Wealth AndCoC

8/8/2019 Wealth AndCoC

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WEALTH MAXIMIZATION AND THE COST OFCAPITAL*

A L A N J. A U E R B A C HI. Introduction, 433.II. The model, 434.III. The prohlem, 435.IV. Wealthmaxim ization: no personal taxes, 436.V. Wealth maximization with personal taxes,438.VI. Firm value and the cost of capital, 440.VII. Financial policy and the costof capital, 442.VIII. Conclusion, 445.

I. I N T R O D U C T I O NIn a simple world of certainty, with perfect capital mark ets, notaxes on capital income, and all investment financed through directownership, a utility-maximizing investor would strive to maximizetbe present value of his investment. Tbis wealth m aximization wouldbe achieved by the acceptance of all investment projects baving apositive p resen t value wben discounted at tbe individual's personalrate of time preference.Once the possibility of corporate finance is introduced, compli-cations arise concerning the optimal choice of financing m ethod andthe appropriate discount rate to use in presen t value calculations. Intbe absence of taxation certain principles are generally accepted. First,it is irrelevant whether equity-source investment funds come fromreta ined earnings or the sale of new shares. Second, firms should usea composite "cost of capita l" in their discounting decisions, a weightedaverage of the interest rate on debt and the rate of time preferenceof stockbolders. Finally, if the cost of capital varies with the degreeof deb t finance, or "leverage," ^ firms should choose the debt-equityratio for which it is minimized. While many authors have arrived attbese results sta rting from tbe objective of wealtb maximization, theexact relationship rem ains somew hat unclear. Inaddition, tbere basbeen considerable d ebate over tbe im pact of corporate and personalincome taxes on firm policy.^In this paper we explore the issue of wealtb m aximization andthe implied behavior of the firm, paying particular attention to the


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434 QUARTERLY JOURNAL OF ECONOMICSof capital income taxes. Our results indicate that a tax structursimilar to that in existence in the United States influences the cosof capital in a very different way than has been assumed previousland that the relative advantages of debt over equity as a method ofinance, and capital gains over dividends as a vehicle for personarealization of corporate profits, may have been greatly overstatedThese findings may help to explain certain aspects of corporate fnancial behavior that have seemed puzzling.

II. THE MODELWe consider the behavior of competitive firms in a discrete-tim

infinite-horizon model. These firms finance their investment projecthrough sales of common stock, retention of earnings, and sales odebt, which, for the sake of simplicity, we take to have a term of onperiod. At the beginning of each period, firms distribute dividendto pre-existing shareholders, pay interest, and repay principal on deboutstanding in the previous period, and sell new shares, ex dividendCorporate profits are taxed at rate r, but interest payments may bdeducted from taxable profits. Dividends are taxed at rate 6 at thpersonal level. We assume that capital gains are taxed upon accruaat a rate c


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WEALTH MAXIMIZATION A ND THE CO ST OF CAPITAL 435If we let xi represent the cash flow of the firm at the beginning ofperiod t, net of corporate tax, but before account is taken of net salesof debt and equity, interest payments, and the tax savings on suchpayments, then dividends paid at the beginning of period t +1 (i.e.,the end of period t) on shares held during period t are(4) Dt [ ]By the very definition of dividends, D must always be nonnegative.

III. THE PROBLEMWe assume that firms seek to maximize the wealth of existing

shareholders. This is accomplished by choosing, at the beginning ofperiod t, an investment policy, represented by the vector x =...),& debt policy, B = (B^, St+1,...), and an equity policy,Vfti, . . . ) , ' ' which, among feasible choices, maximize the value ofshares owned as of the beginning of period t, including the concurrentafter-tax distribution.

Before proceeding any further, we must derive an expression forthe valuation of equity. We assume that shares are equal in value tothe present discounted value of the distributions their owners receive.At the end of period t, the total dividends paid to all stockholders areDt. However, if there are personal taxes, the net distribution issmaller. The personal tax liability at the end of period t is the tax ondividends ODt, plus that on accrued capital gains on existing stock,c( Vf+i Vt). Thus, the net distribution received by all shareholdersat the end of period t is(5) Et = {l-O)Dt-c(Vf ,-Vt).

If new equity is issued between period tand a subsequent period.s, not all of the distribution E^ will go to shares held as of period t. Afraction will go to the shares arising from the new issues. The fractionof shares held during period s that were in existence before the startof period t is, from (2),(6) M-; = ( l - 5 , ) ( l - 5 , + i ) . . . ( l - 5 , ) .


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436 QUARTERLY JOURNAL OFECONOMICSUsing (1) and (2), andapplying (7) for successive values of t, we obtain(8) PtVt =E,-^-{VI, - Vt).Equation (8) states that, for all t, the one-period holding yield oequity, which includes the net distribution and capital gain, musequal the rate of time preference associated with equity holding p

The wealth of existing stockholders that the firm seeks tomaximize is(9) Wf = V? + Et-i.Using (1), (4), and (5), we may rewrite this expression as(10) W?={l-c)Vt + {l-d)

X{Bt-[l + it-i{l-T)]Bt-i + Xt\- {d - c)V^ + cVt-i.

Since it-i, Vt-i, and B^-i are predetermined at the beginning operiod t, W? is maximized if and only if the firm maximizes(11) W: = { 1 - c ) V t + ( 1 - e ) B t + ( 1 - d)xt - { 6 -

We are now ready to examine the characteristics of wealthmaximizing behavior. Because of the complexity of the generproblem, we attack first the simpler case in which 6 = c = 0.

IV. WEALTH MAXIMIZATION: NO PERSONAL TAXESIn this situation the firm's objective function (11) reduces to

(12) W: = Vt + Bt+Xt.Firms desire to maximize the sum of their securities' market value ancurrent cash flow. This is a standard result.

To simplify the exposition in this and succeeding sections, wintroduce the term,


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WEALTH MAXIMIZATION AND THE COST OF CAPITAL 437Solving for Vt, we obtain^

n (1 + Pz)-'Thus, as first argued by Miller and Modigliani [1961], the particularsource of equity funds used to finance investment has no impact onthe firm valuation. Once x and B are determined, so is Vt, since F is,by definition, independent of V . Because of this result, we may ig-nore the question of share sales and repurchases when consideringthe question of wealth maximization in the absence of personaltaxes.

Using expressions (1) and (13) to substitute into (14), and thenrearranging terms, we obtain an expression for shareholder wealth:(16) W: = Xt + {1 + rt)-^WUi,where we define the term,(17) rt = btit{l - T)-i-{I - bt)pt,to be the "cost of capital," an average of the returns the firm must earnon equity holdings and debt, weighted by their respective portionsin the firm financial structure. Because of the deductibility of interestpayments, the net cost of debt is t((l r), rather than the gross in-terest rate.

Solving (16) for W *, we obtain^(18) a- In

Shareholder wealth may be expressed as the present value of thestream of after-tax corporate cash flows, where the financial policydetermines the appropriate discount rate.

It is clear that the cost of capital, defined above, determines thecutoff point for investment projects. Given the choice of financialpolicy and, hence, the value of r = (r , rt-n,. ..), the firm will increaseW't by accepting those projects having positive present value whenthe cost of capital in each period is used as the discount rate.Up until this point we have said very little about the determi-


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438 QUARTERLY JOURNAL OF ECONOMICS

nation of p and i, or about the place of risk in our model. A completelyadequate treatment of risk is beyond tbe scope of tbis paper. It is evenquestionable wbetber firms sbould always seek to maximize sbarebolder wealtb wben tbere is uncertainty as to tbe particular state onature tbat will occur in eacb period. One fairly simple, albeit imperfect, approacb is to assume tbat tbe market evaluation of a firm'"riskiness" increases witb tbe degree of leverage, and tbat tbe requirerates of return p and i must increase;'' tbat is,(19) Pt = p{bt), it = i{bt), p',i'>0.If we assume tbat pt and it are determined in tbis way, it follows from(17) and (18) tbat, for any given stream x, tbe firm maximizes W't bcboosing 6,, for eacb period s >t to minimize tbe cost of capital. Tbustbe overall wealth maximization, in this case, may be viewed as two-stage procedure in wbicb firms first determine tbe financial polictbat minimizes tbe cost of capital faced in eacb period, and tben ustbis cost of capital as tbe discount rate in determining tbe optimainvestment strategy.

V. WEALTH MAXIMIZATION WITH PERSONAL TAXESA more interesting, and more difficult, problem arises once w

remove tbe assumption tbat 0 = c = 0. Many of tbe familiar resultof tbe previous case must be altered to take account of tbe existencof personal taxes.

Tbe first difference, seen in tbe definition of W* (in equatio(11)) is tbat tbe firm sbould not strive to maximize tbe sum of currencasb flow and tbe market value of its securities, unless 0 = c. Tbicondition is, in our model, equivalent to tbe integration of corporatand personal income taxes, since all corporate source income, wbetberetained or distributed, would be taxed at tbe same rate; T' = T += T +C.

Following tbe same procedure used to obtain equation (15), wget, after various substitutions.

0 -c


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WEALTH MAXIMIZATION AND THE CO ST OF CAPITAL 439of X and B , and hence F , a decrease in the n et issue of new equity, V^,at any tim e s>t, increases Vt and hence W], since d> c. Th us , it willnever be optim al to issue new shares and pay dividends at th e sametime, since Vt may he increased hy an equal decrease in Ds-i andyN' .s * The same logic would compel firms to continue decreasing V^helow zero, repurchasing equ ity, as long as dividends were positive,if such hehavior were not otherwise h indered. T his dominance of sharerepurchases over dividends as a method of distrihution has heenrecognized in the past hy many authors.^ However, share repurchases

have never constituted an imp ortant pa rt of firm distrihutions. Pa rtof the answer prohahly lies in Section 302 of the Internal RevenueCode, which proh ihits firms from repurchasing shares in lieu of dis-tributing dividends.Recent U . S. da ta confirm that new share issues have constitutedonly a small portion of new equity finance, and th at repurchases havebeen a very unimportant method of distribution, compared to divi-dend s. For example, in 1976, ne t issues of de bt by U. S. nonfinancialcorporations amounted to $47.8 billion, retained earnings were $44.3billion, and dividends were $32.2 billion. Net issues of equity were only$10.5 billion,^ and share repu rchases were, in tu rn , only a small per-centage of this number.^*'Given the above discussion, we shall henceforth assume thatfirms neither issue new shares nor repurchase existing ones. Whileno t greatly abstrac ting from reality, this restriction con stitutes a veryhelpful simplification. Assuming that Vf = 0, we see that (20) be-comes(21) Vt=Y. " " - " ' ' ''^1 - c 1 - cFollowing the same procedure used in deriving (18), we ohtain(22) w: = {i-e)xt+ Y. f ' n a + '- j

s = t+l \z = twhere the cost of capital is


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(23) rt = [(1 - c) - {6 - c)bt]-^btit{l - T ) ( 1 - 0) H- (1 - bt)pt]As in the case without personal taxes, the firm maximizes wealthgiven financial policy, by choosing all projects that have a positivepresent value when discounted by the cost of capital, and if i and pare determined by (19), should choose the degree of leverage thaminimizes the cost of capital.

The formula for the cost of capital given in (23) differs from results obtained previously, and there are a numher of interesting observations to be made concerning this expression. If the firm is financed solely through debt, then r reduces to the net of tax rate ofinterest, t(l r), which the firm must pay on its debt. At the othextreme, an equity-financed firm faces a cost of capital of p/(l cThere are two rather surprising aspects of this result. First, the cosof capital does not depend on the dividend payout rate. Second, thrate of dividend taxation does not enter at all. Equity gains are effectively taxed at rate c, which may be very small compared to thestatutory rate of taxation of dividends. Even for the general case inwhich there is both debt and equity in the firm, the payout rate doenot directly influence the cost of capital.The next section contains an example that should aid in theunderstanding of the above results.

VI. FIRM VALUE AND THE COS T OF CAPITALIn this section we consider the particular case of an economy in

which there is no inflation, in which i, p, b, and hence r, are all constant, and assume that firms produce output using homogeneousnondepreciating capital and labor, subject to a constant-returnsto-scale technology. These assumptions are made to facilitate thexposition and do not influence the characteristics of the results.

The investment in an additional unit of capital at any given timt will decrease concurrent cash flow hy the purchase price, and increase the cash flow in each succeeding period by the marginal producof such capital, less corporate taxes, /'(I T ) . Such a project increas^*t by [/'(I T)/r] 1. Perfect competition ensures that firms w


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WEALTH MAXIMIZATION AND THE COST OF CAPITAL 441t if no new investment occurs is, by Euler's Theorem , / '( I T)Kt-\,where Kt-i is the capital on hand during period t 1. The net cashflow at the heginning of period t is therefore this amount less newinvestment:(25) Xt='{l-r)K-i-{K-K-i).Using (22), (24), and (25), we ob ta in(26) W\ = {l- s = t - l

= {1 -e)[xt + KtiSu bs titu ting (26) into (11), we obtain(27) V i

\ 1 -cwhich is similar to results ohtained by Auerbach [1978] for the caseof all equity finance and Bradford [1978] for the case in which T = c= 0. Since 6 > c, the market value of equity is lower than the repro-duction cost of its capital stock, less the market value of its debt. Inthe language of Tobin [1969], "q" is less than unity.

Since this capitalization occurs, the presumed superiority ofcapital gains over dividends disappears, and the effective tax on equityincome is lower than a weighted average of the statutory taxes ondividends and capital gains. For example, imagine an all-equity firmconsidering taking one dollar out of current dividends for reinvest-ment. The net loss to stockholders in the current period is (1 0),since their dividend tax liability is also reduced. The capital stock isincreased by one unit, and by (27) the stock's market value goes upby (1 0}/{l c). Therefore, current capital gains taxes are increasedby c(l d)/{l c). If the stockholders then sell off the value of thestock corresponding to the new investment, they receive a capital gainof (1 6)/{l c), and the value of their remaining stock is the sameas it would have been if the dividend had been paid out. But the totaldistribution is (1 6)/{l c) c(l d)/{l c) = (1 6), which equalsthe dividend foregone. Thus, payout policy is irrelevant from the point


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a total of (1 - 0)/{l - c). The net distribution in each period is themarginal product of capital, less corporate and dividend taxes, /'(I T ) ( 1 6). Thus, this marginal asset has zero present value, dis-counted at p, when /'(I r) = p/(l ~ c), the all-equity cost of capitaderived in the previous section. ^

VII. FINANCIAL POLICY AND THE COST OF CAPITALThus far, we have had little to say about the choice of financial

mix hetween deht and equity. One important question to ask iswhether, in the absence of any notion of risk or bankruptcy, firms wilfinance with hoth equity and debt or, as has been suggested by Stiglitz[1973], will finance with only debt at the margin.

If all future streams are certain, then the degree of leverage ofa firm has no impact on the values of p and i that it faces. Shares inall firms are perfect substitutes, as are bonds, so that firms areprice-takers with respect to p and i.

Differentiating (23) with respect to b, for i and p constant, weobtain (dropping subscripts)( 2 8 ) ^ = [{1 -c) - {0- c ) 6 ] - 2 ( l - 9)[ i { l - T ) ( 1 - C ) - p]abw h i c h is either a l ways posi t ive, a lw ays nega tive, or a lw ays zero. T h u sfirms wi l l desire all d e h t if i( l T ) (1 C ) < p, all e qu i ty if j( l T ) ( c)> p, and be indifferent if i(l r)(l c) = p. The determinationof j and p depends, in turn, on the characteristics of individual investors. We consider two cases, one in which all investors face the samtax schedule, the second in which they do not. In each case we let (denote the individual tax rate on debt income.Case 1: One Class of Investors

Without risk, debt and equity should be perfect substitutes, andinvestors will hold whichever yields a higher net rate of return, holdingboth only if these returns are equal. For an equilihrium to exist withdebt and equity present, both firms and individuals must be indifferent between the alternatives. From our discussion ahove, the


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WEALTH MAXIMIZATION AND THE COST OF CAPITAL 443income tax, they receive f (1 0) from holding debt. Thus, individualswill be indifferent if and only if i{l (p) = p. Combining these twoconditions, we have the resu lt that deb t and equity will exist togetherin equilibrium if and only if (1 T)(1 C ) = (1 ).Now assume tha t we are initially in a position in which both debtand equity are held, so th at t( l (p) = p, and suppose tha t (1 T)(1 c) > (1 (/)). Then i ( l T)(1 c) > p, and firms will shift entirelyto equity. Similarly, if ( l T)(1 c ) < ( l ), they will shift entirely. to deh t. Thu s, the equilibrium will be characterized by all equity, alldebt, or both according to whether (1 T)(1 c) is greater th an , lessthan , or equal to (1 (p).^"^It is interesting that the tax on dividends 6 does not enter directlyinto the process at all.'^ Since the question of which method of financewill be used depends on c, it would be helpful to relate this hypo-thetical tax on accrued capital gains to the currently existing tax onrealized gains. Bailey [1969] has shown tha t the effective tax rate onaccrued capital gains approximately equals the sta tuto ry ra te on re-alizations, which cannot exceed 0.56*, times the ratio of realizationsto accruals, which he estimated to be 0.2 for the long run in the UnitedStates. If we le t c = 0.19 and assume tha t 9 = cp, then the equilibriumwill be characterized by all equity, all debt, or both according towbether 9 is greater tha n, less tha n, or equal to 9, where(29) 9 = T/(0.9 -I- O.IT).If we tak e r to be the U . S. sta tutory rate 0^46, the n 9 = 0.486. Thatis, an all-deh t equilibrium will occur only if ^ < 0.506. In com parison,the maximum statutory rate on personal capital income in the U nitedStates is presently 0.7.In the m odel jus t presented , an equilibrium will have all equ ityor all debt according to whether or not " th e " tax ra te on personal in-come exceeds 9. In reality, however, the marginal tax ra tes faced hyindividuals vary greatly.Case 2: Two Classes of Investors

Suppose th a t there are investors of two types, differing only by


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4 4 4 QUARTERLY JOURNAL OF ECONOMICSclass holding equity and the other debt. For this to occur, firms mustbe indifferent between debt and equity, equity-holders must preferequity, and debt-holders must prefer debt. Without any loss of gen-erality, we take class 1 to be the equity-holders.

In equilibrium the rate of return received by equity investorsmust equal their rate of time preference. Combining (5) and (8), weobtain(30) pi = ( l - ^ i ) ^ + ( l - c

where p ^ is the rate of time preference of class 1 mem bers and A V ithe capital gain they receive. Similarly, the net return to holde rs ofdeb t m ust equal their rate of time preference:(31) p2 = (1 _ ^2) j .The potential return to equity holders from holding debt is(32) pi' = (1 - (/)i)i,and the potential return to debt holders from holding equity is(33) P^' = {l-92)^+{l-cThe conditions for an equilibrium with debt and equity are(34.1) pi > pi*(34.2) p2 > p2*(34.3) t ( l - T ) ( l - c i ) = p i ;that is, debt-holders prefer debt, equity-holders prefer equity, andfirms are indifferent between debt and equity. Combining (30)-(34)(and substituting 9^ for 0i and O2 for ^2), we obtain(35) 1 - 6*1 (d - ci)/(l - ^i))(l - a) + a > (1 - T) >\l-ci) \l-cilwhere is the fraction of gross equity gains received through divi-


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WEALTH MAXIMIZATION AND THE COST OF CAPITAL 445must be held by those who receive a relatively greater tax advantagein the statutory treatment of capital gains. This result is similar tothat of the simple Ricardian trade model which dictates that countrieswill specialize in the production of the good for which they have acomparative advantage.

If (1 - ci)/(l - 9i) = (1 - C2)/(l - 92), then condition (35) willbe met if and only if (1 - r) = (1 - 9^)/{l - cj) = (1 - 92)/{l - Cg), inwhich case both classes will be indifferent between debt and equity.Thus, the two-class example collapses to the one-class example, andno segmentation will occur. Suppose, at the other extreme, that class2 consists of tax-exempt investors, sucb as nonprofit institutions.Then (35) becomes(37) ._ , _ _ . _ . , _ U-Ci /Unless a is near unity,^'' the first inequality in (37) will be satisfied,and tbe condition reduces to tbat necessary for tbe existence of equityin a one-class economy. Thus, in a two-class world, witb one classtax-exempt, and tbe other facing a marginal tax rate on regular capitalincome between 0.51 and 0.70, debt and equity would coexist, witbtbe former group bolding debt and tbe latter bolding equity.

VIII. CONCLUSIONTbis paper bas reviewed tbe investment and financial bebavior

of corporations seeking to maximize tbe wealtb of tbeir sbarebolders.We bave focused on tbe impact of personal income and capital gainstaxes, finding tbat, in tbe presence of differential taxation of dividendsand capital gains, wealtb maximization does not imply maximiza-tion of firm market value and tbe source of equity financing is notirrelevant.

Tbe appropriate cost of capital in the presence of personal taxesdoes not depend directly on either the dividend payout rate or the taxon dividends. Equity shares have a market value lower than the dif-ference between the reproduction cost of a firm's assets and themarket value of its debt obligations. Because of tbis capitalization,


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4 4 6 QUARTERLY JOURNAL OF ECONOM ICSREFERENCES

Auerbach, A. J., "Share Valuation and C orporate E quity Policy," Journal of PublicEconomics, 1979.Bailey, M. J., "Capital Gains and Income Taxation," in A. C. Harberger and M. J.Bailey, eds.. The Taxation of Income from Capital (Washington, D.C : BrookingsInstitution, 1969).Bierman, H., and R. W est, "The Acquisition of Common Stock by the Corporate Is-suer," Journal of Finance, XXI (Dec. 1966), 687-96.Bradford, D., "T he Incidence and A llocation Effects of a Tax on Corporate Distribu-tions," mimeographed, 1978.Farrar, D . F., and L. L. Selwyn, "Taxes, Corporate Financial Policy, and Re turn toInvestors," National Tax Journal, XX (Dec. 1967), 444-54.Feldstein, M . S., J. Green, and E. Sheshinski, "Corporate Financial Policy and Taxationin a Growing Econom y," this Journal, XCIII (Aug. 1979), 411-32.King, M. A., "Taxation and the Cost of Cap ital," Review of Economic Studies, XLI(Jan. 1974), 21-35.Miller, M. H., "De bt and Tax es," Journal of Finance, XXXII (May 1977), 261-75., and F. Modigliani, "Dividend Policy, Growth, and the Valuation of Shares "Journal of Business, XX XIV (Oct. 1961), 411-33.Modigliani, F., and M. H. Miller, "The C ost of Capital, Corporation Finan ce, and theTheory of Investment," American Economic Review, XLVIII (June 1958),261-97.Pye, G., "Preferential Tax T rea tm ent of Capital Gains, Optimal Dividend Policy, andCapital Budgeting," this Journal, LX XX VI (May 1972), 226-42.Stapleton, R. C , "Taxes, the C ost of Capital, and the Theory of Investment," EconomicJournal. LX XXII (Dec. 1972), 1273-92.Stiglitz, J. E ., "Taxation, Corporate F inancial Policy, and the Cost of Capital," Journalof Public Econornics, II (Feb. 1973), 1-34., "The Corporation Tax," Journal of Public Economics, V (April 1976), 303-Tobin, J., "A General Equilibrium Approach to M onetary Theory," Journal of Money,Credit and Banking, I (Feb. 1969), 15-29.U.S. Federal Reserve System, Board of Governors, Flow of Funds Accounts., Federal R eserve Bulletin, LX III (Dec. 1977).


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Wealth AndCoC