THE CONSEQUENCES OF COSTLY DEFAULT RICHARD HARRIS* This paper investigates the consequences of assuming that default on loans or corporate debt is costly; that is, the act of default imposes a deadweight cost on the economy. Thv analysis deals with tuo simple capital market models. Con- ditions for capital market equilibrium nominal intcrcst rates and probability of dcfault to exist are gicen and the conzpara- changes in expectations, productivity of incestment, cost of default and riskless intcvst rates are examined. In many cases these comparative static results are unambiguous in sign. tive statics of these equilibrium uariables with r(, )T p rct to I. INTRODUCTION The analysis of default on contracts in capital markets is a subject of increasing interest to economists. I It is generally recognized that in many financial markets the principal uncertainty in a t>orrowcr-lcndcr relation is the possibility that the borrower may default on his contract; that is, repayment of the loan will not occur at terms originally agreed to in the contract. This feature has a number of interesting implications, both from the point of view of economic theory and the marc practical analysis of actual capital markets. In this paper we focus on one aspect of the default problcm. Thc act of default may induce some significant costs which do not come under the usual heading of transaction costs associatcd with capital markets. What makes these costs unique to the act of default is the transfer of ownership of assets that results. In the specific context of a capital market, default implies that the ownership of the investment project undertaken by the borrower with the proceeds of the loan is transferred to the lender. The investment 'project' may be anything from a consumer durable in the context of the personal consumer loan market to an entire corporation in the context of the corporate securities market. Of course, default may occur on contracts that do not involve a transfer of owner- ship in the event of default, in which case our analysis does not apply. We could argue that in many cases, however, one of the distinguishing features of default is the resulting transfer of assets. Economic Inquiry Vol. XVI.Oct. I978 477
Transcript
THE CONSEQUENCES OF COSTLY DEFAULT
RICHARD HARRIS*
This paper investigates the consequences of assuming that default on loans or corporate debt is costly; that is, the act of default imposes a deadweight cost on the economy. Thv analysis deals with t u o simple capital market models. Con- ditions for capital market equilibrium nominal intcrcst rates and probability of dcfault to exist are gicen and the conzpara-
changes in expectations, productivity of incestment, cost of default and riskless in tcvs t rates are examined. In many cases these comparative static results are unambiguous in sign.
tive statics of these equi l ibrium uariables w i th r ( , ) T p r c t to
I. INTRODUCTION
The analysis of default on contracts in capital markets is a subject of increasing interest to economists. I I t is generally recognized that in many financial markets the principal uncertainty in a t>orrowcr-lcndcr relation is the possibility that the borrower may default on his contract; that is, repayment of the loan will not occur at terms originally agreed to in the contract. This feature has a number o f interesting implications, both from the point of view of economic theory and the marc practical analysis of actual capital markets.
In this paper we focus on one aspect of the default problcm. Thc act of default may induce some significant costs which do not come under the usual heading of transaction costs associatcd with capital markets. What makes these costs unique to the act of default is the transfer of ownership of assets that results. In the specific context of a capital market, default implies that the ownership of the investment project undertaken by the borrower with the proceeds of the loan is transferred to the lender. The investment 'project' may be anything from a consumer durable in the context of the personal consumer loan market t o an entire corporation in the context of the corporate securities market. Of course, default may occur on contracts that do not involve a transfer of owner- ship in the event of default, in which case our analysis does not apply. We could argue that in many cases, however, one of the distinguishing features of default is the resulting transfer of assets.
Economic Inquiry V o l . XVI.Oct . I978
477
478 ECONOMIC INQUIRY
The costs involved in such a transfer are of at least three sorts. First, there are the transaction costs involved in the administration of the transfer. These include legal costs, temporary shutdown costs, agency costs and so forth.2 Second, and more significantly, there are efficiency losses in such a transfer; these are losses which arise either because of ditferences between the borrower and lender in their utility valuation of an asset, or, in the case of an investment project, the borrower, or the agent acting on his behalf, may not possess the necessary expertise in managing the particular project and consequently will make inefficient decisions. Third, there may be losses due to shut downs; if the original investment is abandoned by the lender after the transfer of ownership, the losses could be as much as the entire output lost from shutting down the project. This will be the case if the original investment was irrevers- ible or of the putty-clay variety and consequently has no alternative use. Hencetorth we shall refer to a situation in which the act of default induces deadweight costs on the economy as a result of the transfer of ownership of assets as one of ‘costly default.’
The purpose of this paper is to examine the consequences of costly default in the context of two simple capital market models. These models have the feature that some or all of the investment taking place is financed with a type of loan instrument. The return on the investment project is uncertain and there is a positive probability of default on any loan. Our primary concern is to establish conditions under which these capital markets have an equilibrium and, more importantly, to develop the comparative static implications of the models. In the first model all investment is financed with a single type of debt contrait, or bond, issued by borrowers. In the second model real investment may be financed with either debt or a limited liability security. The proposition of Miller- Modigliani (1958) on the irrelevance of financial policy is examined in the context of the second model.3
II. LOAN MARKET EQUILIBRIUM: SUPPLY DETERMINED LOAN PRICE
In this section we consider a partial equilibrium model of a competi- t ive capital market. Borrowers in this market constitute the demand side and can be thought of as either firms or individuals who undertake real investment in a project yielding uncertain returns. In order to finance this investment they must borrow from lenders who constitute the supply side of the market. The only financing instrument available in this market
HARRIS: COSTLY DEFAULT 479
FIGURE 1
t E x p e c t e d R a t e o f
Q Y Q u a n t i t y o f L o a n s
is a simple one-period debt contract which is repaid in the second period, provided the borrower does not default.
The suppliers of investment funds are assumed to be risk neutral and to be sufficiently large in number that the market is considered to be competitive. These lenders may put their funds in either a riskless security paying a gross rate of interest e , or loans to individuals or firms having access to risky technologies. These loans are risky in that the borrowers may choose to default. Should default occur, the rights to the investment project and the capital associated with it are transferred to the lender. Deadweight costs o f default occur as we assume that the investment project to the lender is worth less than to the original borrower.
The assumption of large numbers of risk neutral investors together with the assumption of a fixed rate of return e on a riskless security are sufficient to imply that the supply curve of loans in this capital market is perfectly elastic at the expected rate of return e . See Figure 1 . The adjustment mechanism in this market is the standard arbitrage process of capital market theory. The gross nominal interest rate on debt and the probability of default adjust such that the expected return on loans equals the exogenous riskless return e . Independent of the demand curve, which can be influenced by a number of considerations including the number of risky investment projects available, the expectations and attitudes towards risk of borrowers and the effect of other markets, in equilibrium all expected rates of return must be eyual t o 4. The eyuilib- rium quantity o f loans will depend upon the demand side of the market. In this model, however, we ignore explicit consideration of the demand side and hence the determination of equilibrium quantities. The focus is on the relationship between the ecpilibrium loan price, the equilibrium probability of default and the exogenous parameters of the model.
Borrowers have access to a constant-returns-to-scale technology or investment project. For a real investment or input in the first period,
4x0 E(:ONOMIC I N Q U I R Y
clcnotcd by x, the invcstmcwt projcct gives ;I return in the second period of y(s, 81, where 8 clenotes a state o f nature variablc~, an element of the state space R . hlorc specifically i t is assunwtl
r(8) = rs, r > 0
'I'hc positivc scalar s paranietcrizcs the state space and takes valiws in the interval [0, MI.
Borrowers may choosc to default o n their loans i f they wish,' the only penalty being that they lose all claims to returns from the investnrent project, i t being assumed that the lenders in the market have acquired these claims. Let y denote thc interest rate (gross) a borrower pays on a one dollar loan. Given the assumption of stochastic constant returns to scale a borrower will earn negative rcctlipts in any state s for which rs < y. I n this case we assunie the borrower automatically defaults wit h o 11 t penalty .
We shall term the value of the investment project to the lender, should he hilve to take i t ovcr, the collatrral t'alur. The collateral return function is c(s, s) = cxs, where c > 0. The assumption that the collateral value of the project is always less than its investment value implies that c is less than r. All lenders are assumed to have the same collateral function. An investor autoniatically receives the collateral value of the project should borrowers default; i.e., in the events s < y/r.
The expectations of lenders are rcprescnterl by a distribution function F ( s ) with support in the interval [O, All lenders have the same expectations. Changes in expectations will be represented by changes in t , where F ( s , t ) is a family of distributions indexcd with the parameter t .
Competition forces all suppliers of investment funds to charge the same price q on risky loans, and capital-market equilibrium requires that the lowest y compatible with the cxpccted return on risky loans being equal to 4 be charged. Should this not be true, for example i f the expected return to these loans exceeded e , then there would be an excess supply of fiintls for risky investment projects driving down the loan rate so as to restore capital market equilibrium.
HARRIS: COSTLY DEFAULT 48 1
The return per dollar loaned out by lenders is denoted by B ( q , s) where
9 if s 2 9/r
cs if s < q/r. Bfq, s) =
Therefore the expected return per dollar loaned out is givcn by
Let z Thus we have
9/r. Capital market equilibrium requires that ( I ) be cqiial to e .
as the basic equilibrium equation. .4n equilil)rium is characterized as ;i solution z * L [ O , MI, to the equation g ( z * ) = e . i\ solution z * will l)c> referred to as the equilibrium tlcfault point. We have the following propo- sition which provides a characterization of the eciciilibrium.6
Proposition 2. I : If an equilibrium exists thcm there exists a unique equilibrium default point z *, and in the ncighl)oiirhootl of this equilibrium the expected return on risk!, loans, g ( z ) , is increasing with z .
Thus in a neighbourhoocl of the eqiiilibriuni point, incbrcnscs in the loan price raise the expected return to investment in risky projects. .4ltcrnu- tively stated, in the neighboiirhood o f eciiiilihriiirii investing i n risk), projects is not a n **inferior" activity.
The equilibrium is prcsentcd graphicall!' in Figure 2. The wrtic;il axis corresponds to expected rates of return and the horizontal axis to default points. The function g ( z ) which is graphcd rcprcscmts thc c~xpc~ctctl rate of return on loans as a function of thc tlcfault point z . S dcnotcs the mean value of the state-of-nature parameter s. (;iven ;in cxogrnoiis rate of return e l on riskless securities the tlcfaiilt point will adjust through changes in 9 such that the expected rate of return on loans eqiials e l , Not(. that as the loan rate q increases the probabilit). of tlcfault also inc*r('asw. The condition for an cqiiilibrium t o exist in this market is that t h c exogenous rate of return e 1 x 1 Icss than thc ni;ixiniuni r%xpectcd rate of return on risky loans, g( , zmor) , in Figure 2. I t nia)' I)c the case that no equilibrium in this market will exist. For the rcmaintlcr of this section we assume an equilibrium exists.
The comparative static prcdictions of this model arc sumniurizctl in propositions 2.2 through 2.5.
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FIGURE 2
t E x p e c t e d R a t e s o f R e t u r n
* t
z, = q , / r ‘2 z: q / r D e f a u l t p o i n t
Proposition 2.2: An increase in the exogenous rate of return causes an increase in the equilibrium loan rate and an increase in the equilibrium probability of default.
This proposition follows immediately from proposition 2.1 and an inspection of Figure 2. Notice that if we have a downward sloping demand curve for risky loans, then an increase in the riskless interest rate would cause a decline in the equilibrium quantity of investment. The model thus predicts a positive correlation between high interest rates and high levels of default. Considering changes in the productivity of risky investment we have
Proposition 2.3: An increase in the productivity of investment causes a) a decrease in the equilibrium default rate and b) a decrease in the equilibrium loan rate,
Since increased productivity implies, at any fixed loan price, that the probability of default diminishes, this raises the expected return and hence the nominal interest rate on loans must fall in order to restore equilibrium. As the nominal interest rate falls, this necessarily diminishes the probability of default. The two effects reintorce each other and in equilibrium there must be both a lower probability of default and a lower cost of funds to borrowers.
We turn now to a consideration of changes in the collateral value of projects. If there is an increase in the collateral value of the risky projects, then the deadweight costs of default will diminish. An obvious question is the equilibrium relationship between nominal rates of interest on risky loans and the costs o f default. The following proposition provides an answer to this question.
Proposition 2.4: An increase in the collateral value of projects causes a decrease in the equilibrium probability of default and a decrease in the equilibrium loan rate.
HARRIS: COSTLY DEFAULT 483
This result makes a great deal of intuitive sense. It seems reasonable that the higher the collateral value of the project the lower the loan rate should be. In fact, this is a rather common phenomena observed in capital markets.
Finally we consider a change in the expectations of lenders. We define an increase in default risk as a mean preserving increase in risk as defined by Rothschild-Stiglitz (1970) with the added provision that at the equilib- rium default point the change in the distribution of returns does not decrease the probability of default. Thus we have
Proposition 2.5: A change in the expectations of lenders, repre- sented by an increase in default risk, does not decrease the equilibrium loan rate and does not decrease the equilibrium probability of default.
Thus a collapse in the expectations of lenders may cause an increase in the equilibrium loan rate. If the demand curve for loans were downward sloping as a function of the loan rate this would cause a reduction in the equilibrium quantity of risky investment.
This proposition reaffirms the belief held by many economists, includ- ing Keynes (1936), that adverse changes in the expectations of investors in the sense of increased riskiness will cause a decrease in the amount of investment undertaken. We stress the term “increased riskiness”; recall that the mean of the distribution is held constant and increased weight is added to the tails of the distribution.
This completes our analysis of the single security market. We turn now to a model with both equity and debt.
111. THE INTRODUCTION OF EQUITY
Our analysis so far has concentrated on a capital market with a single type of contract which facilitates the savings-investment process. In this section we consider a more complicated capital market which provides an alternative means of investing in risky investment projects - the purchase of equity or limited liability stocks in risky firms.
The model in this section will be a simple partial equilibrium model of the standard two-period variety, investment taking place in the first period and output (random) occurring in the second period. The economy consists of two types of firms with all firms of each type having identical technologies. The riskless firms have a constant-returns-to-scale tech- nology which fixes the riskless interest rate (gross) e provided some positive investment occurs in the riskless technology. Alternatively, we could assume the riskless interest rate is the interest rate paid on high grade government securities and is fixed for the purposes of the model. Risky firms have technologies identical to the investment projects out- lined in section 11. We shall require, however, that in states in which the
484 ECONOMIC INQUIRY
firm goes bankrupt, i.e., defaults on its payments to bondholders, a change in technique occurs. That is, in states of bankruptcy the firm does not operate with the same technology as in those states in which the firm is solvent. The technology in bankrupt states will be identical to the collateral return function of section 11, and has the property that its out- put, for the same input, is less than the output of the technology in solvent states.
The reason for making the above assumption is that it introduces a deadweight cost to the economy of bankruptcy. In previous analyses the bankruptcy occurs at zero resource costs to the e ~ o n o m y . ~
As in the model of section I1 equilibrium is characterized by equality of expected rates of return on bonds and equity to the exogenous interest rate 4. In this model, however, equilibrium is attained by adjustments in the nominal interest rate on bonds and by adjustments in the debt/equity ratio. Both of these variables are equilibrium determined.
As a result of bankruptcy inducing real costs, we shall show in this section that it is possible for the economy to have an equilibrium with a determinate debt-equity ratio, while maintaining the assumptions of identical expectations and risk neutrality on the part of investors. This is in contrast to the well-known Modigliani-Miller ( 1958) proposition, which states that provided the debt issued has no default risk the economy as a whole is indifferent between alternative debt/equity ratios of the firms in the economy.x We shall show that, even with the strong simplifying assumptions made, the debtlequity ratio of the economy may be a market determined variable.
In addition to providing an equilibrium determination of the debt/ equity ratio, the model is of interest because one would expect that changes in various exogenous parameters would have qualitatively dif- ferent effects on equilibrium variables when more than one source of financing is available. As we shall see this turns out to be the case.
Recall that the gross return on an investment project, or to a risky firm’s technology, is given by rsx, where x is total investment, provided the firm does not default. If the firm does default on its bond payments, then in those states s in which it is bankrupt output is given by csx. We have assumed that bankruptcy uses real resource costs, or equivalently induces deadweight losses which is expressed by the assumption that c is less than r. Let B stand for dollars worth of bonds (measured in units of investment goods) and E the dollars amount of equity for a particular firm. Thus x = B + E , or the amount invested by a single firm is equal
7. See f o r example Milne (1975) and Stiglitz (1972). Stigler (1967) has an interesting discussion of transaction costs in capital markets and their relation to the over worked term “capital market imperfect ions.”
8. The original M & M theorem was a partial equilibrium proposition. See Miller and Modigliani ( I 9%). Stiglitz ( I 969). (1974) gives a general equilibrium version of the theoreni.
HARRIS: COSTLY DEFAULT 485
to the value ot its debt plus eyuity. This assumes all firms start up in the first period and are dissolved in the second period. Bondholders are to receive 9B in the second period where 9 is the nominal interest rate (gross) on bonds. Then n (9, B , s, x), the return to total equity E , is given by
Again let 9 l r f z . Let Blx f tc, where w is the proportion of investment which is debt financed and ( 1 - w ) the proportion which is equity financed; an increase in u' will correspond to an increase in the debt/ equity ratio. Using this notation the return on a dollar's worth of invest- ment in equity, R,(z, w ) , is given by
\
Note that the constant-returns assumption makes the return to equity independent of the scale of investment. The default point is given by zw in this model. Therefore, using (4), the expected rate of return on invest- ing in the equity of risky firms is given by
where F ( s ) is distribution function representing the expectations of investors.
The return to a dollar's investment in the bonds of a risky firm is
9 if s 2 zw
CS if s < zw. w
(6) R B ( z , u:) =
\
Hence the expected return to investment in risky bonds is given by
486 ECONOMIC INQUIRY
The rate of return to investing in either the bonds or equity of the risk- less firm is given by e . Capital market equilibrium, givcn the assumption of risk neutrality, requires that the expected rates of return on all securi- ties in which positive investment takes place bc equal. Supposing that invcstrnent occurs in all securities the market equilibrium conditions can t ie written as
Dividing both of these equations through b j r r, and letting elr c, wc' get
- - c,
in which We wish t o investigate thr possibilit)- of a n equilibrium investnient occurs in both risky Imntls and eqiiit), and thus the cqu;itions (9) and ( 1 0 ) havc a solution ( z * , t c ' * ) . If this is the case, then capitul market cqiiilit>riuni determines both a noniinal intcrest ratc on d d ) t and an equilit)rium debtlequity ratio.
Note that if an investor holds a portfolio with thr fraction 11' ht3ld in risky I)onds and thr remaintlrr ( 1 - u ) in cyuities of the risky firm. his expected return on the portfolio is given b y
.4 ga in , p rov i ded invest men t oc c 11 r s in a I I scc 11 r i t i es , ca p i t ;1 I - 111 a r k c t cciiiilit>rium requircs ER,(au>) = e . Let zu* = y, then thc function h(y) 5 c J,YsdF(s) + r JYsdF(s), can be shown t o take all valucs in tho interval [cS, TS] and to be uniquely invertiblc on [0, MI.!' Thcrc4orc, ;I necessary condition for such an cquilil)riuni t o exist is that e E (cT, K), and we assume this condition holds henceforth. Onc possil)lc cquiIi1)- rium is for no invcstmcnt to occur in the l>ontls ot' risky firms, in \\ hich cme the expccted return on all portfolios is givcn 1)). ,f = e .
HARRIS: COSTLY DEFAULT 487
The case w* = 1 is excluded in this section as it corresponds to a completely debt financed equilibrium, which we treated in section 11. Such an equilibrium is possible as demonstrated in that section. In order to prove that an intermediate case is possible, we must show there exists a w* , 0 < w* < 1 and z * > 0, 0 < z*w* < M , such that (9) and (10) hold. This is possible, as the following proposition shows.
Proposition 3.1: If 4 E (cS, rF) there exists a capital market equilibrium with 0 < w * < 1 , i.e., a strictly positive debt equity ratio.
Thus we have established that, provided the riskless firms’ technolo- gies have a productivity of an appropriate value, there exists a capital market equilibrium with a determinate debtiequity ratio. Notice that the debt/equity ratio of this economy is an equilibrium one, and any attempt by firms to change their debt/ecluity ratios will result in arbitrage taking place in the capital market such as to force the debt/equity ratio back to its equilibrium value. In equilibrium the market value of any firm investing an amount x is x.II The costly bankruptcy assumption is clearly the key factor in explaining this result. If bankruptcy were cost- less, i.e., c = r, then investors would be indifferent between holding debt and equity in the risky industry. Furthermore, the higher the cost o f bankruptcy, i.e., the larger the discrepancy between c and r , the more probable it is such an equilibrium will be established. The analysis clearly suggests that the deadweight costs of bankruptcy are probable cause for financial irrelevance not to hold.
We turn now to an examination of the comparative statics of this model with respect to the two endogenous variables, the nominal rate of interest on risky debt and the debt/equity ratio. These results are surn- marized in propositions 3 . 2 through 3.5 . The first of these deals with changes in the riskless interest rate.
Proposition 3.2: An increase in the riskless interest rate e (the productivity of the riskless firm) causes a) a decrease in the nominal rate of interest on risky bonds and b) a decrease in the equilibrium debt/equit)r ratio.
This proposition states that as a result of market equilibrium the nominal interest rate on risky bonds is negatively correlated with the riskless
1 I. Notice that ~e do not attempt to tlrscril)r a firm’s c,hoicc of its tlrl)t/c.cliiit! r:itio, H;ithcr this is an eqiiilibrium-(lrteriiiinrtl \:irial)Iv, 1:irnis .ire a s s u m c ~ t l to hc ;imarr ot thcb fact tliat this is parameter to thrni. not a choice \:iri,il)lr. It 11‘ is grcx;itc.r than I ) . * . at the cqiiilil)ririm noiiiiiial rate of interest on tlcbt, ( I * . then me can s w that thr niarhd vii lu(~ of thew tirnis, Iwr unit ot i n \ c s \ t i i i c m t will be less than one. ( ~ o n s c q i i c ~ n t l ~ , no in\rstnrrnt \ \o~ i Id occur in risk! tirnir ; i n d thrrc \\oiiId I)? tlisrquilihrium in the niiirhct tor thr wciiritirs ot thc rkk! iiitlustr!. .\ sirnil.ir \lor! can I)c toltl it u is less than t c * . I n telling thr t l is~~.c~~i i l i l~r iui i i story then, i t nright be usr fu l to tliiiih ol lirnis iicljiistiiig their dcbt/rquit!. ratios in :in attempt to c lwr thr sc.curitic.s nriirket. It is nn;ilogoris to thc stor! \ \ c * often te l l of how a conilwtitive l i rm scts its product pricr.
488 ECONOMIC INQUIRY
interest rate. Furthermore, risky firms will tend to have relatively highly levered capital structures when interest rates are low, even though the nominal interest rates on risky bonds will tend to be high.
I t is interesting to note the effect on the nominal interest on risky bonds of a change in the interest rate in this model which allows for both debt arid eyuity financing, compared to the model of section I1 in which all investment was debt financed. The reason for this difference is straight- forward.
In the model of section I1 the only way to raise the expected rate of return to holding risky debt was to raise the nominal interest paid on these bonds. In an economy with both debt and equity financing of investment projects changes in the ciebt/eyuity ratio affect not only the probability of default but also the share of the income from the project in default states going to a dollar invested in risky bonds. See equation (7 ) . The higher the debt/equity ratio the higher the probability of default and the smaller the share of total project income in default states per dollar invested in risky bonds. Consequently a lower debt/equity ratio implies a substantially higher expected return t o holding risky bonds. A lower tlebt/equity ratio, however, implies a lower return to holding a share ot equity because of obvious dilution effects and thus the interest paid on bonds must fall in order to raise the return on eyuity to its eyuilibriuni level. The equilibrating mechanism in a capital market with debt and equity is yuite different than in a capital market with debt only.
The next proposition shows that the technological characteristics ot an economy will affect the type of security mix which is observed. Thus there is a close relationship between “real” and “financial” variables.
Proposition 3.3: An increase in the productivity of risky firms ca uses an increase in equ i I i b riu m deb t/eq 11 i ty ratio.
Thus economies with more productive risky technologies will have more highly levered financial structures. The effect of changes in productivity on the price of bonds is ambiguous in this model. What of changes in the cost of bankruptcy?
Proposition 3.4: A decrease in the cost of bankruptcy causes a) an increase in the nominal interest rate on risky bonds and 1)) an increase in the equilibrium dcbt/eyuity ratio.
As the costs of default to bontiholders diminishes, 110th the rate of interest p‘iitl on bonds goes up, and the economy’s relative security mix shifts so as to raise the proportion of risky investment financed by issuing debt. Note that in the limiting caw of zero costs to bankruptcy, c = r , the Modigliani-Miller theorem is true and there is no equilibrium determined dtWtquity ratio for the economy, in the sense that any debt/equity ratio is compatible with an investor earning an expected rate of return of r.T- on his portfolio.
HARRIS: COSTLY DEFAULT 489
Finally, we inquire a s t o the effect of various parameter changes on the equilibrium probability of bankruptcy. If wc‘ consider this model as one of an economy in long r u n equilibrium then the equilibrium proba- bility of bankruptcy can be thought of a s the long run relative frequency of firms failing who undertake risky investment. The long-run expected social costs Of bankruptcy will vary directly with the equilibrium probability of tlefaul t.
Proposition 3..5: The cqiiilibriuni probability of default or bankruptcy of risky firms increases with a) tlecrc~ases in the riskless interest rate, t)) incrcascs in the productivity of risky investment and c) decrc~ases in the cost of bankruptcy.
Therefore low interest rates on riskless securities are associated with high bankruptcy rates on risky firms, the reason being that, from proposition 3.2, with low interest rates firms tend t o have highly levered capital structures and therefore bankruptcy becomes more likely. Note also that economies with more highly productive technologies tend to have higher bankruptcy r n tes.
The major proposition of this section (Proposition 3.1) is a very strong statement about the properties of the equilibrium of the capital market considered here. I t states that, provided certain technological conditions are met, there is an equilibrium rieterniined debtiequity ratio for risky firms which is inclependent of the relative quantities of debt and equity supplied, providcd both asscts are available in positive finite amounts. This result is in strong contrast to the well known Miller-Motligliani (1958) theorem, anti thus it is worth considering to what extent we may or may not relax the assumptions and still maintain the proposition in its present form.
Costly Bankruptcy: This assumption is critical, as assuming risk neutrality and dropping the costly bankruptcy ‘issumption will yield an indeterminate debtiequity ratio. Note, however, that if costless bankruptcy is maintained and either the assumptions of risk neutrality or homogeneous expectations are dropped then the Miller-Modigliani theorem will not go through. Milne ( 1975) and Stiglitz (1972) have treated these issues. Risk Neutrality: It’ this assumption is dropped, and instead we maintain investors are risk averse then diversification of portfolios in order to reduce risk becomes an important tactor in explaining individual behaviour. But as the individual investor cannot “undo” changes in the debtlequity ratio by undertaking some homemade leverage we will still get an equilibrium determined debt/eyuity ratio. Homogeneous Expectations: If individuals have ditterent expecta- tions this would merely complicate the algebra, i t would not change the basic equilibrium conditions.
490 ECONOMIC INQUIRY
d) Constant Returns Stochastic Technology: In the simple economy considered here the firm does not have to make a decision about either its choice of technique or scale of investment, as there is only one technique available and the constant returns assumption makes scale a matter of indifference. If the firm has either a scale or choice of technique decision to make it is quite likely this cleci- sion will be closely related to the choice of a debt/equity ratio given the assumptions of risky debt and costly bankruptcy. Whether the capital market equilibrium will determine the debtlequity ratio as in the present model is an open question. It is clearly an impor- tant issue to consider, however, as real decisions become closely related to financial decisions, and the presumed independence of these as in neoclassical theory is no longer valid.
There is one final point which the above analysis raises. The fact that an equilibrium can exist in which deadweight costs (in an expected value sense) of bankruptcy are being imposed on the economy raises the more fundamental question as to why firms and individuals investing in the firm are complacent with such a state of affairs. There is a very strong competitive myopia on the part of both firms and investors in this model. The treatment of this question is beyond the scope of this paper, but it is hopeful that the above analysis at least suggests the problem may be of some importance.
IV. CONCLUSION
We have considered two models of a capital market that has a type of debt contract through which investment is made on a risky investment project. The distinguishing feature of this contract is that there is some positive probability that the borrower will default on the loan. When default occurs real assets are transferred from the borrower to the lender and real resource costs are incurred as the result of default. In each case we considered the determination of the equilibrium interest rate on the loan and the equilibrium probability of default. A number of compara- tive static results were developed from models with the simplifying assumption of identical, risk neutral investors. We would suggest, how- ever, that the comparative static predictions of these “polar cases” a re useful for isolating the factors that would explain the influence of various market parameters, and furthermore they provide a basis for comparison with more general models that deal with the costs of default.
APPENDIX
This appendix presents a proof of all propositions given in the main text. Proposition 3.1: Note that g(z) is continuous and differentiable, g(0) =
0 and g ( M ) = cS. Evaluating the derivative of g(z) we have
HARRIS: COSTLY DEFAULT 49 1
= (c - r ) z d F ( z ) + rJzMdF(s) .
In Figure 2 g(z) is graphed. For an exogenous rate of return e the equation g(z) = 4 has two solutions, z1 and z2. Now, we shall argue z , is the only possible equilibrium solution. Suppose for example all lenders are charging 92 = rzz . Then any lender could charge q”,where 91 < 6 < 92, attract a large number of loan customers and earn an expected rate of return above 4. Notice a t z1 that g’(z,) > 0. Is i t possible that at some equilibrium z * , g’(z*) < O? The answer is negative. Notice first that from (12) g’(0) = rJ,MdF(s) = r > 0. Thus the function g(z) always starts a t 0 with a rising segment. Now suppose there existed an equilib- rium z* such that g’(z*) < 0. But then by the mean value theorem there would exist a 2 < z * , such that g(2) = e , and g’(2) > 0. Thus z * could not have been an equilibrium. Q.E.D.
T h e proposition excludes the degenerate case g ’ ( z * ) = 0 a t a n equilibrium default point g ( z * ) = e . This case is degenerate in the sense that the set of points for which it may occur is of measure zero by Sard’s theorem.
Proposition 2.2: Differentiate (2) and use g ’ ( z * ) > 0. Q.E.D. Proposition 2.3: Differentiating (2) with respect to r and z * we get
the equation
As g ’ (z* ) > 0, we have dz*/dr < 0. Differentiating the identity 9 = z * r we have c
and from
Therefore
:*/dr = r fdz* /dr ) + z* . From (13)
-z * Jz?dF(s) dz* /dr = g ’fz *)
12)
g’fz*) = fc - r ) z * d F f z * ) + r(,?dF(s).
d9*/dr = -rz*J,?dF(s) + g ’ f z *) 2 * g l f z *) g ’(2 *)
- - -rz*J,?dF(s) + ( c - r ) z * 2 d F ( z * ) + r z*dF(z* ) g’fz *)
492 ECONOMIC INQUIRY
Proposition 2.4: Differentiating (2) with respect to c and z * we get
g'fz*)dz* + { J ; ' s d F ( s ) } d c = 0
which implies dz*/dc < 0. Proposition 2.5: We wish to consider an increase in risk in the Roths-
child-Stiglitz (1970) sense, such that probability of default does not decrease a t the equilibrium default point. Following Diamond and Stiglitz (1 974) we characterize this by the conditions
Q.E.D.
( i ) JiF,(s, t)ds Z 0 for 0 Q x Q M
( i i ) J,MdF(s, t)ds = 0
and
(iv) F, ( z* , t ) 2 0
where F(s , t ) is a family of distribution functions with support in [O, M ] and indexed by t .
Using integration by parts (2) may be re-written as
(14) g(Z*, t ) = c [ z * F ( z * , t ) - J;*Ffs, t )ds] + r z * [ 1 - Ffz*, t ) ] = 4 .
Now g,(z *, t ) > 0 from proposition 2.1. Differentiating (14) with respect to t we have
where the inequality follows from the definition of an increase in default risk. Thus differentiation of (14) with respect to z * and t implies dz * /d t 2 0. Now F ( z * , t ) is the probability of default as viewed by lenders, and hence dF(z *, t ) /dt = F,(z *, t ) fdz * /d t ) + F,fz *, t ) Z 0. Q.E.D.
Proposition 3.1: (Proof): The proof is based on the fact that
(15) wrH,(z, w ) + (1 - w ) r H , ( z , w ) = ERp(zw) .
Therefore of the three equations (9), (10) and ( 1 1 ) only two are independ- ent, the third being determined by ( 1 5 ) . As e L (cS, rS) there exists a uniquey*,O < y * < M,suchthat
HARRIS: COSTLY DEFAULT 493
As y * = zw, we may write z as an implicit function of w , z (w) , with the property that z (w)w = y*. Taking the solution y * from (16) and sub- stituting into HI ( z , w ) we have
We shall show now there exists a w * , 0 < w * < 1 such that H,[z(w*) , w * ] = u. Consider the expression in brackets on the right- hand side of ( 1 7 ) ,
Dividing through by [ 1 - F (y *)] > 0 we get
E ( s l s > y*) - y * > y * - y * = 0.
Hence the expression in brackets is strictly positive in ( 1 7). Now from (16 )
Subtracting ( 18) from ( 19), we have
(20) c/r sdF(s) + y * J,”idF(s) > 0 ,
which implies that (18) is strictly less than u. Letting ( 1 8) equal a we have shown 0 < a < u. ( 1 7 ) can be written as
H , [ z ( w ) , W ] = a / l - W .
Since H , [ z ( w ) , w ] is a continuous function of w and a < u, there exists a w * , 0 < w * < 1 such that H,[z (w*) , w * ] = u. Using (15) and substi- tuting for w * and z * = z ( w * ) we have that H,(z*, w * ) = u, and the proposition is proved. Q.E.D.
In order to do the comparative statics of the equilibrium in section I11 we must examine the Jacobian of the relevant equilibrium equations. It is necessary to work with only two of the three equilibrium equations. We shall use (10 ) multiplied b y r a n d (16 ) to give us
G,(z*, w * ) = c J””’ ,, sdF(s) + rz* I,”;’,. dF(s) = q , W*
(21)
494 ECONOMIC INQUIRY
Evaluating the partial derivatives of these equations we have
G,,(z, w) = (c - r)zw dF(zw) + r JzzdF(s),
c G,,(z, w) = - J;"sdF(s) + (c - r)z'dF(zw) < 0,
W 2
Gzl(zr w) = (c - r)zw2dF(zw) < 0,
and
GZ2(z, w) = (c - r)z2wdF(zw) < 0,
where G, denotes the partial derivative of G i , i = 1 , 2 with respect to j = z , w . The last three inequalities follow from the fact that c < r . G,, is of indeterminate sign as the first term is negative and the second positive. Recall, however, that G,(z, w ) is the expected return to investing in the bonds of risky firms at the nominal interest rate of q = rz. Now, just as was argued in section I1 with regard to proposition 2.1, competi- tion amongst lenders will force q = rz*, the interest rate on bonds, to the lowest one compatible with (21) for any value of w , which to the individual investor is taken as given. Note that Cl(O, w) = 0, Gl(M, w) > 0 and G,,(O, w) = 1 > 0. Thus by the same reasoning as used to prove proposition 2.1, it must be the case that a t an equilibrium (z*, w*) G11(z*, w') > 0.
Therefore the Jacobian matrix of (21) and (22) has the sign pattern
[' I] at equilibrium values of ( z* , w * ) . Letting [GI denote the determinant of this matrix we see that I G I < 0.
Proposition 3.2: Differentiating (21) and (22) with respect to z , w and e at the equilibrium point (z *, w *) we get
