W CHICAGO JOURNALS Dividend Policy, Growth, and the valuation of Shares Author(s): Merton H. Miller and Franco Modigliani Source: The Tournal of Business, Vol 34, No 4(Oct, 1961), pp. 411-433 Published by: The University of Chicago Press StableurL:http://www.jstor.org/stable/2351143 Accessed:11/09/20130204 Y of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available http://www.jstor.org/page/info/about/policies/terms.jsp is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@ jstor. org The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 02: 04: 42 AM All use subject to STOR Terms and Conditions
Dividend Policy, Growth, and the Valuation of Shares Author(s): Merton H. Miller and Franco Modigliani Source: The Journal of Business, Vol. 34, No. 4 (Oct., 1961), pp. 411-433 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2351143 . Accessed: 11/09/2013 02:04 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. http://www.jstor.org This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS The Graduate School of Business of the University of Chicago VOL. XXXIF OCTOBER 1961 No 4 DIVIDEND POLICY, GROWTH, AND THE VALUATION OF SHARES* MERTON H MILLERT AND FRANCO MoDIGLIANIt [E effect of a firms dividend policy of dividend policy. Lacking such a state on the current price of its shares is a ment, investigators have not yet been matter of considerable importance, able to frame their tests with sufficient not only to the corporate officials who precision to distinguish adequately be must set the policy, but to investors tween the various contending hypothe planning portfolios and to economists ses. Nor have they been able to give a seeking to understand and appraise the convincing explanation of what their test functioning of the capital markets. Do results do imply about the underlying companies with generous distribution process of valuation. policies consistently sell at a premium In the hope that it may help to over- over those with niggardly payouts? Is the come these obstacles to effective empiri reverse ever true? If so, under what con- cal testing, this paper will attempt to fill ditions? Is there an optimum payout the existing gap in the theoretical litera- ratio or range of ratios that maximizes ture on valuation. We shall begin, in Sec the current worth of the shares? tion I, by examining the effects of differ Although these questions of fact have ences in dividend policy on the current been the subject of many empirical stud- price of shares in an ideal economy char ies in recent years no consensus has yet acterized by perfect capital markets,ra been achieved. One reason appears to be tional behavior, and perfect certainty the absence in the literature of a com- Still within this convenient analytical plete and reasonably rigorous statement framework we shall go on in Sections II of those parts of the economic theory of and iii to consider certain closely related valuati on bearing directly on the matter issues that appear to have been respon- The authe who read and comment pns of.an sible for considerable misunderstanding their thanks of the role of dividend policy. In d especially to lar. Section II will focus on the long onsiderable simplification of a number of the proofs. standing debate about what investor Professor of finance and economics, University "really "capitalize when they buy shares Professor of economics, Northwestern Univer- and Section III on the much mooted rela tions between price, the rate of growth of his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS The Graduate School of Business of the University of Chicago VOL. XXXIV OCTOBER 1961 No. 4 DIVIDEND POLICY, GROWTH, AND THE VALUATION OF SHARES* MERTON H. MILLERt AND FRANCO MODIGLINIt Tz i~xeffect of a firm's dividend policy on the current price of its shares is a matter of considerable importance, not only to the corporate officials who must set the policy, but to investors planning portfolios and to economists seeking to understand and appraise the functioning of the capital markets. Do companies with generous distribution policies consistently sell at a premium over those with -niggardly payouts? Is the reverse ever true? If so, under what conditions? Is there an optimum payout ratio or range of ratios that maximizes the current worth of the shares? Although these questions of fact have been the subject of many empirical studies in recent years no consensus has yet been achieved. One reason appears to be the absence in the literature of a complete and reasonably rigorous statement of those parts of the economic theory of valuation bearing directly on the matter of dividend policy. Lacking such a statement, investigators have not yet been able to frame their tests with sufficient precision to distinguish adequately between the various contending hypotheses. Nor have they been able to give a convincing explanation of what their test results do imply about the underlying process of valuation. In the hope that it may help to overcome these obstacles to effective empirical testing, this paper will attempt to fill the existing gap in the theoretical literature on valuation. We shall begin, in Section I, by examining the effects of differences in dividend policy on the current price of shares in an ideal economy characterized by perfect capital markets, rational behavior, and perfect certainty. Still within this convenient analytical framework we shall go on in Sections II and III to consider certain closely related issues that appear to have been responsible for considerable misunderstanding of the role of dividend policy. In particular, Section II will focus on the longstanding debate about what investors "really" capitalize when they buy shares; and Section III on the much mooted relations between price, the rate of growth of * The authors wish to express their thanks to all who read and commented on earlier versions of this paper and especially to Charles C. Holt, now of the University of Wisconsin, whose suggestions led to considerable simplification of a number of the proofs. t Professor of finance and economics, University of Chicago. t Professor of economics, Northwestern University. 411 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
412 THE JOURNAL OF BUSINESS profits, and the rate of growth of divi- vestor as to the future investment pro- dends per share. Once these fundamen- gram and the future profits of every cor- tals have been established, we shall pro- poration. Because of this assurance ceed in Section IV to drop the assump- there is, among other things, no need tion of certainty and to see the extent to distinguish between stocks and bonds as which the earlier conclusions about divi sources of funds at this stage of the anal- dend policy must be modified. Finally, in ysis. We can, therefore, proceed as if Section V, we shall briefly examine the there were only a single type of financial implications for the dividend policy instrument which, for convenience, we problem of certain kinds of market im- shall refer to as shares of stock The fundamental principle of valua- lion.Under these assumptions the valu I. EFFECT OF DIVIDEND POLICY WITH PER- ation of all shares would be governed by FECT MARKETS, RATIONAL BEHAVIOR, the following fundamental principle: the AND PERFECT CERTAINTY price of each share must be such that the The meaning of the basic assumptions. rate of return(dividends plus capital S lthough the terms"perfect markets, gains per dollar invested)on every share rational behavior, and"perfect cer- will be the same throughout the market tainty"are widely used throughout eco- over any given interval of time. That is y spelling out the precise me. to start if we let nomic theory, it may be helpft of these assumptions in the present context d,0=dividends per share paid by firm j du capital markets, "no P:(0 ice(ex any dividend buyer or seller(or issuer)of securities is of a share in firmi at the start of large enough for his transactions to have orice. All traders have equal and costless we must have access to information about the ruling d, (2)+p:(+1)-pi (2) price and about all other relevant charac teristics of shares ( to be detailed spe- =p(t) independent of j cifically later). No brokerage fees, trans- fer taxes, or other transaction costs are or, equivalently, incurred when securities are bought sold,or issued, and there are no tax dif- p (0)-1+p(ld (0)+p; ( +1)1(2) ferential either between distributed and undistributed profits or between divi- for each i and for all t. Otherwise, holders dends and capital gains. of low-return(high-priced)shares could 2. " Rational behavior"means that increase their terminal wealth by selling investors always prefer more wealth to these shares and investing the proceeds less and are indifferent as to whether a in shares offering a higher rate of return given increment to their wealth takes the This process would tend to drive down form of cash payments or an increase in the prices of the low-return shares and the market value of their holdings of drive up the prices of high-return shares until the differential in rates of return 3.“ Perfect certainty olies com- had been eliminated plete assurance on the part of every in The effect of dividend policy. The im- his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
412 THE JOURNAL OF BUSINESS profits, and the rate of growth of dividends per share. Once these fundamentals have been established, we shall proceed in Section IV to drop the assumption of certainty and to see the extent to which the earlier conclusions about dividend policy must be modified. Finally, in Section V, we shall briefly examine the implications for the dividend policy problem of certain kinds of market imperfections. I. EFFECT OF DIVIDEND POLICY WITH PERFECT MARKETS, RATIONAL BEHAVIOR, AND PERFECT CERTAINTY The meaning of the basic assumptions. -Although the terms "perfect markets," "rational behavior," and "perfect certainty" are widely used throughout economic theory, it may be helpful to start by spelling out the precise meaning of these assumptions in the present context. 1. In "perfect capital markets," no buyer or seller (or issuer) of securities is large enough for his transactions to have an appreciable impact on the then ruling price. All traders have equal and costless access to information about the ruling price and about all other relevant characteristics of shares (to be detailed specifically later). No brokerage fees, transfer taxes, or other transaction costs are incurred when securities are bought, sold, or issued, and there are no tax differentials either between distributed and undistributed profits or between dividends and capital gains. 2. "Rational behavior" means that investors always prefer more wealth to less and are indifferent as to whether a given increment to their wealth takes the form of cash payments or an increase in the market value of their holdings of shares. 3. "Perfect certainty" implies complete assurance on the part of every investor as to the future investment program and the future profits of every corporation. Because of this assurance, there is, among other things, no need to distinguish between stocks and bonds as sources of funds at this stage of the analysis. We can, therefore, proceed as if there were only a single type of financial instrument which, for convenience, we shall refer to as shares of stock. The fundamental principle of valuation.-Under'these assumptions the valuation of all shares would be governed by the following fundamental principle: the price of each share must be such that the rate of return (dividends plus capital gains per dollar invested) on every share will be the same throughout the market over any given interval of time. That is, if we let dj(t) = dividends per share paid by firm j during period t pj(t) = the price (ex any dividend in t - 1) of a share in firm j at the start of period t, we must have dj(t) +pj(t+ 1) -pj(t) pj(t) ~~~(1) = p ( t ) independent of j; or, equivalently, pj( t)= [dj(t)+pj(t+)] (2) for each j and for all t. Otherwise, holders of low-return (high-priced) shares could increase their terminal wealth by selling these shares and investing the proceeds in shares offering a higher rate of return. This process would tend to drive down the prices of the low-return shares and drive up the prices of high-return shares until the differential in rates of return had been eliminated. The effect of dividend policy.-The imThis content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARE 413 ications of this principle for our prob- able information as to what that future lem of dividend policy can be seen some- dividend policy would be. The first possi what more easily if equation (2)is re- bility being the relevant one from the stated in terms of the value of the enter- standpoint of assessing the effects of divi- prise as a whole rather than in terms of dend policy, it will clarify matters to as- the value of an individual share. Drop- sume, provisionally, that the future divi ping the firm subscript i since this will dend policy of the firm is known and ad to no ambiguity in the present con- given for t+ 1 and all subsequent peri text and letting ods and is independent of the actual divi- n(= the number of shares of record dend decision in 4. Then V(+ 1) will at the start of t also be independent of the current divi- m(t+ 1)=the number of new shares(if dend decision, though it may very well any) sold during t at the ex be affected by D(t+1)and all subse- dividend closing price p(+ 1), quent distributions. Finally, current div idends can influence v through the v(2=n(0)p(t)= the total third term, -m(t+1)p(t+1), the val the enterprise and ue of new shares sold to outsiders during D(=n(ed(o= the total the period. For the higher the dividend id during t to hold ord at the start of t, rout in capital that must be raised from external we can rewrite(2) sources to maintain any desired level of investment 1+p() D()+#()(+1)] The fact that the dividend decision effects price not in one but in these two 1+p(4)[D()+v(4+1) conflicting ways-directly via D()and inversely via -me)p(t-+ 1)-is, of m(L+1)P(L+1)].(3)course, precisely why one speaks of there being a dividend policy problem. If the The advantage of restating the funda- firm raises its dividend in t, given its in- mental rule in this form is that it brings vestment decision, will the increasein the into sharper focus the three possible cash payments to the current holders be routes by which current dividends might more or less than enough to offset their affect the current market value of the lower share of the terminal value? Which firm V(), or equivalently the price of its is the better strategy for the firm in individual shares, P. Current divi- financing the investment: to reduce divi- dends will clearly affect v(o via the first dends and rely on retained earnings or to term in the bracket, D(. In principle, raise dividends but float more new current dividends might also affect v shares? indirectly via the second term, V(E+ 1), In our ideal world at least these and the new ex dividend market value. since related questions can be simply and v(t+1)must depend only on future mediately answered: the two dividend and not on past events, such could be the effects must always exactly cancel out so case, however, only if both(a)v(+ 1)that the payout policy to be followed in t were a function of future dividend policy will have no effect on the price at t. and(b)the current distribution D() We need only express m(+1). P(t+1) served to convey some otherwise unavail- in terms of d( to show that such must his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 413 plications of this principle for our problem of dividend policy can be seen somewhat more easily if equation (2) is restated in terms of the value of the enterprise as a whole rather than in terms of the value of an individual share. Dropping the firm subscript j since this will lead to no ambiguity in the present context and letting n(t) = the number of shares of record at the start of t m(t + 1) = the number of new shares (if any) sold during t at the ex dividend closing price p(t + 1), so that n(t + 1) = n(t) + m(t + 1) V(t) = n(t) p(t) = the total value of the enterprise and D(t) = n(t) d(t) = the total dividends paid during t to holders of record at the start of t, we can rewrite (2) V(t l +,) 1[D(t)+n(t)p(t+1) I 1+0 -1+ (t) [ D(t) + V(t+ 1) -m (t+ 1) p (t+ 1)I. (3) The advantage of restating the fundamental rule in this form is that it brings into sharper focus the three possible routes by which current dividends might affect the current market value of the firm V(t), or equivalently the price of its individual shares, p(t). Current dividends will clearly affect V(t) via the first term in the bracket, D(t). In principle, current dividends might also affect V(t) indirectly via the second term, V(t + 1), the new ex dividend market value. Since V(t + 1) must depend only on future and not on past events, such could be the case, however, only if both (a) V(t + 1) were a function of future dividend policy and (b) the current distribution D(t) served to convey some otherwise unavailable information as to what that future dividend policy would be. The first possibility being the relevant one from the standpoint of assessing the effects of dividend policy, it will clarify matters to assume, provisionally, that the future dividend policy of the firm is known and given for t + 1 and all subsequent periods and is independent of the actual dividend decision in t. Then V(t + 1) will also be independent of the current dividend decision, though it may very well be affected by D(t + 1) and all subsequent distributions. Finally, current dividends can influence V(t) through the third term, -m(t + 1) p(t + 1), the value of new shares sold to outsiders during the period. For the higher the dividend payout in any period the more the new capital that must be raised from external sources to maintain any desired level of investment. The fact that the dividend decision effects price not in one but in these two conflicting ways-directly via D(t) and inversely via -m(t) p(t + 1)-is, of course, precisely why one speaks of there being a dividend policy problem. If the firm raises its dividend in t, given its investment decision, will the increase in the cash payments to the current holders be more or less than enough to offset their lower share of the terminal value? Which is the better strategy for the firm in financing the investment: to reduce dividends and rely on retained earnings or to raise dividends but float more new shares? In our ideal world at least these and related questions can be simply and immediately answered: the two dividend effects must always exactly cancel out so that the payout policy to be followed in t will have no effect on the price at t. We need only express m(t+l) 1 p(t+1) in terms of D(t) to show that such must This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS indeed be the case. Specifically, if I( once you think of it. "It is, after all, is the given level of the firms invest- merely one more instance of the genera ment or increase in its holding of physical principle that there are no "financial il- assets in t and if X( is the firms total lusions""in a rational and perfect econom- net profit for the period, we know that ic environment. Values there are deter- the amount of outside capital required mined solely by "real"considerations- will be in this case the earning power of the m(+1)φ(t+1)=I(4) firms assets and its investment policy IX(E)-D(/. (4) and not by how the fruits of the earning power are "packaged"for distribution Substituting expression(4)into(3), the, Obvious as the proposition may be, D( cancel and we obtain for the value however, one finds few references to it in of the firm as of the start of t the extensive literature on the problem.' It is true that the literature abounds with V()≡()夕(4) statements that in some <theoretical 1+p(IX(O-I(0+V\+/ C 5)sense, dividend policy ought not to count; but either that sense is not clearly specified or, more frequently and espe D (does not appear directly cially among eco onomists, it is(wrongly) the arguments and since X o, identified with a situation in which the r (t+ 1)and p(o are all independ- firm's internal rate of return is the same ent of D((either by their nature or by as the external or market rate of re ssumption) it follows that the current turn 2 value of the firm must be independent of A major source of these and related the current dividend decision misunderstandings of the role of the divi- Having established that V( is unaf- dend policy has been the fruitless concern fected by the current dividend decision and controversy over what investors it is easy to go on to show that v()must "really" capitalize when they buy shares also be unaffected by any future dividend We say fruitless because as we shall now decisions as well. Such future decisions proceed to show, it is actually possible to can influence V(O)only via their effect on derive from the basic principle of valua V(t+1). But we can repeat the reason- tion(1) not merely one, but several valu- ing above and show that V(t+ 1-and ation formulas each starting from one of hence V(O-is unaffected by dividend the "classical"views of what is being policy in t+1; that V(+ 2)-and capitalized by investors. Though differ- hence V(t+ 1)and v(t-is unaffected ing somewhat in outward appearance, by dividend policy in t+ 2; and so on the various formulas can be shown to be for as far into the future as we care to equivalent in all essential respects in look. Thus, we may conclude that given a cluding, of course, their implication that firm's investment policy, the dividend dividend policy is irrelevant. While the payout policy it chooses to follow will af 1 Apart from the references to it in our earlie fect neither the current price of its shares papers, especially [161, the closest nor the total return to its shareholders seems to be that in Bodenborn [1 Like many other propositions in ec his treatment of the role of divi policy is not nomics, the irrelevance of dividend pol mpletely explicit. (The numbers in brackets refer references listed below, pp. 432-33 icy, given investment policy,is“obⅵious, See below p 424 his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
414 THE JOURNAL OF BUSINESS indeed be the case. Specifically, if I(t) is the given level of the firm's investment or increase in its holding of physical assets in t and if X(t) is the firm's total net profit for the period, we know that the amount of outside capital required will be m(t+1)p(t+1) = I(t) (4) - [X (t) -D (t) ]. Substituting expression (4) into (3), the D(t) cancel and we obtain for the value of the firm as of the start of t V (t)-n (t) p (t) (5) = +p(t)[X 1 (t)-I(t) + V(t+ 1) Since D(t) does not appear directly among the arguments and since X(t), I(t), V(t + 1) and p(t) are all independent of D(t) (either by their nature or by assumption) it follows that the current value of the firm must be independent of the current dividend decision. Having established that V(t) is unaffected by the current dividend decision it is easy to go on to show that V(t) must also be unaffected by any future dividend decisions as well. Such future decisions can influence V(t) only via their effect on V (t + 1). But we can repeat the reasoning above and show that V(t + 1)-and hence V(t)-is unaffected by dividend policy in t + 1; that V(t + 2)-and hence V(t + 1) and V(t)-is unaffected by dividend policy in t + 2; and so on for as far into the future as we care to look. Thus, we may conclude that given a firm's investment policy, the dividend payout policy it chooses to follow will affect neither the current price of its shares nor the total return to its shareholders. Like many other propositions in economics, the irrelevance of dividend policy, given investment policy, is "obvious, once you think of it." It is, after all, merely one more instance of the general principle that there are no "financial illusions" in a rational and perfect economic environment. Values there are determined solely by "real" considerationsin this case the earning power of the firm's assets and its investment policyand not by how the fruits of the earning power are "packaged" for distribution. Obvious as the proposition may be, however, one finds few references to it in the extensive literature on the problem.' It is true that the literature abounds with statements that in some "theoretical" sense, dividend policy ought not to count; but either that sense is not clearly specified or, more frequently and especially among economists, it is (wrongly) identified with a situation in which the firm's internal rate of return is the same as the external or market rate of return.2 A major source of these and related misunderstandings of the role of the dividend policy has been the fruitless concern and controversy over what investors "really" capitalize when they buy shares. We say fruitless because as we shall now proceed to show, it is actually possible to derive from the basic principle of valuation (1) not merely one, but several valuation formulas each starting from one of the "classical" views of what is being capitalized by investors. Though differing somewhat in outward appearance, the various formulas can be shown to be equivalent in all- essential respects including, of course, their implication that dividend policy is irrelevant. While the 1 Apart from the references to it in our earlier papers, especially [16], the closest approximation seems to be that in Bodenborn [1, p. 4921, but even his treatment of the role of dividend policy is not completely explicit. (The numbers in brackets refer to references listed below, pp. 432-33). 2 See below p. 424. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 415 controvery itself thus turns out to be an as T approaches infinity so that( 7)can empty one, the different expressions do be expressed as ha intrinsic intere highlighting different combinations of v(o)=Jim >1 variables they provide additional insights (1+p) (8) [X(t)-I()], open alternative lines of attack on some of the problems of empirical testing which we shall further abbreviate to Ⅱ. WHAT DOES THE MAR3Ay”V(h(+mx(0-(0).(9) In the literature on valuation one can The discounted cash flow approach nd at least the following four more or Consider now the so-called discounted less distinct approaches to the valuation of shares: (1)the discounted cash flow cash flow approach familiar in discus- approach; (2) the current earnings plus sions of capital budgeting. There, in val future in estment opportunities ap- uing any specific machine we discount at proach;(3)the stream of dividends ap- the market rate of interest the stream of proach; and(4)the stream of earnings cash receipts generated by the machine approach. To demonstrate that these ap- plus any scrap or terminal value of the proaches are, in fact, equivalent it will be machine; and minus the stream of cash helpful to begin by first going back to outlays for direct labor, materials,re- quation(5)and developing from it a pairs, and capital additions. The same valuation formula to serve as a point of approach, of course, can also be applied reference and comparison. Specifically, if to the firm as a whole which may be we assume, for simplicity, that the mar- thought of in this context as simply a ket rate of yield p()=p for all 4,then, large, composite machine. 5 This ap- setting t=0, we can rewrite(5)as 3 More general formulas in which p() is allowed V(0)-1+Ix(0)-I(0) (6)presented here merely by substituting the cum some product 1+pV(1) Since(5)holds for all t, setting t= 1 per- I1+p()]for(1+p)+1 mits us to express V(1)in terms of V (2) hich in turn can be expressed in terms tion that the remainder vanishes is of v(3)and so on up to any arbitrary only and is in no way essential to the argument substitutions. we obtain m of the two terms in(7), be finite, but this can always be safely assumed (1+)+X(0)-1(0) (7) mally taken in eacn mic theory when discussing te value of the assets of an enterprise, but much mor V(T) ty side. One of the few to apply the approach In general, the remainder term(1+)-.to the sha res as wellas the assets s Bodenorm in /I, V(r)can be expected to approach zero above 's it to derive a formula closely similar to his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 415 controvery itself thus turns out to be an empty one, the different expressions do have some intrinsic interest since, by highlighting different combinations of variables they provide additional insights into the process of valuation and they open alternative lines of attack on some of the problems of empirical testing. II. WHAT DOES THE MARKET "REALLY" CAPITALIZE? In the literature on valuation one can find at least the following four more or less distinct approaches to the valuation of shares: (1) the discounted cash flow approach; (2) the current earnings plus future investment opportunities approach; (3) the stream of dividends approach; and (4) the stream of earnings approach. To demonstrate that these approaches are, in fact, equivalent it will be helpful to begin by first going back to equation (5) and developing from it a valuation formula to serve as a point of reference and comparison. Specifically, if we assume, for simplicity, that the market rate of yield p (t) = p for all t,3 then, setting t = 0, we can rewrite (5) as V (O) 1 IX (O)-I (0) ] + 1 +p ( (6) +-- V (1). Since (5) holds for all t, setting t = 1 permits us to express V(1) in terms of V(2) which in turn can be expressed in terms of V(3) and so on up to any arbitrary terminal period T. Carrying out these substitutions, we obtain T-1 V(O) = E(l+p)t+l[X(t)I(t)] +(1+p) V(T). In general, the remainder term (1 + P)-T. V(T) can be expected to approach zero as T approaches infinity4 so that (7) can be expressed as T-1 v (O) = rnim (8) X [X(t)-I(t)], which we shall further abbreviate to c 1 V(O) = 2 (1-+ I1 [X(t)-I(t)]. (9) t- (I+ P)t The discounted cash flow approach.- Consider now the so-called discounted cash flow approach familiar in discussions of capital budgeting. There, in valuing any specific machine we discount at the market rate of interest the stream of cash receipts generated by the machine; plus any scrap or terminal value of the machine; and minus the stream of cash outlays for direct labor, materials, repairs, and capital additions. The same approach, of course, can also be applied to the firm as a whole which may be thought of in this context as simply a large, composite machine.5 This ap- 3 More general formulas in which p(t) is allowed to vary with time can always be derived from those presented here merely by substituting the cumbersome product 1L [l+p(r)] for (1+p)t+' TO0 4 The assumption that the remainder vanishes is introduced for the sake of simplicity of exposition only and is in no way essential to the argument. What is essential, of course, is that V(O), i.e., the sum of the two terms in (7), be finite, but this can always be safely assumed in economic analysis. See below, n. 14. 5 This is, in fact, the approach to valuation normally taken in economic theory when discussing the value of the assets of an enterprise, but much more rarely applied, unfortunately, to the value of the liability side. One of the few to apply the approach to the shares as well as the assets is Bodenhorn in [1], who uses it to derive a formula closely similar to (9) above. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
416 THE JOURNAL OF BUSINESS proach amounts to defining the value of investments in real assets that will yield the firm as more than the"normal"(market)rate of return. The latter opportunities, fre V(0)= 2(1+p) quently termed the“ good will’ of the (10) business, may arise, in practice, from any [(-0(4)1+①+p(T), of a number of circumstances (ranging all the way from special locational advan where R()represents the stream of cash tages to patents or other monopolistic outlays, or, abbreviating, as above, to To see how these opportunities affect the value of the business assume that in some future period t the firm invests I() v(0)f-0(1+p)i* [R(-0(01(11) dollars. Suppose, further, for simplicity, that starting in the period immediately But we also know, by definition, that following the investment of the funds [X(o-I(D]=[R(-0(0] since, X( the projects produce net profits at a con differs from R( and I( differs from stant rate of p*() per cent of I( )in each (and also by the depreciation expense if worth as of t of the (perpetual) stream of rather than gross profits and invest- the "good will "of the projects(i.e, the ment). Hence(11)is formally equivalent difference between worth and cost)will to(9), and the discounted cash flow ap- be proach is thus seen to be an implication I(PD-1(0)=I([P*(D) of the valuation principle for perfect The investment opportunities approach. ture"good will", as of now of this fu- markets given by equation(1) The present worth -Consider next the approach to valua- tion which would seem most natural I() (1+p)-(+), from the standpoint of an investor pro- posing to buy out and operate some al- and the present value of all such future ready-going concern. In estimating how opportunities is simply the sum much it would be worthwhile to pay for the privilege of operating the firm, the I() (1+p)-(+) to be paid is clearly not relevant, since the new owner can, Adding in the present value of the(uni- within wide limits, make the future divi- form perpetual)earnings, X(O), on the as dend stream whatever he pleases. For him the worth of the enterprise, as such The assumption that I() yields a uni is not restrictive in the present o will depend only on:(a) the"normal since it is always possible by rate of return he can earn by investing his capital in securities (i. e, the market the time shape of its actual ety p note also that the physical assets currently held by the the firm are behaving rationally, they will, o cogens rate of return); (b)the earning power of p"( is the auerage rate of return. If the i their cut-off criterion(cf. below p. 418) firm; and( ) the opportunities, if any, in this event we would that the firm offers for making additional mulas remain valid, however, even where"0<e his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
416 THE JOURNAL OF BUSINESS proach amounts to defining the value of the firm as T-1 V(O) = E t=O (0 P) (10) X [E (t-co() +(+p Tv (T), where IR(t) represents the stream of cash receipts and ()(t) of cash outlays, or, abbreviating, as above, to co v ( ?) = E 1+p teRW [st-(t I . ( 1 1) ,_O (1+p),+'(11 But we also know, by definition, that [X(t) -I(t)] = [IR(t) -()(t)] since, X(t) differs from IR(t) and 1(t) differs from CO(t) merely by the "cost of goods sold" (and also by the depreciation expense if we wish to interpret X(t) and I(t) as net rather than gross profits and investment). Hence (11) is formally equivalent to (9), and the discounted cash flow approach is thus seen to be an implication of the valuation principle for perfect markets given by equation (1). The investment opportunities approach. -Consider next the approach to valuation which would seem most natural from the standpoint of an investor proposing to buy out and operate some already-going concern. In estimating how much it would be worthwhile to pay for the privilege of operating the firm, the amount of dividends to be paid is clearly not relevant, since the new owner can, within wide limits, make the future dividend stream whatever he pleases. For him the worth of the enterprise, as such, will depend only on: (a) the "normal" rate of return he can earn by investing his capital in securities (i.e., the market rate of return); (b) the earning power of the physical assets currently held by the firm; and (c) the opportunities, if any, that the firm offers for making additional investments in real assets that will yield more than the "normal" (market) rate of return. The latter opportunities, frequently termed the "good will" of the business, may arise, in practice, from any of a number of circumstances (ranging all the way from special locational advantages to patents or other monopolistic advantages). To see how these opportunities affect the value of the business assume that in some future period I the firm invests 1(t) dollars. Suppose, further, for simplicity, that starting in the period immediately following the investment of the funds, the projects produce net profits at a constant rate of p*(t) per cent of I (t) in each period thereafter.6 Then the present worth as of t of the (perpetual) stream of profits generated will be I(t) p*(t)/p, and the "good will" of the projects (i.e., the difference between worth and cost) will be I(t)fP-22)-I(t) P* =1(t) [P (t) P P* The present worth as of now of this future "good will" is It P* ( ) p] (1 + p)-+ and the present value of all such future opportunities is simply the sum to P Adding in the present value of the (uniform perpetual) earnings, X(O), on the as- 8 The assumption that I(t) yields a uniform perpetuity is not restrictive in the present certainty context since it is always possible by means of simple, present-value calculations to find an equivalent uniform perpetuity for any project, whatever the time shape of its actual returns. Note also that p*(t) is the average rate of return. If the managers of the firm are behaving rationally, they will, of course, use p as their cut-off criterion (cf. below p. 418). In this event we would have p*(t) > p. The formulas remain valid, however, even where p*(t) < p. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES sets currently held, we get as an expres- The first expression is, of course sion for the value of the firm simply a geometric progression summing v(o)- X(O) to X(o/e, which is the first term of(12) +∑I() To simplify the second expression note (12) that it can be rewritten as p*(4) (1+p)-(4+) ∑({°*(∑(1+) To show that the same formula can be derived from(9) note first that our defini (1+p)-(4+1 tion of p*(t )implies the following relation between the X(t Evaluating the summation within the brackets gives x(1)=X(0)+p*(0)I(0) r()p*(t) (1+p) x()=X(t-1)+p*(t-1)I(t-1) (1+p)-(+ and by successive substitution X()=X(0)+∑p*(r)I() =∑r(2)*(4)-P(1+)-( hich is precisely the second term of Substituting the last expression for (12) X(t)in(⑨) yields la(12)has a number of reveal V(0)=[X(0)-I(0)](1+p)-1 widely used in discussions of valuation. 7 For one thing, it throws considerable light on the meaning of those much X(0)+∑p*(r)I(r) abused terms“ growth”and“ growth stocks. As can readily be seen from(12) I(4)(1+p)-((+1) a corporation does not become a"growth stockwith a high price-earnings ratio =X(0)∑(1+p)-4 I(0)(1+p)-1 glamor cae g orer s assets and earnings merely because ire growl (>p. For if p* ever large the growth in assets may be *(T)I(T)-I()the second term in(12)will be zero and X(1+p)-(+1) the firm's price-earnings ratio would not rise above a humdrum 1/p. The essence =X(0)∑(1+p)-4 of“ growth, ort, is not but the existence of opportunities to in- vest significant quantities of funds at ∑[∑?*(n)()-l(4-1) higher than"normal"rates of return f=1=0 i A valuation formula analogous to(12)though derived and interpreted X(1+p)(1+p)-(+1). is found in Bodenhorn(1]. Variants of(12)for certai special cases are discussed in Walter[201 his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 417 sets currently held, we get as an expression for the value of the firm V(O) =(O) + E I (t) P t=O (12) xP* (t) - p +P XP()----( 1 + p)-(t+l). p To show that the same formula can be derived from (9) note first that our definition of p*(t) implies the following relation between the X(t): X (1) = X (O) + p* (O) I (O), .................... X (t) = X(t -1) +p* (t -1) I(t -1) and by successive substitution t-1 X (t) = X(O) + Yd p* X () Tr=O t=1,2 ...o . Substituting the last expression for X(t) in (9) yields V(O) = [X(O)-I(O)] (1 + p) +X X(O) +Ep*(r)I (r) t = =X(O)-(O +1p)-1 (1 t1-1 I ( ___ 0 t =1 T=O X ( + p)-t) CO =X(O) f, (I1+ p) -t t =1 + Y. *T) T-It1 t =1 T=O X (+ P) +5 {12 )(t The first expression is, of course, simply a geometric progression summing to X(O)/p, which is the first term of (12). To simplify the second expression note that it can be rewritten as 1:I (t) [p*t E ( 1+ P) -T tO0 T-=t+2 - ( 1 + p)(t+)] Evaluating the summation within the brackets gives E .1(t) , I (t) [p* (t)( + p) -(t+l + t00 - (1+p)-(t+1)] = I(t (t) ]* P +p -t) which is precisely the second term of (12). Formula (12) has a number of revealing features and deserves to be more widely used in discussions of valuation.7 For one thing, it throws considerable light on the meaning of those much abused terms "growth" and "growth stocks." As can readily be seen from (12), a corporation does not become a "growth stock" with a high price-earnings ratio merely because its assets and earnings are growing over time. To enter the glamor category, it is also necessary that p*(t) > p. For if p*(t) = p, then however large the growth in assets may be, the second term in (12) will be zero and the firm's price-earnings ratio would not rise above a humdrum i/p. The essence of "growth," in short, is not expansion, but the existence of opportunities to invest significant quantities of funds at higher than "normal" rates of return. 7A valuation formula analogous to (12) though derived and interpreted in a slightly different way is found in Bodenhorn [1]. Variants of (12) for certain special cases are discussed in Walter [201. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE JOURNAL OF BUSINESS so that if p( <p ment in real assets by the firm will ac- Even without raising questions of bias tually reduce the current price of the in the coefficients, it should be apparent shares. This should help to make clear that such a conclusion is unwarranted among other things, why the "cost of since formula( 12)and the analysis un capital"to the firm is the same regard- derlying it imply only that dividends will less of how the investments are financed not count given current earnings and or how fast the firm is growing The func- growth potential. No general prediction is tion of the cost of capital in capital made (or can be made) by the the budgeting is to provide the"cut-off rate" about what will happen to the dividend n the sense of the minimum yield that coefficient if the crucial growth term investment projects must promise to be omitted. 10 worth undertaking from the point of The stream of dividends approach view of the current owners. Clearly, no From the earnings and earnings oppor proposed project would be in the interest tunliLles approach we turn next to the of the current owners if its yield were ex- dividend approach, which has, for some pected to be less than p since investing in reason, been by far the most popular one such projects would reduce the value of in the literature of valuation. This ap- their shares In the other direction, every proach too, properly formulated, is an project yielding more than p is just as entirely valid one though, of course, not learly worth undertaking since it will the only valid approach as its more en- necessarily enhance the value of the en- thusiastic proponents frequently sug- terprise. Hence, the cost of capital or cut- gest. It does, however, have the disad off criterion for investment decisions is vantage in contrast with previous ap simply p proaches of obscuring the role of dividend Finally, formula( 12)serves to em- policy. In particular, uncritical use of the phasize an important deficiency in many recent statistical studies of the effects of e serious bias problem in tests using current Durand [4, 5]). These studies typical briefly by us in [161 dividend policy(such as Walter [19] or suggesting that recent statistical studie involve fitting regression equations in have not controlled adequately for growth we do not which price is expressed as some function his tests contain an explicit "growth"variable, but ing that the dividend coefficient is sig- in general provide s%e 8) financing Furthermore, ained earnings to book value. This ratio would nificant-as is usually the caseis then the"growth"vari interpreted as a rejection of the hypothe- firms resorted to externa even if by a sample was found in whic sis that dividend policy does not affect all firms relied entirely on retained earnings, his The same conclusion could also have been tests then could not settle the question of dividend (or used external financing only in strict ty any new shares nternal financing as Gordon assumes in [8)then floated to finance investment must be priced to effects of dividend policy and investment policy ia there would be no way to distinguish between yield p; and withholding funds from the stockhold- below P of the chance to earn p on these funds by investing u See lassic statement of their dividends in other shares. The advantage of in J B. wi [21]. The equivalence thinking in terms of the cost of capital as the cut-off dend app criterion is that it minimizes the danger of confusing proaches is no pledge only in costs" with mere“outl and, by implication, in Bodenhorn [1] his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
418 THE JOURNAL OF BUSINESS Notice also that if p*(t) < p, investment in real assets by the firm will actually reduce the current price of the shares. This should help to make clear among other things, why the "cost of capital" to the firm is the same regardless of how the investments are financed or how fast the firm is growing. The function of the cost of capital in capital budgeting is to provide the "cut-off rate" in the sense of the minimum yield that investment projects must promise to be worth undertaking from the point of view of the current owners. Clearly, no proposed project would be in the interest of the current owners if its yield were expected to be less than p since investing in such projects would reduce the value of their shares. In the other direction, every project yielding more than p is just as clearly worth undertaking since it will necessarily enhance the value of the enterprise. Hence, the cost of capital or cutoff criterion for investment decisions is simply p.8 Finally, formula (12) serves to emphasize an important deficiency in many recent statistical studies of the effects of dividend policy (such as Walter [19] or Durand [4, 5]). These studies typically involve fitting regression equations in which price is expressed as some function of current earnings and dividends. A finding that the dividend coefficient is significant-as is usually the case-is then interpreted as a rejection of the hypothesis that dividend policy does not affect valuation. Even without raising questions of bias in the coefficients,9 it should be apparent that such a conclusion is unwarranted since formula (12) and the analysis underlying it imply only that dividends will not count given current earnings and growth potential. No general prediction is made (or can be made) by the theory about what will happen to the dividend coefficient if the crucial growth term is omitted."0 The stream of dividends approach.- From the earnings and earnings opportunities approach we turn next to the dividend approach, which has, for some reason, been by far the most popular one in the literature of valuation. This approach too, properly formulated, is an entirely valid one though, of course, not the only valid approach as its more enthusiastic proponents frequently suggest." It does, however, have the disadvantage in contrast with previous approaches of obscuring the role of dividend policy. In particular, uncritical use of the 8 The same conclusion could also have been reached, of course, by "costing" each particular source of capital funds. That is, since p is the going market rate of return on equity any new shares floated to finance investment must be priced to yield p; and withholding funds from the stockholders to finance investment would deprive the holders of the chance to earn p on these funds by investing their dividends in other shares. The advantage of thinking in terms of the cost of capital as the cut-off criterion is that it minimizes the danger of confusing "costs" with mere "outlays." I The serious bias problem in tests using current reported earnings as a measure of X(O) was discussed briefly by us in [161. 11 In suggesting that recent statistical studies have not controlled adequately for growth we do not mean to exempt Gordon in [81 or [9]. It is true that his tests contain an explicit "growth" variable, but it is essentially nothing more than the ratio of retained earnings to book value. This ratio would not in general provide an acceptable approximation to the "growth" variable of (12) in any sample in which firms resorted to external financing. Furthermore, even if by some chance a sample was found in which all firms relied entirely on retained earnings, his tests then could not settle the question of dividend policy. For if all firms financed investment internally (or used external financing only in strict proportion to internal financing as Gordon assumes in [81) then there would be no way to distinguish between the effects of dividend policy and investment policy (see below p. 424). 11 See, e.g., the classic statement of the position in J. B. Williams [211. The equivalence of the dividend approach to many of the other standard approaches is noted to our knowledge only in our [16] and, by implication, in Bodenhorn [1]. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES dividend approach has often led to the The summation term in the last expres- unwarranted inference that since the sion can be written as the difference be vestor is buying dividends and since div- tween the stream of dividends accruing idend policy affects the amount of divi- to all the shares of record as of t+ 1 and dends, then dividend policy must also that portion of the stream that will ac- affect the current price. crue to the sharesnewly issued in t, thatis, Properly formulated, the dividend proach defines the current worth of yD4(4++1)=(1-m(+1) share as the discounted value of the =(1+p)+I dividends to be paid on the (16) share in perpetuity. That is D4+1(t+T r+12 力(t)= d(t+ t) ×∑-(1+p) But from(14)we know that the second To see the equivalence between this summation in( 16)is precisely V(t+ 1) proach and previous ones, let us first so that(15)can be reduced to restate( 13)in terms of total market alue as (°+n[D V()= D(+r) (1+p)r (14) m(计+1)(+1) n(+1)(+1) re D(t+r)denotes that portion (17) the total dividends D(t+ r) paid during V(t+1) period t+T, that accrues to the shares of record as of the start of period t(indi 1+p [D(4)+v(+1) cated by the subscript). That equation (14)is equivalent to(9)and hence also m(+1)(+1)], which is(3)and which has already special case in which no outside financing shown to imply both(9)and(12).12 is undertaken after period 4, for in that There are, of course, other ways in which the equivalence of the dividend D(t+T)=D(t+T) approach to the other approaches might X(+t)-I(t+T) To allow for outside financing, note that (14)are equivalent must be qualified to allow for we can rewrite(14)as certain pathological cases, fortunately of ficance. An obyious exa such a case is the legendary company that is expect V(4)= D (t) ed never to pay a dividend. If this were literally true then the value of the firm by( 14)would be zero; by +∑:(4+) dividends rule out X(o>I( but not x(0 per cent of its earnings, however small the value of t his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 04: 42 AM All use subject to JSTOR Terms and Conditions
THE VALUATION OF SHARES 419 dividend approach has often led to the unwarranted inference that, since the investor is buying dividends and since dividend policy affects the amount of dividends, then dividend policy must also affect the current price. Properly formulated, the dividend approach defines the current worth of a share as the discounted value of the stream of dividends to be paid on the share in perpetuity. That is co p (t) = 1 d +(13) -r=o( 1 + P ) 7+1 To see the equivalence between this approach and previous ones, let us first restate (13) in terms of total market value as V (t)- (t + ) V(t) =2. 14) (+ I p ) +1'(4 where Dt(t + r) denotes that portion of the total dividends D(t + r) paid during period t + r, that accrues to the shares of record as of the start of period t (indicated by the subscript). That equation (14) is equivalent to (9) and hence also to (12) is immediately apparent for the special case in which no outside financing is undertaken after period t, for in that case -I(t+=r). -X(t+r) To allow for outside financing, note that we can rewrite (14) as V(t) D(t) 1 + P [ ( + E DtcoD(t +1 +E ( 1 +p)7~~~~~~ The summation term in the last expression can be written as the difference between the stream of dividends accruing to all the shares of record as of t + 1 and that portion of the stream that will accrue to the shares newly issued in t, that is, 1:Dt (t+ r+ 1) I m (t+ 1)0 (16) c Dt+l (t+T+ 1) X ( I1 + p)rl+ But from (14) we know that the second summation in (16) is precisely V(t + 1) so that (15) can be reduced to V(t) =_ l [D (t) [D(o+l)V(t+ 1)> I(t) but not X(t) 0 per cent of its earnings, however small the value of e. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:04:42 AM All use subject to JSTOR Terms and Conditions
