The Personal-Tax Advantages of Equity

Transcription

1 The Pesonal-Tax Advantages of Equity Richad C. Geen and Buton Hollifield Gaduate School of Industial Administation Canegie Mellon Univesity Decembe 23, 999 Discussions with Piee Collin Dufesne, Bob Dammon, John Gaham, Nathalie Moyen, Jim Poteba, and Byan Routledge have been vey helpful to us. We would also like to thank semina paticipants at GSIA, Rocheste and Texas A & M fo helpful comments.

2 Abstact We compute the value of a fim that pays its cash flows each peiod though shae epuchases in a dynamic envionment whee pesonal taxes ae paid on ealized capital gains and dividends. These esults povide a measue of the pesonal tax advantages of equity financing elative to debt financing, which ae often cited as inceasing the cost of debt. The initial pice of the fim depends on the pesent value of the taxes paid, which, in tun, depends on the initial pice. We solve this valuation poblem in closed fom in a deteministic setting and numeically in a stochastic setting. We find significant valuation effects fom the tax potection affoded by the equity basis. The tax savings ae on the ode of 40-50% of the taxes paid by the shaeholdes of fim that distibutes cash though dividends, and the cost of capital is educed by appoximately.8 to.2 pecentage points though the use of epuchases elative to dividends.

3 Intoduction The Modigliani and Mille (963) model of capital stuctue choice unde taxation still foms the theoetical basis fo most pedagogy and pactice in moden Finance, despite its obvious empiical and theoetical limitations. This theoy pedicts a cone solution of all debt financing fo all fims, due to the deductibility of inteest payments at the copoate level. Such an outcome, howeve, appeas gossly at vaiance with obseved pactices and has neve been taken seiously as a policy ecommendation. To explain this discepancy, textbooks have followed eseaches in pointing to thee consideations that ae ignoed in the Modigliani and Mille (heeafte, M&M) valuation of the tax shields fom debt:. Costs of financial distess 2. Redundancy of copoate tax shields 3. Tax advantages to equity at the level of pesonal taxes. While consideable eseach has been devoted to all thee of these lines of inquiy, and this eseach has eniched consideably ou qualitative undestanding of the tadeoffs involved, it has poved difficult to geneate the sot quantitative esults that could help eseaches evaluate the empiical evidence o help pactitiones detemine how much debt is too much debt. With egad to the costs of distess, the M&M model takes as fixed and exogenous the fim s opeating policies, assets, and net cash flows. It thus ignoes the bankuptcy costs and incentive poblems that may distot eal choices when the fim is in, o close to, bankuptcy. A geat deal of eseach in the last two decades has been devoted to this issue, and much has been leaned about the ways incentive and infomation poblems can influence capital stuctue. Examples of this wok include Jensen and Meckling (976), Myes (977), Leland (998), Paino and Weisbach (999) and Moyen (999). The stylized natue of the models used to captue these tadeoffs, howeve, makes thei quantitative impotance difficult to assess. Ae these costs sufficiently lage to explain the low levels of debt finance fims employ? O, ae they elatively insignificant when compaed to the tax benefits of debt, as suggested by the analogy in Mille (977) to the ecipe fo hose and abbit stew? Second, the M&M model suely ovestates the tax benefits of debt at the copoate level. The tax shields fom inteest may, in some states of the wold, be edundant, which lowes thei expected value.

4 They may also be isky, which lowes thei pesent value. Models that qualitatively descibe the tadeoffs this would impose upon the fim go back to DeAngelo and Masulis (980), but the dynamic natue of the teatment of tax shields in the tax code has made it difficult to evaluate the quantitative impotance of these consideations. Consideable pogess has been made on this font ecently, using methods based on simulation, in Gaham (996), and especially Gaham (998). In this pape, we focus on the thid facto that investigatos have cited in aguing that M&M ovestate the tax benefits of debt financing. Debt is tax-disadvantaged at the pesonal level. Inteest payments ae taxed as odinay income. Much of the compensation eceived by equity holdes, on the othe hand, comes in the fom of capital gains. Mille (977) agued that this may aise the isk-adjusted cost of debt to the fim, elative to equity, sufficiently to neutalize the tax advantages of debt at the copoate level. The pesonal tax advantages of equity ae lagely attibutable to the option to defe capital gains. The value of this option, and thus the cost of equity capital to the fim, depends on the timing of the fim s cash distibutions and the way those distibutions ae split between dividends and shae epuchases. In the inteest of obtaining tactable expessions, howeve, eseaches who have studied this poblem in the past have appoached it with static models, making it vey difficult to evaluate the quantitative ealism o impotance of the effects they descibe. Mille (977), fo example, simply assumes payments to equity ae tax exempt. This is clealy not the case. Even if all cash is distibuted though epuchases, investos may have to ealize capital gains in ode fo the distibution to take place. DeAngelo and Masulis (980) assume a constant, exogenous pesonal tax ate on equity. Gaham (998) assumes the pesonal tax ate on equity is a simple linea function of the dividend payout. Ou pupose hee is to povide moe quantitative guidance as to the magnitude and deteminants of the pesonal tax advantages of equity financing to the fim in a dynamic model. The model follows the oiginal M&M appoach of taking as exogenous the fim s pe-tax net opeating cash flows. Thus, we abstact fom bankuptcy costs and incentive poblems. We also ignoe the complications at the copoate level studied by Gaham (998). Ou model is also patial equilibium in the sense of taking as exogenous the picing kenel. Despite these obvious simplifications, this model allows us to pose and answe the following questions: In an envionment whee the fim can issue debt o equity, and whee it can distibute cash though epuchases o dividends, how high is the isk-adjusted cost of equity elative to the isk-adjusted cost of debt? How does this depend on the mix between debt and equity? How does it depend on the fim s dividend policy? 2

5 We addess these questions, fist, in a setting that assumes cetain and pepetual cash flows. This allows fo closed-fom expessions fo the fim s value, cost of capital, and the gains to leveage. These expessions can be diectly compaed to the taditional analyses in M&M, and Mille (977). The cetainty case also illustates the logic we employ in ou numeical analysis of the fim s poblem unde uncetainty. Next we descibe methods fo obtaining numeical solutions to the model when the fim s net cash flows ae andom, but positive. Using these methods we calculate the value of the fim when all cash flows ae distibuted though epuchases. Compaisons of this value to the value unde full dividend payout povides measues of the effects of the pesonal tax advantages of equity on the fim s cost of funds. When net cash flows can be negative, the state space fo the valuation poblem expands quickly, making it difficult to solve the model numeically. New shaes issued to finance negative net cash flows establish new basis values, and the numbe of shaes outstanding at all basis values must be tacked. To evaluate the impotance of these effects, we povide appoximations by simulating the pesonal taxes paid assuming the fim s aggegate value evolves exogenously. We then compute the expected pesent value the taxes paid, and numeically adjust the tax ate in the exogenous value pocess so that the basis values at which new shaes ae issued appoximately eflects the pesent value of futue pesonal tax liabilities. Fo the cases whee we have full solutions to the model, this method povides easonably accuate appoximations. We then use this method to evaluate the magnitude of the tax advantages povided by issuing new shaes, at highe basis values, as the fim moves though time. Ou esults show that the pesonal-tax advantage of epuchases ove dividends o inteest payments is substantial. The pesent value of tax savings ae in the ange of pecent of the pesent value of the pesonal tax liability that would be incued using only dividend payouts. The implicit tax ates we calculate, ates which yield the same pesent value fo the fim if all distibutions wee fully taxed, ae geneally aound 60 pecent of the tax ate faced by the the fim s shaeholdes. To focus diectly on the pesonal-tax consequences of the fim s cash distibution policies, we abstact fom a numbe of consideations that have been impotant in othe eseach on capital gains taxation. We ignoe liquidity needs o potfolio ebalancing that may lead individual investos to ealize gains even if it is disadvantageous fom a tax standpoint to do so. The only motive in ou model fo ealizing gains is to facilitate the fim s need to disgoge cash, and the fim epuchases shaes at a pice that fully compensates the selling shaeholde fo suendeing the oppotunity to continue defeing. In contast, a 3

6 lage liteatue in public economics studies the effects taxation based on ealization athe than an accual has on the effective tax ate faced by individual investos. (See Poteba (999) fo a ecent summay of these esults in the context of the measuement of afte-tax etuns fo investos.) Fo example, Bailey (969) finds that the option to defe gains educes thei effective taxation by oughly a half, and the oppotunity to step up the basis on death educes it by anothe half, leading to effective taxation at a ate one-fouth of the statutoy ate. This appoximation has, in tun, been caied though by many authos, though Balce and Judd (987) citicizes this appoach. In thei dynamic model no constant, popotional, implicit ate exists that gives ise to the same investment and consumption behavio as capital gains taxation on a ealization basis. Ou esults, since they ignoe any othe motives fo ealization, can be viewed as estimating the effect of the fim s need to distibute cash on the implicit ate at which capital gains ae taxed. To the extent that othe motives exist, ou esults will undestate the total tax buden bon by equity holdes, and hence the advantage of equity epuchases elative to dividends o debt. Similaly, in ode to focus puely on the effects of the fim s distibution policies, we assume the ates at which dividends, inteest, and capital gains ae taxed ae identical, and abstact fom the distinction between long- and shot-tem gains. These assumptions will lead us to undevalue the pesonal-tax benefits of equity financing. We also take the net cash flows of the fim as exogenous. This abstacts fom the fim s ability to etain funds to delay taxation at the pesonal level. The pape is oganized as follows. The next section we evaluate the value the fim and detemine its cost of funds unde cetainty. In Section 3 we povide numeical solutions to the valuation poblem unde uncetainty. In Section 4 we develop an appoximation to the value when thee ae negative cash flows by simulating the pesonal tax liability when the dynamics of the fim s value ae taken as exogenous. Section 5 povides a bief summay of ou conclusions. All poofs ae contained in the appendix. 2 The Cetainty Case Befoe studying envionments whee closed-fom solutions ae inaccessible, it is helpful to fist undestand the simplest cases and examples. We begin, theefoe, by evaluating the dynamics of the stock pice, numbe of shaes outstanding, fim value, and cost of capital unde a cetainty model fo the fim s net cash flows. We assume these cash flows ae constant and pepetual, as in M&M, o that the cash flows gow at a constant ate. We will povide solutions fo each of the following cases, in ode of complexity. 4

7 . Full, o zeo, taxation of all distibutions. This coves the cases of full dividend payout, no taxation at the pesonal level, and taxation of all capital gains whethe ealized o defeed. 2. Pesonal taxes paid only when gains ae ealized. 3. Debt is tax deductible at the copoate level, while pesonal taxes ae paid on capital gains only on ealization. 4. Cash flows gow at a constant ate, while pesonal taxes paid only when gains ae ealized. Thoughout, we assume that inteest, dividends, and ealized capital gains ae taxed at the same ate at the pesonal level. Fo cases and 2, we assume the fim is entiely equity financed. The fim s cost of capital, then, is the cost of equity. The cash flows the fim pays out should be viewed as net cash flow, afte copoate taxes. When we conside the thid case, we teat copoate taxes explicitly. The fim can be viewed as simply a sequence of cetain, pepetual cash flows, C t, paid at peiodic intevals, indexed by t 0. This cash flow, and the investment and opeating activities that poduce it, ae viewed as exogenous. Thus, we conside only the financial activities of the fim, and ignoe the possibility that it may wish to etain funds to shield o defe taxation at the pesonal level. Again, these simplifying assumptions ae in keeping with the benchmak case povided by M&M. Let the value of the fim at time t be V t, the pice pe shae be p t, and the numbe of shaes outstanding n t. When the fim is initially established, n 0 shaes ae issued. The fim s cash flows will be paid out eithe as dividends o though epuchases of the shaes initially issued. Afte this date, as the fim s net cash flows ae always positive, no new shaes ae issued. We ignoe consumption o potfolio ebalancing as motives fo tade. We also abstact fom the distinction between long-tem and shot-tem capital gains. Futhe, all investos face the same constant afte-tax discount ate, 0. Thus, investos can be viewed as homogeneous, and will not tade pivately in equilibium as long as the pice sequence is inceasing though time. It will neve benefit an individual shaeholde ealize a capital gain, pay taxes in the cuent peiod, and then eset the basis to shield income in the futue. Defeing the gain instead povides a tax-fee loan fom the govenment. By simila easoning, a buye will not be willing to pay what a selle would demand fo he shaes, unless the buye was motivated by some othe consideation such as desie fo consumption o potfolio ebalancing. Since taxes incued though a tansaction will be eflected in the pice at which the tansaction is executed, if the taxes epesent a net 5

8 loss to the tades, one of them must find the tade unattactive at any pice the othe finds acceptable. The next lemma establishes this esult fomally. It will be the basis fo ou aguments late that we can focus exclusively on the tades between the fim and its initial shaeholdes. The poof, which is povided in the Appendix, simply elies on the agument that fo any feasible ealization stategy, the benefits associated with futue tax deductions fom the step up of the basis will have a lowe pesent value than the taxes that must be paid now to achieve that incease in the basis. Lemma Suppose the cuent pice exceeds the initial pice at which the shaes wee issued, p t p o, and that all cuent equity claimants hold the shaes with a tax basis of p o. Then a new shaeholde would be willing to buy shaes at p t only if that pice is less than o equal to the minimum pice an existing shaeholde would accept fo he shaes. 2. Benchmak Cases The next poposition povides fomulas fo the fim s value and shae pice dynamics, when thee ae no taxes o when taxation is symmetic fo capital gains and dividends. In keeping with the M&M model, we view the cash flows as constant: C t C. The algeba leading to these fomulas, is detailed in the Appendix. Let pt c denote the cum-dividend pice, and let pt e be the ex-dividend pice. Poposition When capital gains ae taxed each peiod, whethe ealized o defeed, and dividends ae taxed as odinay income, the pice pe shae and numbe of shaes obey the following dynamics:. When cash flows ae paid as dividends, the numbe of shaes, cum-dividend pice, and ex-dividend pice ae constant though time. p c t n 0 C τ p C τ p () p e t C τ p n 0 (2) and n t n 0. 6

9 2. When cash flows ae used fo epuchases, the pice pe shae gows geometically and the numbe of shaes shinks geometically, at the pe-tax ate of inteest, τ p. p t p t τ p (3) n t n t (4) τ p 3. In both cases the aggegate, ex-distibution value, denoted V t, is constant though time, V t C τ p (5) These fomulas include the special case of no taxation when τ p 0, in which case the numbe of shaes shinks, and the pice gows, at ate. The poposition shows that, when capital gains ae taxed whethe ealized o defeed, the decision to distibute cash by dividend o capital gain has no effect on aggegate fim value o aggegate pesonal tax liability. The shaes actually epuchased in the distibution ae only patially taxed, due to potection fom the basis. In the M&M setting, howeve, the aggegate capital gain on all outstanding shaes is pecisely equal to the amount of the distibution, and hence the aggegate tax liability is the same as it would be unde full dividend payout. 2.2 Repuchases when Taxes ae Paid On Realized Gains Only When capital gains ae not taxed when eaned, but athe when ealized, thee is a wedge between the value of a shae to a puchase, who establishes a new basis at the cuent pice, and the value to a shaeholde who holds the shae at a lowe basis. In ou model, thee is no motive to tade othe than to distibute cash fom the fim. Lemma implies that if the pice is inceasing, the pice demanded by an existing shaeholde, with an embedded gain, exceeds the pice a new, pivate buye would be willing to pay. Thus, the fim issues shaes only at its founding, and all shaeholdes hold the shaes with the same basis, p o. We will assume that the pice the fim pays fo the shaes it epuchases fully compensates the shaeholde fo the loss of the oppotunity to defe. Then Lemma implies that this pice does, in fact, exceed what a pivate buye 7

10 would be willing to pay. Note that the one point whee a new buye and a selle would view the shaes symmetically is at the initial date, since selling the shaes at p o tigges no tax liability. We nomalize the numbe of shaes oiginally issued to n 0. At any point we know that C t n t n t p t (6) since the value of the shaes epuchased must equal the cash used to epuchase them. Thus, n t is the numbe of shaes pevailing between the distibution at time t and the distibution at t. If an investo sells a shae at time t, she would ealize a cash flow of p t τ p τ p p o and if she holds and sells next peiod, she ealizes a cash flow of p t τ p τ p p o Equating the pesent values of these two altenatives gives us a diffeence equation fo the pice. p t τ p τ p p o τ p p t τ p p o (7) Given a pice seies satisfying this equation at each point in time, a shaeholde is, by ecusive agument, indiffeent between selling shaes back to the fim immediately o defeing the gain to an abitay point in the futue. The indiffeence condition, (7), can be eaanged to give p t τ p p t p o τ p (8) Solving (8) yields the shae pice p t θ t τ p τ p p o whee θ and the initial value of the equity, p o, ae coefficients to be detemined. Evaluating the stock pice 8

11 at time 0, and solving fo θ, θ p o τ p and so p t p o t τ τ p (9) p We can now establish the initial value of the fim, p o. Since buyes and selles will view the shaes symmetically at this point, this pice must simply be the pesent value of the aggegate afte-tax cash flows to the equity holdes. The afte-tax cash flows accuing to initial equity holdes at time t ae the afte-tax poceeds fom the shaes epuchased at that time, n t n t p t τ p τ p p o C t τ p n t n t τ p p o p C t τ p n t o n t τ p p t p t p C t o τ p C t τ p p t τ p C t τ p t τ p t C t τ p (0) t τ p The fist line uses the constaint that the value of the shaes epuchased must equal the cash distibuted, equation (6). The thid line uses this constaint again, and the fouth line uses the solution fo the pice, (9). The limit, as t, of the ight-hand side of (0) is C t τ p. As time passes, the aggegate value of the potection affoded by the initial basis becomes tivially small. The shae pice ises at close to a geometic ate as shaes ae extinguished though epuchases. In the limit, theefoe, the distibution is fully taxed. The deivation to this point allows fo any deteministic patten of positive cash flows ove time. When these flows ae constant, it is simple to compute the initial shae pice, p o, as the pesent value of afte tax flows p o s C τ p s C τ p s s τ p () s s τ p 9

12 The following poposition summaizes the solution. Poposition 2 Suppose that the pesonal tax ate is τ p, the afte tax discount facto is and that the fim distibutes C each peiod by epuchasing shaes. Then, the initial value of fim is given by The pice at any point in time is given by p 0 C τ p (2) s s τ p p t C t τ p The numbe of shaes outstanding ae given by s s τ p (3) n t s t s τ p s s τ p (4) The value of the fim is V t C t τ p s t s τ p (5) and the value of the fim conveges to the no tax value of the fim, lim t V t C C A counteintuitive featue of the solution is that the value of fim conveges to the no tax value ove time, while the afte tax cash flows the shaeholdes eceive conveges to τ p C, the full tax cash flows. At the equilibium pices, the investo must be indiffeent between selling his shaes today and paying taxes on the accumulated capital gains, o holding the shaes indefinitely. The highe the cuent pice, the highe the pice the investo will equie to be indiffeent between selling today and holding foeve. In the limit, as the accumulated capital gains ise, they must eceive a pice fo thei holdings equal to what the shaes ae woth to an investo in an wold without taxes. The pice with epuchases is lage than the initial value unde dividends, which is C τ p C τ p s s (6) 0

13 The pesonal-tax advantage of equity, in this setting, is manifest in the fact that the pesonal tax ate is subtacted in the discount factos in equation (2), but not in (6). By tuncating the infinite sum in (2), these expessions can be compaed quantitatively. Figue plots the pesent value of taxes paid unde epuchases, which is C p o as a faction of the pesent value of taxes paid unde dividends, τ p C, fo a fim with cash flow of 00. The value of the fim, p o is computed with 500 tems in the summation. The figue plots these atios against the pesonal tax ate, using thee diffeent values fo the afte-tax discount ate, 3% (+), 6% (*), and 9% (x). In each case, epuchases save the fim s shaeholdes 40-50% in the pesent value of thei tax liability. The fim s cost of equity capital is simply the discount ate that equates the initial value given above to the pesent value of the pe-pesonal-tax cash flows. That is, the cost of equity, ˆ, solves p o C ˆ (7) Figue 2 plots this cost of equity capital, assuming pe-tax etun of 6%, fo vaious levels of the tax ate fom 0 to 40%. The solid line is the cost of equity capital, in this case whee C just p o. The dotted line is the cost of capital pevailing when distibutions ae paid as dividends, τ p. 2.3 Allowing fo Gowth Ou fomulas fo the initial value of the fim genealize in a staightfowad manne to the case of constant gowth in the cash flows. Up to equation (0), the deivation applies to any positive, deteministic seies of net cash flows. If these cash flows gow at a constant ate, g, with an initial cash flow at date t= of C, we have, as in equation (), p o C τ p s g s (8) s τ p Table povides measues of the tax advantage to epuchases vesus dividends unde diffeent gowth ates. In the table, the implict tax ate is the ate that equates the pesent value of the fully taxed cash flows to the initial value of the fim: τ solving p o C τ g. The pecent of PV taxes is computed by calculating the pesent value of taxes paid as a pecentage of the pesent value of taxes paid unde full, popotional taxation. The table shows quite clealy that the advantages of epuchases ove dividend payments decease with the ate of gowth in the cash flows, despite the fact that the gowth in the fim s

14 cash flows is eflected in the pesent value that detemines the initial basis. Gowth ates 0% 2% 4% Panel A: Tax ate of 28% Implicit tax ate (%) Pecent of PV taxes (%) Panel B: Tax ate of 35% Implicit tax ate (%) Pecent of PV taxes (%) Table : Tax Advantage to Repuchases Unde Cetainty: The implicit tax ate is the ate that, unde full taxation, would set the pesent value of afte-tax cash flows equal to thei calculated value. The pecent of PV taxes is the pesent value of expected taxes paid, as a pecentage of the pesent value of taxes paid unde full taxation. The afte-tax discount ate is 6%, and the value of the fim is computed using 200 peiods. 2.4 Allowing fo Debt Financing To account fo debt financing, and quantify the tax advantage of debt net of pesonal taxes, we must allow fo taxation at the copoate level. Thus, we now intepet C as the fim s eanings befoe inteest and taxes, EBIT, as in M&M. The net afte-copoate tax opeating cash flow is then C τ c, whee τ c is the copoate tax ate. We assume inteest expense and dividend payments ae constant and pepetual. Let i i t be the inteest paid by the fim in peiod t, and let d d t be the dividend payment. A fim paying inteest of i each peiod will have C i τ c available to distibute to equity holdes. This will be divided between dividends and epuchases. The valuation of dividends and inteest payments, which ae both fully taxed at the pesonal level, is staightfowad. The value of the fim s debt is the pesent value of the inteest payment, net of pesonal tax, i τ p. Similaly, the dividends have value d τ p. The analysis of the pevious section, specifically equation (2), gives the value of the pepetual steam of epuchases as a constant multiple of the cash distibuted. Thus, the value of the leveed fim is: V L C i τ c d! τ p s s τ p d τ p i τ p (9) 2

15 This can be witten as the value of an unleveed fim, with all cash distibuted though epuchases, as in equation (2), plus the net advantages/disadvantages of dividends and debt, V L C τ c τ p i τ p s s d τ τ p p " τ c s s τ p s s τ p (20) This equation is linea in both i and d, so that dividend policy and capital stuctue policy will be chaacteized by cone solutions in this envionment. Dividends ae dominated. The tem multiplying d τ p in equation (20) is unambiguously negative fo any τ p 0. The elative attactiveness of debt vesus equity (with epuchases) is detemined by the elative magnitudes of, τ p, and τ c. Assuming d the contibution of debt to the value of the fim as 0, we can ewite i τ p τ c s s τ p D L τ c s s τ p (2) whee D L is the value of the debt. The tem multiplying D L is the gain to leveage, and can be instuctively compaed to the gain to leveage in Mille (977), which teats the equity flows as fully taxed, but at a pefeential ate, τ s, τ c τ s τ p In ou setting the flows to equity ae not taxed at a pefeential ate. Rathe the pesonal-tax advantage to equity comes fom the pesent value of the tax shields fom the initial basis. While poviding a quantitative benchmak fo the pesonal tax advantages of equity, this analysis will not, of itself, ationalize inteio capital stuctues at ealistic values of the paametes. Ou intent is, athe, to gain a bette undestanding of how lage othe costs of debt, such as losses in aggegate value associated with financial distess, would have to be to geneate inteio optima. We can ewite the gain to leveage, pe dolla of debt issued, as follows: # τ c s s τ p s s τ c s τ p s τ c τ p s s s τ p (22) 3

16 The denominato in each tem in this summation is positive. Since s, the following lemma follows by inspection of the numeato. Lemma 2 When τ p $ τ c the gain to leveage is positive. Inspection of (22) suggests, indeed, that the pesonal tax ate must be substantially highe than the copoate ate to offset the benefits of deductions at the copoate level. Figue 3 povides a quantitative sense of this. The figue plots the copoate tax ate at which the gain to leveage is zeo, fo diffeent values of the pesonal tax ate. Fo a given pesonal tax ate, at copoate ates above the cuve plotted in the figue, the fim would pefe debt financing ove equity financing. Equation (9) fo the value of the fim can also be used to evaluate the effects of capital stuctue and dividend policy on the fim s oveall cost of capital. In a manne analogous to M&M, we ask what discount ate sets the pesent value of futue afte-copoate-tax, opeating cash flow equal to the value of the fim. We solve fo ˆ in V L C τ c ˆ (23) whee V L is computed using (9). Table 2 epots the esults of these computations fo a fim with τ c 34%, 6%, and two values of the pesonal tax ate, 28% and 35%. 3 Uncetain Cash Flows When cash flows ae cetain, in a setting paallel to that envisioned by M&M, we can solve fo the value of the fim unde epuchases in closed fom. The advantage epuchases povide ove dividends is attibutable, in ou setting, entiely to the tax shield supplied by the initial basis. It takes a mathematically simple fom as a eduction in the discount factos attibutable to afte-tax cash flows (see equations (2) and (6)). We found that this advantage to epuchases is quantitatively substantial fo easonable paamete values, 40-50% of the value of the pesonal tax liability. It was not, howeve, sufficiently lage to completely offset the advantages at the copoate level of debt that, as assumed by M&M, offes full, non-edundant tax shields at the copoate level. Ou pupose in this section is to evaluate whethe these implications hold when cash flows ae uncetain. We continue to make assumptions analogous to M&M. We view the fim as a pepetual steam of pe-tax 4

17 Dividend Payout Ratio Inteest ove EBIT 0% 20% 40% 60% 80% 00% Panel A: τ p 28% 0% % % % % % Panel B: τ p 35% 0% % % % % % Table 2: Cost of Capital Calculations: The body of the table epots the pecentage etun that equates the value of the fim to pesent value of afte-tax opeating cash flows. Rows epesent diffeent capital stuctues, paameteized by the pecentage of pe-tax income paid in inteest. Columns epesent diffeent dividend policies, paameteized by the pecentage of afte-tax cash flow paid in dividends. The copoate tax ate is assumed to be 34%, the pesonal ate is 28% o 35%, and the afte-tax discount ate is 6%. net cash flows that ae constant in expectation. Thus, when cash flows ae paid as dividends, o when all capital gains and losses ae taxed symmetically, whethe ealized o not, the value of the fim, denoted V F, is given by V F Ĉ τ p whee Ĉ is the expected cash flow unde the isk-neutal measue. Let Ct s denote cash flow in state s at date t. Fo simplicity, we take the state space to be finite and discete, and assume net cash flows ae independent and identically distibuted though time. Denote the isk-neutal pobability of cash flow Ct s as π s. When net cash flows ae always positive, we solve numeically fo the value of the fim. With positive net cash flows, as unde cetainty, shaes ae only issued once, and so the basis is the same fo all outstanding shaes. If the pice always ises though time, as shaes ae etied, then this basis will always be the initial pice. The only state vaiables on which pices will depend ae the cuent cash flow, C t, and the numbe of shaes outstanding, n t. Fo the no-tax case, it is simple to veify analytically that the pice always ises 5

18 & & &,,, when net cash flows ae positive. While we have not found a poof fo the geneal case, we can veify numeically that along the wost sample paths, whee the lowest (but still positive) cash flow is ealized, the pice inceases though time with the epuchases. While the decision ules fo the fim ae mechanical in this model, the shae pice must be detemined endogenously. The pice a shaeholde will demand when the fim offes to epuchase shaes will depend on the shaeholde s basis, and on the pesent value of futue taxes paid if she decides not to sell he shaes back to the fim, which in tun will depend on the futue pice path. Suppose at date t we ente the peiod with n t shaes outstanding. If we know the pice next peiod will be set so that a shaeholde with basis p o will be just indiffeent between continuing to hold the shaes and selling into the epuchase, then the pice the shaeholde would demand in the cuent peiod, p t, must satisfy: p t τ p τ p p o E t p t! τ p τ p p o (24) In addition, all cash must be distibuted though the epuchase n t n t p t C t (25) Finally, as thee is no distibution at the initial date, t 0, the initial pice must be set to satisfy p o E 0 p! τ p τ p p o (26) Let n t% be the numbe of shaes outstanding upon enteing peiod t. The pice at date t will satisfy the equalities: p t & n t% ' C t ( E t * s) C t+ s ( - s.0/ & n t ' E t * s) C t+ s ( - s.0/2 Theefoe, p t+ 3 p t n t ' n t 4 C t+ ( C t+ ( n t C t+ E t+ * s) Ĉ 3 Ĉ 5 C t+ + s, ( - s. 3 E t * s) C t+ s ( - s.0/ 6

19 o p o E 0 p! τ p τ p (27) These thee equations, (24), (25), and (27), must be solved to detemine the value of the fim, and the paths of the pice and the numbe of shaes. As the fim poceeds though time, the numbe of shaes is shinking and the pice pe shae is gowing, both at oughly geometic ates. Theefoe, it is moe staightfowad numeically to deal with analogous equations fo the value of the fim. Examining the value also povides some insight into the dynamics of the tax liability and its impact on pices. On enteing peiod t, the fim knows the numbe of shaes outstanding, n t, and the initial basis, p o. Once the andom cash flow is known, p t and n t ae detemined. We will deive a ecusion fo the value of the fim, afte the cuent cash flow is known, but befoe the payment is made. The fim value is given by n t p t, befoe the distibution, and by n t p t immediately afte. Denote the numbe of shaes epuchased ove the peiod by t 6 n t n t and so C t t p t. Then, V t n t p t n t t p t n t p t C t n t E t p t 7 τ p τ p p o C t E τ t p V t 8 n t p o C t (28) τ p whee the fouth line follows fom substituting fom the indiffeence equation fo pices, (24), fo p t. Let V c n be the value of the fim at the beginning of the peiod, with cuent cash flow c and cuent numbe of shaes n, let c n be the numbe of shaes epuchased ove the peiod, and let n9 denote the numbe of shaes next peiod and c9 the cash flow next peiod. Using this ecusive notation, equation (28) can be ewitten as V c n E : V c9 n9 <; ; c n=># n n c τ p τ p p o c (29) 7

20 The fim s cash constaint can be eaanged to give c n c p c n cn V c n (30) whee the second equality follows since p c n V c n n. Substituting equation (30) into equation (29) yields V c n E : V c9 n9 ; ; c n=> n cn V c n τ p τ p p o c (3) Equation (3) is a functional equation that must be satisfied by V c n, and so the solution is a fixed point to equation (3). Assuming the fim begins with one shae, and noting that thee is no distibution between t 0 and t, then Using this in the expession fo p o, (27), yields E 0 p? E : V c9 A@ n = π V C s (32) s p o E V c9b A@ n < τ p (33) τ p Thus, we can solve fo the value of the fim and the initial pice by seeking a fixed point, V c n, to equations (3) and equation (33). To solve fo this fixed point numeically, we pick a gid fo the numbe of shaes, C n n n 2 D, and stat with a guess fo E V c9b n9 0@ c n fo each of these points and fo each cash flow outcome. We set E V c9e n9 0@ c n equal to the no tax value fo n F n, since as we show late, the value of the fim must convege to the no tax value as the numbe of shaes goes to zeo. We then guess the initial pice, and iteate as follows. Denote the guesses at the i th iteation as ψ i c n and p i 0 espectively. Using these on the ight-hand side of equation (3), we solve fo the fim value, which we denote V i c n. This step involves solving a quadatic equation at each point c n. We can then use equation (30) to calculate the numbe of shaes epuchased at each point i c n cn V i c n. Given these solutions, we compute new guesses fo 8

21 the conditional expectations: ψ i c n π s V i C s n i n c (34) s fo each pai c n, and use equations (32) and (33) to compute a new guess of the initial pice. p i o s π s V i C s τ p τ p (35) These steps ae epeated until we aive at an appoximate fixed point. In pactice, this pocedue conveges elatively quickly. Table 3 povides numeical esults obtained using this pocedue fo a fim with expected cash flows of 00. In each case thee ae two possible cash flows, and pobability of the highe cash flow is 2 3. The afte-tax discount ate is 6%, and two values fo τ p ae used, 28% and 35%. The table povides vaious desciptive measues of the effects of pesonal taxation on the value. The second column gives the tax ate that sets the pesent value of afte-tax cash flows, assuming they wee fully taxed, equal to the actual value. That is, we solve p o Ĉ τ (36) fo τ. In evey case, this implicit tax ate is substantially lowe than the tax ate on dividends. The thid ow measues the impact of the tax advantages to equity on the cost of funds to the fim. We find the discount ate, ˆ, that solves: Ĉ ˆ p o (37) The diffeence between this cost of capital and the pe-tax cost of capital unde dividends is in evey case substantial, between 0.79 and.2 pecentage points. The final ow epots the pesent value of pesonal taxes paid as a faction of the pesent value of the tax liability unde full dividend payout. Fom this table we see that, as in the cetainty case, the use of epuchases ove dividends educes the pesent value of the shaeholdes pesonal tax liability by 40% to 50%. Thee is vey little vaiation in the quantities of inteest acoss cash-flow outcomes of diffeent volatility. This is not supising, given the natue of the tax liability. Though ecusive substitution, and use of the budget constaint, we can ewite 9

22 Cash Flow Outcomes Full tax C 20 60D C 4 8D No tax Panel A: Tax ate of 28% Value,200.00,403.73,402.0, Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Panel B: Tax ate of 35% Value,083.33,327.7,325.45, Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Table 3: Numeical Measues of Tax Advantage to Equity: The implicit tax ate is the ate that, unde full taxation, would set the pesent value of afte-tax cash flows equal to thei calculated value. The cost of capital is the discount ate that sets the pesent value of the pe-tax, expected cash flows equal to the calculated value. The pecent of PV taxes is the pesent value of expected taxes paid, as a pecentage of the pesent value of taxes paid unde full taxation. The tax ate is assumed to be 35%, the expected net cash flow befoe pesonal tax is 00, and the afte-tax discount ate is 6%. the equation fo the fim s value, (28), as: V t Ĉ C t HGI n t C E t J tk i p tk i L τ p i i MN p o (38) τ p Inceases in volatility would affect the value though the effects of coelation between C t i and p t i on the expectation of thei quotient, and though Jensen s inequality. Since the pice is gowing geometically, howeve, these effects become elatively unimpotant vey quickly. The fim s value is not sufficiently nonlinea in the cash flows fo the volatility to have a significant effect, when the cash flows ae unifomly positive. Equation (38) also makes clea that the value of the fim must, as time advances, appoach the no tax value. The numbe of shaes is steadily deceasing, as cash is distibuted though epuchases, so the middle tem involving the basis eventually becomes negligible. This implies that lim t V t Ĉ C t 20

23 This may seem cuious given what is happening to the afte-tax cash flows. They can be witten as t p t τ p τ p p o C t τ p n t n t τ p p o (39) Fom this expession it is obvious that as the numbe of shaes outstanding appoaches zeo, the distibution becomes fully taxed. The pice at which shaes ae being epuchased gows, and a smalle faction of the distibution is shielded by the basis. These outcomes ae consistent, howeve. The fim epuchases shaes fom its ownes at a pice that leaves them indiffeent between selling, and incuing capital gains tax, and defeing. Since the option to defe indefinitely is always available, the highe the immediate tax liability on ealization, the highe a pice the shaeholdes demand until, in the limit, they eceive a pice equal to what the shaes would be woth to new, pivate buyes in an untaxed wold. 4 Uncetain, Negative Net Cash Flows We next investigate the effect of negative net cash flows on the pesonal tax liability of the fim s shaeholdes. When the fim has negative net cash flow, and must issue new shaes, these shaes will be issued at the pevailing pice, which may be consideably highe than the initial pice. This, in tun, will ceate new tax shields, though highe basis shaes, and lowe the subsequent aggegate tax buden fo shaeholdes. Unfotunately, a full numeical solution to this poblem is computationally unmanageable, because of the path dependencies and expansion of the state space due to the need to tack the tax basis at which all outstanding shaes ae held. In ode to evaluate quantitatively the value of pesonal taxes paid in this situation, we appoximate the dynamics of the quantities of inteest and estimate thei values though simulation. The execise is simila in spiit to, although consideably less complicated than, the appoach employed in Gaham (996). This method can also be easily adapted to accomodate gowth in the fim s pe-tax net cash flows. 4. The M&M Case of Constant Expected Cash Flows Fist, we take the aggegate ex-distibution value of the fim to be fixed and exogenous, at V F τ o Ĉ τ o 2

24 fo a given tax ate τ o. Using this ex-distibution value, we simulate a sample path fo the fim s net cash flows befoe pesonal tax unde the isk-neutal measue, and ceate a coesponding sample path fo the stock pice and numbe of shaes. The the pice pe shae at date t is given by the following laws of motion fo the pice pe shae, The numbe of shaes obeys p t n t C t τ o V F τ oo (40) n t n t C t p t (4) whee, if C t F 0, shaes will be issued. To detemine the taxes paid along the sample path, we assume that if the cuent pice is less than the basis at which any shaes ae cuently held, all those shaes have thei basis eset at the cuent pice and pay negative taxes of τ p times the associated capital loss. If shaes ae epuchased, we assume the highest basis shaes ae tendeed fist and epuchased. The only taxes paid along a sample path ae the positive taxes associated with gains ealized in a epuchase, and the negative taxes paid when the pice dops below the level of the basis on shaes outstanding, occasioning a capital loss. If the fim expeiences a negative cash flow, but the pice exceeds all outstanding basis values, new shaes ae issued but taxes ae zeo fo that date. Since the pobabilities used in dawing cash flows ae those associated with the isk neutal measue, we can daw many sample paths, and aveage the taxes paid acoss them fo at a given date t, to estimate the expected taxes paid unde the isk-neutal measue. We then discount this aveage fo t peiods to detemine a pesent value of taxes paid in peiod t. Adding these pesent values gives the pesent value of expected taxes paid. Denote this quantity as PVTAX τ o. Note that this quantity depends on the tax ate used to compute the exogenous ex-distibution value we assumed initially. Fo example, if we use τ p to compute V F, we will undestate the ex-distibution value and, thus, undestate the initial basis and ovestate subsequent taxes tiggeed by epuchases. To minimize these effects, we adjust the tax ate used to compute V F τ. Unde full taxation at ate τ the pesent value of taxes paid is just τĉ. We solve fo the tax ate at which this quantity is equal to the estimated pesent value of expected taxes paid. That is, we find τ PVTAX τ o Ĉ 22

25 We then epeat the simulation, calculate PVTAX τ, and iteate in this way until we find that τ k P τ k. We find the tax ates convege vey quickly. In effect, we ae iteatively solving the equation Ĉτ PVTAX τ (42) This appoximation will give aveage taxes paid that ae simila in magnitude to what we would expect unde a full solution fo the equilibium value of the fim. It will distot somewhat, howeve, the time path of taxes paid. A check on the accuacy of the method can be made by implementing the appoximation in the cetainty case, whee we have an analytical solution fo the equilibium value, and in the case whee cash flows ae all positive, whee we have a numeical solution fo the initial pice. This execise suggests the appoximation is quite accuate. Table 4 povides esults on the taxes paid unde this appoximation fo the dynamics of fim value. We assume thee ae two states fo the pe-tax cash flow, which has an expected value of 00 in evey peiod. Thus, the model confoms to the M&M assumption of constant, pepetual expected cash flow. The afte-tax discount ate is assumed to be 6%. The table epots esults fo two tax ates and two levels of volatility fo the cash flows. It epots the same measues of the tax benefits of equity with epuchases as peviously epoted fo the cases of cetainty and positive cash flows. The diffeence between the cost of capital unde epuchases and the pe-tax cost of capital unde dividends is in evey case substantial. The implicit tax ate on equity flows is, again, substantially lowe than the pesonal tax ate, and the diffeence inceases as the pesonal tax ate inceases. The estimated pesent value of pesonal taxes paid as a faction of the pesent value of the tax liability unde full dividend payout vaies between 55% and 65%. Ove all, the magnitudes in Table 4 ae simila to those epoted fo the cetainty case, and fo the case whee cash flows ae stictly positive. The effective tax ate on flows to equity holdes is, fo example, close to 20% when dividends ae taxed at 35%. Thus, it appeas that the tax shields affoded by the oiginal basis educe the pesonal tax buden quite damatically, but the equity is also fa fom tax fee as assumed in Mille (977). The pesent value of the tax liability appeas to incease somewhat with the volatility of the cash flows. This may seem supising given that shaeholdes, who have feedom to defe o ealize gains and losses, have an option-like claim on thei shaes. The shape of the equity holdes tax liability is ambiguous, howeve. When the fim has a positive cash flow, the afte-tax cash flow to equity is given by the ight-hand 23

26 Cash Flow Outcomes Full tax C 20 20D C 80 60D No tax Panel A: Tax ate of 28% Value,200.00,392.76,402.52, Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Panel B: Tax ate of 35% Value,083.33,35.79,324.50, Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Table 4: Cost of Capital and Implicit Tax Rates with Negative Cash Flows:The implicit tax ate is that ate that, unde full taxation would set the pesent value of the afte tax cash flows equal to thei calculated value. The cost of capital is the discount ate that sets the pesent value of of the pe tax, expected cash flows equal to the calculated value. The pecent of PV taxes is the pesent value of the of taxes paid, as a pecentage of the the pesent value of the taxes paid unde full taxation. taxes paid assuming fully taxed dividends. Calculations assume an afte-tax discount ate of 6%, and that the pobability of the highe cash flow is two-thids. The simulations used to estimate the pesent value of taxes paid wee caied out ove 200 peiods, with 2,000 sample paths fo each case. side of (39). It is linea in the cash flow, with a slope of τ p and an intecept of n t n t τ p p o. If the fim has a negative cash flow, and the pice exceeds the basis values of outstanding shaes, the afte-tax cash flow is just C t. When the pice dops below basis values, thee ae negative taxes. In the simulations epoted in Table 4, negative taxes ae only paid % to 2% of the time. Conside, theefoe, the othe two cases. As n t n t 0, the afte-tax claim of the shaeholdes becomes minc C t τ p C t D, which is concave and theefoe will have a value that deceases with the volatility of the cash flows. This effect appeas to dominate in the numeical esults. Figues 4 and 5 depict the time path of taxes paid. Figue 4 shows the taxes paid in each peiod, as a faction of the cash flow, unde cetainty, using C t 00, τ p 35%, and othe paametes the same as in Table 4. The taxes ae small initially because of the tax shield supplied by the oiginal basis. As shaes ae epuchased, the shae pice gows, and the tax shield fom the basis becomes negligible. Within 50 peiods, the flows ae being taxed at close to the full 35% ate. Figue 5 shows that with uncetainty and the possibility of negative cash flows a simila patten emeges. In it we plot the aveage taxes (unde the 24

27 matingale measue) paid in each peiod, acoss sample paths in the simulation epoted in the thid line of Table 4. Finally, in Table 5 we povide infomation about the elative accuacy of ou diffeent methods of calculating o estimating the pesent value of the tax liability. When cash flows ae positive, we can compae the appoximation in this section to the numeical solutions epoted in Section 3. Similaly, fo the cetainty case, we can compae both the numeical solution and the appoximation to the closed-fom solution. The table shows that the appoximation undestates the pesent value of the tax liability, but this bias is of simila magnitude to the numeical eos associated with the methods fom Section 3. Cash Flow Outcomes C 00 00D C 20 60D Appox. Numeical Closed-Fom Appox. Numeical Panel A: Tax ate of 28% Value,407.99,404.8,400.40,407.06, Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Panel B: Tax ate of 35% Value,336.5,328.39,323.70,336.09,327.7 Implicit tax ate (%) Cost of capital (%) Pecent of PV taxes (%) Table 5: Cost of Capital and Implicit Tax Rates with Negative Cash Flows: The implicit tax ate is that ate that, unde full taxation would set the pesent value of the afte tax cash flows equal to thei calculated value. The cost of capital is the discount ate that sets the pesent value of of the pe tax, expected cash flows equal to the calculated value. The pecent of PV taxes is the pesent value of the of taxes paid, as a pecentage of the the pesent value of the taxes paid unde full taxation. taxes paid assuming fully taxed dividends. Calculations assume an afte-tax discount ate of 6%, and that the pobability of the highe cash flow is two-thids. The simulations used to estimate the pesent value of taxes paid wee caied out ove 200 peiods, with 2,000 sample paths fo each case. The column maked Appox. gives esults obtained by applying the numeical method descibed in Section 4, the column maked Numeical gives esults fom applying the ecusive method descibed in Section 3 and the column maked Closed Fom gives the esults fom applying Equation 2. 25