Corporate Income Taxes and the Cost of Capital: A Correction

American Economic AssociationCorporate Income Taxes and the Cost of Capital: A CorrectionAuthor(s): Franco Modigliani and Merton H. MillerSource: The American Economic Review, Vol. 53, No. 3 (Jun., 1963), pp. 433443 Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1809167Accessed: 10/09/2009 09:53Your use of the JSTOR archive indicates your acceptance of JSTOR’s Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR’s Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. 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For more information about JSTOR, please contact support@jstor.org.American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to TheAmerican Economic Review.http://www.jstor.orgCOMMUNICATIONS433equanimity a writing-down of the value of their reserves, or unless one isprepared to forego the possibility of exchange-rate adjustment, any majorextension of the gold exchange standard is dependent upon the introductionof guarantees. It is misleading to suggest that the multiple key-currency system is an alternative to a guarantee, as implied by Roosa [6, pp. 5-7 and 912 ].IV. ConclusionThe most noteworthy conclusion to be drawn from this analysis is that thesuccessful operation of a multiple key-currency system would require bothexchange guarantees and continuing cooperation between central bankers ofa type that would effectively limit their choice as to the form in which theyhold their reserves. Yet these are two of the conditions whose undesirabilityhas frequently been held to be an obstacle to implementation of the alternative proposal to create a world central bank. The multiple key-currency proposal represents an attempt to avoid the impracticality supposedly associatedwith a world central bank, but if both proposals in fact depend on the fulfillment of similar conditions, it is difficult to convince oneself that the sacrificeof the additional liquidity that an almost closed system would permit isworth while. Unless, of course, the object of the exercise is to reinforcediscipline rather than to expand liquidity.JOHN WILLIAMSON*REFERENCES1. R. Z. ALIBER, “Foreign Exchange Guarantees and the Dollar: Comment,”Am. Econ. Rev., Dec. 1962, 52, 1112-16.2. S. T. BEZA AND G. PATTERSON, “Foreign Exchange Guarantees and theDol lar,” Am. Econ. Rev., June 1961, 51, 381-85.3.—AND—, “Foreign Exchange Guarantees and the Dollar:Reply,”Am. Econ. Rev., Dec. 1962 , 52, 1117-18.4. F. A. LuTz, The Problem of International Equilibrium. Amsterdam 1962.5. R. NuRKSE, International Currency Experience. Geneva 1944.6. R. V. RoosA, “Assuring the Free World’s Liquidity,” Business ReviewSup plement, Federal Reserve Bank of Philadelphia, Sept. 1962.* The author is instructor in economics at Princeton University. He acknowledges thehelpful comments of Fritz Machlup. Views expressed are those of the author alone.Corporate Income Taxes and the Cost of Capital:A CorrectionThe purpose of this commu nication is to correct an error in our paper”The Cost of Capital, Corporation Finance and the Theory of Investment”(this Review, June 1958). In our discussion of the effects of the presentmethod of taxing corporations on the valuation of firms, we said (p. 272) :The deduction of interest in computing taxable corporate profits willprevent the arbitrage process from making the value of all firms in agiven class proportional to the expected returns generated by their434THE AMERICAN ECONOMIC REVIEWphysical assets. Instead, it can be shown (by the same type of proofused for the original version of Proposition I) that the market valuesof firms in each class must be pro portional in equilibrium to their expected returns net of taxes ( that is, to the sum of the interest paid andexpected net stockholder income). (Italics added.)The statement in italics, unfortunately, is wrong. For even though onefirm may have an expected return af ter taxes (our X’ ) twice that of anotherfirm in the same risk-equivalent class, it will not be the case that the actualreturn af ter taxes (our X’) of the first firm will always be twice that of thesecond, if the two firms have different degrees of leverage.1 And since thedistribution of returns af ter taxes of the two firms will not be proportional,there can be no “arbitrage” process which forces their values to be proportional to their expected af ter-tax returns.2 In fact, it can be shown-andthis time it really will be shown-that “arbitrage” will make values withinany class a function not only of expected af ter-tax returns, but of the taxrate and the degree of leverage. This means, among other things, that thetax advantages of debt financing are somewhat greater than we originallysuggested and, to this extent, the quantitative difference between the valuations implied by our position and by the traditional view is narrowed. Itstill remains true, however, that under our analysis the tax advantages ofdebt are the only permanent advantages so that the gulf between the twoviews in matters of interpretation and policy is as wide as ever.I. Taxes, Leverage, and the Probabilit y Distribution of After-T ax ReturnsTo see how the distribution of af ter-tax earnings is affected by leverage,let us again denote by the random variable X the (long-run average) earnings before interest and taxes generated by the currently owned assets of agiven firm in some stated risk class, k.3 From our definition of a risk class itfollows that X can be expressed in the form X Z , where X is the expectedvalue of X , and the random variable Z =X / X , having the same value forall firms in class k , is a drawing from a distribution, say fk( Z). Hence the1 With some exceptions, which will be noted when they occur, we shall preserve here boththe notation and the terminology of the original paper. A working knowledge of both on the partof the reader will be presumed.2 Barring, of course, the trivial case of universal linear utility functions. Note that in deference to Professor Durand (see his Comment on our paper and our reply, this Review, Sept.1959,49, 639-69) we here and throughout use quotation marks when referring to arbitrage.3 Thus our X corresponds essentially to the familiar EBIT concept of the financeliterature. The use of EBIT and related “income” concepts as the basis of valuation isstrictly valid only when the underlying real assets are assumed to have perpetual lives. Insuch a case, of course, EBIT and “cash flow” are one and the same. This was, in effect, theinterpretation of X we used in the original paper and we shall retain it here both to preservecontinuity and for the considerable simplification it permits in the exposition. We shouldpoint out, however, that the perpetuity interpretation is much less restrictive than mightappear at first glance. Before tax cash flow and EBIT can also safely be equated evenwhere assets have finite lives as soon as these assets attain a steady state age distribution inwhich annual replacements equal annual depreciation. The subject of finite lives of assetswill be further discussed in connection with the problem of the cut-off rate for investmentdecisions.COMMUNICATIONS435random variable XT , measuring the af ter-tax return, can be expressed as:(1) XT= (1 -T)( X – R)+ R = (1 -T) X + TR = (1 – T)X Z+TRwhere T is the marginal corporate income tax rate (assumed equal to theaverage), and R is the interest bill. Since E( XT ) =XT = (1-T)X+TR we cansubstitute XT -TR for (1-T)X in (1) to obtain:(2)Thus, if the tax rate is other than zero, the shape of the distribution of XTwill depend not only on the “scale” of the stream XT and on the distributionof Z, but also on the tax rate and the degree of leverage (one measure ofwhich is R/ XT ). For example, if Var (Z) = u2 , we have:Var ( XT ) = u2 (XT) 2 (1 – T T Yimplying that for given XT the variance of af ter-tax returns is smaller, thehigher T and the degree of leverage.4II. The Valuation of After-Tax ReturnsNote from equation (1) that, from the investor’s point of view, the long runaverage stream of af ter-tax returns appears as a sum of two com ponents: (1)an uncertain stream (1-T) XZ ; and (2) a sure stream TR.6 This suggeststhat the equilibrium market value of the combined stream can be found bycapitalizing each component separately. More precisely, let pT be the rate atwhich the market capitalizes the expected returns net of tax of anunlevered company of size X in class k , i.e.,PT =(1 T)XorVu =(1 – T) XPT;6Vu’ It may seem paradoxical at first to say that leverage reduces the variability ofoutcomes, but remember we are here discussing the variability of total returns, interest plusnet profits. The variability of stockholder net profits will, of course, be greater in thepresence than in the absence of leverage, though relatively less so than in an otherwisecomparable world of no taxes. The reasons for this will become clearer after the discussionin the next section.5 The statement that TR-the tax saving per period on the interest payments-is a surestream is subject to two qualifications. First, it must be the case that firms can always obtainthe tax benefit of their interest deductions either by offsetting them directly against othertaxable income in the year incurred; or, in the event no such income is available in any givenyear, by carrying them backward or forward against past or future taxable earnings; or, in theextreme case, by merger of the firm with (or its sale to) another firm that can utilize the deduction. Second, it must be assumed that the tax rate will remain the same. To the extent thatneither of these conditions holds exactly then some uncertainty attaches even to the taxsavings, though, of course, it is of a different kind and order from that attaching to the streamgenerated by the assets. For simplicity, however, we shall here ignore these possible elementsof delay or of uncertainty in the tax saving; but it should be kept in mind that this neglectmeans that the subsequent valuation formulas overstate, if anything, the value of the taxsaving for any given permanent level of debt.6 Note that here, as in our original paper, we neglect dividend policy and “growth” in the436THE AMERICAN ECONOMIC REVIEWand let r be the rate at which the market capitalizes the sure streams generated by debts. For simplicity, assume this rate of interest is a constantindependent of the size of the debt so thatRr = – orDDR7= -·rThen we would expect the value of a levered firm of size X , with a permanent level of debt DL in its capital structure, to be given by:(1 – T)X-+(3)= Vu + T DL.8TRrPTIn our original paper we asserted instead that, within a risk class, marketvalue would be proportional to expected af ter-tax return X’ (cf . our originalequation [11]), which would imply:X'(4)VL = -=p’TRr+– = Vu + -T(1 – T) Xp’p’p’We will now show that if (3) does not hold, investors can secure a moreefficient portfolio by switching from relatively overvalued to relativelyundervalued firms. Suppose first that unlevered firms are overvalued or thatVL – T DL < Vu.An investor holding m dollars of stock in the unlevered company has a rightto the fraction m/ Vu of the eventual outcome, i.e., has the uncertain incomeYu = (;:) (1 - T) X Z.Consider now an alternative portfolio obtained by investing m dollars asfollows: (1) the portion,is invested in the stock of the levered firm, SL; and (2) the remaining portion,sense of opportunities to invest at a rate of return greater than the market rate of return.These subjects are treated extensively in our paper, "Dividend Policy, Growth and theValuation of Shares," Jour. Bus., Univ. Chicago, Oct. 1961, 411-33.7 Here and throughout, the corresponding formulas when the rate of interest rises with leverage can be obtained merely by substituting r(L) for r , where L is some suitable measure ofleverage.8 The assumption that the debt is permanent is not necessary for the analysis. Itis employedhere both to maintain continuity with the original model and because it gives an upper boundon the value of the tax saving. See in this connection footnote 5 and footnote 9.COMMUNICATIONS437is invested in its bonds. The stock component entitles the holder to a fraction,mSL + (1 - r) DLof the net profits of the levered company orm) [(1 - r)( X Z (RL) ] . S L + (1 - r) DLThe holding of bonds yieldsHence the total outcome isand this will dominate the uncertain income Yu if (and only if)SL+ (1 -r) DL= SL + DL -r DL= VL -r DL< Vu.Thus, in equilibrium, Vu cannot exceed VL-T DL, for if it did investorswould have an incentive to sell shares in the unlevered company and purchase the shares (and bond s) of the levered company.Suppose now that VL-r DL> Vu. An investment of m dollars in the stockof the levered firm entitles the holder to the outcomeYL= (m/SL) [(l = (m/SL) (l -r)( X Z – RL) ]r) X Z – (m/SL) (l – r)RL,Consider the following alternative portfolio: (1) borrow an amount ( m/SL)( l -r) DL for which the interest cost will be ( m/ SL)( l -r)RL (assuming,of course, that individuals and corporations can borrow at the same rate, r) ; and (2) invest m plus the amount borrowed, i.e.,m +m( l – r) DLDL SL=msL + ( l – r) DLSL[= (m/SL) VL – Tin the stock of the unlevered firm. The outcome so secured will beSubtracting the interest charges on the borrowed funds leaves an income ofwhich will dominate YL if (and only if) VL-T DL> Vu. Thus, inequilibrium, both VL-r DL> Vu and VL-r DL < Vu are ruled out and (3)must hold.43STHE AMERICAN ECONOMIC REVIEWIII. Some Implications of Formula (3)To see what is involved in replacing (4) with (3) as the rule of valuation,note first that both expressions make the value of the firm a function ofleverage and the tax rate. The diff erence between them is a matter of thesize and source of the tax advantages of debt financing. Under our originalformulation, values within a class were strictly proportional to expectedearnings af ter taxes. Hence the tax advantage of debt was due solely to thefact that the deductibility of interest payments implied a higher level ofaf ter-tax income for any given level of before-tax earnings (i.e., higher bythe amount TR since XT= (1-T)X+TR). Under the corrected rule (3), however, there is an additional gain due to the fact that the extra af ter-taxearnings, TR, represent a sure income in contrast to the uncertain outcome(1-T)X. Hence TR is capitalized at the more favorable certainty rate,1/r,rather than at the rate for uncertain streams, 1/pT.9Since the difference between (3) and (4) is solely a matter of the rate atwhich the tax savings on interest payments are capitalized, the requiredchanges in all formulas and expressions derived from (4) are reasonablystraightforward. Consider, first, the before-tax earnings yield, i.e., the ratioof expected earnings before interest and taxes to the value of the firm. 10Dividing both sides of (3) by V and by (1-T) and simplifying we obtain:(31.c)XV= _f_1-[1 - DJTTVwhich replaces our original equation (31) (p. 294). The new relation differsfrom the old in that the coefficient of D/ V in the original (31) was smallerby a factor of r/pConsider next the af ter-tax earnings yield, i.e., the ratio of interest payments plus profits af ter taxes to total market value.11 This concept was discussed extensively in our paper because it helps to bring out more clearlythe differences between our position and the traditional view, and becauseit facilitates the construction of empirical tests of the two hypotheses aboutthe valuation process. To see what the new equation (3) implies for thisyield we need merely substitute XT -TR for (1-T)X in (3) obtaining:9 Remember, however, that in one sense formula (3) gives only an upper bound on the valueof the firm since TR/r=rD is an exact measure of the value of the tax saving only where boththe tax rate and the level of debt are assumed to be fixed forever (and where the firm is cer tainto be able to use its interest deduction to reduce taxable income either directly or via transfer ofthe loss to another firm). Alternative versions of (3) can readily be developed for cases in whichthe debt is not assumed to be permanent, but rather to be outstanding only for somespecified finite length of time. For reasons of space, we shall not pursue this line of inquiry herebeyond observing that the shorter the debt period considered, the closer does the valuationformula approach our original (4). Hence, the latter is perhaps still of some interest if only asa lower bound.1 Following usage common in the field of finance we referred to this yield as the "averagecost of capital." We feel now, however, that the term "before-tax earnings yield" would be preferable both because it is more immediately descriptive and because it releases the term "costof capital" for use in discussions of optimal investment policy (in accord with standard usagein the capital budgeting literature).11 We referred to this yield as the "after-tax cost of capital." Cf. the previous footnote.°(5)vxr =COMMUNICATIONSTRp'439xr+ T D = - + T -p'--p'rD,p'from which it follows that the af ter-tax earnings yield must be:x,V(11.c)= p' - T ( p' - r) D/ V.This replaces our original equation (11) (p. 272) in which we had simplyX'/ V = p'. Thus, in contrast to our earlier result, the corrected version(11.c) implies that even the af ter-tax yield is affected by leverage. Thepredicted rate of decrease of X'/ V with D/ V, however, is still considerablysmaller than under the naive traditional view, which, as we showed, impliedessentially X'/ V =p '-(p '-r)D/ V. See our equation (17) and the discussionimmediately preceding it (p. 277) .12 And, of course, ( 11.c) implies that theeffect of leverage on X'/ V is solely a matter of the deductibility of interestpayments whereas, under the traditional view, going into debt would lowerthe cost of capital regardless of the method of taxing corporate earnings.Finally, we have the matter of the af ter-tax yield on equity capital, i.e.,the ratio of net profits af ter taxes to the value of the shares.13 By subtracting D from both sides of (5) and breaking X' into its two componentsexpected ter simplif ying:obtain af net profits af ter taxes, fr', and interest payments, R =r D-we(6)S = V - D =- fr'p'- (1 - T )(p' -')D.p'From (6) it follows that the af ter-tax yield on equity capital must be:fr'(12.c)-s= p'+(1 - T) [ p' - r ] D/ Swhich replaces our original equation (12), fr '/S =p '+(p'-r)D/ S (p. 272).The new (12.c) implies an increase in the af ter-tax yield on equitycapital as leverage increases which is smaller than that of our original(12) by a factor of (1-T) . But again, the linear increasing relation of thecorrected (12.c) is still fundamentally diff erent from the naivetraditional view which asserts the cost of equity capital to becompletely independent of leverage (at least as long as leverageremains within "conventional" industry limits).IV. Taxes and the Cost of CapitalFrom these corrected valuation formulas we can readily derive correctedmeasures of the cost of capital in the capital budgeting sense of the minimum prospective yield an investment project must offer to be just worth12 The ik* of (17) is the same as p' in the present context, each measuring the ratio of netprofits to the value of the shares (and hence of the whole firm) in an unlevered company ofthe class.13 We referred to this yield as the "after-tax cost of equity capital." Cf. footnote 9.THE AMERICAN ECONOMIC REVIEW440undertaking from the standpoint of the present stockholders. If we interpret earnings streams as perpetuities, as we did in the original paper, thenwe actually have two equally good ways of defining this minimu m yield:either by the required increase in before-tax earnings, d X , or by the required increase in earnings net of taxes, dX(l -r ).14 To conserve space,however, as well as to maintain continuity with the original paper, weshall concentrate here on the before-tax case with only brief footnoterefer ences to the net-of-tax concept.Analytically, the derivation of the cost of capital in the above senseamounts to finding the minimum value of d X / dl for which dV =dl,where I denotes the level of new invest ment.15 By differentiating (3) wesee that:(7)1dV1- rdXdD-= -- -+r ->dlp’dld){dD1 – r dlif — >df -dl —— p’.1-THence the before tax required rate of return cannot be defined withoutref erence to financial policy. In particular, for an investment considered asbeing financed entirely by new equity capital d D/ dl=O and the requiredrate of return or marginal cost of equity financing (neglecting flotationcosts) would be:prPS =— .1-TThis result is the same as that in the original paper (see equation [32], p.294) and is applicable to any other sources of financing where the remuneration to the suppliers of capital is not deductible for tax purposes. It applies,therefore, to pref erred stock (except for certain partially deductible issuesof public utilities) and would apply also to retained earnings were it notfor the favorable tax treatment of capital gains under the personal incometax.For investments considered as being financed entirely by new debt capitaldl = d D and we find from (7) that :(33.c)which replaces our original equation (33) in which we had:(33)T— r.1- TPD = PS -14 Note that we use the term “earnings net of taxes” rather than “earnings after taxes.”We feel that to avoid confusion the latter term should be reserved to describe what willactually appear in the firm’s accounting statements, namely the net cash flow including thetax savings on the interest (our X’). Since financing sources cannot in general be allocated toparticular investments (see below), the after-tax or accounting concept is not useful for capitalbudgeting purposes, although it can be extremely useful for valuation equations as we saw inthe previous section.ia Remember that when we speak of the minimum required yield on an investment we arereferring in principle only to investments which increase the scale of the firm. That is, the newCOMMUNICATIONS441Thus for borrowed funds (or any other tax-deductible source of capital) themarginal cost or before-tax required rate of return is simply the marketrate of capitalization for net of tax unlevered streams and is thus independ entof both the tax rate and the interest rate. This required rate is lower thanthat implied by our original (33), but still considerably higher than thatimplied by the traditional view (see esp. pp. 276-77 of our paper) underwhich the before-tax cost of borrowed funds is simply the interest rate, r.Having derived the above expressions for the marginal costs of debt andequity financing it may be well to warn readers at this point that these expressions represent at best only the hypothetical extremes insofar as costsare concerned and that neither is directly usable as a cut-off criterion forinvestment planning. In particular, care must be taken to avoid falling intothe famous “Liquigas” fallacy of concluding that if a firm intends to float abond issue in some given year then its cut-off rate should be set that yearat pD ; while, if the next issue is to be an equity one, the cut-off is p 8• Thepoint is, of course, that no investment can meaningfully be regarded as 100per…