)0.5] = 1.6276d2 = 1.6276-[.7(4)0.5]

= 0.2276Therefore:

N(d1)=0.9482;N(d2)=0.5901

Thevalueoftheequityistherefore:Ve = 120(0.9482)-(40/e0.2)(0.5901) = 94.459millionThevalueofthedebtisthereforeVd = 120-94.459 = 25.541millionThusbyincreasingthefirm’svolatilityshareholderscanincreasetheirwealthattheexpenseofthedebtholders.AssumeGoskaplcincreasesgearingbyissuingafurther£40millionzerocoupondebtrepayablein4yearsandrepurchasessomeequity:d1 = {ln(120/80)+[.05+.5(.3)2]4}/[.3(4)0.5]= 1.3091d2 = 1.3091-[.3(4)0.5] = 0.7091Therefore:

N(d1)=0.9047;N(d2)=0.7608

Thevalueoftheequityistherefore:Ve = 120(0.9047)-(80/e0.2)(0.7608) = 58.733millionThevalueofthedebtisthereforeVd = 120-58.733 = 61.267millionThenewdebthasidenticalrightstotheolddebtthereforeValueofnewdebt:

61.267(40/80) = 30.6335million

Valueofolddebt: 61.267(40/80) = 30.6335million

Thevalueoftheolddebthasfallenby£1.9255million.Theshareholders’wealthwillbe:Valueofshares

58.733Proceedsofdebtissue

30.6335TotalWealth

89.3665Thisrepresentsanincreaseof1.9255million.AGENCYCOSTSDebtholderswillseektoprotectthemselvesagainsttheseproblemse.g.usingprotectivecovenants