The American Economic Review VOLUME XLVIII JUNE 1958 NUMBER THREE THE COST OF CAPITAL,CORPORATION FINANCE AND THE THEORY OF INVESTMENT By FRANCO MODIGLIANI AND MERTON H.MILLER* What is the "cost of capital"to a firm in a world in which funds are used to acquire assets whose yields are uncertain;and in which capital can be obtained by many different media,ranging from pure debt instru- ments,representing money-fixed claims,to pure equity issues,giving holders only the right to a pro-rata share in the uncertain venture? This question has vexed at least three classes of economists:(1)the cor- poration finance specialist concerned with the techniques of financing firms so as to ensure their survival and growth;(2)the managerial economist concerned with capital budgeting;and (3)the economic theorist concerned with explaining investment behavior at both the micro and macro levels.1 In much of his formal analysis,the economic theorist at least has tended to side-step the essence of this cost-of-capital problem by pro- ceeding as though physical assets-like bonds-could be regarded as yielding known,sure streams.Given this assumption,the theorist has concluded that the cost of capital to the owners of a firm is simply the rate of interest on bonds;and has derived the familiar proposition that the firm,acting rationally,will tend to push investment to the point The authors are,respectively,professor and associate professor of economics in the Grad- uate School of Industrial Administration,Carnegie Institute of Technology.This article is a revised version of a paper delivered at the annual meeting of the Econometric Society,Decem- ber 1956.The authors express thanks for the comments and suggestions made at that time by the discussants of the paper,Evsey Domar,Robert Eisner and John Lintner,and subse- quently by James Duesenbercy.They are also greatly indebted to many of their present and former colleagues and students at Carnegie Tech who served so often and with such remark- able patience as a critical forum for the ideas here presented. 1 The literature bearing on the cost-of-capital problem is far too extensive for listing here. Numerous references to it will be found throughout the paper though we make no claim to completeness.One phase of the problem which we do not consider explicitly,but which has a considerable literature of its own is the relation between the cost of capital and public utility rates.For a recent summary of the"cost-of-capital theory"of rate regulation and a brief dis- cussion of some of its implications,the reader may refer to H.M.Somers [20J. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
The American economic Revlew VOLUME XLVIII JUNE 1958 NUMBER THREE THE COST OF CAPITAL, CORPORATION FINANCE AND THE THEORY OF INVESTMIENT By FRANCO MODIGLIAN1 AND MERTON H. MILLER* What is the "cost of capital" to a firm in a world in which funds are used to acquire assets whose yields are uncertain; and in which capital can be obtained by many different media, ranging from pure debt instru- ments, representing money-fixed claims, to pure equity issues, giving holders only the right to a pro-rata share in the uncertain venture.? This question has vexed at least three classes of economists: (1) the cor- poration finance specialist concerned with the techniques of financing firms so as to ensure their survival and growth; (2) the managerial economist concerned with capital budgeting; and (3) the economic theorist concerned with explaining investment behavior at both the micro and macro levels.' In much of his formal analysis, the economic theorist at least has tended to side-step the essence of this cost-of-capital problem by pro- ceeding as though physical assets-like bonds-could be regarded as yielding known, sure streams. Given this assumption, the theorist has concluded that the cost of capital to the owners of a firm is simply the rate of interest on bonds; and has derived the familiar proposition that the firm, acting rationally, will tend to push investmnent to the point * The authors are, respectively, professor and associate professor of economics in the Grad- uate School of Industrial Administration, Carnegie Institute of Technology. This article is a revised version of a paper delivered at the annual meeting of the Econometric Society, Decem- ber 1956. The authors express thanks for the comments and suggestions made at that time by the discussants of the paper, Evsey Domar, Robert Eisner and John Lintner, and subse- quently by J'ames Duesenberry. They are also greatly indebted to many of their present and former colleagues and students at Carnegie Tech who served so often and with such remark- able patience as a critical forum for the ideas here presented. 1 The literature bearing on the cost-of-capital problem is far too extensive for listing here. Numerous references to it will be found throughout the paper though we make no claim to completeness. One phase of the problem which we do not consider explicitly, but which has a considerable literature of its own is the relation between the cost of capital and public utility rates. For a recent summary of the "cost-of-capital theory" of rate regulation and a brief dis- cussion of some of its implications, the reader may refer to H. M. Somers [201. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
262 THE AMERICAN ECONOMIC REVIEW where the marginal yield on physical assets is equal to the market rate of interest.?This proposition can be shown to follow from either of two criteria of rational decision-making which are equivalent under certain- ty,namely (1)the maximization of profits and(2)the maximization of market value. According to the first criterion,a physical asset is worth acquiring if it will increase the net profit of the owners of the firm.But net profit will increase only if the expected rate of return,or yield,of the asset exceeds the rate of interest.According to the second criterion,an asset is worth acquiring if it increases the value of the owners'equity,i.e.,if it adds more to the market value of the firm than the costs of acquisi- tion.But what the asset adds is given by capitalizing the stream it gen- erates at the market rate of interest,and this capitalized value will exceed its cost if and only if the yield of the asset exceeds the rate of interest.Note that,under either formulation,the cost of capital is equal to the rate of interest on bonds,regardless of whether the funds are acquired through debt instruments or through new issues of common stock.Indeed,in a world of sure returns,the distinction between debt and equity funds reduces largely to one of terminology. It must be acknowledged that some attempt is usually made in this type of analysis to allow for the existence of uncertainty.This attempt typically takes the form of superimposing on the results of the certainty analysis the notion of a "risk discount"to be subtracted from the ex- pected yield (or a"risk premium"to be added to the market rate of interest).Investment decisions are then supposed to be based on a com- parison of this"risk adjusted"or"certainty equivalent"yield with the market rate of interest.3 No satisfactory explanation has yet been pro- vided,however,as to what determines the size of the risk discount and how it varies in response to changes in other variables. Considered as a convenient approximation,the model of the firm constructed via this certainty-or certainty-equivalent-approach has admittedly been useful in dealing with some of the grosser aspects of the processes of capital accumulation and economic fluctuations.Such a model underlies,for example,the familiar Keynesian aggregate invest- ment function in which aggregate investment is written as a function of the rate of interest-the same riskless rate of interest which appears later in the system in the liquidity-preference equation.Yet few would maintain that this approximation is adequate.At the macroeconomic level there are ample grounds for doubting that the rate of interest has 2 Or,more accurately,to the marginal cost of borrowed funds since it is customary,at least in advanced analysis,to draw the supply curve of borrowed funds to the firm as a rising one. For an advanced treatment of the certainty case,see F.and V.Lutz [13]. The classic examples of the certainty-equivalent approach are found in J.R.Hicks [8]and O.Lange [11]. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
262 THE AMERICAN ECONOMIC REVIEW where the marginal yield on physical assets is equal to the market rate of interest.2 This proposition can be shown to follow from either of two criteria of rational decision-making which are equivalent under certain- ty, namely (1) the maximization of profits and (2) the maximization of market value. According to the first criterion, a physical asset is worth acquiring if it will increase the net profit of the owners of the firm. But net profit will increase only if the expected rate of return, or yield, of the asset exceeds the rate of interest. According to the second criterion, an asset is worth acquiring if it increases the value of the owners' equity, i.e., if it adds more to the market value of the firm than the costs of acquisi- tion. But what the asset adds is given by capitalizing the stream it gen- erates at the market rate of interest, and this capitalized value will exceed its cost if and only if the yield of the asset exceeds the rate of interest. Note that, under either formulation, the cost of capital is equal to the rate of interest on bonds, regardless of whether the funds are acquired through debt instruments or through new issues of common stock. Indeed, in a world of sure returns, the distinction between debt and equity funds reduces largely to one of terminology. It must be acknowledged that some attempt is usually made in this type of analysis to allow for the existence of uncertainty. This attempt typically takes the form of superimposing on the results of the certainty analysis the notion of a "risk discount" to be subtracted from the ex- pected yield (or a "risk premium" to be added to the market rate of interest). Investment decisions are then supposed to be based on a com- parison of this "risk adjusted" or "certainty equivalent" yield with the market rate of interest.3 No satisfactory explanation has yet been pro- vided, however, as to what determines the size of the risk discount and how it varies in response to changes in other variables. Considered as a convenient approximation, the model of the firm constructed via this certainty-or certainty-equivalent-approach has admittedly been useful in dealing with some of the grosser aspects of the processes of capital accumulation and economic fluctuations. Such a model underlies, for example, the familiar Keynesian aggregate invest- ment function in which aggregate investment is written as a function of the rate of interest-the same riskless rate of interest which appears later in the system in the liquidity-preference equation. Yet few would maintain that this approximation is adequate. At the macroeconomic level there are ample grounds for doubting that the rate of interest has 2 Or, more accurately, to the marginal cost of borrowed funds since it is customary, at least in advanced analysis, to draw the supply curve of borrowed funds to the firm as a rising one. For an advanced treatment of the certainty case, see F. and V. Lutz [131. a The classic examples of the certainty-equivalent approach are found in J. R. Hicks [8] and 0. Lange [11]. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER:THEORY OF INVESTMENT 263 as large and as direct an influence on the rate of investment as this analysis would lead us to believe.At the microeconomic level the cer- tainty model has little descriptive value and provides no real guidance to the finance specialist or managerial economist whose main problems cannot be treated in a framework which deals so cavalierly with uncer- tainty and ignores all forms of financing other than debt issues. Only recently have economists begun to face up seriously to the prob- lem of the cost of capital cum risk.In the process they have found their interests and endeavors merging with those of the finance specialist and the managerial economist who have lived with the problem longer and more intimately.In this joint search to establish the principles which govern rational investment and financial policy in a world of uncer- tainty two main lines of attack can be discerned.These lines represent, in effect,attempts to extrapolate to the world of uncertainty each of the two criteria-profit maximization and market value maximization- which were seen to have equivalent implications in the special case of certainty.With the recognition of uncertainty this equivalence vanishes. In fact,the profit maximization criterion is no longer even well defined. Under uncertainty there corresponds to each decision of the firm not a unique profit outcome,but a plurality of mutually exclusive outcomes which can at best be described by a subjective probability distribution. The profit outcome,in short,has become a random variable and as such its maximization no longer has an operational meaning.Nor can this difficulty generally be disposed of by using the mathematical expecta- tion of profits as the variable to be maximized.For decisions which affect the expected value will also tend to affect the dispersion and other characteristics of the distribution of outcomes.In particular,the use of debt rather than equity funds to finance a given venture may well in- crease the expected return to the owners,but only at the cost of in- creased dispersion of the outcomes. Under these conditions the profit outcomes of alternative investment and financing decisions can be compared and ranked only in terms of a subjective "utility function"of the owners which weighs the expected yield against other characteristics of the distribution.Accordingly,the extrapolation of the profit maximization criterion of the certainty model has tended to evolve into utility maximization,sometimes explicitly, more frequently in a qualitative and heuristic form.5 The utility approach undoubtedly represents an advance over the certainty or certainty-equivalent approach.It does at least permit us 4 Those who have taken a "case-method"course in finance in recent years will recall in this connection the famous Liquigas case of Hunt and Williams,19,pp.193-96]a case which is often used to introduce the student to the cost-of-capital problem and to poke a bit of fun at the economist's certainty-model. 5 For an attempt at a rigorous explicit development of this line of attack,see F.Modigliani and M.Zeman [14]. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER: THEORY OF INVESTMENT 263 as large and as direct an influence on the rate of investment as this analysis would lead us to believe. At the microeconomic level the cer- tainty model has little descriptive value and provides no real guidance to the finance specialist or managerial economist whose main problems cannot be treated in a framework which deals so cavalierly with uncer- tainty and ignores all forms of financing other than debt issues.4 Only recently have economists begun to face up seriously to the prob- lem of the cost of capital cum risk. In the process they have found their interests and endeavors merging with those of the finance specialist and the managerial economist who have lived with the problem longer and more intimately. In this joint search to establish the principles which govern rational investment and financial policy in a world of uncer- tainty two main lines of attack can be discerned. These lines represent, in effect, attempts to extrapolate to the world of uncertainty each of the two criteria-profit maximization and market value maximization- which were seen to have equivalent implications in the special case of certainty. With the recognition of uncertainty this equivalence vanishes. In fact, the profit maximization criterion is no longer even well defined. Under uncertainty there corresponds to each decision of the firm not a unique profit outcome, but a plurality of mutually exclusive outcomes which can at best be described by a subjective probability distribution. The profit outcome, in short, has become a random variable and as such its maximization no longer has an operational meaning. Nor can this difficulty generally be disposed of by using the mathematical expecta- tion of profits as the variable to be maximized. For decisions which affect the expected value will also tend to affect the dispersion and other characteristics of the distribution of outcomes. In particular, the use of debt rather than equity funds to finance a given venture may well in- crease the expected return to the owners, but only at the cost of in- creased dispersion of the outcomes. Under these conditions the profit outcomes of alternative investment and financing decisions can be compared and ranked only in terms of a subjective "utility function" of the owners which weighs the expected yield against other characteristics of the distribution. Accordingly, the extrapolation of the profit maximization criterion of the certainty model has tended to evolve into utility maximization, sometimes explicitly, more frequently in a qualitative and heuristic form.5 The utility approach undoubtedly represents an advance over the certainty or certainty-equivalent approach. It does at least permit us 4 Those who have taken a "case-method" couirse in finance in recent years will recall in this connection the famous Liquigas case of Hunt and Williams, 19, pp. 193-961 a case which is often used to introduce the student to the cost-of-capital problem and to poke a bit of fun at the economist's certainty-model. 6 For an attempt at a rigorous explicit development of this line of attack, see F. Modigliani and M. Zeman [141. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
264 THE AMERICAN ECONOMIC REVIEW to explore(within limits)some of the implications of different financing arrangements,and it does give some meaning to the"cost"of different types of funds.However,because the cost of capital has become an essentially subjective concept,the utility approach has serious draw- backs for normative as well as analytical purposes.How,for example, is management to ascertain the risk preferences of its stockholders and to compromise among their tastes?And how can the economist build a meaningful investment function in the face of the fact that any given investment opportunity might or might not be worth exploiting depend- ing on precisely who happen to be the owners of the firm at the moment? Fortunately,these questions do not have to be answered;for the alter- native approach,based on market value maximization,can provide the basis for an operational definition of the cost of capital and a workable theory of investment.Under this approach any investment project and its concomitant financing plan must pass only the following test:Will the project,as financed,raise the market value of the firm's shares?If so,it is worth undertaking;if not,its return is less than the marginal cost of capital to the firm.Note that such a test is entirely independent of the tastes of the current owners,since market prices will reflect not only their preferences but those of all potential owners as well.If any current stockholder disagrees with management and the market over the valuation of the project,he is free to sell out and reinvest elsewhere, but will still benefit from the capital appreciation resulting from man- agement's decision. The potential advantages of the market-value approach have long been appreciated;yet analytical results have been meager.What ap- pears to be keeping this line of development from achieving its promise is largely the lack of an adequate theory of the effect of financial struc- ture on market valuations,and of how these effects can be inferred from objective market data.It is with the development of such a theory and of its implications for the cost-of-capital problem that we shall be con- cerned in this paper. Our procedure will be to develop in Section I the basic theory itself and to give some brief account of its empirical relevance.In Section II, we show how the theory can be used to answer the cost-of-capital ques- tion and how it permits us to develop a theory of investment of the firm under conditions of uncertainty.Throughout these sections the approach is essentially a partial-equilibrium one focusing on the firm and "industry."Accordingly,the "prices"of certain income streams will be treated as constant and given from outside the model,just as in the standard Marshallian analysis of the firm and industry the prices of all inputs and of all other products are taken as given.We have chosen to focus at this level rather than on the economy as a whole because it This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
264 THE AMERICAN ECONOMIC REVIEW to explore (within limits) some of the implications of different financing arrangements, and it does give some meaning to the "cost" of different types of funds. However, because the cost of capital has become an essentially subjective concept, the utility approach has serious draw- backs for normative as well as analytical purposes. How, for example, is management to ascertain the risk preferences of its stockholders and to compromise among their tastes? And how can the economist build a meaningful investment function in the face of the fact that any given investment opportunity might or might not be worth exploiting depend- ing on precisely who happen to be the owners of the firm at the moment? Fortunately, these questions do not have to be answered; for the alter- native approach, based on market value maximization, can provide the basis for an operational definition of the cost of capital and a workable theory of investment. Under this approach any investment project and its concomitant financing plan must pass only the following test: Will the project, as financed, raise the market value of the firm's shares? If so, it is worth undertaking; if not, its return is less than the marginal cost of capital to the firm. Note that such a test is entirely independent of the tastes of the current owners, since market prices will reflect not only their preferences but those of all potential owners as well. If any current stockholder disagrees with management and the market over the valuation of the project, he is free to sell out and reinvest elsewhere, but will still benefit from the capital appreciation resulting from man- agement's decision. The potential advantages of the market-value approach have long been appreciated; yet analytical results have been meager. What ap- pears to be keeping this line of development from achieving its promise is largely the lack of an adequate theory of the effect of financial struc- ture on market valuations, and of how these effects can be inferred from objective market data. It is with the development of such a theory and of its implications for the cost-of-capital problem that we shall be con- cerned in this paper. Our procedure will be to develop in Section I the basic theory itself and to give some brief account of its empirical relevance. In Section II, we show how the theory can be used to answer the cost-of-capital ques- tion and how it permits us to develop a theory of investment of the firm under conditions of uncertainty. Throughout these sections the approach is essentially a partial-equilibrium one focusing on the firm and "industry." Accordingly, the "prices" of certain income streams will be treated as constant and given from outside the model, just as in the standard Marshallian analysis of the firm and industry the prices of all inputs and of all other products are taken as given. We have chosen to focus at this level rather than on the economy as a whole because it This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER:THEORY OF INVESTMENT 265 is at the level of the firm and the industry that the interests of the vari- ous specialists concerned with the cost-of-capital problem come most closely together.Although the emphasis has thus been placed on partial- equilibrium analysis,the results obtained also provide the essential building blocks for a general equilibrium model which shows how those prices which are here taken as given,are themselves determined.For reasons of space,however,and because the material is of interest in its own right,the presentation of the general equilibrium model which rounds out the analysis must be deferred to a subsequent paper. I.The Valuation of Securities,Leverage,and the Cost of Capital A.The Capitalization Rate for Uncertain Streams As a starting point,consider an economy in which all physical assets are owned by corporations.For the moment,assume that these corpora- tions can finance their assets by issuing common stock only;the intro- duction of bond issues,or their equivalent,as a source of corporate funds is postponed until the next part of this section. The physical assets held by each firm will yield to the owners of the frm一its stockholders--a stream of“profits'”over time;but the ele- ments of this series need not be constant and in any event are uncertain. This stream of income,and hence the stream accruing to any share of common stock,will be regarded as extending indefinitely into the future. We assume,however,that the mean value of the stream over time,or average profit per unit of time,is finite and represents a random vari- able subject to a(subjective)probability distribution.We shall refer to the average value over time of the stream accruing to a given share as the return of that share;and to the mathematical expectation of this average as the expected return of the share.6 Although individual inves- tors may have different views as to the shape of the probability distri. These propositions can be restated analytically as follows:The assets of the ith firm gener- ate a stream: X(1),X(2)···X(T) whose elements are random variables subject to the joint probability distribution: xX:(1),X(2)···X(0 The return to the ith firm is defined as: Xi=lim xi(). X;is itself a random variable with a probability distribution(Xi)whose form is determined uniquely by x.The expected return X;is defined as X;=E(Xi)=/x;X;(Xi)dXi.If N:is the number of shares outstanding,the return of the ith share is x;=(1/N)X:with probability distribution (xdxs=(Nxi)d(Nx;)and expected value=(1/N)Xi. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER: THEORY OF INVESTMENT 265 is at the level of the firm and the industry that the interests of the vari- ous specialists concerned with the cost-of-capital problem come most closely together. Although the emphasis has thus been placed on partial- equilibrium analysis, the results obtained also provide the essential building blocks for a general equilibrium model which shows how those prices which are here taken as given, are themselves determined. For reasons of space, however, and because the material is of interest in its own right, the presentation of the general equilibrium model which rounds out the analysis must be deferred to a subsequent paper. I. Tihe Valuation of Securities, Leverage, and tihe Cost of Capital A. T'he Capitalization Rate for Uncertain Streams As a starting point, consider an economy in which all physical assets are owned by corporations. For the moment, assume that these corpora- tions can finance their assets by issuing common stock only; the intro- duction of bond issues, or their equivalent, as a source of corporate funds is postponed until the next part of this section. The physical assets held by each firm will yield to the owners of the firm-its stockholders-a stream of "profits" over time; but the ele- ments of this series need not be constant and in any event are uncertain. This stream of income, and hence the stream accruing to any share of common stock, will be regarded as extending indefinitely into the future. WTe assume, however, that the mean value of the stream over time, or average profit per unit of time, is finite and represents a random vari- able subject to a (subjective) probability distribution. We shall refer to the average value over time of the stream accruing to a given share as the return of that share; and to the mathematical expectation of this average as the expected return of the share.6 Although individual inves- tors may have different views as to the shape of the probability distri 6 These propositions can be restated analytically as follows: The assets of the ith firm gener- ate a stream: Xi (I), Xi (2) ... Xi (T) whose elements are random variables subject to the joint probability distribution: Xi [Xi (1), Xi (2) .. *X\i (t)J. The return to the ith firm is defined as: liT Xi-= lim - Xsit). 7--co T t= Xi is itself a random variable with a probability distribution diW(Xi) whose form is determined uniquely by Xi. The expected return Xi is defined as Xi=E(Xi) =fxXib(X,)dX;. If Ni is the number of shares outstanding, the return of the ith share is xi= (1/N)X; with probability distribution Oi(xi)dx1=4i(Nxi)d(Nxi) and expected value 9i=(1/N)X,. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
266 THE AMERICAN ECONOMIC REVIEW bution of the return of any share,we shall assume for simplicity that they are at least in agreement as to the expected return.7 This way of characterizing uncertain streams merits brief comment. Notice first that the stream is a stream of profits,not dividends.As will become clear later,as long as management is presumed to be acting in the best interests of the stockholders,retained earnings can be regarded as equivalent to a fully subscribed,pre-emptive issue of common stock. Hence,for present purposes,the division of the stream between cash dividends and retained earnings in any period is a mere detail.Notice also that the uncertainty attaches to the mean value over time of the stream of profits and should not be confused with variability over time of the successive elements of the stream.That variability and uncer- tainty are two totally different concepts should be clear from the fact that the elements of a stream can be variable even though known with certainty.It can be shown,furthermore,that whether the elements of a stream are sure or uncertain,the effect of variability per se on the valua- tion of the stream is at best a second-order one which can safely be neg- lected for our purposes(and indeed most others too).8 The next assumption plays a strategic role in the rest of the analysis. We shall assume that firms can be divided into "equivalent return" classes such that the return on the shares issued by any firm in any given class is proportional to (and hence perfectly correlated with)the return on the shares issued by any other firm in the same class.This assumption implies that the various shares within the same class differ, at most,by a "scale factor."Accordingly,if we adjust for the difference in scale,by taking the ratio of the return to the expected return,the probability distribution of that ratio is identical for all shares in the class.It follows that all relevant properties of a share are uniquely char- acterized by specifying (1)the class to which it belongs and (2)its expected return. The significance of this assumption is that it permits us to classify firms into groups within which the shares of different firms are "homoge- neous,"that is,perfect substitutes for one another.We have,thus,an analogue to the familiar concept of the industry in which it is the com- modity produced by the firms that is taken as homogeneous.To com- plete this analogy with Marshallian price theory,we shall assume in the 7 To deal adequately with refinements such as differences among investors in estimates of expected returns would require extensive discussion of the theory of portfolio selection.Brief references to these and related topics will be made in the succeeding article on the general equilibrium model. 8 The reader may convince himself of this by asking how much he would be willing to rebate to his employer for the privilege of receiving his annual salary in equal monthly installments rather than in irregular amounts over the year.See also J.M.Keynes [10,esp.pp.53-541. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
266 THE AMERICAN ECONOMIC REVIEW bution of the return of any share, we shall assume for simplicity that they are at least in agreement as to the expected return.7 This way of characterizing uncertain streams merits brief comment. Notice first that the stream is a stream of profits, not dividends. As will become clear later, as long as management is presumed to be acting in the best interests of the stockholders, retained earnings can be regarded as equivalent to a fully subscribed, pre-emptive issue of common stock. Hence, for present purposes, the division of the stream between cash dividends and retained earnings in any period is a mere detail. Notice also that the uncertainty attaches to the mean value over time of the stream of profits and should not be confused with variability over time of the successive elements of the stream. That variability and uncer- tainty are two totally different concepts should be clear from the fact that the elements of a stream can be variable even though known with certainty. It can be shown, furthermore, that whether the elements of a stream are sure or uncertain, the effect of variability per se on the valua- tion of the stream is at best a second-order one which can safely be neg- lected for our purposes (and indeed most others too).8 The next assumption plays a strategic role in the rest of the analysis. We shall assume that firms can be divided into "equivalent return" classes such that the return on the shares issued by any firm in any given class is proportional to (and hence perfectly correlated with) the return on the shares issued by any other firm in the same class. This assumption implies that the various shares within the same class differ, at most, by a "scale factor." Accordingly, if we adjust for the difference in scale, by taking the ratio of the return to the expected return, the probability distribution of that ratio is identical for all shares in the class. It follows that all relevant properties of a share are uniquely char- acterized by specifying (1) the class to which it belongs and (2) its expected return. The significance of this assumption is that it permits us to classify firms into groups within which the shares of different firms are "homoge- neous," that is, perfect substitutes for one another. We have, thus, an analogue to the familiar concept of the industry in which it is the com- modity produced by the firms that is taken as homogeneous. To com- plete this analogy with Marshallian price theory, we shall assume in the 7To deal adequately with refinements such as differences among investors in estimates of expected returns would require extensive discussion of the theory of portfolio selection. Brief references to these and related topics will be made in the succeeding article on the general equilibrium model. 8 The reader may convince himself of this by asking how much he would be willing to rebate to his employer for the privilege of receiving his annual salary in equal monthly installments rather than in irregular amounts over the year. See also J. M. Keynes [10, esp. pp. 53-541. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER:THEORY OF INVESTMENT 267 analysis to follow that the shares concerned are traded in perfect mar- kets under conditions of atomistic competition. From our definition of homogeneous classes of stock it follows that in equilibrium in a perfect capital market the price per dollar's worth of expected return must be the same for all shares of any given class.Or, equivalently,in any given class the price of every share must be propor- tional to its expected return.Let us denote this factor of proportionality for any class,say the kth class,by 1/p&.Then if denotes the price and is the expected return per share of the jth firm in class k,we must have: 1 (1) =一元； Pk or,equivalently, (2) =p&a constant for all firms j in class k. 帖 The constants pr(one for each of the k classes)can be given several economic interpretations:(a)From (2)we see that each pa is the ex- pected rate of return of any share in class k.(b)From (1)1/p&is the price which an investor has to pay for a dollar's worth of expected re- turn in the class k.(c)Again from(1),by analogy with the terminology for perpetual bonds,P can be regarded as the market rate of capitaliza- tion for the expected value of the uncertain streams of the kind gen- erated by the kth class of firms.10 B.Debt Financing and Its Efects on Security Prices Having developed an apparatus for dealing with uncertain streams we can now approach the heart of the cost-of-capital problem by drop- ping the assumption that firms cannot issue bonds.The introduction of debt-financing changes the market for shares in a very fundamental way.Because firms may have different proportions of debt in their capi- Just what our classes of stocks contain and how the different classes can be identified by outside observers are empirical questions to which we shall return later.For the present,it is sufficient to observe:(1)Our concept of a class,while not identical to that of the industry is at least closely related to it.Certainly the basic characteristics of the probability distributions of the returns on assets will depend to a significant extent on the product sold and the tech- nology used.(2)What are the appropriate class boundaries will depend on the particular prob- lem being studied.An economist concerned with general tendencies in the market,for example, might well be prepared to work with far wider classes than would be appropriate for an inves- tor planning his portfolio,or a firm planning its financial strategy. 10 We cannot,on the basis of the assumptions so far,make any statements about the rela- tionship or spread between the various p's or capitalization rates.Before we could do so we would have to make further specific assumptions about the way investors believe the proba- bility distributions vary from class to class,as well as assumptions about investors'preferences as between the characteristics of different distributions. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER: THEORY OF INVESTMENT 267 analysis to follow that the shares concerned are traded in perfect mar- kets under conditions of atomistic competition.9 From our definition of homogeneous classes of stock it follows that in equilibrium in a perfect capital market the price per dollar's worth of expected return must be the same for all shares of any given class. Or, equivalently, in any given class the price of every share must be propor- tional to its expected return. Let us denote this factor of proportionality for any class, say the kth class, by l/Pk. Then if pi denotes the price and sj is the expected return per share of the jth firm in class k, we must have: (1) pj =-xj; Pk or, equivalently, (2) = Pk a constant for all firms j in class k. pi The constants Pk (one for each of the k classes) can be given several economic interpretations: (a) From (2) we see that each Pk iS the ex- pected rate of return of any share in class k. (b) From (1) l/Pk is the price which an investor has to pay for a dollar's worth of expected re- turn in the class k. (c) Again from (1), by analogy with the terminology for perpetual bonds, Pk can be regarded as the market rate of capitaliza- tion for the expected value of the uncertain streams of the kind gen- erated by the kth class of firms.10 B. Debt Financing and Its Effects on Security Prices Having developed an apparatus for dealing with uncertain streams we can now approach the heart of the cost-of-capital problem by drop- ping the assumption that firms cannot issue bonds. The introduction of debt-financing changes the market for shares in a very fundamental way. Because firms may have different proportions of debt in their capi- 9 Just what our classes of stocks contain and how the different classes can be identified by outside observers are empirical questions to which we shall return later. For the present, it is sufficient to observe: (1) Our concept of a class, while not identical to that of the industry is at least closely related to it. Certainly the basic characteristics of the probability distributions of the returns on assets will depend to a significant extent on the product sold and the tech- nology used. (2) What are the appropriate class boundaries will depend on the particular prob- lem being studied. An economist concerned with general tendencies in the market, for example, might well be prepared to work with far wider classes than would be appropriate for an inves- tor planning his portfolio, or a firm planning its financial strategy. 10 We cannot, on the basis of the assumptions so far, make any statements about the rela- tionship or spread between the various p's or capitalization rates. Before we could do so we would have to make further specific assumptions about the way investors believe the proba- bility distributions vary from class to class, as well as assumptions about investors' preferences as between the characteristics of different distributions. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
268 THE AMERICAN ECONOMIC REVIEW tal structure,shares of different companies,even in the same class,can give rise to different probability distributions of returns.In the language of finance,the shares will be subject to different degrees of financial risk or "leverage"and hence they will no longer be perfect substitutes for one another. To exhibit the mechanism determining the relative prices of shares under these conditions,we make the following two assumptions about the nature of bonds and the bond market,though they are actually stronger than is necessary and will be relaxed later:(1)All bonds(in- cluding any debts issued by households for the purpose of carrying shares)are assumed to yield a constant income per unit of time,and this income is regarded as certain by all traders regardless of the issuer. (2)Bonds,like stocks,are traded in a perfect market,where the term perfect is to be taken in its usual sense as implying that any two com- modities which are perfect substitutes for each other must sell,in equi- librium,at the same price.It follows from assumption(1)that all bonds are in fact perfect substitutes up to a scale factor.It follows from as- sumption(2)that they must all sell at the same price per dollar's worth of return,or what amounts to the same thing must yield the same rate of return.This rate of return will be denoted by r and referred to as the rate of interest or,equivalently,as the capitalization rate for sure streams.We now can derive the following two basic propositions with respect to the valuation of securities in companies with different capital structures: Proposition I.Consider any company j and let X;stand as before for the expected return on the assets owned by the company (that is,its expected profit before deduction of interest).Denote by D;the market value of the debts of the company;by S;the market value of its com- mon shares;and by V;=S;++D;the market value of all its securities or, as we shall say,the market value of the firm.Then,our Proposition I asserts that we must have in equilibrium: (3) Vi=(Sj+Di)=Xi/pe,for any firm j in class k. That is,the markel value of any firm is independen of its capital structure and is given by capitalizing its expected relurn at the rale p&appropriate to its class. This proposition can be stated in an equivalent way in terms of the firm's"average cost of capital,"Xi/Vi,which is the ratio of its expected return to the market value of all its securities.Our proposition then is: 又 X (4) -pa,for any firm j,in class k. (S;+D)V That is,the average cosl of capital to any firm is complelely independent of This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about jstor.org/terms
268 THE AMERICAN ECONOMIC REVIEW tal structure, shares of different companies, even in the same class, can give rise to different probability distributions of returns. In the language of finance, the shares will be subject to different degrees of filancial risk or "leverage" and hence they will no longer be perfect substitutes for one another. To exhibit the mechanism determining the relative prices of shares under these conditions, we make the following two assumptions about the nature of bonds and the bond market, though they are actually stronger than is necessary and will be relaxed later: (1) All bonds (in- cluding any debts issued by households for the purpose of carrying shares) are assumed to yield a constant income per unit of time, and this income is regarded as certain by all traders regardless of the issuer. (2) Bonds, like stocks, are traded in a perfect market, where the term perfect is to be taken in its usual sense as implying that any two com- modities which are perfect substitutes for each other must sell, in equi- librium, at the same price. It follows from assumption (1) that all bonds are in fact perfect substitutes up to a scale factor. It follows from as- sumption (2) that they must all sell at the same price per dollar's worth of return, or what amounts to the same thing must yield the same rate of return. This rate of return will be denoted by r and referred to as the rate of interest or, equivalently, as the capitalization rate for sure streams. We now can derive the following two basic propositions with respect to the valuation of securities in companies with different capital structures: Proposition I. Consider any company j and let Xi stand as before for the expected return on the assets owned by the company (that is, its expected profit before deduction of interest). Denote by Di the market value of the debts of the company; by Sj the market value of its com- mon shares; and by Vj=Sj+Dj the market value of all its securities or, as we shall say, the market value of the firm. Then, our Proposition I asserts that we must have in equilibrium: (3) Vi (Sj + Dj) = Xjl/pk, for any firm j in class k. That is, the market value of any firm is indepezdentt of its capital structure and is given by capitalizinzg its expected return at the rate Pk appropriate to its class. This proposition can be stated in an equivalent way in terms of the firm's "average cost of capital," Xj/Vj, which is the ratio of its expected return to the market value of all its securities. Our proposition then is: xj Xj (4) - = Pk, for any firm j, in class k. (Sj + Di) Va That is, thec average cost of capital, to any firm 'IS comipletely independent of This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER:THEORY OF INVESTMENT 269 its capital structure and is egual to the capitalization rate of a pure equity stream of its class. To establish Proposition I we will show that as long as the relations (3)or(4)do not hold between any pair of firms in a class,arbitrage will take place and restore the stated equalities.We use the term arbitrage advisedly.For if Proposition I did not hold,an investor could buy and sell stocks and bonds in such a way as to exchange one income stream for another stream,identical in all relevant respects but selling at a lower price.The exchange would therefore be advantageous to the inves- tor quite independently of his attitudes toward risk.As investors exploit these arbitrage opportunities,the value of the overpriced shares will fall and that of the underpriced shares will rise,thereby tending to eliminate the discrepancy between the market values of the firms. By way of proof,consider two firms in the same class and assume for simplicity only,that the expected return,X,is the same for both firms. Let company 1 be financed entirely with common stock while company 2 has some debt in its capital structure.Suppose first the value of the levered firm,V2,to be larger than that of the unlevered one,Vi.Con- sider an investor holding sa dollars'worth of the shares of company 2, representing a fraction a of the total outstanding stock,S2.The return from this portfolio,denoted by Y2,will be a fraction a of the income available for the stockholders of company 2,which is equal to the total return X2 less the interest charge,rDa.Since under our assumption of homogeneity,the anticipated total return of company 2,X2,is,under all circumstances,the same as the anticipated total return to company 1,Xi,we can hereafter replace X2 and Xi by a common symbol X. Hence,the return from the initial portfolio can be written as: (5) Y2=a(X-rD2). Now suppose the investor sold his aS:worth of company 2 shares and acquired instead an amount s=(S2+D)of the shares of company 1. He could do so by utilizing the amount aS2 realized from the sale of his initial holding and borrowing an additional amount aDa on his own credit,pledging his new holdings in company 1 as a collateral.He would thus secure for himself a fraction s1/S1=a(S2+D2)/S1 of the shares and earnings of company 1.Making proper allowance for the interest pay- ments on his personal debt aDa,the return from the new portfolio,Yi,is given by: u In the language of the theory of choice,the exchanges are movements from inefficient points in the interior to efficient points on the boundary of the investor's opportunity set;and not movements between efficient points along the boundary.Hence for this part of the analysis nothing is involved in the way of specific assumptions about investor attitudes or behavior other than that investors behave consistently and prefer more income to less income,ceferis paribus. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
MODIGLIANI AND MILLER: THEORY OF INVESTMENT 269 its capital structure and is equal to the capitalization rate of a pure equity stream of its class. To establish Proposition I we will show that as long as the relations (3) or (4) do not hold between any pair of firms in a class, arbitrage will take place and restore the stated equalities. We use the term arbitrage advisedly. For if Proposition I did not hold, an investor could buy and sell stocks and bonds in such a way as to exchange one income stream for another stream, identical in all relevant respects but selling at a lower price. The exchange would therefore be advantageous to the inves- tor quite independently of his attitudes toward risk.1' As investors exploit these arbitrage opportunities, the value of the overpriced shares will fall and that of the underpriced shares will rise, thereby tending to eliminate the discrepancy between the market values of the firms. By way of proof, consider two firms in the same class and assume for simplicity only, that the expected return, X, is the same for both firms. Let company 1 be financed entirely with common stock while company 2 has some debt in its capital structure. Suppose first the value of the levered firm, V2, to be larger than that of the unlevered one, Vi. Con- sider an investor holding S2 dollars' worth of the shares of company 2, representing a fraction a of the total outstanding stock, S2. The return from this portfolio, denoted by Y2, will be a fraction ac of the income available for the stockholders of company 2, which is equal to the total return X2 less the interest charge, rD2. Since under our assumption of homogeneity, the anticipated total return of company 2, X2, is, under all circumstances, the same as the anticipated total return to company 1, XI, we can hereafter replace X2 and Xi by a common symbol X. Hence, the return from the initial portfolio can be written as: (5) Y2- a(X - rD2). Now suppose the investor sold his aS2 worth of company 2 shares and acquired instead an amount Sl= a(S2+D2) of the shares of company 1. He could do so by utilizing the amount aS2 realized from the sale of his initial holding and borrowing an additional amount aD2 on his own credit, pledging his new holdings in company 1 as a collateral. He would thus secure for himself a fraction sl/S = a(S2+?D2)/S, of the shares and earnings of company 1. Making proper allowance for the interest pay- ments on his personal debt aD2, the return from the new portfolio, Y1, is given by: 11 In the language of the theory of choice, the exchanges are movements from inefficient points in the interior to efficient points on the boundary of the investor's opportunity set; and not movements between efficient points along the boundary. Hence for this part of the analysis nothing is involved in the way of specific assumptions about investor attitudes or behavior other than that investors behave consistently and prefer more income to less income, ceteris paribus. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms
270 THE AMERICAN ECONOMIC REVIEW (6) a(Sa+D2x raDa ay:X-raDy. Y1= S1 Comparing (5)with (6)we see that as long as Va>Vi we must have Yi>Ya,so that it pays owners of company 2's shares to sell their hold- ings,thereby depressing Sa and hence Va;and to acquire shares of com- pany 1,thereby raising S and thus Vi.We conclude therefore that levered companies cannot command a premium over unlevered com- panies because investors have the opportunity of putting the equivalent leverage into their portfolio directly by borrowing on personal account. Consider now the other possibility,namely that the market value of the levered company V:is less than Vi.Suppose an investor holds ini- tially an amount s of shares of company 1,representing a fraction a of the total outstanding stock,S1.His return from this holding is: 1=X=ax. Suppose he were to exchange this initial holding for another portfolio, also worth si,but consisting of sa dollars of stock of company 2 and of d dollars of bonds,where sa and d are given by: g D2 (7) S2= S1,d=s1. V2 V2 In other words the new portfolio is to consist of stock of company 2 and of bonds in the proportions S2/V2 and D2/V2,respectively.The return from the stock in the new portfolio will be a fraction sa/Se of the total return to stockholders of company 2,which is(X-rDa),and the return from the bonds will be rd.Making use of (7),the total return from the portfolio,Y2,can be expressed as follows: -K-则+d-元X-0+,= D2 S1 S1 -X a V2 (since s1=aS1).Comparing Y2 with Yi we see that,if V2<S=Vi,then Ya will exceed Y1.Hence it pays the holders of company 1's shares to sell these holdings and replace them with a mixed portfolio containing an appropriate fraction of the shares of company 2. The acquisition of a mixed portfolio of stock of a levered company j and of bonds in the proportion Si/V;and D;/V;respectively,may be regarded as an operation which"undoes"the leverage,giving access to an appropriate fraction of the unlevered return Xi.It is this possibility of undoing leverage which prevents the value of levered firms from be- ing consistently less than those of unlevered firms,or more generally prevents the average cost of capital Xi/V;from being systematically higher for levered than for nonlevered companies in the same class. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/terms
270 THE AMERICAN ECONOMIC REVIEW (6) - cx~(S2 + D2) V2 (6) Y, = t(S2 X - raD2 = a - X - raD2. Si V1 Comparing (5) with (6) we see that as long as V2> V1 we must have Y1 > Y2, so that it pays owners of company 2's shares to sell their hold- ings, thereby depressing S2 and hence V2; and to acquire shares of com- pany 1, thereby raising Si and thus V1. We conclude therefore that levered companies cannot command a premium over unlevered com- panies because investors have the opportunity of putting the equivalent leverage into their portfolio directly by borrowing on personal account. Consider now the other possibility, namely that the market value of the levered company V2 is less than V1. Suppose an investor holds ini- tially an amount s1 of shares of company 1, representing a fraction cx of the total outstanding stock, Si. His return from this holding is- Si Y -S X = agx. Si Suppose he were to exchange this initial holding for another portfolio, also worth s1, but consisting of S2 dollars of stock of company 2 and of d dollars of bonds, where s2 and d are given by: S2 D2 (7) S2=- 1, d =-s. V2 V2 In other words the new portfolio is to consist of stock of company 2 and of bonds in the proportions S2/V2 and D2/V2, respectively. The return from the stock in the new portfolio will be a fraction S2/S2 of the total return to stockholders of company 2, which is (X- rD2), and the return from the bonds will be rd. Making use of (7), the total return from the portfolio, Y2, can be expressed as follows: S2 D2 s1 S Y2= - (X - rD2) + rd = - (X - rD2) + r V-S =- X = - X S2 V2 V2 V2 V2 (since si = aSi). Comparing Y2 with Yi we see that, if V2 <SI V1, then Y2 will exceed Y1. Hence it pays the holders of company l's shares to sell these holdings and replace them with a mixed portfolio containing an appropriate fraction of the shares of company 2. The acquisition of a mixed portfolio of stock of a levered company j and of bonds in the proportion Sj/Vj and D1/Vj respectively, may be regarded as an operation which "undoes" the leverage, giving access to an appropriate fraction of the unlevered return Xj. It is this possibility of undoing leverage which prevents the value of levered firms from be- ing consistently less than those of unlevered firms, or more generally prevents the average cost of capital jl/Vj from being systematically higher for levered than for nonlevered companies in the same class. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms