CreditRisk andDynamicCapitalStructureChoice

�ThomasDangl†

Dept.of BusinessStudiesUniversityof Vienna

JosefZechner‡

Dept.of BusinessStudiesUniversityof Vienna

August14,2001

�Bothauthorsacknowledgeresearchassistanceby MichaelHalling.

†Departmentof BusinessStudies,Universityof Vienna,BrunnerStraße72,A-1210Vienna,e-mail: Thomas.Dangl@univie.ac.at,Tel:+43- 1 - 427738107,Fax: +43- 1 - 427738054,ThomasDanglwassupportedby theAustrianScienceFund(FWF) undergrantSFB010andby CCEFM(Centerfor CentralEuropeanFinancialMarkets).

‡Departmentof BusinessStudies,University of Vienna,BrunnerStraße72, A-1210 Vienna,e-mail: Josef.Zechner@univie.ac.at,Tel:+43- 1 - 427738071,Fax: +43 - 1 - 427738074,JosefZechnergratefullyacknowledgesthefinancialsupportthrougha grantfrom theAustrianCentralBank’s (OeNB)Jubilaumsfonds.

1

CreditRiskandDynamicCapitalStructureChoice

Abstract

This paperpresentsan analysisof the effect of dynamiccapitalstruc-

ture adjustmentson credit risk. Firms may optimally adjusttheir leverage

in responseto stochasticchangesin firm value. This is shown to influence

a bond’s expecteddefault frequency and its fair credit spread. Generally

capitalstructuredynamicssignificantlyincreasebothcreditspreadsandand

expecteddefault probabilities.Numericalexamplesdemonstratethat there

exists a u-shapedrelationshipbetweenthe traditional distanceto default

measureandexpecteddefault frequencies.The magnitudeof the effect of

capitalstructuredynamicsis shown to dependonfirm characteristicssuchas

assetvolatility, thegrowth rate,theeffective corporatetax rate,call features

andtransactionscosts.Theresultsthereforesuggestacross-sectionalvaria-

tion of therelationshipbetweenthedistanceto default andexpecteddefault

frequencies.

Finally we extend the analysisto include the estimationof the firm’s

assetvalueandits volatility from observed equityprices. We find that the

underestimationof creditspreadsandexpecteddefault frequenciesis exac-

erbatedwhentheassetvalueandvolatility areinferredfrom a modelwhich

ignorestheopportunityto recapitalize.

2

1 Introduction

Measuringandmanagingcredit risk hasbecomeof centralimportancefor finan-

cial institutions.In mostcountries,banks’equityrequirementsarealreadytied to

their exposureto creditrisk. Accordingto theproposedBaselAccordII, thelink

betweencreditrisk andcapitalrequirementwill beregulatedin muchmoredetail.

Bankswill beallowedto calculatetheir creditrisk exposureandthustheir equity

requirementson thebasisof their internalratingmodels.

Perhapsevenmoreimportantly, thesearchfor shareholdervaluerequiresthat

bankscanaccuratelyquantify their exposuresto unexpectedcredit losses.This

is a prerequisitefor a correctallocationof economiccapital to variouslending

activitiesandthusfor optimizingthecapitalbudgetingdecisions.

Despitetheir importancefor regulationandthemanagementof financialinsti-

tutions,existing creditrisk modelsarestill unableto capturesomeimportantrisk

factors.For example,mostexistingcreditrisk modelsassumethatthefirm’sdebt

level remainsconstantover time or changesin a deterministicway. In practice

firms’ adjusttheir financialstructuresin responseto stochasticchangesin their

economicenvironment.Thismayhavesignificantinfluenceoncreditrisk.

In this paperwe show how firms’ dynamiccapital structurechoicescanbe

integratedinto a creditrisk model.We analyzetheeffect of intertemporalcapital

structurechoiceson a corporatebond’s fair creditspread,on estimateddistances

to default,andon expecteddefault frequencies.

Wepresentamodelwherethefirm’s freecashflow followsageometricBrow-

nian motion. This cashflow is partly usedto pay the couponon the firm’s debt

andtheremainderis paidoutasadividendto equityholders.

Debt is advantageousfor tax reasons.The net tax advantageof debt is the

3

differencebetweenthecorporatetax advantageof debt(interestis corporatetax

deductible)and the personaltax disadvantageof debt (interestincomeis taxed

moreheavily thancapitalgainsor dividends).1

Recapitalizationsareassociatedwith transactionscosts.As a resultfirms do

not adjusttheir capitalstructurescontinually. If thefreecashflow increasesby a

sufficientamount,thenthefirm mayfind it optimalto issuemoredebt.Sincethe

risk free rateof interestis assumedconstantandsincethe new optimal leverage

ratio is equalto the initially chosenleverageratio the new debtcanbe issuedat

preciselythesametermsastheoriginal debt.

We contrastour model of dynamicrecapitalizationwith the traditional ap-

proachin thespirit of Merton(1974)wherethefacevalueof debtat therisk hori-

zonis assumedto befixed.Wefind thatconsiderationof dynamicrecapitalization

decisionsgenerallyincreasesfair creditspreadsandtheexpecteddefault frequen-

cies. Interestinglywe find a non-monotonic,u-shapedrelationshipbetweendis-

tanceto default andexpecteddefault frequencies.Oneof themajor implications

of ouranalysisis thatit wouldbewrongto estimatedanunconditionalrelationship

betweendistanceto defaultandexpecteddefault frequencies.Our resultsindicate

that onemustconditionon the firm’s assetvolatility, its effective corporatetax

rate,its expectedgrowth rateandestimatedbankruptcy costs.

Our model is relatedto severalpapers.As in Fischer, Heinkel, andZechner

(1989)we explicitly modelthe possibility of dynamiccapitalstructurechanges.

Weextendtheanalysisto focusontheimpactof dynamiccapitalstructureadjust-

mentson fair credit spreadsandexpecteddefault frequencies.Also, we usethe1Interestis taxableat the personallevel whereasthe realizedrateof returnon equity is not.

This is sosinceweassumethattherateof returnonequityis eitherrealizedin theform of tax freecapitalgainsor realizedin theform of dividendswhicharenot taxedbecauseof imputationof thecorporatetax rate.

4

firm’s cashflow asthestatevariable,ratherthanthevalueof thefirm’s unlevered

assets,asin Fischer, Heinkel, andZechner(1989).

Christensen,Flor, Lando,andMiltersen(2000)developa modelof dynamic

capitalstructureadjustments.They explicitly exploretheimpactof renegotiations

betweenequityholdersanddebtholdersin timesof financialdistresswhereaswe

do not allow for renegotiation. The main focusof our paperis the comparative

staticanalysisof theimpactof dynamicrecapitalizationoncreditrisk.

Collin-DufresneandGoldstein(2000)analyzewhetheror not credit spreads

reflectstationaryleverageratios. In their model,leverageratiosaremeanrevert-

ing. Consistentwith empiricalevidencethey find that in comparisonto a model

with constantleveragedebtissuedby low-leveragefirmshaslargercreditspreads

andthatthetermstructureof debtis upwardslopingfor low-gradedebt.

We extend the analysisof the effects of dynamic leverageadjustmentson

credit spreadsin Collin-Dufresneand Goldstein(2000) by explicitly modeling

equityholders’optimal capitalstructurechoices.This allows us to explore how

theeffect of capitalstructuredynamicson creditspreadsis relatedto thecharac-

teristicsof issuingfirms. By modelingcapitalstructurechoicesendogenouslywe

alsorecognizethatleverageadjustmentsareasymmetric.A firm increasesits debt

level whenits firm valueincreasesbut debttendsto be “sticky”when firm value

decreases.This featureof capitalstructuredynamicsinfluencescreditspreadsand

expecteddefault frequencies.

Theremainderof thepaperis organizedasfollows. Section2 introducesthe

model. Theresultsof theanalysisarepresentedin Section3. Section4 summa-

rizesandconcludes.

5

2 The Model

We assumethat thefirm’s instantaneousfreecashflow aftercorporatetax ct fol-

lowsa geometricBrownianmotiongivenby

dct

ct

� µdt � σdWt � (1)

wheretheexpecteddrift rateandtheinstantaneousvarianceof theflow processare

determinedby µ andc2t σ2 respectively (seeTable1 for thenotationusedthrough-

out thepaper),dWt is theincrementto astandardWienerprocess.Hence,if r and

µ denoterisklessinterestrateandtherisk adjusteddrift rateof thecashflow pro-

cessrespectively andτp thepersonaltax rateon ordinaryincomethenthecurrent

valueof theunleveredflow is givenby ctr � 1 � τp � � µ.

In our modelwe assumethat the effective corporatetax rateτc exceedsτp.

Thus,giventhatcouponpaymentsaretax deductibleandconsideringthatissuing

debtcausesdefault risk, firms have an incentive to issuedebtandto maintainan

optimalcapitalstructure.Allowing for dynamiccapitalstructure,Bt denotesthe

facevalueof outstandingdebtat time t, i.e.,Bt is endogenouslydeterminedby the

decisionmakerswithin thefirm. We definethefirm’s inverseleverageratioyt as

yt� 1

Bt

ct

r � 1 τp µ � (2)

However, sincewe assumethat it is costly to call outstandingdebt(call pre-

mium λ) aswell asto issuenew debt(proportionaltransactionscostsk) it is not

optimalto adjustthecapitalstructureinstantaneously. Ratherthanreactingto any

changein the firm’s leverageratio, only sufficient deviationsfrom the optimum

6

Table1: Notation

afirm’s instantaneousfreecashflow aftercorporatetax ct

expectedrateof changeof ct µrisk adjusteddrift of thecashflow process µrisklessrateof interest rinstantaneousvarianceof thecashflow process c2

t σ2

facevalueof debt Bvalueof equity Evalueof debt Dtotal valueof thefirm Vinstantaneouscouponrate ifirm’s inverseleverageration yt

personaltax rateonordinaryincome τp

corporatetax rate τc

proportionalbankruptcy costs gproportionaltransactionscostsassociatedwith issuingnew debt kproportionalcall premium λ

satisfytheexpensesassociatedwith areorganizationof debt(seeFischer, Heinkel,

andZechner(1989)). Consequently, the inverseleverageratio yt is drivenby the

dynamics

dyt

yt

� �� �� µdt � σdWt : nodebtreorganizationat time t �Bt

B�t 1 : debtis restructuredfrom Bt to B�t at time t,(3)

that is, during periodswherethe amountof debt issuedis constant,the inverse

leverageratio follows the samegeometricBrownian dynamicsasthe cashflow

processct . Whenever thefirm’s managementfinds it optimal to reorganizedebt,

thefacevalueBt jumpsto theamountof newly issueddebtandanalogouslythere

is a jump in theinverseleverageratioyt .

7

�����

� ����

Figure 1: One particularrealizationof yt . When the firm reorganizesits debt(whenyt hits y) the inverseleverageratio jumps to y0. This jump reducesthedistanceof yt to the critical default thresholdy, and thus, increasesthe defaultprobabilityof thefirm thatdynamicallyadjustsits capitalstructure.

In thesequelwe interpretthefirm’s equityE anddebtD asclaimscontingent

on yt andBt ratherthanasclaimscontingenton the firm’s profit flow ct . This

constructionallows usto formulatetheentiremodelto behomogenousin B, i.e.,

E � y� B � BE � y� 1 andD � y� B � BD � y� 1 . The reasonis the fact that both the

cashflow ct� � r � 1 τp µ ytBt andthecouponflow iBt aswell aspaymentsin

thecaseof debtrestructuringareproportionalto Bt . Theassumptionof propor-

tionalbankruptcy costspreservesthis homogeneity.

Following Fischer, Heinkel, andZechner(1989),we considerreorganization

strategiesdeterminedby anupperthresholdy anda lower thresholdy for the in-

8

verseleverageratio. Thismeans,wheneveryt reachesy theamountof outstanding

debtis increasedby calling existing debtandissuingnew contracts.Wheneveryt

reachesy equityholdersdecideto default. As aconsequenceof thehomogeneityit

is – in thecaseof areorganization– alwaysoptimalto establishacertainoptimum

leverageratio, denotedby y�0. This means,thecurrentamountof debtBt is kept

constantaslong asyt is in therangebetweeny andy. Only if yt hits y, B jumps

to yy�0Bt . If equityholdersdefault, theownershipis transferredto thebondholders.

After payingbankruptcy coststhey will optimally leverthefirm, i.e.,y will imme-

diately jump to y�0. Of course,in orderto bea consistentreorganizationstrategy,

we requirey � y�0 � y. Figure1 plotsoneparticularrealizationof yt which illus-

tratesthecharacteristicsof thedynamicsof thefirms inverseleverageratio. On a

reorganizationof debt(whenyt hitsy) yt jumpsto y0 therebyreducingthedistance

to thecritical default triggery.

2.1 The Value of Equity and Debt

In this subsectionwe considera given (not necessarilyoptimal) reorganization

strategy � y � y0 � y and determinethe value of equity and debt. Basedon these

resultswesubsequentlydeterminetheoptimalstrategy in Subsection2.2.

SinceB is kept constantin the interval � y � y we canapply standardcontin-

gentclaimsvaluationtechniquesto determinethevalueof equityE � y� B anddebt

D � y� B . More precisely, whenthe facevalueof debtissuedis constant,Bt� B,

thevalueof equityanddebtmustsatisfy

12

σ2y2Eyy � µyEy r � 1 τp E �� 1 τc iB ��� r � 1 τp µ ytB � 0 � (4)

12

σ2y2Dyy � µyDy r � 1 τp D ��� 1 τp iB � 0 � (5)

9

Thesolutionsto thesesecondorderordinarydifferentialequationsare

E � y� B � B E1ym1 � B E2ym2 � 1 τc i� 1 τp r B � ytB � (6)

D � y� B � B D1ym1 � B D2ym2 � irB � (7)

wherem1 andm2 arethepositiveandthenegativerootof thecharacteristicquadratic

polynomial,i.e.,

m1 � 2 � 12 µ

σ2 ��� � 12 µ

σ2 2 � 2r � 1 τp σ2 � (8)

andE1 � 2 andD1 � 2 areconstantsthathave to bedeterminedby thefollowing con-

ditions.

E � y � B � 0 � (9)

E � y � B � V � y0 � B yy0 kB

yy0 ! "� 1 � λ B � (10)

whereV denotesthe total valueof thefirm, V � E � D. Equation(9) statesthat

equityis worthlessin thecaseof default. Whenthefirm is recapitalized(Equation

(10)),it first buysbacktheoutstandingdebtsecurities,paying � 1 λ B. After that

thefirm immediatelyreleveresoptimally, i.e., it issuesnew debtwith a facevalueyy0

B. This is coveredby thefirst termof Equation(10).

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Theboundaryconditionsfor debtvaluationare

D � y � B � V � y0 � B y

y0 kB

y

y0 ! � 1 g�� (11)

D � y � B � � 1 � λ B � (12)

On default (Equation(11)) the bondholdersbecomeownersof the firm which

they immediatelyrelever optimally. Theproportionalbankruptcy costsareborne

by thenew ownersof thefirm. Whenthefirm is recapitalized(Equation(12)) the

outstandingdebtis calledbackat theprice � 1 � λ B.

Sincewe assumethat debt is alwaysissuedat par we determinethe coupon

ratei endogenously:

choosei suchthat D � y0 � B � (13)

2.2 Optimal Recapitalization

In theprevioussubsectionwe have derived thevalueof equity anddebtundera

given recapitalizationstrategy � y � y0 � y . Now we wish to determinethe optimal

choiceof thesecritical values.Whenthefirm decideseitherto recapitalize(at y)

or to default(aty), it is aleveredfirm and,thus,thesevaluesarebothdeterminedin

orderto satisfyequityholders.In contrast,aftercallingdebt,thefirm is unlevered,

andtherefore,the amountof newly issueddebtis determinedfrom a firm value

optimizing point of view. Precisely, for a given y0 equityholderswill optimize

their decisionvariablesy � y� � y0 andy � y� � y0 which simultaneouslysatisfy

thefirst orderconditionsof optimality (seeDixit (1993)for adiscussionof theso

called‘smoothpasting’conditions)

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∂E∂y

� y � B � 0 � (14)

∂E∂y

� y � B � 1y0

E � y0 � B � B � 1 k � ydEdy

� y0 � B ! � (15)

Whenissuingnew debt– therebyfixing thecouponratei to issuethebondat

par– theownerof theunleveredfirm anticipatestherecapitalizationstrategy and

choosestheoptimalinitial capitalstructureby solving

maxy0

V � y0 � B kB � (16)

subjectto

B � 1y0

c0

r � 1 τp µ �y � y� � y0 #�y � y� � y0 #�i : D � y0 � B � B �

Therefore,thefirst orderconditionthathasto besatisfiedby theoptimalinitial

inverseleverageratio y�0 is

∂V∂y $ � y�0 � B� � ∂V

∂y0 $ � y�0 � B� � ∂V∂y� ∂y�

∂y0 $ � y�0 � B� � ∂V∂y� ∂y�

∂y0 $ � y�0 � B�� ∂V∂i

∂i∂y0 $ � y�0 � B� 1

y�o � V � y�0 � B kB � 0 � (17)

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2.3 A Benchmark: The Case of a Constant Debt Level

As abenchmarkweusethecasewherethefirm is notallowedto reorganizedebt,

i.e., Bt� B0, the initially chosenamountof debtcannotbe changed.As a con-

sequence,the setof decisionvariablescontainsonly the initial capitalstructure

y0 andthe lower critical valuey. The absenceof the reorganizationopportunity

is further reflectedin a changein theboundaryconditions.Sincethereexistsno

upperthresholdthattriggersajumpin thecapitalstructuretheconditions(10)has

to besubstitutedby

limy% ∞

E � y� B � � 1 τc i� 1 τp r B � ytB � (18)

andcondition(12)hasto besubstitutedby

limy% ∞

D � y� B � irB � (19)

.

SinceE andV arenow independentof y theoptimality condition(15) hasto

bedropped,and ∂V∂y� 0 canbesubstitutedinto condition(17).

3 Results and Comparative Statics

Presentingtheresultsof themodelanalysisthissectionis composedof threesub-

sections.Eachof themstartsfrom a commonbasecasescenario(seeTable2 for

theparametervalues)anddiscussescomparativestatics.Thefirst (Subsection3.1)

focusesonthefirm’soptimalcapitalstructurechoicewhenit is allowedto dynam-

ically reorganizedebtcomparedto the‘Merton like’ benchmarkmodelwith static

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Table2: BasecaseparametersParameter Valuerisklessrateof interestr 5%personaltax rateτp 35%corporatetax rateτc 50%varianceσ2

y 5%risk adjusteddrift µ 0%transactionscostsk 1%call premiumλ 0%bankruptcy costsg 25%

debtlevel. Sincewe endogenouslydeterminethefirm’s capitalstructurechoice,

we areableto explore the impactof firm characteristics(like thevarianceof the

cashflow process,the growth rate,or the tax advantageof debt)on fair credit

spreadsandthe optimal initial leverageratio. The second(Subsection3.2) con-

centrateson modelrisk from ananalyst’s point of view. Specifically, we estimate

creditrisk from observedequitytimeseriesandexaminetheimpactof themodel

choiceon creditspreads.In thethird (Subsection3.3)we determineexpectedde-

fault frequenciesimplied by a dynamiccapitalstructurechoice. Comparingthe

resultsto the benchmarkmodelwe show that assumingstaticdebt level signifi-

cantlyunderestimatesexpecteddefault frequencies.Furthermore,we discussthe

impactof firm characteristicson therelationshipbetweendistanceto default and

expecteddefault frequencies.

3.1 The Firm’s Optimal Capital Structure Choice

How doestheoptionto dynamicallyadjustthecapitalstructureaffect a firm’s fi-

nancingdecisions?Table3 shows theoptimal recapitalizationstrategy of a firm

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Table3: Optimalcapitalstructurechoice(basecase)dynamic static

optimalinitial leverageratio 1& y�0 58 % 70 %maximumleverageratio 1& y� 208 % 205 %minimumleverageratio1& y� 39 % 0 %couponratei � y�0 7.7% 7.4%

with dynamiccapital structuretogetherwith the optimal choiceof a firm with

staticdebtlevel. Themostevidentdifferenceis thata dynamicfirm initially uses

much lessdebt than the staticdoes. The dynamicfirm anticipatesthe fact that

it will increasedebtin the casethe firm valueevolveswell. It finds theoptimal

choiceby balancingthe tax benefitsof debtagainstthe costsof capital(includ-

ing couponpaymentsandthe costsassociatedwith recapitalization).Whenthe

staticfirm wantsto take thefull advantageof thetax benefitsin thecasethefirm

evolveswell it initially hasto takea largeramountof debt.Counterintuitively, the

fair couponrateof the dynamicfirm exceedsthat of the staticfirm. The reason

is thefact thatthedynamicfirm callsbackexisting debtandwhenthefirm value

increasessufficiently andissuesa largeramountof debt.Thisactionincreasesde-

fault risk becauseit decreasesthedistanceof critical default threshold(seeFigure

1). Or, from anotherpoint of view it eliminatesthechancefor debtholdersthat

thevalueof their contractgrowssignificantlyabovepar.

Table4 lists comparative staticson σ2, τc τp, k, g, µ, andλ. We seethat

theopportunityto recapitalizereducestheoptimalinitial leverageratioandthatit

generallyincreasescredit spreads.Theseeffectsarestrongerfor high-riskfirms,

for firmswith largetaxadvantageof debtandfor high-growth firms. Theseeffects

arelesspronouncedfor firmswith highcostsof recapitalizationandfor firmswith

15

Table4: Comparativestaticanalysisσ2

y 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i0.05 58.6% 7.7 % 70.0% 7.4 % -11.4% 30 bp0.04 60.6% 7.3 % 71.8% 7.06% -11.2% 24 bp0.02 67.9% 6.35% 77.9% 6.23% -10.0% 12 bp

τc τp 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i0.15 58.6% 7.7 % 70.0% 7.4 % -11.4% 30 bp0.11 45.0% 7.75% 56.0% 7.44% -11% 31 bp0.05 22.0% 5.9 % 30.0% 5.9 % -8% ' 0 bp

k 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i1% 58.6% 7.7 % 70% 7.4 % -11.4% 30 bp2% 58.0% 7.57% 68% 7.35% -10.0% 22 bp4% 55.5% 7.26% 65% 7.18% -9.5% 8 bp

g 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i20% 65.0% 8.05% 75.8% 7.62% -10.8% 43 bp25% 58.6% 7.7 % 70.0% 7.4 % -11.4% 30 bp30% 53.6% 7.53% 65.0% 7.3% -11.4% 23 bp

µ 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i-2% 54.7% 8.56% 66.7% 8.39% -12% 17 bp0% 58.6% 7.7 % 70.0% 7.4 % -11.4% 30 bp2% 74.0% 7.04% 74.0% 6.67% ' 0% 37 bp

λ 1& y�0 (dyn.) i (dyn.) 1& y�0 (stat.) i (stat.) ∆ � 1& y�0 ∆i0% 58.6% 7.7% 70.0% 7.4 % -11.4% 30 bp5% 61.5% 7.4% 70.0% 7.4% -8.5% 0 bp

10% 63.8% 7.3% 70.0% 7.4% -6.2% -10 bp

highbankruptcy costs.

3.2 Model Risk

In the previous sectionwe have explored the effect of dynamicrecapitalization

on leveragechoiceandcredit spreads.We have therebyassumedthat y andσy

areobservableto all parties. For practicalcredit risk managementapplications,

16

however, thetotalvalueof thefirm’sassetsanditsvolatility andthusy andσy must

beinferredfrom theobservablemarketvalueof equity, E andσE. To evaluatethe

importanceof capital structuredynamicson credit risk estimateswe therefore

extendour analysisto incorporatethis estimationstep.

Weanalyzeacreditrisk managerwhoobservesthemarketvalueof equityand

its volatility. From this observation shewishesto infer the fair credit spreadof

a corporatebond. Our benchmarkis the staticMerton-typemodelwherefirms

cannotadjustleverage.For agivenE andσE thismodelis first usedto infer y and

σy andthento calculaterequiredcreditspreads.We comparethis with theresults

from our dynamiccapitalstructuremodel. Thus,for thesameE andσE we use

themodelof Section2 to infer y andσy andthento calculatefair creditspreads.

We proceedasfollows. For a givensetof parametervalueswe calculatethe

optimal initial leverageandfair creditspreadbasedon thedynamiccapitalstruc-

turemodelof Section2. For this initial leverageratio thismodelalsogeneratesan

equityvalueE andanequityvolatility σE.

In a secondstepwe usethesevaluesfor E and σE in the static debt level

model(seeSubsection2.3). In particularwe usethestaticmodelto numerically

calculatethe V and σV consistentwith the initially generatedequity valueand

volatility. Finally, giventheseparameters,weusethestaticmodelto calculatethe

resultingcreditspread.Table5 summarizestheresultsof ournumericalexamples.

Wefirst notethat“consistent”useof thestaticleveragemodelto infer thevalue

andthevolatility of theunderlyingandthento calculatecreditspreadsunderesti-

matesthetruefair creditspreadin all examples.However, theerrordueto ignor-

ing capitalstructuredynamicssignificantlydependson thefirm’s characteristics.

First, theunderestimationof requiredcredit spreadsincreaseswith thevolatility

17

Table5: Comparativestaticanalysis(modelrisk)σ2

y i (dyn.) i (stat.) ∆i0.02 6.35% 6.04% 31 bp0.04 7.30% 6.73% 57 bp0.06 8.14% 7.34% 80 bp0.08 8.85% 7.85% 100bp

λ i (dyn.) i (stat.) ∆i0% 7.75% 7.07% 68 bp5% 7.40% 7.06% 34 bp

10% 7.28% 7.10% 18 bp25% 7.22% 7.20% 2 bp

τc τp i (dyn.) i (stat.) ∆i15% 7.75% 7.07% 68 bp11% 7.03% 6.92% 11 bp5% 5.97% 5.97% 0 bp

of thefirm’s cashflows. Themorevolatile thefirm’s underlyingcashflows are,

the more likely it is that the option to increaseleverageis exercised. Thus,for

firms in risky industriesit is more importantto take capitalstructuredynamics

into account.

Second,theunderestimationof requiredcreditspreadsdependson thedegree

to which original debtholdersareprotectedfrom leverageincreases.This is cap-

turedby theparameterλ in our model.If issuingadditionaldebtrequiresthatthe

old debtholdersreceive thefacevalue,thentheunderestimationof thefair credit

spreadthatresultsif oneignoresleverageadjustmentsis 71 basispoints.By con-

trast, if existing debtmustbe repurchasedat a premiumof 25 percentbeforea

capitalstructureadjustmentcantakeplace,thentheerroris negligible, i.e. 2 basis

points.

Anotherimportantparameterthat influencesthemagnitudeof theerror is the

18

effective tax advantageof debt,i.e. τc τp. If this differenceis 15 percent,then

the resultingunderestimationof credit spreadsin a staticcredit risk modelis 71

basispoints.If thenettaxadvantageof debtis reducedto 5 percent,thentheerror

is essentiallyzero.

3.3 Theoretical Expected Default Frequency

In thissubsectionweexaminetheimpactof adynamiccapitalstructureonthede-

faultprobability. Wecalculatethetheoreticalexpecteddefault frequency (TEDFs)

of afirm, whichwedefineastheprobabilitythatthefirm defaultswithin acertain

time periodof lengthT. SinceTEDFsarecomputedwith respectto the objec-

tiveprobabilitymeasure,it is µ which determinesthedrift of theunderlyingpro-

cessyt in therespectivecalculations.However, whenchoosingthereorganization

thresholds,equityholderswill find theiroptimaldecisionapplyingtherisk neutral

valuationpresentedin previoussections.Thismeans,thecritical thresholdsy, y0,

andy aredeterminedwith respectto therisk adjusteddrift µ while theprobability

of hitting y (which triggersdefault) within theperiodT dependson theobjective

drift µ. ???

If no recapitalizationis allowed, the probability that the firm with initial in-

verseleverageratioy doesnotdefault within thenext T yearsis givenby

P0 � y� T � N ( ln � y) y�+* � µ � 12σ2 � T

σ , T -/. yy 0 2 1 µ2 1

2σ2 3σ2

N ( ln � y) y�+* � µ� 12σ2 � T

σ , T - � (20)

wherethe subscript0 indicates,that recapitalizationis not allowed(The formu-

19

lae in this subsectionarederivedusingtheresultsfor standardBrownianmotion

with drift tabulatedin BorodinandSalminen(1996)). Therefore,the theoretical

expecteddefault frequency in thestaticcaseis

TEDF� y� T � 1 P0 � yt � (21)

For a firm thatdynamicallyadjustsits capitalstructureusingtheoptimal re-

capitalizationstrategy � y � y0 � y we proceedin several steps. Assumingthat the

initial inverseleverageratio is y, we first calculatethe probability that the firm

neitherdefaultsnor recapitalizeswithin thenext T years.This probabilityof sur-

viving wherebykeepingstablecapitalstructureis givenby

Ps � y� T � ∞

∑k 45� ∞ 6 e� 2 � ln 1 y7 y3 1 µ2 1

2σ2 3 kσ2 �98N � d1 N � d2;: e� 2 � ln 1 y7 y3=< ln 1 y7 y3 1 µ2 1

2σ2 3 kσ2 �>8N � d3 N � d4?:+@"� (22)

whered1, d2, d3, andd4 aredeterminedby

d1 � ln � y& y 2ln � y& y k ��� µ 12σ2 T

σ A T�

d2 � ln � y& y 2ln � y& y k ��� µ 12σ2 T

σ A T�

d3 � ln � y& y 2ln � y& y k ��� µ 12σ2 T

σ A T�

d4 � 2ln � y& y � 1 � k ��� µ 12σ2 T

σ A T

andN(.) denotesthecumulativedistribution functionof thestandardnormaldis-

20

tribution.

In a secondstepwe determinethe recapitalizationdensity f � y� t , i.e., 1& dt

timestheprobabilitythatafirm with initial inversecapitalratioy will recapitalize

within thetime interval 8 t � t � dt : . This is givenby

f � y� t � e�+� µ� 12σ2 � ln � y) y� �B� µ � 1

2σ2 � 2 t2 �C ∞

∑k4 0

ln � y& y � 2ln � y& y kA 2πt2) 3 e� 1 ln 1 y7 y3D< 2ln 1 y7 y3 k 3 22σ2t

(23)

In a last stepwe determinethe probability that a firm which is allowed to

recapitalizen timeswill not defaultwithin T yearsusingtheiterationrule

Pn � y� T � Ps � y� T �FE T

0Pn � 1 � y0 � T t f � y� t dt (24)

Two mutuallyexclusiveevents,contributeto this probability. Eitherthefirm sur-

viveswithout recapitalization(representedby the first term). Or it recapitalizes

at sometime t andsurvivesanotherT t yearsstartingfrom y0, thesecondterm

of Equation(24) integratesoverall possiblerecapitalizationtimes.However, this

integrationhasto beperformednumerically. Theprobability thata firm with dy-

namiccapitalstructuredefaultswithin thenext T yearsis therefore

TEDF� y� T � limn% ∞

8 TEDFn � y� T � 1 Pn � y� T ;: � (25)

Equation(25) allows to examinethe contribution of multiple recapitalizationto

TEDFs.Ourstudiesshow, thatthiscontributionconvergesveryfastto zero.How-

ever, to calculateTEDFsover threeyears,thefirst threerecapitalizationoptions

significantlycontributeto thedefault probabilities.

21

2.8 3 3.2 3.4 3.6 3.8 4

5

10

15

20

25

30

35

40

GIHKJMLON PRQTS

UTVXWRY#Z\[^]_a` Y ` [^] b cdcFigure2: Expecteddefault frequencieswith dynamicandstaticdebtlevel plottedagainstthedistanceto default. While staticdebtlevel leadsto adecreasingrelationbetweenDD andTEDF, dynamiccapitalstructureleadsto au-shapedrelationship.Debtreorganizationaty leadsto areductionin thedistanceto defaultandthereforeincreasesthedefault probabilitywheny approachesy.

In the following we computeTEDFsover a time horizonof treeyears,i.e.,

T � 3. Figure2 comparesthe threeyearTEDF of a firm (we take thebasecase

parametersfrom Table2 andsetµ � 0) with dynamiccapitalstructureto thatof

a firm with static debt level. Thesedefault frequenciesare plotted againstthe

‘distanceto default’ (DD) whichwedefinewith respectto aoneperiodcreditrisk

modelas

DD � y � ln � y& y ��� µ 12σ2 T

σ A T� (26)

That is, in a modelwheredefault occursonly at the endof the time horizonT

22

2.8 3 3.2 3.4 3.6 3.8 4

5

10

15

20

25

30

35

40

eIfKgMhOi jRkTl

kRmon�jqpsrXt;uwv#xRy=zkRmon�jqp{rTt;uwv#xTy=z}|�mot�~�v#kqp � �d�Figure3: Expecteddefault frequencieswith dynamiccapitalstructureandtheun-conditionalprobabilitythatdefaultoccursafterrecapitalization,i.e., theprobabil-ity thatthefirm recapitalizesanddefaultssubsequentto thereorganization.Wheny is closeto the recapitalizationthresholdy nearlythe entiredefault probabilitycomesfrom defaultsthatoccursubsequentto a recapitalization.

in the casewherethe time T inverseleverageratio yT is lessthany (Merton?),

the probability of default is given by N � DD � y� . Or in other words, in a one

period credit risk model a distanceto default of 1.65 correspondsto a default

probabilityof N � 1 � 65 � 5% (or DD = 3 correspondsto a default probability of

0.135%respectively). Sincethe benchmarkmodel (seeSubsection2.3) allows

for bankruptcy at any momentwithin the time horizon,theTEDF of a firm with

static debt level exceedsN � DD (seeEquation(20)). However, for static debt

level TEDF is monotonedecreasingin thedistanceto default. Hence,the larger

DD, thelower is thedefault probability.

23

3 3.5 4 4.5 5

20

40

60

80

���D��� �R�R�

���

�q������������q�M����� ��� ��������� ���

Figure4: Theunderestimationof TEDFswhenignoringtheopportunityto recap-italize plottedagainstthedistanceto default for differentrisk levels. Theunder-estimationis higherfor high-riskfirms.

Whenthe firm dynamicallyadjustsits capitalstructure,the monotonicityof

TEDF is lost. Firms that areperformingwell have a high probability that they

will recapitalizein thenearfuture. Whenrecapitalizing,the inverseleveragera-

tio jumpsfrom y� to y�0 (seeFigure1), with theconsequencethat for firms with

dynamiccapitalstructurewe have TEDF� y� � � = TEDF� y�0 � � . That is, thecorre-

spondencefrom distanceto default to TEDF is u-shaped.

Figure3 plotsTEDFof thedynamicfirm togetherwith theprobabilityof paths

that first hit the recapitalizationtrigger anddefault afterwards. It confirmsthat

for well performingfirms nearlytheentiredefault probability comesfrom these

recapitalization-paths.

24

3 3.25 3.5 3.75 4 4.25 4.5

5

10

15

20

25

30

35

40

���D �¡ ¢R£R¤

¥�¥

¦¨§ª©¦«§�¬�­�¬¯®¦«§ª©¦«§F°}¬�­ ¬�®¦«§ª©¦¨§�¬�­ ¬¯¬

Figure5: Theunderestimationof TEDFswhenignoringtheopportunityto recap-italizeplottedagainstthedistanceto default for differentgrowth rates.Theeffectis morepronouncedfor high-growth firms.

The last two figures(Figure4 and5) shedlight on theproblemof underesti-

matingTEDF whenignoringthefirm’s opportunityto adjustits capitalstructure.

Figure4 shows that the underestimationis moreseverefor high-risk firms than

for low risk firms. Figure 5 illustratesthat this effect is more pronouncedfor

high-growth firms.

3.4 Value-at-Risk of Risky Debt

This subsectionfocuseson the Value-at-Risk(VaR) of a debt contract. In this

context we examinetheriskinessof a long-terminvestmentinto debt,wherewe

understandlong-termin the senseof an investmentin a consolebondthat even

25

3 3.25 3.5 3.75 4 4.25 4.5

5

10

15

20

25

30

35

40

±�²D³�´ µR¶R·

¸�¸

¹«º�»�¼ »�½ ¾¹«ºO»R¼�»�½

¾¹«ºÀ¿�»R¼�»¯½¾¹«º�»�¼ »¯»Figure6: Impactof nonzerorisk premium,i.e., µ � µ.

outlastspossiblereorganizationsof thefirm’sownershipor capitalstructure.That

means,after bankruptcy as well as after debt is called the entire payoff is re-

investedinto consolebondsof thefirm. After sucha reorganizationthefirm con-

tinuesoperationasan optimally leveredfirm, however, at a differentscale(see

Section2).

Calculatingthevalueatrisk of adebtcontract,thedynamicsof theunderlying

yt (seeEquation(3)) hasto be translatedinto the dynamicsof D � yt in orderto

determinethe respective quantileof the lossdistribution. Furthermore,it hasto

beregardedthattheoutstandingprinciple(andthusthecouponflow) maychange

dueto re-investmentafterbankruptcy or call of theexisting bond(whencall pre-

mium is positive,λ Á 0). It is immediatelyevidentthatdynamiccapitalstructure

26

0 2 4 6 8

0.5

0.6

0.7

0.8

0.9

1

1.1

ÂÄÃ#Å

ÂÆÂ

ÇÉÈËÊ#ÈÍÌÏÎÐRÑTÒ Ê�ÓdÌÏÎ

Figure7: Thevalueof thedebtcontractfor thebasecaseparametersasfunctionof thedistanceto default for staticanddynamicdebtlevel. Dynamicadaptationofthecapitalstructureleadsto aflat/downward-slopingvaluefunctionathighvaluesof DD.

adaptationaffectstheprobabilityof eventsthatcauserestructuringof thefirm, and

consequentlyrescalethecoupon.However, theentireshapeof D � y changeswhen

firms cancall debtat somecritical thresholdy (seeFigure7), i.e., thedynamics

of thevalueof thedebtinvestmentchanges.A low call premiumleadsto a rela-

tively flat valuefunctionandthus,thevalueof thebondis relatively insensitiveto

changesin y.

Figure8 plots the typical shapeof the oneyear99%Value-at-Riskof an in-

vestmentinto corporatedebtfor dynamicandstaticdebtlevel versusthedistance

to default. Whenthe distanceto default is low, the probability that the firm de-

27

1 2 3 4 5 6 7

-10

10

20

30

40

50

60

ÔBÕ¯Ö�×Í×ÙØ^ÚÜÛÝRÞ ÚßÕ¯àdØ^á âÉã Õ ã Ø^á

äÆäFigure8: Theone-yearValue-at-Riskat a 99%confidenceinterval of a debtcon-tract for thebasecaseparametersasfunctionof thedistanceto default for staticanddynamicdebtlevel. It is implicitly assumedthat the investorstayswith thefirm loyally even after reorganization,i.e., after debt is called as well as afterbankruptcy theentirepayoff is re-investedinto bondsof thethenewly (andopti-mally) leveredfirm.

faultswithin thenext yearis greaterthanonepercent,which meansthatthe99%

VaR is determinedby thosescenarios,wherethe firm runsinto bankruptcy and

is restructuredat a smallerscalebut optimally levered. Sincethe investmentis

down-scaledin thesecases(and hencethe varianceis reduced)and since the

newly purchasedbondof the optimally leveredfirm is lessrisky (seeFigure7,

D � y0 is flatter thanD � y ), bankruptcy definessomethinglike a ‘catch tray’ for

the valueof the bond. That meansaslong asthe probability of default exceeds

onepercent,the Value-at-Riskis simply definedby the lossthat is given in the

28

casethefirm runsinto bankruptcy. Therefore,theValue-at-Riskincreasesfor low

DD (for both,thestaticaswell asthedynamicdebtstructurefirm). WhenDD is

suchthat theprobabilityof default is lessthatonepercent(which is thecasefor

DD Áæå 2 � 4) bankruptcy is not longerthedominanteventandtheVaRdecreases.

This is the casebecauseD � y becomesflatter for higherDD which reducesthe

sensitivity of thebondvalueto changesin y. Sinceflatteningis morepronounced

in the caseof dynamiccapitalstructure,the reductionin VaR is greaterfor this

firm. However, whenthebondcanbecalledat thecritical restructuringlevel, the

inverseleverageratio jumpsbackto y0, a level at which thedownsiderisk is sig-

nificantly increased.Therefore,for high DD theVale-at-Riskis—for thecallable

bond—anincreasingfunction,whereastheflatteningof D persiststhedominant

factorrducingtheVaRfor largeDD in thecaseof staticdebtlevel.

4 Conclusions

Thispaperhasderivedastructuralcreditrisk modelwhichexplicitly accountsfor

the possibility that firms canalter their capitalstructureover time. The results

show thatthedynamicsof firms’ financingdecisionshave importantimplications

for thechosenleverageratiosandtherequiredcreditrisk spreads.

Wefind thattheoptionto adjustcapitalstructureovertimemakesfirmschoose

a lower initial leverageratio. Despitethis fact,bondholdersrequirehighercredit

spreadsto compensatethemfor therisk of futureleverageincreases.

Thenumericalanalysisalsoproducesestimatesfor thesignificanceof model

risk, i.e. the mistake that is madeby using a static model. We find that, if a

staticmodel is usedto infer the valueof the firm’s assetsandits volatility from

29

observed equity prices,then this can lead to substantialunderestimationof fair

credit spreads.Themagnitudeof themistake increaseswith thevolatility of the

underlyingassetvalueand with the tax benefitof debt and decreaseswith the

premiumthatmustbepaidto old debtholdersbeforea leverageincrease.

Wealsoanalyzetherelationshipbetweenthedistanceto default andexpected

default frequencies.We find that this relationshipis non monotonic. While the

expecteddefault frequency initially decreaseswith thedistanceto default, it ac-

tually increasesfor high valuesof thedistanceto default. This happenssincethe

probabilityof a leverageincreasingcapitalstructureadjustmentincreaseswith the

firm’s distanceto default. As a result the relationshipbetweenexpecteddefault

frequency anddistanceto default is u-shaped.Comparedto the resultsfrom the

dynamicmodel,thestaticmodelsignificantlyunderestimatescreditrisk for large

distanceto default values.This result is consistentwith the observation that the

empiricaldefault frequency decreasesmuchslower thanthe theoreticalrelation-

shipwould imply.2.

An importantimplicationof our numericalresultsis that the relationshipbe-

tweena firm’s distanceto default and its expecteddefault frequency crucially

dependson firm characteristics. In particular the relationshipdependson the

volatility of the underlyingcashflow process,the expectedgrowth of the firm’s

cashflow, andthecostsof recapitalization,includingthecall premium.Thus,the

analysisstronglysuggeststhatoneshouldconditiononthesecharacteristicswhen

estimatingtheempiricalrelationshipbetweenthedistanceto defaultandexpected

default frequencies.

Severalof ourresultscouldbetestedempirically. First,expecteddefaultprob-2See,for example,thegraphin Crouhy, Galai,andMark (2000)

30

abilitiesarepredictedto benon-monotonicin thefirms’ distancesto default. Sec-

ond,thenon-monotonicityshouldbeparticularlypronouncedfor high-riskfirms

andhigh-growth firms. Also, for firms with low effective corporatetax rates,i.e.

firms with largenon-debttax shieldsthenon-monotonicityshouldbefoundto be

lesssignificant.

References

Andrei N. BorodinandPaavo Salminen.Handbookof BrownianMotion – Facts

andFormulae. Birkhauser, 1996.

PeterOve Christensen,ChristianRiis Flor, David Lando, andKristian R. Mil-

tersen. Dynamiccapitalstructurewith callabledebtanddebt renegotiations.

workingpaper, 2000.

PierreCollin-DufresneandRobertS.Goldstein.Do creditspreadsreflectstation-

ary leverageratios?forthcoming,Journalof Finance, 2000.

MichaelCrouhy, DanGalai,andRobertMark. A comparativeanalysisof current

creditrisk models.Journalof BankingandFinance, pages59–117,2000.

AvinashK. Dixit. TheArt of SmoothPasting. Harwood AcademicPublishers,

Reading,1993.

Edwin O. Fischer, RobertHeinkel, andJosefZechner. Dynamiccapitalstructure

choice:Theoryandtests.Journal of Finance, 44:19–40,1989.

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31