stiglitz (1969) (a re-examination of the modigliani-miller theorem)

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American Economic Association

A Re-Examination of the Modigliani-Miller TheoremAuthor(s): Joseph E. StiglitzSource: The American Economic Review, Vol. 59, No. 5 (Dec., 1969), pp. 784-793Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1810676

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A Re-Examinationf t h e Modigliani-

M i l l e r Theoremn

By JOSEPH E. STIGLITZ*

In their classic paper of 1958, FrancoModigliani and Merton H. Miller demon-strated that the cost of capital for a firmwas independent of the debt-equity ratio[13]. Although much of the subsequentdiscussion has focused on the realism ofparticularassumptions [3], [7], there havebeen few attempts to delineate exactly theclass of assumptionsunderwhich the M-Mtheoremobtains.' In particular, ive limita-tions of the M-M proof may be noted:

1. It depended on the existence of riskclasses.

2. The use of risk classes seemed toimply objective rather than subjectiveprobability distributions over the possible

outcomes.3. It was based on partial equilibriumrather than general equilibrium analysis.

4. It was not clearwhether the theoremheld only for competitive markets.

5. Except under special circumstances,it was not clearhow the possibility of firmbankruptcy affected the validity of thetheorem.

In Section I, we show in the context of ageneralequilibriumstate preferencemodelthat the M-M theorem holds under muchmore general conditions than those as-sumedin their originalstudy. The validityof the theorem does not depend on theexistence of risk classes, on the competi-tiveness of the capital market, or on theagreement of individuals about the prob-ability distribution of outcomes.2

The two assumptions which do appearto be important for our proof are (a) indi-viduals can borrowat the same marketrateof interest as firms and (b) there is nobankruptcy.3But it is these assumptionswhich appear to be the center of much of

the criticism of the M-M analysis. In Sec-tion II, we show that the M-M resultsmay still be valid even if there are limita-tions on individual borrowing, and inSectionIII, we show that the possibility ofbankruptcy raises more serious problems,although the M-M theorem can still beshown to hold under somewhat morestringent conditions.

I. The Basic Theorem

Consider a firm whose gross returns, X(beforepaying bondholders but after pay-ing all non-capital factors of production)are uncertain. We can consider X as afunction of the state of the world 0. Onedollar invested in a perfectly safe bondyields a grossreturn of r*, so that r*-1 is

* Associateprofessor,CowlesFoundation,Yale Uni-versity.This is a revisedversionof CowlesFoundationDiscussionPaperNo. 242, presentedat the December1967meetingsof theEconometric ociety.The researchdescribed n this paper was carriedout undergrantsfrom the National ScienceFoundationand from theFord Foundation. am deeply indebted to D. Cass,A. Klevorick,M. Miller, and W. Nordhaus or exten-sive discussionson these problemsand detailed com-ments on a previousdraft.

1 Exceptionsare the work of Hirschleifer[91, [101and Robichekand Meyers [19],who used the Arrow-Debreu model (whichassumes at least as many se-curitiesasstates of the world)and thedoctoraldisserta-tion of G. Pye [18].More recentlySher [211has con-cerned himself with some of the difficulties aisedby

bankruptcy.For other general equilibriumportfolio(stock-market)models,see Sharpe[201,Lintner [11],Mossin[17],and Diamond[6].

2 Except that they must agree that there is zeroprobability fbankruptcy. ee discussionn text.

3 It shouldbe clear that these assumptionsare notcompletely ndependent.Presumably,one of the most

importantreasonsindividuals cannot borrowat thesamerate as firms s that there is a higherprobabilityof default.

784



STIGLI1Z: MODIGLIANI-MILLER THEOREM 785

the market rate of interest. If there is anychance of bankruptcy, the nominal rate P

which the firm must pay on its bonds will

depend on the number issued. If princi-pal payments plus interest exceed grossprofits, X, the firm goes bankrupt,and thegross profits are divided among the bond-holders.4Thus the gross return on a dollarinvested in the bonds of the firm dependson state 0

f if PB < X(a)

(1) f(0) x(o)if PB > X(a).

B

Earnings per dollar invested in equity instate 0 are given by

([X(o) - PB]/E if rB < X(o)(2) e(0) = ifP-?X0

O if PB> X(6)

where E is the value of the firm's equity.The value of the firm is

(3) V=E + B.

Individuals will be assumed to evaluate

alternative portfolios in terms of their in-come patterns across the states of nature.

We now prove the following proposi-

tion.

Assume there is no bankruptcy andindividuals can borrow and lend at themarket rate of interest. If there exists ageneral equilibrium with each firm hav-ing a particular debt-equity ratio and aparticular value, then there exists an-other general equilibrium solution forthe economy with any firm having any

other debt-equity ratio but with thevalue of all firms and the market rate ofinterest unchanged.

Proof: Let wi be the jtA individual'swealth, Ei the value of his shares of thejth firM,B' the number of bonds he owns.5

Assume the ith firm, whose value is Vi,issues B, bonds. The jth individual's bud-get constraint may be written

(4) w= E + B.

If we let ai, =Eji/Ei, the share of the jth

firm's equity owned by the jth individual,(4) becomes

(5) w =EaiEi + B.

Then his income in state 0 may be written

Ys(O) (Xi - r*Bi)a.i'=

(6) + r*(w - c(V, - B))

3E Xia,?+ r*(w - E axvs).1=1 i=1

If, as B; changes, V; remains unchanged,

the individual's opportunity set does notchange,and the set of axi-whichmaximizesthe individual's utility is unchanged. If

Z =

-1

before, i.e., demand for shares equalledsupply of shares, it still does. The totalnet demand for bonds is

Ew-E a'i(Vi - Bs) + BEB

= W-EVi.i 5

If the market was in equilibrium nitially,

E w- Evi =

j i

i.e., excess demand equalledzero. If as thedebt equity ratio changes, all V, remainunchanged, excess demand remains atzero.

4Throughout he discussion,we limit ourselves toa two-periodmodel. In a two-periodmodel, a firmeither makes its interestpayments or goes bankrupt.In a multiperiodmodel, he firmcan,in addition,deferthe interestorprincipalpayments. f there s a positiveprobabilityof such deferral, he marketwill force thefirm to pay a highernominalrate of interest. If thereare largetransaction osts involvedin bankruptcyordeferral, he M-M theoremwouldnot hold. Through-out the discussionwe shall assume that there are no

flotation ostsand no taxes. 5By convention,one bond costs one dollar.



786 THE AMERICAN ECONOMIC REVIEW

An alternative way of seeing this is the

following. We may rewrite (6) as

(6') Y '(6)= , ei(O)E+ r* (w - EE.t i~-1/

Assume now that the first firm say, issues

no bonds. If we let carets denote the values

of the various variables in this situation,the opportunity set is given by

(6") YfT(9) = ee(G)E. + *e-i~~~ /

Assume r*=r*

Ei=Efi, i>2. Then from(2), ei(0)= ei(6), i>2. If EA=E1+B1, then

the opportunity sets described by (6') and

(6") are identical. To see this, assume

that for each dollar of equity he owned in

the first firm in the initial situation, the

individual borrows B1/E1 in addition to Bi

so B -B + Ei-E1

With the proceeds of the loan he increases

his holdings of equities in the first firm, so

(7) E,-EE + E1 E E )E

His income in state 0 is then given by

X1E

IAO() -- --+ Eei(6)EjEl i=2

(8) i=2 E

_ (Xi r*B) +EBl

+ r* -E E>

which is identical to (6').Since his opportunity set has not been

changed as a result of the change in thedebt-equity ratio of the firm, if he was

maximizing his utility in the initial situa-

tion, the optimal allocation in the new

situation is identical to that in the initial

situation with the one modification given

above.

We now need to show that the markets

for the firm's equities and the market for

bonds will clear. Summing (7) over all

individuals, we obtain

B1jEl j

Thus the demand for equities has in-

creased by a factor V1/E1. But since

E/B1 = V1/E1, the supply has increased by

exactly the same proportion, so if demand

equalled supply before it also does now.

Similarly, the increase in the demand for

bonds by individuals equals (B1/E1) EE- =

B1. But this exactly equals the decrease in

the demand for bonds by the first firm.

It should be emphasized that in this

proof, X(0) is subjectively determined;

moreover no assumptions about the size of

firms, the source of the uncertainty, andthe existence of risk classes have been

made. The only restriction on the indi-

vidual's behavior is that he evaluates

alternative portfolios in terms of the in-

come stream they generate. The two

crucial assumptions were (a) all indi-

viduals agree that for all firms Xi(0) > r*B

for all 0 (see Section III); and (b) indi-

viduals can borrow and lend at the market

rate of interest. This assumption is con-

siderably weaker than the assumption ofa competitive capital market, since no

assumption about the number of firms has

been made: the market rate of interest

need not be invariant to the supply of

bonds by any single firm.

1f1.Limitations on Individual Borrowing

One of the main objections raised to the

M-M analysis is that individuals cannot

borrow at the same rate of interest as

firms. First, it should be noted (see [13])



STIGLITZ: MODIGLIANI-MILLER THEOREM 787

that the analysis does not require that

individuals actually borrow from the mar-

ket, but only that they change their hold-

ings of bonds. A problem can arise then

only if an individual has no bonds in his

portfolio.

Although the requirement that all indi-

viduals hold bonds does place restrictions

on the possible debt-equity ratios of differ-

ent firms, there still need not be an optimal

debt-equity ratio for any single firm. As-

sume we have some general equilibrium

situation where Bi>O for all j. Then so

long as Bi satisfy the inequalities

(9) a'Bi > w - aiVi for all ji i

all individuals will be lenders. If there were

two firms, the constraints (9) would imply

that (B1, B2) lie in the shaded area shown

in Figure 1. For any pair of (B1, B2) in the

region, there will exist a general equilib-

rium in which the values of both firms are

identical to that in the original situation.

So far, none of our results have depended

on the existence of risk classes.6 The follow-

ing two results depend on more than one

firm having the same pattern of returns

across the states of nature.

We shall first show that if there are two

(or more) firms with the same pattern of

returns and individuals can sell short, then

the two firms must have the same value,

independent of the debt-equity ratio.

We follow M-M in assuming for sim-plicity that one of the two firms has no

outstanding debt, so V1=Ei. The second

firm issues B2 bonds, so V2=B2+E2.

Consider first an individual who owns a,of the shares of the first firm, yielding an

income pattern aiXi(6). If instead he pur-

chases a, of the shares of the second firm,

at a cost of a1E2, and buys a1B2 bonds, his

B2

FIGURE1

income in state 0 is a1(X2(0)- r*B2)+

alr*B2- aIX2(0) which is identical to his

income in state 0 in the previous situation.

But the cost of purchasing a, of the shares

of the first company is a1V, which is

greater than ai(E2+B2)= a,V2 if V1> V2.

Accordingly, if V1 were greater than V2,all holders of shares in the first company

would sell their shares and purchase shares

in the second firm, driving the value of the

second firm up and that of the first down.

Now consider an individual who wishes to

lend money. If he sells short a2 of the

shares of the second firm and buys a2 of

the shares of the first firm, he receives a

perfectly safe return of7 -a2(X2-r*B2) +

a2Xi= a2r*B2 at a net cost of -a2(V2-

B2)+ao2Vi so the return per dollar is

B2 1r*

Vl -(V2- B2) ?V1 - V2

B2

If V1< V2, the individual cain obtain a per-

fectly safe return in excess of r*. It follows

6 Two firms, i and j are in the same risk class if

Xi(O)=XXj(o) for all 0. In the remainder of the dis-cussion we shall assume, for convenience, that X= 1.

, As usual, we assume niotransactions costs anidthat

there is no cash margin requirementon short sales.

(See fn. 9.)




immediately that equilibriumin the capi-tal market requires V1= V2.8

Similararguments can be used to showthe following.

If therearethree or morefirms n thesame risk class, and the firmswith thehighest and lowest debt equity ratioshave the samevalue, then the value ofall otherfirmsmustbe the same.

This is true whether individuals canborrow or can sell securities short. Thisresult rules out the possibility of a U-shaped curverelatingthe value of the firmto the debt-equity ratio.9

III. BankruptcyBankruptcy presents a problem for the

usual proofs of the M-M theorem on twoaccounts: first, it means that the nominalrate of interestwhichthe firmmust pay onits bonds will increase as the number ofbonds increases. (M-M have treated thecase where it increases at exactly the samerate for all firmsand individuals.) Second,if a firmgoes bankrupt,it is no longerpos-

sible for an individual to replicate theexact patterns of returns, except if he can

buy on margin, using, the security as col-lateral; and if he defaults, he only forfeitsthe security and none of his other assets.To see this, consider the two alternative

policies considered n Section I; in the onecase, the firm issues no bonds (hence nochance of default) and in the other itissues B bonds. We have shown how the

individual by buying stock on margin inthe latter case can exactly replicate the re-turns in the former situation in thosestates where the firmdoes not go bankruptif the value of the firm is the same in thetwo situations. But if the firm goes bank-rupt in some state, 0', in the one case his

return is zero, while in the other his returnper dollar invested is

X(O') B B

If, however,he canforfeitthe securitythenhis returnwill againbe zero.

Of course, f the firmhas a positive prob-ability of going bankrupt, it will have topay a higher nominal rate of interest. But

if the individual is to use the security ascollateral, he, too, will have to pay ahigher nominal rate of interest. And in-deed, it is clearthat the two will be exactlythe same, since the pattern of returns onthe bonds in bankruptcywill be the same.Thus, we have shown that

if a firm has a positiveprobabilityofgoing bankrupt,and an individual canborrow using those securities as col-lateral (so that if his returnfrom the

securities s less than his borrowings, ecan forfeit the securities)the value ofthe firmis invariantto the debt-equityratio.

It should be noted that the validity of

this propositiondoes not require 100 per-cent margins.The requiredmarginis onlyB/V.

Individuals may, of course, not be able

to make the limited liability arrangementsor to obtain the level of margin required

8This proofhas the advantagethat no restrictionson the sign of X(O)need be made.Although he case ofX(O) 0 is not veryinterestingroman economicpointof view, some authors(e.g. [211)have drawnattentionto the difficultieswhich arise in the originalM-MproofwhenX(O) 0.

9Taxes and bankruptcymay alter this conclusion.Recently,Baumoland Malkeil[2]have argued hat ifthere are costs of transactions, he leveredcompanymay have a highervalue than the unlevered ompany.They argue hat if, in order o undertakehe arbitrage

operationsrequiredby the M-M analysis,the indi-vidual had to borrow, he total value of transactionswouldbe greater han if the companyprovided he de-siredleverage.If thereare sizeabletransactions osts,in order orthe net income romthe twofirms o bethesame, the leveredcompanymust have a highervaluethan the unlevered irm.Transactions osts cannotbeadequatelyanalyzed n terms of the two-periodmodelthat they (andwe) use, but even in the contextof atwo-periodmodel, t is not clearthat theirpointis cor-rect. If the individualhas bondsin his portfolio,or ifthere are two companies,one with high leverageandone with lowleverage, he individual ansimplychangehisportfoliocomposition.




by the above analysis. Then, a firm bypursuing alternative debt-equity policiesmay be able to offer patterns of returnswhich the individual cannot obtain in anyothermanner (i.e. by purchasingshares inone or more other firms), and the value ofthe firm may consequently vary as thefirm changes its debt-equity ratio. In thefollowing subsections, we consider somespecial situations in which M-M resultsmay still be valid, even though there is afinite probability of bankruptcy.

Risk Classes

If there are a large number of firms inthe same risk class, then potentially theycan all supply the same pattern of returns.If all firms maximize their value, then inmarket equilibrium all firmswill have thesame value.10Firms may have differentdebt-equity ratios and the same value fora number of reasons. For instance, assumethat some individuals, for some reason orother, prefer a low debt equity-ratio, andsomeprefera highdebt-equity ratio.Then,

some firms may have a high debt-equityratio, some a low one. If one firmobservesanother firm in the same risk class with adifferent debt-equity ratio but a highervalue, it will change its debt-equity ratio.Thus the observation that all firms in agiven risk class have the same value butpossiblydifferentdebt-equity ratios can betaken as evidence that firms are valuemaximizersand are in market equilibrium.It is not necessarily evidence that the

arbitrage activities described by Modi-gliani and Miller have occurred, or thatthe value of the firmwould be the same atsome debt-equity ratio other than thoseactually observed.

Assume the market is in equilibrium,with V= pEX for all membersof the riskclass.The securitiessoldby a firmare com-pletely described by the risk class and the

debt-equity ratio. A new, small firm is

created, belonging to the same risk class,with mean return X. If it chooses a debt-equity ratio used by other firms in the

same risk class, the priceof its sharesmustbe the same as those of the other firms(since they are identical) so its value will

be Xp. But, if it chooses some other debt-

equity ratio, its value may be lower (if, for

instance, there is a positive probability of

bankruptcy)."

Mean-VarianceAnalysis and the Separa-

tion Theorem

In this subsection we consider thespecial case where all individuals evaluatealternative income patterns in terms oftheir mean and variance.For simplicty, let

us assume that only the first firm issues

enough bonds to go bankrupt. If all indi-viduals agree on the probability distribu-tion of returns for each firm, it can be

shown that the

... total marketvalue of any stock inequilibriums equalto the capitalization

at the risk-free nterestrate r*, of thecertaintyequivalent .. of its uncertainaggregatedollar return; . . the differ-ence . . . betweenthe expectedvalue ofthese returnsand their certainty equi-valent is proportionalor eachcompanyto its aggregateisk representedby thesumof the variance f these returnsandtheir total covariancewith those of allother stocks, and the factor of propor-tionality is the samefor all companiesin the market.[11,pp. 26-27].

This implies that

(10) E + Bi=

{X- k (X -X ) (Xi X)} /r*

i=2,.. ,n

10 Recallthat we have assumed orexpositional on-venience hat

Xi(O)=Xi()for all firms n theriskclass.

11Thisalsomayoccurwithtaxes f interestpaymentsare tax deductibleand if capital gains are treatedpreferentially. ee [8].




(1 1) E1 =

kz_, ;Z-S (Xj_ 1j)} r*

(12) B1=

r-lSl-2 k 8(f -r) B,(X.-- Xj)}

whvere? is the expections operator,

Z -nax(XI -rBl1, 0), &Z ?

6XAi7= ,xr.

and

k =r (XA _-

i~~~~~~~

EZ ;(Xi - Xi) (Xi -Xi)I J

'Then adding B1 aind El, ((11) and (12)),we obtain

VI = E,+ Bil -vi - k E3x9<- y)

1 YY}

(Xj - XD} /r*


The intuitive reason for this should be

clear: it is well-known that if all individ-

uals agree on the probability distribution

of the risky assets, if there exists a safe

asset, and if individuals evaluate income

patterns in terms of mean and variance,

the ratio in which different risky assets arepurchased will be the same for all indi-

viduals, i.e. all the relevant market oppor-

tunities can be provided by the safe asset

and a single mutual fund which (in market

equilibrium) will contain all the risky

assets, including the risky bonds. More

generally, whenever the ratio in which

different risky assets are purchased is the

same for all individuals, then the M-M

theorem will be true even with bank-

ruptcy. For a complete discussion of the

conditions under which the separation

theorem obtains, see Cass and Stiglitz [4].

If, however, (a) all individuals do not

agree on the probability distribution of Xi

(6) or, (b) the conditions under which the

separation theorem is valid do not obtain,

then the value of the firm will in general

depend on the debt-equity ratio.12

Arrow-Debreu Securities

Arrow [1] and l)ebreu [5] have formu-

lated a model of general equilibrium under

uncertainty in which individuals can buy

and sell promises to pay if a given state ofthe world occurs. See also Hirshleifer [10].

A stock market security and a bond can

be viewed as a bundle of these Arrow-

Debreu securities. If there is a sufficient

number of different firms, equal to or

greater than the number of states of na-

12 For then, issuing a risky bond (a high debt-equity

ratio) changes the relevant market opportunities avail-

able to the individual. For the M-M result to be valid,

the debt-equity ratio can have no real effects on the

economy. But it is easy to show that the assumptions(a) marginal utility of income in each state of nature

is independent of the debt-equity ratio, and (b) the

value of the firm is independent of the debt-equity

ratio are in general inconsistent with the first order

conditions for expected utility maximization being

satisfied by all individuals (if bankruptcy may occur).

To see this, observe that if an individual chooses his

portfolio to maximize 8U(Y(O)), where U"<O, then a

necessary and sufficient condition for the optimal al-

location (assuming short sales are allowed) may be

written 8U'e,-=8U'r*, or from (2), [8U'(Xi-PyBjj/

U'r*= Ei where 8 is defined as the set of states of

nature for which Xi(O)>r;Bj. Assume U'(Y(O)) is in-

variant for all 0 and for all individuals to the jth firm'sdebt-equity ratio. Then,

d /dB1 - ,(1 + d l(IU'B8U');

and if the value of the firm is to be unchanged, dEi/dBi= -1. But unless all individuals have identical utility

functions and identical assessments of the probability

of bankruptcy (8 U'/ U') will differ for different ndi-S

viduals, so dEi/dBj = -1 only if marginal utilities in

some states of nature change for some individuals.

It should be observed that when the actions of a firm

can change the opportunity set, there is no reason that

firms necessarily will maximize market value.




ture, then the market opportunities avail-

able to the individual (by purchasing or

selling short different amounts of the mar-

ket securities) are identical to those of a

corresponding Arrow-Debreu market. If a

promise to pay one dollar in state 0 has a

price p*(O),13 then the value of the firm's

equity is

E = (X(8) - rB)p*(O).

If

[zX(O)]/Z1E (0)]/Ep*(O3

where 8 a | X(O)>f B}, i.e. the states ofnature in which the firm does not go

bankrupt, and 8'=- { | X(O) rB }, then

E , X(O)p*(O) B

i.e.

V=E+B- B 3X(O)p*(O)


Three observations are in order: First,

individuals do not need to agree on the

probability of different states of nature oc-

curring, i.e. they may disagree on the pro-

bability distribution of the returns to any

firm.'4 Second, if there are fewer firms than

states of nature, whether there are as

many securities as states of nature is a

function of the debt-equity ratio. If thereare four states of nature and two firms,

and if neither firm issues enough securities

to go bankrupt, then there will only be

three securities, but if one of the firms

goes bankrupt, there will be four. Al-

though the latter situation will be Pareto

optimal (the marginal rate of substitution

between consumption in any two states

identical for all individuals), the value of

the firm which goes bankrupt may be

larger or smaller in the former situation

than in the latter.15

Third, if we take literally the Arrow-

Debreu definition of a state of nature,

there undoubtedly will be more states ofnature than firms. Yet, in some sense,

most of these states are not very different

from one another. For example, much

of the variation in the return on stocks can

be explained by the business cycle. If in

any given business cycle state, the variance

of the return were very small, and there

were a small number of identifiable busi-

ness cycle states, then the economy might

look very much as if it were described by

an Arrow-Debreu securities market.16

Bankruptcyand Perfect CapitalMarkets

The usual criterion for a perfectly com-

petitive market is that the price of a

commodity or factor an individual (or

firm) buys or sells be independent of the

amount bought or sold and be the same for

all individuals in the economy. On this

basis, it has been argued that the capital

market is imperfectly competitive: (a) as a

firm issues more bonds the rate of interestit pays may go up; (b) individuals may

have to pay a higher interest rate than

firms, and some firms higher than others;

(c) lending rates may differ from borrow-

ing rates. In this section, we have, how-

ever, considered perfectly competitive

'3,If there are no Arrow-Debreu securities on themarket, p*(o) is the net cost to the individual of in-creasing his income in state 8 by one dollar, i.e. bybuying and selling short different securities. If thereare more securities than states of nature, marketequilibrium requires that the set of market pricesgenerated by considering any subset of market securitieswhich span the states of nature be independent of theparticular subset chosen. For a more thorough discus-sion of these problems, see [4].

14They must, however, not assign zero probabilitiesto different states of nature occurring.

15 In this situation we cannot assume that firms willnecessarily maximize market value. (See fn. 12.)

16 The point is that under these conditions the in-dividual, by diversification of his portfolio, can essen-

tially eliminate the variations in returns within a givenbusiness cycle state.




capital markets (with bankruptcy) inwhichall three of these would be true.'7 Seealso [22]. Thus the possibility of bank-

ruptcymakes somewhat questionable the

interpretationof much of this evidence ofan imperfect capital market. The crucialfallacy lies in the implicit assumptionthatone firm's bond is identical to anotherfirm's bond, and that bonds a firm issueswhen it has a low debt-equity ratio andthose which it issues when it has a highdebt-equity ratio are the same. But theyare not. They give differentpatterns of re-turns. If there is any chance of default, a

bond gives a variablereturn (i.e. is a riskyasset). Just as there is no reason to expectbutter and cheese, even though they arerelated commodities, to have the sameprice, so there is no reason to expect thenominal rate of interest where there is alow debt-equity ratio to be the same aswhen there is a high debt-equity ratio.Even the discrepancybetween borrowingand lendingrates does not imply imperfectcapital markets, for when a person lends

to the bank and the account is insuredbyFDIC, he can assume there is a zero prob-ability of bankruptcy, but when thebank lends back to the same individual,it cannot make the same assumption.

REFERENCES

[1] K. J. ARROW, "The Role of Securities inthe Optimal Allocation of Risk Bear-ing," Rev. Econ. Stud., Apr. 1964, 31,91-96.

[2] W. BAUMOL AND B. MALKIEL, "TheFirm's Optimal Debt-Equity Combina-

tion and the Cost of Capital," Quart.J.Econ., Nov. 1967, 18, 547-78.

[3] D. E. BREWER AND J. B. MICHAELSON,

"The Cost of Capital, Corporation Fi-nance, and the Theory of Investment:Comment," Amer. Econ. Rev., June1965, 55, 516-23.

[4] D. CASS AND J. E. STIGLITZ, "TheStructure of Preferences and Returnsand Separability in Portfolio Allocation:A Contribution to the Pure Theory of

Mutual Funds," Cowles FoundationDiscussion Paper, May 1969.

[5] G. DEBREU, The Theoryof Value, NewYork 1959.

[6] P. DIAMOND, "The Role of a StockMarket in a GeneralEquilibrium Modelwith Technological Uncertainty," Amer.Econ. Rev., Sept. 1967, 57, 759-76.

[7] D. DURAND, "Cost of Capital, Corpora-tion Finance, and the Theory of Invest-ment: Comment," Amer. Econ. Rev.,

Sept. 1959, 49, 639-55.[8] D. E. FARRAR AND L. L. SELWYN,

"Taxes, Corporate Financial Policy,and Return to Investors," Nat. Tax J.Dec. 1967, 20, 444-54.

[9] J. HIRSHLEIFER, "Investment Decisionunder Uncertainty: Choice TheoreticApproaches," Quart. J. Econ., Nov.1965, 79, 509-36.

[10] , "Investment Decision underUncertainty: Applications of the State-Preference Approach," Quart. J. Econ.,May 1966, 80, 237-77.

[11] J. LINTNER, "The Valuation of Risk As-sets and the Selection of Risky Invest-ments in Stock Portfolios and CapitalBudgets," Rev.Econ. Statist., Feb. 1965,47, 13-37.

[12] H. MARKOWITZ, Portfolio Selection,New York 1959.

[13] F. MODIGLIANI AND M., H. MILLER,

"The Cost of Capital, Corporation Fi-nance, and the Theory of Investment,"

Amer. Econ. Rev., June 1958, 48, 261-97.

[14] , "Reply to Rose and Durand,"Amer. Econ. Rev., Sept. 1959, 49, 665-69.

[15] "Corporate Income Taxes andthe Cost of Capital: A Correction,"Amer. Econ. Rev., June 1963, 53, 433-43.

[16] "Reply to D. E. Brewer andJ. B. Michaelson," Amer. Econ. Rev.,June 1965, 55, 524-27.

1? Transactions osts may also partly explain (b) and(c).




[17] J. MOSSIN,"Equilibrium in a CapitalAsset Market," Econometrica, Oct.1966, 34, 768-83.

[18] G. PYE, "Investment Rules for Corpor-ations," doctoral dissertation, M. I. T.1963.

[19] A. A. ROBICHEK ND S. C. MYERS,"Problems in the Theory of OptimalCapital Structure," J. Finance Quant.Anal., June 1966, 1, 1-35.

[20] W. F. SHARPE, Capital Asset Prices:

A Theory of Market Equilibrium underConditions of Risk," J. Finance, Sept.1964, 19, 425-42.

[21] W. SHER, "The Cost of Capital andCorporation Finance Involving Risk."A paper presented at the winter meet-ings of the Econometric Society, Evans-ton, Illinois, Dec. 1968.

[22] G. STIGLER, "Imperfections in theCapital Market," J. Polit. Econ., June1967, 75, 287-93.

stiglitz (1969) (a re-examination of the modigliani-miller theorem)

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