American Finance Association
A Mean-Variance Theory of Optimal Capital Structure and Corporate Debt Capacity
Author(s): E. Han Kim
Source: The Journal of Finance, Vol. 33, No. 1 (Mar., 1978), pp. 45-63
Published by: Wiley for the American Finance Association
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THE JOURNAL OF FINANCE
VOL. XXXIII, NO. I
MARCH 1978
A MEAN-VARIANCE THEORY OF OPTIMAL CAPITAL
STRUCTURE AND CORPORATE DEBT CAPACITY
E. HAN KIM*
I.
INTRODUCTION
AN ISSUE OF CONCERN to the theoryof businessfinance over the past two decades
has been the effect of financialstructureon the valuationof firms.The traditional
presumptionis that a firm'svalue is a concave function of its financialleverage,
and that an optimal financial leverage exists where the slope of the function is
zero.' This argumentis suspect to the extent that it attempts to value a firm's
securitiesin isolationfrom the rest of the capitalmarket.The pathbreakingworks
by Modiglianiand Miller (MM) have providedthe foundationsfor studyingthe
effect of financial structureon the valuationof firms in equilibrium.MM (1958,
1969) establishthat the total value of the firm, in the absence of taxes, remains
constantacross all degreesof financialleverage.Buildingon the foundationslaid
by MM, numerousauthors2have confirmedthe MM no-tax thesis using a variety
of equilibriumapproaches.MM (1963) and some of these authors have shown
further that a proportionalcorporate income tax provides sufficient economic
incentive for firms to maximize their use of debt financing. However, in the
five-yearperiod from 1966 to 1970 the capital needs of nonfinancialcorporations
in the United States were financed approximatelyby two-thirdsequity and onethird debt.3 Furthermore,the average corporate debt ratio (which reflects the
valuationof equityat marketvalue) is only approximately20 percent.4Even these
highly aggregatedfiguressuggestthat an elementof majorimportanceto financial
managersand the investingpublicis missingfrom the MM theory.
Robicheckand Myers (1965, p. 20) and Hirshleifer(1970, p. 264) suggest that
bankruptcycosts may representthe majormissingelement and that incorporating
these costs within the foundationslaid by MM may support the concept of an
optimalcapitalstructure.The importanceof bankruptcycosts was particularlywell
demonstratedby Miller (1962) when he explicitly utilized bankruptcycosts to
*VisitingAssociateProfessorof Finance,PurdueUniversity,on leavefromthe Ohio StateUniversity.
This paper is based on ChapterFive of my Ph.D. thesis [Kim (1974)]at State
Acknowledgement.
Universityof New Yorkat Buffalo.I am gratefulto my thesiscommittee,F. Jen (Chairman),A. Chen,
and the late H. Samuelsson.I would also like to thank J. Boness, M. Gordon, R. Hagerman,R.
Hamada,E. Kane, S. Kon, J. McConnell,and StewardC. Meyers,a refereeof this Journal,for their
helpfulcomments.
1. For example,see Gordon(1962)and Solomon(1963).
2. The authorsinclude Robichekand Myers (1966), Hamada (1969), Stiglitz(1969, 1974), Schall
(1972),Rubinstein(1973),Baron(1974),Merton(1974),and Stapleton(1975a, 1975b).
3. See FederalReserveBulletin(1971,p.A. 71.4).
4. See Friend(1974,pp. 3-4).
45
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46
The Journal of Finance
explain the phenomenonof credit-rationingon the basis of rational economic
self-interestby lenders.Millerhas stated:
The substantialcosts and delaysnormallyincurredin case of defaultand the fact that compensating increasesin rates actuallyincreasethe probabilitythat these costs will be incurredmakes the
loan contacta relativelyinefficientinstrument.(1962,pp. 487-488)
Explicit treatmentof bankruptcycosts in the theory of capital structureis
limited. Kraus and Litzenberger(1973) provide a state-preferencemodel with
wealthtaxes and bankruptcycosts, and suggesta stochasticdynamicprogramming
approach to search for an optimal capital structure.In a recent paper which
appearedafter this study was completed,Scott (1976) shows that, if investorsare
indifferentto risk, imperfectmarkets for physical assets (along with a constant
liquidationvalue of the firm's assets in bankruptcy)imply the existence of an
optimalcapital structure.Althoughthese studiesprovideinsightinto the theoryof
optimalcapitalstructure,theirmodels are eithertoo complexto implement(Kraus
and Litzenberger)or ignorerisk-aversionin the capitalmarket(Scott).
More importantly,these studies fail to recognize that, if there are bankruptcy
costs, debt capacity,definedas the maximumamountof borrowingallowedby the
debt financcapital market,occurs well before the point of one-hundred-percent
ing.5Thus, the existenceof bankruptcycosts presentsanotherissue that was best
characterizedby Myersand Pogue(1974,p. 589): It is unclearwhether"thelenders
chickenout first"(i.e., debt capacityoccurs first) or "the shareholderschickenout
first" (i.e., optimal capital structureoccurs first). For example, if "the lenders
chickenout first,"the optimalamountof borrowingwould not be obtainableand
the question of an optimal capital structurewould become irrelevant.Or, if the
optimaldebt level coincideswith the firm'sdebt capacity(i.e., the shareholdersand
the lenderschickenout together),the implicationis the same as that of the MM tax
model-the firm should simply borrow as much as possible. It is only when the
optimalamountof debt is strictlyless than the debt capacitythat firmsmust search
for the optimal trade-off between the tax advantage of debt and the costs of
bankruptcy.Therefore,a logical progressionrequiresanalysis of the problem of
debt capacitybeforeconsiderationof the questionof optimalcapitalstructure.
This paper examines the issues of debt capacity and optimal capital structure
when firms are subjectto stochasticbankruptcycosts and corporateincometaxes
(accompaniedby a parallelanalysiswith wealth-taxesin footnotes) in the context
of the Sharpe (1964)-Lintner(1965)-Mossin
(1966) Capital Asset Pricing Model
(CAPM). After discussingthe nature and the magnitudeof bankruptcycosts in
Section II, we analyze the effects of bankruptcycosts and corporate income
taxation on firm valuationin Section III. Section IV shows that when firms are
subject to bankruptcy costs their debt capacities will be reachedprior to one-hundredpercent debtfinancing. Section V makes it clear that optimal capital structuresinvolve
less debt financing than the maximum amount of borrowing allowed by the capital
5. For example,Miller (1962) has shown that when bankruptcycosts exist, lenders will impose
credit-rationingbefore bankruptcybecomes certain.Although the Miller model is restrictedto the
behavioracrossall investors(lenders)
individuallender'sbehavior,aggregationof such credit-rationing
for corporateborrowing.
in the capitalmarketmay implythe existenceof "credit-rationing"
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Mean-Varianceof Optimal Capital Structureand CorporateDebt Capacity
47
market, and, hence, shareholder-wealth-maximizingfirms will search for optimal
capital structuresrather than simply maximize their borrowing.
Althoughless general,the CAPM is more amenableto implementationthan the
state-preferenceapproach,and in Section VI we show that it provides a much
simpler method for approximatingoptimal capital structuresthan Kraus and
Litzenberger'sapproach. Since, unlike Scott's risk indifference approach, the
CAPM incorporatesrisk-aversion,our method is more realistic but no more
complex than the method suggestedby Scott for determiningoptimalleverage.A
numericalexample,employingthe method suggestedin this paper,illustratesthat
not only is a firm's value a strictly concave function of its end-of-perioddebt
obligationswith a uniqueglobal maximum,but also that the maximumis reached
priorto the firm'sdebt capacity.
This paper also provides some interestingresults in other related areas. An
explicit pricing model that values corporatedebt directly with or without bankruptcycosts is derivedin Section VI. This model providesadditionalinsightsinto
the operationsof bond markets.In Sections III and VI, we compareand contrast
differenttax schemesand theireffects on firmvaluation.The resultsimply that the
tax advantageof debt financingassumesmuch less importancein financialstructure decisionsthan generallysuggested.
II.
COSTS
BANKRUPTCY
The cost of bankruptcycan be thoughtof as comprisingthree majorcomponents.
First, dependingon whetherbankruptcytakes the form of liquidationor reorganization, there may be either the "short-fall"arising from the liquidation or the
"indirect"cost of reorganization.Second, arisingin the course of the bankruptcy
proceedings,variousadministrativeexpensesmust be paid to third parties.Third,
firmslose tax creditswhich they would have receivedhad they not gone bankrupt.
If bankruptcy takes the form of liquidation, the first type of cost is the
"short-fall"arising from the liquidationof physical assets below their economic
values at "distress"prices. This is mainly due to the imperfectionof secondary
marketsfor physicalassets.In a recentpaper,Van Horne (1975) argues:
In a distresssale, a finishedgood frequentlybringsonly 30 to 70 per cent of the wholesaleprice.
Depending on market conditions,a fixed asset may bring even less. While most bankruptcy
auctionsarehonest,the verynatureof the processcoupledwith some shadypracticesdo not augur
well for the seller.(p. 15-16)
If bankruptcytakesthe form of reorganization,the first type of cost is what Baxter
(1967) describesas the "indirect"costs of bankruptcy.The indirectcosts include
reductionin futuresales due to customers'doubts of the reliabilityof the bankrupt
firm as a supplier;difficultyin obtainingtradecredit;higherproductioncosts due
to dislocationswithin the companyand renegotiationof contractsfor employees;
and the time lost by executivesin the reorganizationprocedure.In either form of
bankruptcy,these costs are difficult to documentalthoughthey may be the most
importantcomponent of bankruptcycosts. To our knowledge, no one has attemptedto measurethe magnitudeof these costs.
Embodiedin the total "short-fall"of the liquidationor the total "indirect"costs
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48
The Journal of Finance
of the reorganizationare seriousdelays in bringingabout the liquidationand the
reorganization.For example, Warner (1976) reports that bankruptciesfor 11
railroadcompaniestook on average12.5years to settle in the courts.These delays
impose additionalcosts to debtholderswith provenclaims.
The second type of cost involves fees and other compensationto third parties
(i.e., the lawyers,trustees,auctioneers,referees,accountants,appraisers,etc.) and
representsthe administrativeexpenses of bankruptcy.Payment for these costs
receives the highest priorityin a bankruptcyproceeding under the Bankruptcy
Act.6 In a BrookingsInstitutionstudy, Stanley and Girth (1971) examine these
bankruptcycosts in a large sampleof case analysesand interviews.They estimate
that total administrativeexpenses in a business bankruptcyapproximate20 per
cent of the estate and that about half of these expenses go to attorneys.7This
inferenceis supportedby bankruptcystatisticsfrom the AdministrativeOffice of
the U.S. Courtswhich show that in fiscal 1969 total administrativeexpenseswere
23.4 per cent of the total realizationfrom bankruptcies.8
However,Warner's(1976)
analysisof 11bankruptrailroadcompaniesreportsthat the administrativeexpenses
are on average 5.3 per cent of the market value of the firm at the time of
bankruptcy.Warner attributesthis discrepancyto the fact that he deals with
entities of greaterdollar size than Stanley and Girth's sample. Warner'ssample
shows that there are economiesof scale with respectto bankruptcycosts. Collectively, these studies suggest that while some administrativeexpenses, such as the
attorneyfees,9may not declineon a relativebasis as the size of the estateincreases,
thereare fixed costs associatedwith the bankruptcyprocessthat do so decline.
The third type of bankruptcycosts is due to the tax court'srefusalto grant tax
creditsfor the tax losses of a bankruptfirm. Even if tax laws were lenient in this
regard,bankruptcyusuallyis the result of severalsuccessiveyears of unprofitable
operations,which lessens the possibilityof carryingback these tax losses against
previouslypaid taxes. Thus, in order for the bankruptfirm to receive even a
fractionof tax creditsfor its losses, either the firm must merge with a profitable
firm or it must carry-forward
its tax losses afterthe bankruptcy.Section269 of the
U.S. Internal Revenue Code of 1954 prohibitsfirms from merging for the sole
purposeof taking advantageof the tax law.'0Also, sections 269, 371, and 382 of
the U.S. Internal Revenue Code of 1954 provide formidablebarriersto a postreorganizationfirm from carryingforwardthe losses incurredby a bankruptfirm,
and past court cases have not allowedcarry-oversof such tax losses." Hence, it is
most likely that the creditorsof a bankruptfirm will lose the tax credits to which
the firmwouldhave been entitledhad it not gone bankrupt.
6. See Lusk,Hewitt,Donnell,and Barnes(1974,p. 961).
7. For the breakdownof specificexpenses,see eitherStanleyand Girth(1971)or Van Horne(1975).
8. The breakdownof specificexpensesis reportedin Van Horne(1975).
9. Van Horne pointed out to the author that, as the size of the estate increases,the economic
incentivesto hire lawyersfor legal suits increase,and hence attorney'sfees tend to increase.
10. For an extensivediscussionon this subject,see Holzman(1955)and Tobolowsky(1960).
11. See Holzman(1955),Krantz(1961),Testa (1963)and Tobolowsky(1960).
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Mean-Varianceof Optimal Capital Structureand CorporateDebt Capacity
III.
49
FIRM VALUATION
A. Assumptionsand Definitions
To distinguishthe effect of financialfrom investmentdecisions,we assume that
the firm alreadyhas selected its investmentsbut has not yet decided on how to
finance them.'2The cost of acquiringphysical assets for investmentis A dollars.
The investmentpromisesa stochasticterminalvalue of X dollars after paying all
non-capitalfactors of production.To separatethe firm valuationeffects of financial leveragefrom other extraneousissues (such as dividendpolicy, asset depreciation, etc.), it is assumedthat the firm is to be dissolved at the end of the period.
The firm'soperatingearningswill be X-A, and the residualearningsafter deducting interestpaymentswill be subjectto an income tax rate T.
If the firm chooses to finance the entireinvestmentby equity alone, the market
value of the firm will be Vu-Su, where Su is the marketvalue of the unlevered
firm's equity. Its taxable earningsat the end of the period will equal operating
earnings,X-A. Thus, one-plus-the-rate-of-return
on a dollar investedin the firm's
equitywill be:
RU= [X- T(X-A)]ISu
=-[(1-
T)i+AT]/SU(1
If the firmchooses to financepart of the investmentby borrowingD dollars,the
marketvalue of the firm will be V- Sr + D,-whereSe is the marketvalue of the
levered firm's equity. The levered firm will be bankruptif it fails to meet its
obligationsto debtholders(the principalD plus promisedinterestpayments)at the
end of the period.'3The promisedinterestrate,rP-1, will dependon the amountof
borrowing,D.
If the levered firm does not go bankrupt, its taxable earnings will be X- A - (r -
1)D, and one-plus-the-rate-of-return
on a dollar invested in its equity will be
{X- T[X-A
-(P- 1)D ]-D
}/ Se = [(1 - T)(X-PD) + T(A -D)]/ Se. One-plus-
on a dollarinvestedin the firm'sdebt will be r.
the-rate-of-return
Bankruptcyoccursif the firm'sterminalvalue is less than its total end-of-period
debt obligation,i.e., X < PD.In the event of an actualbankruptcy,stockholderswill
exercisetheir limited liability,and ownershipof the bankruptfirm will be transferredto debtholders.Debtholderswill have to pay the cost of bankruptcyout of
X. Therefore,one-plus-the-rate-of-return
on a dollar invested in the securitiesof
12. This assumption implies that there is no debt outstanding prior to the financial structure decision.
If the firm has already issued some positive amount of debt, it is necessary to assume an effective
"me-first" rule. For an extensive discussion of "me-first" rule, see Fama and Miller (1972) and Kim,
McConnell, and Greenwood (1977).
13. For the sake of brevity, we do not distinguish "bankruptcy" from "default". Whereas "default"
merely describes the borrower's action, bankruptcy often involves lender initiatives. The question of
whether or not bankruptcy proceedings should be initiated by creditors when the firm defaults on its
debt obligation and the optimal timing of such proceedings are examined in detail by Van Horne (1976).
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The Journal of Finance
50
the levered firm may be defined as:
R=(
[(l
O
T)(X
rD)+T(A-D)]1S,
if X>PD
(2)
if X<PD,
and
r
(r
if X >PD(3
((XB)/D
if X&<D,
where Re and r are one-plus-the-rate-of-return on the levered firm's stocks and
debt, respectively, and B equals bankruptcy costs.
Because tax credits are not included in the gross return to debtholders of the
bankrupt firm, the third type of bankruptcy costs (i.e., the loss of tax credits) is
already implicit in (3). Thus, B in (3) represents only the sum of the first two types
of bankruptcy costs identified in Section II, and may be expressed as:
h= =B(X)
if)?D(4)
if X< PD,
where B(X) is an implicit positive function of X, but is no greater than X, i.e.,
B(X) < X. This upper limit on B(X) provides debtholders with limited liability in
a nominal asset case in which the proceeds from liquidation are entirely consumed
in administrative expenses with no distribution to debtholders.
Finally, we assume that risky securities are priced according to the CAPM such
that the equilbrium expected return on any risky security i is: 14
E(Ri)= RF+Xcov(RiiRm)
(5)
where RF is one-plus-the-rate-of-return on the riskfree asset;'5 Rm is one plus the
value-weighted rate of return on all risky securities in the market, and has an
is
expected value of E(Rm) and a standard deviation of arm;X= [E(Rm) -RF]/2
and
and
Rm
between
covariance
is
the
cov(Ri,
the market price of risk; and
Ri
Rrm)
represents the systematic risk of security i.
B. Market Value of the Firm
If we define the bankruptcy operator as:
g 1
if X<D
(6)
14. Since we assumethat firms are subjectto bankruptcycosts, all returnson risky securitiesand
portfolios,includingRm,are definedas returnsafterbankruptcycost. With this definition,Kim (1974,
pp. 100-104)showsthat the exactformof the CAPMholdswhenfirmsare subjectto bankruptcycosts.
15. The assumptionof the existenceof a risk-freesecurityis not substantive,providedzero-beta
portfoliosexist in which case the symbolRF in this papermay be replacedat everypoint by E(RZ),
whereZ is the minimumvarianceportfoliowith returnsuncorrelatedwith the marketreturn.
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Mean-Varianceof Optimal Capital Structureand CorporateDebt Capacity
the market value of the firm may be expressed as:
Ve = Vu+ TD(RF -I)IRF-
51
16
T(A- D )V(b) - (1-T) V(B )
(7)
where V(b)=[E(b)-N
cov(b,Rm)]/RF is the risk-adjusted present value of one
dollar associated with the occurrence of bankruptcy, and
V(B)=[E(B) -N cov(B, Rrn)]/RF is the risk-adjusted present value of the
bankruptcy costs.
In the absence of taxes and bankruptcy costs (i.e., T=O and B=O), but with a
positive probability of bankruptcy, (7) reduces to V, = Vu,i.e., the market value of
the firm is independent of its financial structure. With a positive income'7 tax rate
and positive bankruptcy costs, (7) illustrates that the market value of the levered
16. By combining(2), (3), (4), and (6), we can rewrite(2) as:
Rie= [(I1- T)(X -riD ) + T(A - D )(I1-b ~)-(1-
T)B I/ Se .
(2a)
[To show that (2) and (2a) are identical,one only has to substitute(3), (4), and (6) back into (2a).]
Havingobtained(1) and (2a), derivationof (7) is straightforward.
Substituting(1) into both sides of (5)
gives:
(1- T)E(X )=SRRF-A T+ ( - T)X cov(X, Rm)
Substituting(2a) into both sides of (5), and becauseE()
(1- T)E()=
[ Se + D (I1-T)]RF -T(A+ T(A
-
= RF+ Xcov(W,
Rm),
(a)
it followsthat:
D) + (1- T)X cov(i,Rkm )
D )[E(b )-Xcov(b,Rm )] + (1- T)[E(B ) -Xcov(B,Rm )]
(b)
From(a) and (b), and fromthe definitionsthat:
Ve=Se+D
and Vu=Su,
we can obtain(7). It shouldbe noted that, althoughthe CAPM is necessaryto providea simple,but
practicalmethodto approximatethe optimalcapitalstructure,one does not need to assumethe CAPM
merelyto derive(7). It can be shownthat,given(1) and (2a),a similarresultalso holds for moregeneral
theoreticalmodelssuch as the State Preferenceapproachor Schall's(1972)ValueAdditivityPrinciple.
in whichthe interestpaymentsare tax
17. The tax structureconsideredhere is an income-tax-system
deductiblebut principalpaymentsare not. The alternativetax structurethat has been consideredin
financeliteratureis a net-terminal-wealth-tax
systemin whichboth the interestand principalpayments
of corporatedebt are tax deductible.[For example,see Rubinstein(1973)and Krausand Litzenberger
(1973).]Clearly,an income-taxstructureis more realisticthan a wealth-taxstructurein view of the
United Statestax code.
and lets T be the wealth-taxrate,
Nevertheless,if one assumes a net-terminal-wealth-tax-system
on the unleveredand the levered firms' equities should be redefinedas
one-plus-the-rate-of-return
follows:
R = (1-T)X/S",
(la)
and
if X>rD
R R=T(1-T)(X-rD)/Se
(2b)
if X<D.
O
By followingthe steps that led to (7), it can be showneasily that the marketvalue of the leveredfirm
undera wealth-taxsystemis:
Vwe
= Vee+ TDe- ( heT)wV(B
where V,,' is the market value of the unlevered firm under the wealth-tax system.
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(7a)
52
The Journal of Finance
firmis equalto the marketvalue of the unleveredfirm plus the presentvalue of tax
deductibilityof interestpayments,TD(RF - 1)/RF,18 minusthe-presentvalue of the
loss of tax credits in the event of bankruptcy,T(A - D) V(b), and the fraction
(1 - T) of the present value of bankruptcy costs, V(B).19
With corporateincome taxes and bankruptcycosts, the ex-ante marketvalue of
X, which depends only on the firm'sinvestmentdecisions,is divided among four
parties: stockholders,debtholders,the government,and bankruptcycosts. Since
V(B) increasesas the probabilityof bankruptcyincreasesand the probabilityof
bankruptcyincreases as financial leverage increases, V(B) also will increase as
financial leverage increases.On the other hand, debtholdershave claims to the
future earningsof the firm that are prior to the government'sclaim, and hence
V(G), the value of the government'sclaim to the future earnings of the firm,
decreases with increased financial leverage. Therefore, as the firm's financial
leverageincreases,the increasein V(B) will be offset by a decreasein V(G), and
the sum of V(B) and V(G) will either increase or decrease depending on the
particulardegreeof financialleverage.
The part of the marketvalue of the firm that belongs to the suppliersof the
capital is the differencebetween the market value of X, V(X), and the sum of
V(B)
and V(G), i.e., V = V(X)-[V(B)+
V(G)]. Since V(X)=[E(X)-
Acov(X,Rm)]/RF is independentof financialstructure,the financialstructurethat
minimizes the sum of V(G) and V(B) will maximize Ve in (7).
IV.
DEBT CAPACITY
CORPORATE
Since an optimalcapitalstructureis a meaningfulconcept only if it can be shown
that the optimaldebt is strictlyless than debt capacity,in this section we presenta
formal analysisof corporatedebt capacity.Corporatedebt capacityis defined as
the maximumamount that a firm with given investmentscan borrowin a perfect
capital market.Corporatedebt capacity is denoted by D, and rD representsthe
amountthat the firm must promiseits debtholdersto reach D.
18. Since the availabilityof an interesttax shield is contingentupon the actual payment of the
promisedinterest,with a positiveprobabilityof bankruptcy(and defaultof interest)the interesttax
shieldbecomesrisky.TD(RF- 1) is the certaintyequivalentof the riskytax shieldfor interestpayment
(or the certaintax shieldif thereis no probabilityof bankruptcy).Discountingthis certaintyequivalent
at the risk-freerate yields the present value of the tax shields, TD(RF- l)/RF. This quantity is
substantiallysmallerthan the presentvalue of tax savingsin the originalMM tax model,which states
that Ve= Vu+ TD. While the debt in our model maturesat the end of a single period with no
presumptionabout the levered and the unleveredfirm's future financial structuresbeyond a single
period,the MM tax modelassumesthat debt has no maturityin an infinitehorizonframeworkand that
the presentfinancialstructureof the leveredand the unleveredfirm will be maintainedforever.If we
assertthis perpetuityassumption,the presentvalue of tax deductibilityof interestpaymentsbecomes
00 TD(RF- l)/(RF)' = TD. This is identicalto the presentvalueof tax subsidyin the MM tax model.
I
Note thatthe presentvalueof tax subsidyin (7a) is also TD. [Seefootnote 17.]Hence,a wealth-taxin
a single-periodframeworkyields the same presentvalue of tax savings as an income-taxin MM's
infinitehorizonframeworkdoes.
19. The remainingT percentof V(B) is borne by the government,as the premiumadded to the
promisedinterestrate due to bankruptcycost reducesthe government'sclaim to the futureearningsof
the firmby T percent.
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Mean- Varianceof Optimal Capital Structureand CorporateDebt Capacity
53
In a perfectcapitalmarket,unlessthe firm alreadyhas reachedits debt capacity,
it can borrowmore by promisingto pay more to its potentiallendersat the end of
the period. Once it has reachedits debt capacity,by definition,it can borrowno
more regardlessof how much more it promisesto pay at the end of the period.
Mathematically, it means that (dD/drD) >0 for PD< PD, and (dD/dPD)
=
0 for
rD = rD. Since in a perfect capital market,investors will lend exactly what the
promisefor futurepaymentis worth accordingto the marketpricingmechanism,
dD/ drD will equal zero only when any furtherincreasein PD does not create any
additionalvalue for lenders.
Such a debt capacity occurs before bankruptcybecomes certain if there are
bankruptcy costs.20Although Miller (1962) assumes that lenders display risk
aversionwhen he utilizes bankruptcycosts to explain the phenomenonof creditrationing,21 it is not risk aversion itself that causes occurrenceof debt capacity
before bankruptcybecomes certain. It occurs because: (1) the present value of
bankruptcycosts, V(B), increasesas PD increasesand (2) in the event of bankruptcy,the claimsof bankruptcycosts to X must be satisfiedpriorto the claims of
debtholders.
These points can be demonstratedby assumingthat lendersare risk-neutral.By
assumingrisk-neutrality
we can distinguishthe effect of bankruptcycosts from the
effect of riskaversion.Then the marketvalueof rD is simplythe expectedreturnto
debtholdersdiscountedat RF:22
D = E(rD)/RF
(8)
Substituting(3) and (4) into (8) gives:
D= {D[1-F(rD)1+
Xf(X)dX-f
B(X)f(X)dX
}/RF
(9)
wheref(X) =the probabilitydensity of X with an expected value of E(X) and a
standarddeviationof a
f(X)dX is the probabilitythat the firm will be bankruptat the
F(rD)= f'i%3
end of the period.
Differentiating(9) with respectto PDyields:
dD
d^D = [ 1-F(rD )-B(rD)f(rD)1/RF.
(10)
20. If bankruptciesare costless,corporatedebt capacityis not an operativetermin a perfectcapital
market,as the firm'smaximumborrowingis reachedonly when bankruptcybecomescertain.
21. Milleraddressesthe issue of credit-rationing
by individuallendersfor personalborrowingrather
than the phenomenonof credit-rationing
by the capitalmarketfor corporateborrowing.However,his
resultshave importantimplicationsfor corporateborrowing,because it does not rely on the legal
constraintsof maximumcontractrates of interest.Althoughcredit-rationingfor personalborrowing
may be explainedthroughthe maximumceiling on contractualrates of interest,credit-rationingfor
corporateborrowingmay not. Due to the discountingof corporatedebt securitiesin the capitalmarket,
thereexistsno practicalceilingon the maximumcontractualratesof intereston corporateborrowing.
22. A more completederivationof the marketvalue of corporatedebt within the context of the
CAPMis providedin SectionVI.
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54
The Journal of Finance
(10) illustratesclearly that any change in rD has both a positive and a negative
effect on lender'sexpectedreturn.On the one hand, an increasein rD means an
increasein expectedtotal return.This incrementin expectedreturnis represented
by the term 1- F(PD), which is the probabilitythat the firm will not be bankrupt.
On the other hand, an increasein PD means an increasein expected bankruptcy
costs, because a higherrD means a higherprobabilityof bankruptcy.This increment in expectedbankruptcycosts is representedby the term B(rD)f(PD).
Since the probabilityof bankruptcy,F(PD), increasesas PD increases,the first
term in (10), 1- F(rD), decreasesas PD increases.B(PD) varies directlywith PD
because B(X) is a positive function of X [See (4)].23 If X is normally distributed,
1- F(rD) is equal to B(rD)f(rD) before the probabilityof bankruptcyreaches
one. That is, thereexistsan rD at which(dD/ dPD)= 0 and F(PD)< 1. This resultis
shown in AppendixA. AppendixA also shows that the second-orderconditionis
met at PD. Therefore,the firm's borrowingreachesits maximum(the firm's debt
capacity)whilebankruptcyremainsuncertain.
AppendixA also shows that (dD/dPD) < 0 for PD> PD,which means that firms
will face a decreasing D while PD is increasing once they reach their debt
capacities.The intuitiveexplanationis as follows: Althoughdebtholdersbear the
ex-postcost of bankruptcy[See (3)], in a perfectcapitalmarketlenderspass-onthe
entireex-antecost of bankruptcy,V(B), to stockholdersand the governmentin the
form of higher promisedinterestrates. However,when the firm increasesits PD
beyond PD, the incrementalincrease in V(B) will dominate the incremental
increase in lenders' expected returns.Consequently,lenders cannot pass-on the
entire V(B) unless they reduce the total amount of funds lent, D. In a perfect
capitalmarket,perfectsubstitutesare alwaysavailable.Hence, lenderswill simply
switch to other corporatedebt securitieswhich allow them to pass-on the entire
V(B) to stockholdersand the governmentand earn the marketequilibriumreturn.
These switcheswill reduce the demandfor the firm'sPD, which in turn will force
the marketvalue of rD to decline.That is, D will decreaseas rD exceeds rD.
If there is any chance of solvency at the end of the period,the marketvalue of
commonequityat the beginningof the periodwill be positiveand D/ Ve< 1. Since
debt capacity occurs while bankruptcyremains uncertain, the firm's financial
structureat its debt capacitywill be less than one-hundredpercentdebt financing.
V.
OPTIMALFINANCIALSTRUCTURE
Differentiating(7) with respectto rD gives
dVe
dP=
r
R l
[ T(RF-I)RF]
Dd
drD + TV(b ) drD - T(A-D
dV(b)
) dD -( 1-T)
dV(B)
drD
(11)
23. For example,B(rD)= C+ crD if we assumethat the costs of bankruptcyare the sum of a fixed
chargeof C dollarsand a variablechargeequalto a fractionc < 1 of X (i.e., B(X) = C + cX).
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Mean- Varianceof Optimal Capital Structureand CorporateDebt Capacity
55
(11) illustratesthat any change in FD has both a positive and a negativeeffect on
marketvalue of the firm.On the one hand, an increasein rD means an increasein
the presentvalue of tax savings(PVTS).On the otherhand, it meansan increasein
the presentvalue of bankruptcycosts (PVBC).
When PD approachesthe minimumpossible value of X, (dD/dPD)-(1/RF),
V(b)->O, (dV(b)/dr^D)-*0, and (dV(B)/dPD)--0; hence dVI/dPD would be
strictlypositive.That is, an increasein PD means a greaterincreasein PVTS than
in PVBC,and hence the marketvalue of the firm increases.
When rD is equal to PD, (dD/drPD)= 0, and hence (dV/JdrD) -[T(A -D)
(dV(b)/PdrD)+ ( - T) dV(B)/drD ]. If bankruptcyremainsuncertain,an increase
in PD increases the probabilityof bankruptcy,and thus both V(b) and V(B)
should increase as PD increases,i.e., (dV(b)/drD)>0 and (dV(B)/dPD) >0.24
Since bankruptcyremainsuncertainat the firm's debt capacity,dVl/dPD should
be strictlynegative.25That is, at debt capacityan incrementalPD means a positive
incrementin PVBCbut a zero incrementin PVTS(i.e., the first termin (11) is zero
when(dD/dr^D)= 0), and hence the marketvalue of the firmwill decline.
With Ve risingwhen PDis small, and Ve falling when rD is large,theremust be
an AD,say rD*, at which dV, /d?D equals zero and Ve attains a maximum,V*.
Furthermore,since dV, /dFD is strictlynegativeat PD,rD* shouldbe strictlyless
than r^D.rD* is the optimalend-of-periodamount that the firm should promiseto
pay its debtholdersin order to maximize V,. Likewise,D*, the amount borrowed
by promisingrD*, is the optimalamountto borrow.Since rD* is less than PrD,D*
should also be less than D. That is, the optimal capital structure involves less debt
financing than the firm's debt capacity.
Therefore, a shareholder-wealth-maximizing
firm will not maximize its borrowing.
Instead, it will search for its optimal capital structure to attain V*. By setting (11)
equalto zero,we obtain:26
RF_
1dD
dV(b)
R +V(b)] dD(A-D)
dFD +
I-~T dV(B)
T
dPD
(12)
24. If bankruptcyis certain, V(b)= l/RF and V(B)= {E[B(X)]-X cov[B(X),Rm])/RF, thus, any
further increase in rD will not affect V(b) and V(B), i.e., (dV(b)/drD) = 0 and (dV(B)/drD) = 0.
25. If a net-terminal-wealth-tax-structure
is assumedand if T representsthe tax rate, differentiation
of (7a) [See footnote 17]gives:
dVw Td
d =D TdD -(l-T)
dV(B)i
dD
(Ila)
whichis also negativeat the firm'sdebt capacity.
26. Underthe wealth-tax-system,
the optimalcapitalstructureis the rD which satisfies:
dD
drD
1-T
T
dV(B
drD
withoutmakingboth sides of (12a) be equalto zero.
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(12a)
56
The Journal of Finance
The optimal capital structureis the rD which satisfies (12) without making both
sidesof (12) be equalto zero.27
VI.
NORMAL DISTRIBUTION
We have shown that in a perfect capital market where firms are subject to
corporateincome taxes and costly bankruptcies,debt capacityoccurs at less than
one-hundred-percent
debt financingand the optimalcapitalstructureoccursbefore
debt capacity.In this section we derive an explicit valuationmodel for risky debt
with or withoutbankruptcycosts, and developa simplemethodto approximatethe
optimalcapital structure.We assume that (1) X and Rmare normallydistributed;
and (2) B(X) in (4) is the sum of a fixed chargeof C dollarsand a variablecharge
equalto a fractionc < 1 of X:28
B(X)= C+cX
(13)
rD=PD(1-b)+bX-B.
(14)
Substituting(6) into (3) gives:
Substituting(14) into both sides of (5), and from E(b)=F(PD)
fD Xf(X)dX, it follows that:
f
D=(rD[I-F(D)]+
and E(bX)=
Xf(X)dX+X[PDcov(b,Rm)
-cov(bX,Rm)] /RF-V(B).
(15)
Substituting(B-1), (B-3), and (B-4) of AppendixB into (15) gives:
D= [E(rDO)-X cov(X,Rm)F(PD)]/RF-
V(B)
(16)
whereE(rD?)= rD [1- F(rD)]+ E(X)F(rD)- a2f(JD) is the expectedgross dollar
returnto debtholdersin the absence of bankruptcycosts.
If thereare no bankruptcycosts (i.e., B = 0), (16) providesan explicitpricingmodel
for corporatedebt. It also states explicitlythat the risk premium on corporate debt
equals the borrowingfirm's operating risk premium, X cov(X,Rm), multiplied by the
27. If bankruptcyis certain,both sides of (12) are equalto zero,but the firm'stotal marketvaluewill
be strictly less than V*. This is because dVe/drD stays negative throughoutall rD >rD* until
bankruptcybecomescertain:First,we knowfromthe text thatdVe/dPD <0 for rD* < rD < r^D.Second,
it also has been shown that, for rD > rD, dD/drD <0 and both dV(b)/drD and dV(B)/drD are
positive,and hence,from(11), (dVe /drD) <0 until bankruptcybecomescertain.
28. To be moreprecise,the bankruptcycost may be expressedas:
BB
( C+cX
X
ifC/(l-c)<X<rD
if X<C/(l-c),
which providesthe debtholdersof a bankruptfirm with limited liability. However,for the sake of
brevitywe assume(13).
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Mean- Varianceof Optimal Capital Structureand CorporateDebt Capacity
57
firm's probability of bankruptcy,F(QD).29The explanation for this measure of risk
premium is straightforward. To the extent the firm is not bankrupt, debtholders
receive a fixed amount rD which has no systematic relationship with the market. If
the firm is bankrupt, debtholders receive X which has a systematic risk of
cov(X, Rm) that cannot be diversified away in debtholders' portfolios. Since this
systematic risk is relevant to debtholders only to the extent that the firm is
bankrupt, the relevant risk is the firm's total systematic risk multiplied by the
probability of bankruptcy.
With positive bankruptcy costs, (16) shows that market value of risky debt is
simply what it would have been in the absence of bankruptcy costs minus the
present value of bankruptcy costs. In other words, lenders deduct the ex-ante costs
of bankruptcy from the amount they would have lent had there been no bankruptcy costs. Substituting (6) and (13) into (4) gives B = Cb + cXb, which can be
substituted into the definition of V(B) in (7) to obtain:
V(B)= { CF(PD)+c[E(X
)F(D )-a2f(PD)]
+X cov(X,Rm)[(C+crD)f(rD)-cF(rD)]}/RF.
(17)
By substituting (17) into (16), we obtain the formal expression for the pricing of
corporate debt with linear bankruptcy costs:
D= {r D[1-F(PD)]
-CF(PD)
+(1-c)[E(X)F(rD)-a2f(D)]
-X cov(X,Rm)[(C+crD)f(PD)+(1-c)F(PD)]}/RF.
(18)
Finally, to provide a method that can be used to search for the optimal capital
structure, we solve for the remaining terms in equations (7) and (12). Substituting
(B-3) of Appendix B into the definition of V(b) in (7) yields:
V(b)= [F(D ) + X cov(X, Rm)f(rD )/RF.
(19)
Differentiating (17), (18), and (19) with respect to rD yields:
PD
dV(B)
~DE(X)~
[
D
dV(B) =(C+cPD)
]f(PD)/RF
l-Xcov(X, Rm)
dD
trDD)
-
)[CciD(
Ej
Xco()
(20)
~
)
~~~~~~
A~~~~
+Xcov(XRm)1/RF,
dV(b)
drD
=f(PD)
cov(X mPD1-X COV(XRM)
and
(21)
E(X)R
2
J/RF
(22)
29. As the referee pointed out, this result may not depend on the assumption of normal distribution.
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58
The Journal of Finance
Note that all of the values in (17) through (22) are expressed in terms of f(rD)
and F(rD). Thus, if we know the firm-specific and the market-wide parameters, all
of the values in (17) through (22) for a given rD can be found by using tables of
ordinates and areas of the normal distribution. By trial and error, we can find the
rD* which satisfies (12) without making both sides of (12) be equal to zero. The
corresponding D* and V* can be obtained from (18) and (7).
For a hypothetical bi-variate distribution of X and Rm5 Table 1 contains the
numerical values of V(b), V(B), D, P, Vp, D/ VE, V ,30 and D/Vw for various
levels of PD. Figure 1 depicts the behavior of D, VeJ,and Vw as a function of PD.
From Table 1 and Figure 1 we see, first, that corporate debt capacity occurs
prior to a one-hundred percent debt ratio, and any increase in PD beyond rD only
decreases the firm's borrowing. Hence, from the firm's perspective, PD > PD is
inferior to PD < PD and D > D is infeasible.
Second, regardless of whether one assumes income-taxes or wealth-taxes, the
firm's value is a strictly concave function of its end-of-period debt obligation with a
unique global maximum, and the maximum is reached before debt capacity.
TABLE 1
VALUATION EQUATIONS (19), (17), (18), (7), AND (7a) FOR A HYPOTHETICAL BI-VARIATE
NORMALDISTRIBUTION;E(X)= 1,300,000, Ux = 300,000, E(Rm)= 1.15, am=.20, pxm =5X
A = 1511356365
C=0, c=.4, T=.5, AND RF= 1.05.
rD F(z)b
za
-4.33
-3.5
-3.0
-2.68
-2.5
-2.0
-1.5
-1.02
- 1.0
- .5
0
.5
1.0
1.5
2.0
2.5
3.0
3.5
0
250,000
400,000
496,000
550,000
700,000
850,000
994,000
1,000,000
1,150,000
1,300,000
1,450,000
1,600,000
1,750,000
1,900,000
2,050,000
2,200,000
2,350,000
az= [rD-
.0000
.0002
.0013
.0037
.0062
.0228
.0668
.1539
.1587
.3085
.5000
.6915
.8413
.9332
.9772
.9938
.9987
.9998
f(z)c
V(b)
V(B)
.0000
.0009
.0044
.0110
.0175
.0540
.1295
.2371
.2420
.3521
.3989
.3521
.2420
.1295
.0540
.0175
.0044
.0009
.0000
.0004
.0023
.0061
.0101
.0346
.0945
.2030
.2088
.3776
.5712
.7424
.8589
.9196
.9435
.9506
.9522
.9524
0
12
271
989
1,810
8,069
26,857
67,169
69,450
142,290
237,132
331,083
401,826
442,277
459,627
465,190
466,479
466,672
D
0
238,012
380,445
470818520,985
6545569
769,524
845,613
847,795
874,384
851,277
801,091
752,376
720,789
706,268
701,348
700,137
699,952
d
1.05
1.05
1.05
1.06
1.07
1.10
1.16
1.18
1.32
1.53
1.81
2.13
2.43
2.69
2.92
3.14
3.36
Vee D/ V,
1.1135636 .0000
1,119,120 .2127
1,121,712 .3392
1,122,386 .4195
1,122,138 .4643
1,117,238 .5859
1,102,263 .6981
1,072,973 .7901
1,071,335 .7913
1,018,131 .8588
940,401 .9052
851,144 .9412
775,486 .9702
729,021 .9887
708,456 .9969
701,773 .9994
700,193 .9999
699,968 1.0000
E(X)]/ ox representsthe standardizedrD.
b F(z) =
F(rD) is
the probabilityof bankruptcy.
representsthe standardizednormaldensity.
dr=(PD)/D is one plus the promisedinterestrate.
eNote that Vu= [(1 - T)E(X) + A T- X( - T) cov(X, Rm)]/ RFcf(z)= =afj(D)
fSee footnotes 17, 25, and 26, and note that V7=[(1
-
T)E(X)-(1
- T)COV(X,Rm)]/RF.
30. See footnotes 17, 25, and 26.
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V
D/ Ve
583,333 .0000
702,333 .3389
773,420 .4919
818,248 .5754
842,921 .6181
9065583 .7220
954,667 .8061
972,555 .8695
972,506 .8718
949,380 .9210
890,406 .9561
818,337 .9789
758,608 .9918
722,589 .9975
706,654 .9995
701,412 .9999
700,162 1.0000
699,973 1.0000
Mean-Varianceof Optimal Capital Structureand CorporateDebt Capacity
59
Third, while with wealth-taxesthe differencebetween the marketvalue of the
firmat its optimalcapitalstructureand the marketvalue of the unleveredfirm(i.e.,
Vw) is significantly large, the counterpart with income taxes (i.e., V* - V.) is
-W
ratherinsignificant.While both interestand principalpaymentsof corporatedebt
are tax-deductibleunder wealth-taxes,only interest payments are tax-deductible
under income-taxes.Hence, the tax-advantageof debt financingis much greater
with wealth-taxesthan with income-taxes,which in turn, implies a much steeper
slope for V' than V, before they reach maximumvalues at the optimal capital
structure.On the other hand, government'sshare of the firm is greater with
wealth-taxesthan with income-taxes,and thus the initial value for Vw (i.e., Vu) is
smallerthan the initial value for V, (i.e., V.).
These two alternativetax structuresrepresentthe two opposite extremecases.
w
I.0
_ .?
0.9
iV
0.8
z
0.6
0
-J
0.5
0.4
0.3
0.2
0.I
0
D
0.5
1.0
1.5
2.0
2.5
MILLIONDOLLARS
FIGURE 1. D, Ve, and V, as a functionof rD for a hypotheticalbi-variatenormaldistribution;
E(X)= 1,300,000,AX=300,000, E(Rm)= 1.15,a. =.20, Pxm.5, A=1,113,636, C=0, c=.4, T=.5, and
RF= 1.05
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60
The Journalof Finance
With income-taxesthe market value of- the tax subsidy derives from the taxdeductibilityof only the one-periodinterest payment; but with wealth-taxesthe
marketvalue of the tax,subsidyis equivalentto the value associatedwith perpetual
interestpaymentsunderan income-taxstructure[See footnote 18].Therefore,these
two alternativetax cases in a singleperiodframeworkprovidethe upperand lower
bounds for the marketvalue of the tax subsidyand hence for the optimal capital
structurein a multi-periodworld with an income-tax.
VII.
CONCLUSION
This paperhas shown that, in a perfect capital marketwhere firms are subjectto
income taxes and costly bankruptcies,debt capacity occurs at less than onedebt financingand firmsdo have optimalcapitalstructureswhich
hundred-percent
involveless debt financingthan theirdebt capacities.The marketvalue of the firm
increases for low levels of debt and decreases as financial leverage becomes
extreme. With linear bankruptcy costs, a simple method to approximatethe
optimal capital structurewas derived. A numericalexample using this method
shows that the marketvalue of the firm is a strictlyconcave function of its total
end-of-perioddebt obligationswith a uniqueglobal maximum.
This is essentiallythe same as the traditionalist'sposition on the relationship
between the value of the firm and corporatefinancial leverage.We have shown
that the traditionalist'sargumentfollows from the MM logic by allowing for the
existence of corporateincome taxes and bankruptcycost. However, there are
fundamental differences between the approach taken in this paper and the
traditionalist'sapproach to the valuation of firms: While the traditionalist's
approach is based on the notion that valuation of firms can be explained by
consideringsecuritiesin isolation from the rest of the capital market, our conclusions are derived within a theoretical frameworkbased on capital market
equilibrium.
APPENDIX
A
CorporateDebt Capacity With BankruptcyCosts and Normally DistributedX
To prove that corporatedebt capacity is reached before bankruptcybecomes
certain,we must show that there exists a finite PD at which (10) equals zero (the
first-ordercondition) and the market value of debt, D, is at its maximum(the
second-ordercondition).
Since at very low rD, (dD/dPD)1/RF>0,
if (dD/dPD)<0
for a large finite
rD, (10) must equal zero at a finite PD. As PD increases,B(rD) increases and
a2/[PD
-
E(X)] decreases. Hence, BQ(D)> U2/[PD - E(X)] for a sufficiently large
PD. But for a normally distributedrandom variable, we know the following
inequality[Feller(1968, p. 175)]:
-
E
f (D ) > 1- F(D),
for PD > E(x).
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(A- i)
Mean-Varianceof Optimal Capital Structureand CorporateDebt Capacity
61
Thus, theremust exist a finite rD at which:
<
such that dPD <?
faD>2-(P)dD
-PDE(X) f(rD)> 1-F(rD)
B(rD)f(rD)>
To show that the second-orderconditionis satisfied,we differentiate(10) with
respectto PD. Noting that (df(D)/ dD) =-((rD - E(X))/l2) f(rD) for a normal
distribution,
D
dd2D
f(rD )-
-[
PD -E(X)
(
'R
f (rD ) R
d(PD)
B(PD )f (rD ) + dBP
A2
(A-2)
Since (dB(?D)/dPD)>0 [See (4)], the second derivative is clearly negative if
into
PD< E(X). Substitutingthe first-ordercondition, B(rD)f(rD)=l-F(PD),
(A-2) yields:
d2D
drDI|
dpD2 PD=P [J\f
FDE()dB(?)~
=-[f(rD)_
I
[1-F(PD)]+
\u
f('D)
dPD jrij(A3
(A-3)
If PD > E(X), (A-3) is also negativebecauseof (A-I). Thus, D is maximizedat PD.
APPENDIX
B
Determination of Partial Means and Partial Covariances When X and Rm are
Normally Distributed
From Winkler,Roodman,and Britney'sequation(3.4) (1972, p. 294), we obtain
the partialmean of X,
rD
f
Xf(X )dX = E(X )F(PD ) - 2f (PD).
(B-1)
00
The covariancebetweenb and Rm,
cov(b, Rm)= E(bRm)-E(b
rD
=AfrDf(X
, 0,
)[0
)E(Rm)
Rmg(Rm|X)dRm-E(Rm)
dX.
BXtheorem,the conditionalmean of Rm for a given value of X, fr?ooRmg(Rm X)
[See Mood and Graybill (1963, p.
dRm=E(Rm)+cov(X,Rm)[X-E(X)]/var(X)
202)].Therefore,
rD
d]
Xf(X )dX-
Cov(b,Rm)=cov(X, Rm)A
E(X )F(r^D )/x
-00
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(B-2)
62
The Journal of Finance
Substituting(B-i) into (B-2) gives:
cov(b, Rm)
cov(X, Rm)f(rD).
(B-3)
The covariancebetweenbX and Rmcan be writtenas:
cov(bX, Rm) = E(bXRm) -E(bX )E(Rm)
=
frXff(X)
[fS
-00
Rmg(RmIX)dRm-E(Rm)
Mdi
L-00J
Using the theoremfor conditionalmean again, we obtain:
~ ~~~D ~~~2
X 2f(X)dX-E(X)
cov(bX, Rm) =cov(X ,Rm)
rD
Xf(X)dX
/var(X)
From Winkler,Roodman, and Britney'sequation (3.4) (1972, p. 294), we also
obtain f oX2f(X)dX- E(X)f.Xf(X)dX=var(X)[F(rD)-PDf(JD)].
Therefore,
COV(bX,
Rm) cov(X,Rm )[F(rD )-PDf(D )]
(B-4)
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Theorem,"American
Economic Review (March 1974).
2. N. Baxter."Leverage,Riskof Ruinand The Costof Capital,"Journalof Finance(September1967).
3. E. F. Fama and M. H. Miller. The Theoryof Finance,New York: Holt, Rinehartand Winston,
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