3. korteweg, 2137-70

THE JOURNAL OF FINANCE • VOL. LXV, NO. 6 • DECEMBER 2010

The Net Benefits to Leverage

ARTHUR KORTEWEG∗

ABSTRACT

I estimate the market’s valuation of the net benefits to leverage using panel data from1994 to 2004, identified from market values and betas of a company’s debt and equity.The median firm captures net benefits of up to 5.5% of firm value. Small and profitablefirms have high optimal leverage ratios, as predicted by theory, but in contrast toexisting empirical evidence. Companies are on average slightly underlevered relativeto the optimal leverage ratio at refinancing. This result is mainly due to zero leveragefirms. I also look at implications for financial policy.

THEORIES OF OPTIMAL capital structure typically explain companies’ choice ofdebt versus equity financing by a trade-off: firms choose a leverage ratio thatoptimally weighs the benefits of debt such as interest tax shields (Kraus andLitzenberger (1973)) and agency benefits due to reductions in free cash flow(Jensen (1986)) against the costs of debt, which include the direct costs ofbankruptcy (Warner (1977) and Weiss (1990)) as well as indirect costs such asdebt overhang (Myers (1977)), asset substitution (Jensen and Meckling (1976)),and asset fire-sales (Shleifer and Vishny (1992)). The literature tests this trade-off by running cross-sectional regressions of leverage on a set of variables thatproxy for the benefits and costs (see Harris and Raviv (1991) for an overview,and recent work such as Rajan and Zingales (2003) and Frank and Goyal(2004)).

A shortcoming of the regression approach is that it is not possible to detectwhether firms have too much or too little debt on average. This question isimportant in light of the claim that companies leave a substantial amount oftax benefits on the table and are therefore systematically underlevered (Miller(1977) and Graham (2000)).

A second drawback of the regression method is the implicit assumptionthat firms are always optimally levered, resulting in misleading effects of

∗Arthur Korteweg is from the Graduate School of Business, Stanford University. This paper isbased on my dissertation entitled “The Costs of Financial Distress across Industries,” completed atthe University of Chicago. I thank my committee—Monika Piazzesi, Nick Polson, Morten Sørensen,and Pietro Veronesi—for their guidance and support. This paper has benefited greatly from sug-gestions by an anonymous referee, an associate editor, and Editor Campbell Harvey. I also thankMike Barclay, Alan Bester, Hui Chen, Peter DeMarzo, John Heaton, Dirk Jenter, Anil Kashyap,Paul Pfleiderer, Michael Roberts, Jay Shanken, Robert Stambaugh, Ilya Strebulaev, Amir Sufi,Michael Weisbach, Jeff Zwiebel, and seminar participants at Boston College, Emory, the Univer-sity of Chicago, Georgia, London Business School, Notre Dame, Rochester, Stanford, Wharton, andthe Board of Governors of the Federal Reserve for helpful discussions, comments, and suggestions.All errors remain my own.

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2138 The Journal of Finance R©

profitability and firm size on optimal capital structure. In reality, firms choosetheir capital structures in a dynamic setting. Companies let leverage ratiosdrift until the gain from rebalancing outweighs the cost of adjusting (Fischer,Heinkel, and Zechner (1989) and Leary and Roberts (2005)), so that firms areaway from their optimal capital structures most of the time without behavingsuboptimally. High profits mechanically lower leverage, so that cross-sectionalregressions show a negative relation between profitability and leverage, even ifoptimal-debt ratios are positively related to profitability. Similarly, small firmsface higher issuance costs and therefore wait longer between refinancings, re-sulting in lower average leverage than big firms, even though in theory theymay have higher optimal-debt ratios.

In light of these issues, this paper addresses four main questions in the cap-ital structure literature: (i) how large are the net benefits to debt financing,(ii) how does optimal capital structure vary with firm characteristics, partic-ularly profitability and size, (iii) are firms on average underlevered, and (iv)how large are the friction costs of being away from the optimal leverage and dofirms refinance when these costs become large?

Using a sample of 29,753 firm-months for 290 firms across 30 Fama–Frenchindustries (Fama and French (1997)), I estimate the net benefits to debt financ-ing using a new relation between a firm’s market value, systematic risk (beta),and net benefits to leverage, extending the Modigliani and Miller (1958) result.In this model, net benefits are defined as the (ex ante) present value of all futurebenefits minus the costs of debt. Assuming that firms within an industry havethe same asset beta, cross-sectional differences in equity and debt betas are en-tirely driven by the net benefits. I use this cross-sectional variation to identifythe level of net benefits, and how they vary as a function of firm characteristics.An important benefit of this approach is that it is not susceptible to the twoproblems with cross-sectional regressions described above. The model identi-fies firm-specific optimal capital structures even if firms’ leverage ratios floataround the optimum due to transaction costs, or if companies consistently takeon too much or too little debt. In fact, such cross-sectional spread in leverage isneeded to identify the optimal capital structure.

The paper’s primary findings are as follows. First, I find that the net benefitsto leverage amount to as much as 5.5% of firm value. This means that themedian firm at its value-maximizing leverage ratio is worth 5.5% more thanthe same firm with no debt in its capital structure. The net benefits are higherfor highly profitable firms with low depreciation, stable profits, and low market-to-book ratios, and during economic expansions. Severely distressed firms havenegative net benefits in the range of −15% to −30% of firm value. Assumingthat these companies have no expected benefits of future debt financing, thecosts of financial distress are 15% to 30% for firms in or near bankruptcy. Thesedistress costs represent the present value of future cash flows that are lost dueto the presence of debt in the firm’s capital structure, and include both directand indirect costs. Andrade and Kaplan (1998) estimate ex post distress costsof 10% to 23% of firm value using a small sample of distressed buy-out firms.Since buy-out firms likely have lower distress costs, it is not surprising that Ifind higher numbers.

The Net Benefits to Leverage 2139

Second, net benefits increase in leverage for low-debt firms but decreasewhen leverage becomes high. This result implies that there is some optimalcapital structure that maximizes the benefits net of the costs of debt financing.In a dynamic setting, this is the leverage ratio that firms will choose whenrefinancing. I find that smaller, more profitable firms have higher optimal-debtratios. In contrast, existing empirical evidence, which is based on observedleverage ratios, finds the opposite relation (Graham (2000), Myers (2001),Korajczyk and Levy (2003), and studies cited in Harris and Raviv (1991)). Thisresult highlights the distinction between analyzing observed and optimal lever-age ratios. Similarly, I find that optimal leverage moves procyclically, whereasKorajczyk and Levy (2003) find the opposite result for observed leverage ratios.Bhamra, Kuehn, and Strebulaev (2010) and Chen (2010) show that counter-cyclical observed leverage and procyclical optimal leverage are consistent withdynamic capital structure models in the presence of fixed transaction costs.Finally, consistent with the literature, optimal leverage is positively related tothe proportion of tangible assets and negatively related to depreciation, profitvolatility, and market-to-book ratios.

Third, I find that firms are on average slightly underlevered, relative totheir target (the optimal capital structure upon refinancing in dynamic mod-els of the firm) capital structure. This does not necessarily imply that firmsbehave suboptimally if they face market frictions such as transaction costsof changing leverage. The underleverage result is mainly due to firms thatpay little or no interest (true zero leverage firms). The most puzzling sub-set of firms are those without debt that pay dividends to their shareholdersand should find it easy to lever up (Minton and Wruck (2001) and Lemmonand Zender (2003)). These firms are highly underlevered, and a new resultin this paper is that the market expects them to lever up in the future andcapture some of the benefits to leverage. Conversely, zero leverage firms thatdo not pay dividends are not expected to lever up. These companies act as ifthey face costs to levering up that are much larger than the issuance costsalone.

Firms that are about to be taken private in a leveraged buy-out are sub-stantially underlevered, whereas most noninvestment grade firms have toomuch debt. Overlevered firms are likely to reduce their indebtedness whenthe gains of doing so are moderately large. However, the most overlev-ered companies with the highest potential gains to unlevering are the leastlikely to refinance. This result is consistent with market frictions such asdebt overhang (Myers (1977)) and creditor hold-out problems that preventfirms from refinancing immediately. The cost of reducing debt levels for dis-tressed firms is much higher than the mere cost of issuing new securities(Kane, Marcus, and McDonald (1985) and Gilson (1997)). In contrast, I findthat the cost of being underlevered is much lower than the cost of beingoverlevered.

Two recent papers are related to this work. Almeida and Philippon (2007)use the estimates of ex post costs of financial distress in Andrade and Ka-plan, and calculate the ex ante distress costs using risk-neutral probabilitiesof default in a multiperiod setting. They find that the average firm generally


picks a capital structure that balances the costs of debt with the tax benefitsfrom Graham (2000), but they do not address the cross-section of firms. vanBinsbergen, Graham, and Yang (2010) use a large sample of companies thatthey argue are likely to make close to equilibrium leverage decisions. They es-timate marginal cost of debt functions for individual firm-years from variationsin the tax benefits of debt using three identification approaches that aim to holdthe marginal cost function fixed. They then integrate under and between themarginal cost and simulated tax benefit functions to estimate the net benefitsto debt, and find it to be around 4% of asset value. Overall, it is reassuringthat van Binsbergen, Graham, and Yang and my method yield similar results,given that we use very different empirical approaches.

The paper is organized as follows. Section I explores the relation betweenthe net benefits to leverage and the market values and betas of corporate debtand equity. Section II explains the identification of net benefits by invertingthis relation. Section III describes the estimation methodology that applies themodel to the data. The data are presented in Section IV. I discuss the resultsin Section V, and Section VI concludes.

I. Modigliani–Miller with Net Benefits to Leverage

Modigliani and Miller (1958) consider the firm as a portfolio of all outstandingclaims on the company. The total market value of the company at time t, V L

t ,is the sum of the market values of the individual claims

V Lt = Dt + Et, (1)

where Dt and Et are the market values of corporate debt and equity, respec-tively, at time t.1

A different way of decomposing the same company is

V Lt = V U

t + Bt, (2)

where V Ut is the market value of the unlevered firm. It is equal to the value

of the company at time t if all its debts were repurchased by its shareholders.The differences between V L

t and V Ut represent the net benefits to leverage, Bt,

a fictitious security defined as the expected present value of the benefits minusthe costs of debt financing. The benefits of debt financing include interest taxshields and decreases in agency costs due to the presence of debt in the firm’scapital structure, such as the reduction in free cash flows that managers canspend on perks or unproductive pet projects (Jensen (1986)). The costs of debtrepresent all future cash flows that are lost as a result of the presence of debtin the firm’s capital structure, and represent the direct and indirect (agency)

1 The debt and equity claims can be decomposed further into corporate bonds, bank debt andcapitalized leases, and common and preferred equity, but it is not necessary to do so for the purposeof this paper.


costs that are realized both before and after default.2 Since Bt is a marketexpectation of future cash flows, it also reflects the market’s opinion of futurefinancing policy and transaction costs of adjusting capital structure. I discussthe interpretation of Bt in more detail in the next section.

The company also has systematic risk, βLt , proportional to the (conditional)

covariance of returns to the firm with some risk factor. The decomposition in (1)yields the firm’s systematic risk as a weighted average of the debt and equitybetas βD

t and βEt

βLt = Dt

V Lt

βDt + Et

V Lt

βEt . (3)

Note that while this equation appears to be derived under the Modigliani andMiller (MM) assumption of zero taxes, equation (3) is a mechanical identitythat holds even in a world with taxes, and under any financial policy, sincewe have the levered firm’s beta on the left-hand side. As I show below, with notaxes or distress costs, βL

t = βUt and we find the traditional MM relation

βUt = Dt

V Lt

βDt + Et

V Lt

βEt . (4)

The difference between βLt and βU

t is clearly seen when one writes the leveredfirm’s beta using the decomposition in equation (2):

βLt = V U

t

V Lt

βUt + Bt

V Lt

βBt . (5)

By definition, the systematic risk of the unlevered assets, βUt , is not affected

by the firm’s capital structure. The effect of leverage on the beta of the leveredfirm, βL

t , is driven entirely by the net benefit of debt financing, Bt, and itssystematic risk, βB

t . In the absence of taxes and distress costs, Bt = 0, and themodel reverts to the classic MM no-taxes case, where V L

t = V Ut and βL

t = βUt .

Equate the expressions for V LT in (1) and (2), and the expression for βL

T in (3)and (5).

V Ut + Bt = Dt + Et, (6a)

V Ut

V Lt

βUt + Bt

V Lt

βBt = Dt

V Lt

βDt + Et

V Lt

βEt . (6b)

2 Examples of costs of debt are the impaired ability to do business due to customers’ concerns forparts, service, and warranty interruptions or cancelations if the firm files for bankruptcy (Titman(1984) and Titman and Opler (1994)), investment distortions due to debt overhang (Myers (1977))and asset substitution (Jensen and Meckling (1976)), employees leaving the firm or spending theirtime looking for another job, and management spending much of its time talking to creditors andinvestment bankers about reorganization and refinancing plans instead of running the business.


These are the key equations that I will use in the next section to identify thenet benefits.

To illustrate how βBt accounts for taxes, consider the well-known special case

of a constant marginal corporate tax rate, τ , no nontax benefits to debt, and nocosts of financial distress. Bierman and Oldfield (1979) show that with constantface value of debt, the present value of the tax shield is Bt = τ Dt. Equations (6a)and (6b) then become

V Lt = V U

t + τ Dt, (7a)

βUt = Dt

V Ut

(1 − τ )βDt + Et

V Ut

βEt . (7b)

Equation (7a) shows that the value of the levered firm equals the value of theunlevered firm plus the present value of the interest tax shield, and equation(7b) is the textbook unlevering expression.

With both benefits and costs to leverage, the trade-off theory of optimalcapital structure predicts that the company’s market value becomes a hump-shaped function of leverage. Similarly, the firm’s beta, βL

t , varies with leverageas determined by the systematic risk of the net benefits. In the next section,I show how these two relations identify the net benefits from the variation inlevered firm values and betas within an industry, under certain identifyingassumptions.

II. Identification of the Net Benefits to Leverage

Consider a simple case in which equations (6a) and (6b) hold without error.We can observe the market values of corporate debt and equity, but since wedo not observe unlevered firm values, it is not possible to calculate B directlyfrom equation (6a).3 A similar problem arises when looking at betas instead ofvalues: the right-hand side of equation (6b) can be estimated, but the unleveredasset beta cannot be observed. However, if firms in the same industry have acommon asset beta, βU, then differences in βL within an industry are drivenentirely by the net benefits to leverage. Assume that net benefits are a functionof an observable variable X, B = B(X), and that this function is the same forall firms. All companies in an industry then have the same βL at the same X.Figure 1 illustrates the intuition behind this identification result. Industrypeers fall on the same graph of βL versus X, the shape of which depends onlyon B(X). We can therefore estimate B(X) from firms’ levered betas and X, bothof which are observable.

The variable X can be any characteristic of the firm, but a likely candidate isa firm’s leverage ratio, L. The trade-off theory predicts that net benefits to debt

3 Cutler and Summers (1988) and Andrade and Kaplan (1998) take first differences of (6a) toeliminate V U and achieve identification through a natural experiment. This approach tends toyield very small samples.


Figure 1. Levered firm beta as a function of one covariate. This figure depicts the hypo-thetical relation between a firm characteristic, X, and the beta of the levered firm, βL, when thenet benefits to debt depend only on the characteristic (B = B(X)). Levered firm beta is defined asthe weighted average of debt and equity betas: βL = D

V L βD + EV L βE. The line traces the relation

between βL and X. The dots on the line represent firms with identical (unlevered) asset betas, βU .

financing rise for companies with low debt but decrease as leverage becomeshigh, implying that B is a nonlinear function of L. Assume for now that B is afunction of L only, B = B(L). Figure 1 shows that it is necessary to observe aspread in L within an industry in order to identify B(L).

In a dynamic capital structure world, shocks to the firm exogenously changeleverage (Welch (2004)). This gives rise to two distinct costs as L drifts awayfrom the optimal capital structure, L∗: the cost of moving to L∗, and the costof being away from L∗. The former are transaction costs or, as referred to byLeary and Roberts (2005), adjustment costs. Transaction costs include issuancecosts of new securities, coordination costs (such as creditor hold-out problems),adverse tax consequences of debt concessions, and asset fire-sales. These costsare particularly important for highly levered, distressed firms (Gilson (1997)).Fischer, Heinkel, and Zechner (1989) show that even small transaction costsproduce large variation in observed leverage ratios. Since individual firms haveexperienced different shocks in the past, we observe a spread in leverage ratiosthat can be used to estimate B(L).

Certain market frictions, such as creditor hold-outs and regulations thatdiscourage institutions from writing down debt or exchanging debt for equity,delay refinancings (especially for highly levered firms). Consequently, even iffirms want to refinance, they may not be able to do so immediately. Firms thenincur costs of being away from L∗, such as lost cash flows from financial distresscosts, and these costs are reflected in B. Since out-of-court restructurings cantake months to complete, and bankruptcies even longer, these costs can beespecially large for distressed firms.

At a given leverage ratio, B(L) reflects the present value of all expectedfuture benefits and costs to debt, based on the market’s expectation of the firm’sfinancial policy. The difference between B(L) and B(L∗) is then a measure ofthe transaction costs of moving from L to L∗, and the costs of being away fromL∗. I refer to the total of these costs as “friction costs.”

In the model described above companies optimally allow leverage to vary,yielding a range of optimal leverage ratios, consistent with the empirical find-ings of Leary and Roberts (2005). However, a key feature of the approach in thispaper is that it is not necessary to assume that firms are optimally levered, not


even on average. If management is not maximizing firm value, or perceives opti-mal capital structure to be different from the market, then firms have leverageratios that on average do not maximize net benefits. For example, most firms inFigure 1 are overlevered, having leverage ratios that do not minimize the firm’scost of capital, βL. Nonetheless, as long as a spread in L is observed we willcorrectly identify the shape of B(L).4 We can therefore use observed leverageratios to identify the market’s assessment of B(L), which in turn pins down themarket’s opinion of optimal leverage (or target leverage) as the point where netbenefits are maximized. This market-implied optimum can be compared to av-erage observed leverage ratios to assess whether firms are on average under- oroverlevered.

Under restrictive assumptions, B is indeed a function of leverage alone.5

However, firms within an industry may differ, for example, in terms of growthopportunities and asset tangibility, and these characteristics may vary overtime. We can extend the identification result to cases where firms within anindustry have different net benefits at the same leverage ratio. In general, twoidentification assumptions are necessary:

A1: The unlevered asset beta, βUit , is either (i) the same for some subset of

firms, βUit = βU

t , or (ii) constant over time for the same firm, βUit = βU

i .A2: Net benefits are a function of observable variables and the value and beta

of the unlevered firm only: Bit = B(Xit).

Note that Xit can be a vector of observable variables. Equations (6a) and(6b) have to hold for each firm i in month t, so with N firms and T monthsof data, there are 2NT equations. These equations have to be solved for 4NTunknowns: the value of unlevered assets, V U

it , the net benefit of debt financ-ing, Bit, and their respective betas (βU

it and βBit ), for each firm-month. Under

assumption (A1(i)), the asset beta varies over time but is equal across the Nfirms, eliminating (N − 1)T unknowns. Assumption (A2) reduces the 2NTunknown Bit and βB

it to a set of k parameters that determine the value of B(Xit).Together, assumptions (A1) and (A2) reduce the problem to (N + 1)T + k un-knowns: the NT unlevered firm values, the T unlevered asset betas, and thek parameters. With 2NT equations, where we observe N firms over T timeperiods such that (N − 1)T ≥ k, we can solve for all parameters exactly. Forexample, with three parameters in the function for Bt, it is sufficient to observefour firms for 1 month, or two firms for 3 months. A similar derivation holdsunder assumption (A1(ii)).

4 I confirm this in unreported simulations where I force firms to stay in a suboptimal region ofleverage.

5 Specifically, B = B(L) if all firms within an industry have similar investment opportunities,production technology, and asset tangibility; if they produce similar goods or services (e.g., durableversus nondurable goods); and if these characteristics are stable over time. Structural models (e.g.,Merton (1974) and Leland (1994)) then imply a one-to-one relation between L and tax benefits aswell as a firm’s probability of default. As an example of a structural motivation for the specificationof B, Leland’s (1994) model implies B/V = θ1L+ θ2LX+1 with θ1 = 1+X

Xτ

1−τand θ2 = −( 1+X

Xτ

1−τ+ α),

where α is the loss-given-default, X ≡ 2r/σ 2, and L ≡ VB/V .


The identification argument in this section is based on the model equationsholding exactly. The next section allows for error terms in the model equationsand describes the methodology used to estimate the model.

III. Estimation

The empirical implementation in this paper estimates the following specifi-cation of net benefits relative to total firm value:

Bit/V Lit = X′

0itθ0 + (X′1it · Lit)θ1 + (

X′2it · L2

it

)θ2. (8)

The squared leverage term captures the nonlinear effect of leverage on netbenefits predicted by theory: at low leverage, tax benefits and reductions inagency costs increase firm value, but their marginal benefit declines as leveragegrows. In addition, the costs of financial distress negatively impact B at highL. The vectors X0it, X1it, and X2it include a constant and firm characteristicssuch as profitability, the proportion of intangible assets, and market-to-bookratios. These characteristics interact with leverage to make net benefits growor decline faster as firms change their capital structure. The parameter vectorsθ0, θ1, and θ2 are common to all firms.

Substituting (8) into equations (6a) and (6b), and adding error terms thatallow for temporary deviations, the full model becomes

V Uit

V Lit

= 1 − X′0itθ0 − (X′

1it · Lit)θ1 − (X′

2it · L2it

)θ2 + uit, (9a)

V Uit

V Lit

· βUk(i)t = Lit · βD

it + (1 − Lit) · βEit

− [X′

0itθ0 + (X′1it · Lit)θ1 + (

X′2it · L2

it

)θ2

] · βBit + vit,

(9b)

⎡⎢⎢⎢⎢⎣

rUit − r f

t

rBit − r f

t

rEit − r f

t

rDit − r f

t

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

αUi

αBi

αEi

αDi

⎤⎥⎥⎥⎦ +

⎡⎢⎢⎢⎣

βUk(i),t−1

βBi,t−1

βEi,t−1

βDi,t−1

⎤⎥⎥⎥⎦ · (

rMt − r f

t) + εit, (9c)

βUkt = φ0 + φ1 · βU

k,t−1 + ηkt. (9d)

The error term uit is by assumption orthogonal to the explanatory variablesX0it, X1it, X2it, and Lit, which rules out the potential simultaneity problem of Litand Bit being jointly determined. All variables that affect Bit are included inXit, and the optimal capital structure is determined by B(Xit), so that uit doesnot show up in the first-order condition for optimal leverage. I assume thatboth uit and vit are distributed Normal with mean zero, and are i.i.d. over time.


The errors are allowed to be contemporaneously correlated across firms in thesame industry. Since the beta relation is derived from the value equation, vit isalso allowed to be correlated with ujt for firms i and j in the same industry.

In the discussion of identification, it was assumed that the conditional betasof debt and equity are observed, but in reality they have to be estimated. Theset of equations (9c) augments the model with the regression equations to es-timate the conditional betas with the market portfolio. I define rt as a returnfrom time t − 1 to t, and rM

t − r ft is the return on the market portfolio in excess

of the 1-month risk-free rate. Since the beta relations derived in this paper aremechanical, the regression equations in (9c) do not imply that the CAPM isthe true asset pricing model. The intercepts are therefore not required to equalzero. The 4-by-1 idiosyncratic returns vector εit is by assumption orthogonal tothe excess market return, and distributed Normal with mean zero and constantcovariance matrix. The elements of εit are independent over time but can becontemporaneously correlated across firms in the industry since there is likelyto be substantial cross-sectional correlation between idiosyncratic returns ofdebt, equity, and unlevered assets of the same firm, as well as between firmswithin the same industry. The estimation also allows for εit to be contempora-neously correlated with uit and vit.6

The asset betas, βUkt , are assumed equal for the cross-section of firms within

the same industry, k. This assumption is frequently used in the academic liter-ature (e.g., Kaplan and Stein (1990), and Hecht (2002)). The economic intuitionbehind this assumption is that firms in the same industry have the same mar-ket risk of operations. Hamada (1972) and Faff, Brooks, and Kee (2002) provideempirical support for the hypothesis that asset betas with respect to the mar-ket portfolio are the same within industries (as defined by two-digit SIC codes).Simulations show that minor violations of (A1) increase the standard error ofparameter estimates of the function B(Lt), but do not cause severe inconsistencyin the parameters, even when βU

kt is correlated with Xit.7

The industry asset betas are allowed to vary over time and follow the mean-reverting AR(1) process (9d), with |φ1| < 1.8 The AR(1) restriction on βU

kt , al-though not strictly necessary, helps to smooth the beta process so that resultsare more stable. The error terms ηkt are distributed i.i.d. Normal with meanzero and constant variance, and are uncorrelated with εt. Similarly, it is notnecessary for the estimation to impose a time-series process on the equity anddebt betas, but to ensure smoothness and tighter estimation bounds I run theestimation with an AR(1) on debt and equity betas, with a general correlationstructure.

I estimate the parameters of the model jointly with the conditional betasand the unlevered asset values, using a Markov Chain Monte Carlo (MCMC)

6 Note that βBit is estimated from the time series of Bit as implied by (8), and is therefore implicitly

a function of leverage; the explanatory variables X0it, X1it, and X2it; and parameters θ0, θ1, and θ2.7 The simulation results are reported in the Internet Appendix, available at http://www.afajof.

org/supplements.asp.8 Previous studies (e.g., Berk, Green, and Naik (1999)) argue that betas should be mean-

reverting to ensure stationarity of returns.


algorithm. This simulation-based estimation methodology is explained in detailin Robert and Casella (1999) and Johannes and Polson (2004), and in partic-ular for structural models of the firm in Korteweg and Polson (2009). MCMCprovides a way of sampling the posterior distribution of the model’s parame-ters and unobserved variables (the betas and unlevered asset values), giventhe observed values of debt and equity. Once this sample is obtained, the unob-served variables are numerically integrated out, leaving the distribution of theparameters θ0, θ1, and θ2, conditional on the observed data. This integrationstep needs to be done only once.

At the core of this methodology lies the Clifford–Hammersley theorem, whichallows for a break up of the joint posterior distribution of parameters, betas,and unlevered asset values. Instead of drawing from the joint distribution, thetheorem allows one to separately draw from (i) the distribution of parametersgiven the betas and unlevered asset values, (ii) the distribution of betas givenparameters and unobserved values, and (iii) the distribution of unlevered assetvalues given parameters and betas. These complete conditionals are easy toevaluate and sample from, using simple regressions and basic linear filters.

As an added bonus, MCMC provides a convenient way to deal with missingdata. This is especially useful for companies with infrequently traded bonds. Inessence, missing values are treated as additional model parameters. The sam-pling procedure automatically takes into account the uncertainty over thesevalues, and they are integrated out at the end.9

IV. Data

I construct a sample of monthly debt and equity values for firms in theNational Association of Insurance Commissioners (NAIC) database over theentire coverage period 1994 to 2004. Insurance companies are required to fileall their trades in corporate bonds with the NAIC, which makes these recordsavailable in electronic form. Hong and Warga (2000) report that insurancecompanies account for about 40% of the trades in the investment grade bondmarket, and 25% of the trades in the market for noninvestment grade bonds.With over 1.3 million transactions in total, the NAIC database is the mostcomprehensive source of corporate bond prices currently available. Data onthe amount outstanding, seniority, and security of each bond come from theFixed-Income Securities Database (FISD), compiled by Mergent.

From the NAIC transactions data, I compute month-end bond values foreach outstanding bond issue of every firm. Since not all bonds are traded everymonth, it is not always possible to aggregate the individual bond values toobtain the market value of all publicly traded debt. To mitigate this missingdata problem, I group together bonds of the same firm that are of equal securityand seniority, and that have a maturity within 2 years of one another. Assumingthese bonds have the same interest rate and credit risk, missing values are

9 The algorithm and sampling distributions, and a simulation study analyzing the performanceof the methodology in simulated data are available in the Internet Appendix.


calculated from contemporaneous market-to-book values of bonds in the samegroup that are observed in the same month. For those months in which noneof the bonds in a group trade, the estimation algorithm treats the valuationsas missing model parameters. The large bond issues of a firm trade more oftenthan small issues, and I select those firms for which the largest bond groupsrepresenting at least 80% of the company’s total bond face value trade at least50% of the time. On a face-value weighted basis, the corporate bonds in thesample trade about 71% of the time.

I also include firms from Compustat without any short- or long-term debt intheir capital structure. Even though these firms are currently unlevered, theirequity value is not necessarily equal to their unlevered value. If these firmsare expected to lever up in the future, some net benefits to future leverage willbe impounded into their valuation.10

I supplement the sample with monthly market values of equity (common pluspreferred) from CRSP and accounting data from Compustat, matching compa-nies to the FISD by their CUSIP identifier. I include the monthly dividend andinterest payments in the calculation of returns to debt and equity, to control fordifferences in payout policies that may affect the estimates of firms’ unleveredasset betas.

The assumption of homogeneous industry asset betas requires a working def-inition of industries. I consider two alternatives: (i) two-digit SIC codes (SIC2)and (ii) Fama–French (1997) classifications (FF). The FF classification assignsfirms to 48 industries based on four-digit SIC codes, and avoids some counter-intuitive groupings of firms that occur within two-digit SIC industries. I useonly those industries with data for at least two firms with some debt at anygiven time, a condition required for identification. Table I gives a list of theindustries that are covered and the number of firms and firm-months in eachindustry. The SIC2 sample comprises 232 firms in 22 industries, for a total of24,277 firm-months. Of these firms, 199 had some debt in their capital struc-ture over the sample period. The FF sample spans 290 firms in 30 industries,for a total of 29,753 firm-months. Of the FF firms, 233 had debt outstanding. Ofthe industries covered, the samples each represent about 7% of all firms, anda quarter of total equity market capitalization. The data are biased towardslarger firms, which have more actively traded bonds, but there is no bias to-wards more or less distressed firms. Table II reports summary statistics for theFF sample. The SIC2 sample looks very similar (see the Internet Appendix).

I define leverage as the market value of debt (net of cash) divided by themarket value of debt (net of cash) plus the market value of equity. Note that themarket values of bank debt and capitalized leases are never observed becausethese securities are not publicly traded. On average I observe market prices for61% of a firm’s debt on a book value basis. I calculate leverage and debt returnsusing the face value of the unobserved debt as a proxy for its market value.Model estimates are quantitatively very similar when I calculate the market

10 My thanks go to Ilya Strebulaev for suggesting the addition of zero leverage firms to thesample.


Table ISample Breakdown by Industry

This table presents a breakdown of the 1994 to 2004 sample into industries defined by two-digitSIC codes (SIC2, in Panel A) or Fama–French (1997) classification (FF, in Panel B). The observednumber of firms (# Firms) and firm-months (N) are reported for each industry.

Panel A: SIC2

Name Code # Firms N

Oil & gas 13 11 684Builders 15 5 504Food 20 12 1,416Paper 26 6 768Publishing 27 7 827Chemicals 28 24 2,844Petroleum products 29 4 480Primary metals 33 5 480Machinery 35 16 1,572Electric equipment 36 26 2,275Cars 37 8 972Instruments 38 5 648Transport (air) 45 5 619Telecom 48 17 1,358Utilities 49 17 1,894Wholesale (nondur) 51 9 1,128Retail (misc.) 53 8 1,032Insurance 63 7 495Patent & royalty 67 7 648Hotels 70 2 216Equipment services 73 19 2,132Health 80 12 1,285Total 232 24,277

Panel B: FF Industries

Name Code # Firms N

Food products Food 10 1,224Recreational products Toys 3 360Entertainment Fun 4 515Printing & publishing Books 7 827Consumer goods Hshld 11 1,222Apparel Clths 3 396Health care Hlth 12 1,285Medical equipment MedEq 10 939Pharmaceutical products Drugs 13 1,212Chemicals Chems 10 1,246Construction materials BldMt 3 396Construction Cnstr 9 708Steel works, etc. Steel 5 480Machinery Mach 9 1,044Electrical equipment ElcEq 17 1,387Automobiles & trucks Autos 5 576Aircraft Aero 3 396

(continued)


Table I—Continued

Panel B: FF industries

Name Code # Firms N

Petroleum & natural gas Enrgy 16 1,167Utilities Util 17 1,920Telecommunications Telcm 17 1,366Business services BusSv 18 2,100Electronic equipment Chips 6 732Measuring & control equipment LabEq 6 464Business supplies Paper 6 768Transportation Trans 12 1,181Wholesale Whlsl 12 1,441Retail Rtail 19 2,240Restaurants, hotel, motel Meals 6 514Insurance Insur 9 759Trading Fin 12 888Total 290 29,753

value of unobserved debt using the credit spread of the safest bonds (see theInternet Appendix).

It is important to observe a wide range of leverage ratios within each industryto get a clear picture of how net benefits vary with leverage. The intra-industrystandard deviation of leverage is typically around 0.16, compared to an averageleverage of 0.23, so leverage clearly varies substantially within industries. Interms of credit ratings, most industries contain firms with ratings that varybetween AA and B, and some industries (such as Air Transportation (SIC 45)and Telecom (SIC 48)) have defaulted firms in the sample.

V. Results

In this section, I discuss the estimates of the net benefits to leverage andimplications for optimal capital structure and financial policy, paying specialattention to zero leverage and leveraged buy-out (LBO) firms.

A. Net Benefits to Leverage

Estimates for two specifications of equation (8) are presented in Table III.Specification I groups variables that are associated with the benefits of debtin X1it and variables that drive the costs of debt in X2it. The reason for thisseparation is that at low-leverage ratios, the linear term X′

1it · Lit dominates.Since costs of debt are negligible at low indebtedness, X1it captures the benefitsto debt financing. As leverage increases, so do the probability of distress andthe costs of debt, and the squared leverage term in (8) becomes important.

Table III shows that in this separated model, highly profitable firms (mea-sured by EBITDA/Sales) with stable profits have high present values of net


Table IISummary Statistics

This table presents summary statistics for the 29,753 firm-month observations in the FF sample.Variables are defined as follows: PROF is the most recent EBITDA/Sales (annual Compustat item13 divided by item 12); DEPR is depreciation over book assets (item 14/item 6); VOL is the standarddeviation of (P ROFi,t+1 − P ROFit)/P ROFit; PPE is property, plant, and equipment divided bybook assets (item 8/item 6); MB is equity market capitalization divided by book equity (item 6minus item 181); LN(TA) is the natural logarithm of total assets (item 6, in millions); and L is thenet debt value (market value of debt net of cash) divided by the sum of net debt and market valueof equity, bounded below by zero. The book value of the unobserved portion of debt is used as aproxy for its market value. SD is the standard deviation. Source: FISD, CRSP, and Compustat.

Panel A: Summary Statistics

Percentile

Mean 10 50 90 SD

PROF 0.174 0.045 0.157 0.338 0.123DEPR 0.044 0.015 0.040 0.073 0.033VOL 0.120 0.017 0.045 0.274 0.238PPE 0.328 0.055 0.289 0.663 0.229MB 1.739 0.307 1.014 3.717 3.044LN(TA) 7.508 4.968 7.655 9.684 1.859L 0.234 0 0.179 0.573 0.233

Panel B: Correlation Matrix

DEPR VOL PPE MB LN(TA) L

PROF 0.185 −0.189 0.167 0.091 0.189 0.002DEPR −0.067 0.404 −0.043 −0.060 0.069VOL −0.129 0.181 −0.272 −0.126PPE −0.172 0.178 0.333MB −0.163 −0.346LN(TA) 0.258

benefits at a given leverage ratio. Companies with high depreciation have lowernet benefits, consistent with existing empirical evidence (e.g., DeAngelo andMasulis (1980)). At high leverage, net benefits decline faster for high market-to-book firms, and for companies with few tangible assets. Highly levered firmshave lower net benefits during economic recessions.

The main benefit of the separated model in Specification I is that it clearlyshows the effects of individual variables on the net benefits to debt. However,it is not obvious that the benefits and costs of debt should be linked directlyto the linear and quadratic terms in B. For example, as leverage increases,the marginal tax benefit of debt financing decreases as fewer earnings are leftto be shielded with debt, and the realized marginal tax shield is less certain.Indeed, in the second specification in Table III, profitability enters negativelyinto the quadratic term of (8). The presence of firm characteristics in both thelinear and quadratic parts makes it more difficult to observe the total effect


Table IIIParameter Estimates

This table reports the posterior mean and standard deviation of parameter estimates over the 1994to 2004 sample, defining industries either by their two-digit SIC code (SIC2) or the Fama–French(1997) industry classifications (FF). The model specifies the net benefits to leverage relative to firmvalue (B/V L) as a function of firm characteristics:

Bit/V Lit = X′

0itθ0 + (X′

1it · Lit)θ1 + (

X′2it · L2

it)θ2.

The explanatory variables are as defined in Table II. The dummy variable D RECESS equals onein a recession, as defined by NBER peak-to-trough periods, and zero otherwise. Standard errorsare in parentheses. ***, **, and * denote parameter estimates for which zero falls outside the 99%,95%, and 90% posterior confidence intervals, respectively. N is the number of firm-months, andMSE is the mean squared error of model residuals.

I II

SIC2 FF SIC2 FF

Constant 0.030 0.041 −0.013 −0.012(0.009)∗∗∗ (0.012)∗∗∗ (0.003)∗∗∗ (0.004)∗∗∗

PROF . . 0.126 0.150. . (0.005)∗∗∗ (0.008)∗∗∗

DEPR . . −0.301 −0.288. . (0.046)∗∗∗ (0.034)∗∗∗

VOL . . −0.066 −0.075. . (0.006)∗∗∗ (0.003)∗∗∗

PPE . . 0.110 0.078. . (0.023)∗∗∗ (0.010)∗∗∗

MB . . −0.001 −0.001. . (0.000)∗∗∗ (0.000)∗∗∗

D RECESS . . −0.003 0.004. . (0.002) (0.002)∗∗

LN(TA) . . 0.001 0.001. . (0.001) (0.001)

L∗Constant 0.116 0.140 0.171 0.143

(0.013)∗∗∗ (0.020)∗∗∗ (0.008)∗∗∗ (0.018)∗∗∗PROF 0.218 0.227 0.415 0.539

(0.008)∗∗∗ (0.015)∗∗∗ (0.010)∗∗∗ (0.014)∗∗∗DEPR −0.531 −0.574 −1.970 −1.940

(0.030)∗∗∗ (0.027)∗∗∗ (0.287)∗∗∗ (0.116)∗∗∗VOL −0.103 −0.095 −0.106 −0.086

(0.010)∗∗∗ (0.004)∗∗∗ (0.006)∗∗∗ (0.004)∗∗∗PPE . . 0.243 0.353

. . (0.025)∗∗∗ (0.064)∗∗∗MB . . −0.001 0.002

. . (0.001) (0.005)D RECESS . . −0.035 −0.037

. . (0.003)∗∗∗ (0.005)∗∗∗LN(TA) 0.008 0.003 −0.002 −0.003

(0.003)∗∗∗ (0.004) (0.001)∗∗∗ (0.001)∗∗L2∗Constant −0.193 −0.214 −0.343 −0.346

(0.006)∗∗∗ (0.006)∗∗∗ (0.040)∗∗∗ (0.052)

(continued)


Table III—Continued

I II

SIC2 FF SIC2 FF

PROF . . −0.202 −0.203. . (0.071)∗∗∗ (0.053)∗∗∗

DEPR . . 1.679 1.655. . (0.041)∗∗∗ (0.059)∗∗∗

VOL . . −0.098 −0.184. . (0.004)∗∗∗ (0.004)∗∗∗

PPE 0.164 0.198 0.124 0.149(0.015)∗∗∗ (0.010)∗∗∗ (0.012)∗∗∗ (0.018)∗∗∗

MB −0.096 −0.122 −0.060 −0.063(0.001)∗∗∗ (0.001)∗∗∗ (0.003)∗∗∗ (0.003)∗∗∗

D RECESS −0.060 −0.079 −0.048 −0.052(0.004)∗∗∗ (0.015)∗∗∗ (0.002)∗∗∗ (0.003)∗∗∗

LN(TA) −0.000 0.001 −0.001 −0.006(0.001) (0.000)∗∗∗ (0.001) (0.001)∗∗∗

N 24,277 29,753 24,277 29,753MSE 0.0358 0.0345 0.0241 0.0276

of each variable on the net benefits. Figure 2 therefore plots the net benefitsversus leverage for the FF model.11 All firm characteristics in the model arefixed at their median values except one, which is taken at its 10th, 50th, or90th percentile. Most importantly, the plots in Figure 2 show that as leverageincreases, net benefits invariably increase and then decrease, implying theexistence of an optimal capital structure. Consistent with the results in model1, the plots show that low market-to-book firms with many tangible assets,low depreciation, high profitability and low volatility of earnings have highernet benefits at all leverage ratios. Highly levered firms have lower net benefitsduring recessions compared to expansions.

For the median firm, the maximum attainable net benefits are about 5.5%of firm value, but can be as high as 10% for highly profitable firms or as lowas 1% for companies with few tangible assets. However, in a dynamic settingwith transaction costs, firms do not immediately adjust leverage to maximizenet benefits, and one would expect to see lower net benefits at firms’ observedleverage ratios. The median (average) net benefits across sample firms are 4.0%(4.3%) for the FF model, and 3.6% (3.8%) for the SIC2 model. These estimatesare close to those in Binsbergen et al. (2010), who find net benefits of about4% of firm value using a very different approach. However, their calculationsassume that firms are always at their optimal capital structure, whereas theresults obtained here are based on market expectations of the firm’s futurefinancing policies.

Even at zero leverage ratios, the present value of future net benefits is about2.5% of firm value. If these firms remained unlevered forever, they would not

11 The results for the SIC2 model are quantitatively similar; see the Internet Appendix.


Figure 2. Net benefits to leverage. This figure shows how net benefits to leverage, as a fractionof total firm value (B/V L), varies with leverage (L, on the horizontal axis). The plots are based onparameter estimates of the full model (Specification II) and using the FF industries, as reportedin Table III. The graphs compare the median firm (in an economic expansion) to the same firmbut with one characteristic at either its 10th or 90th percentile of the sample distribution. Firmcharacteristics are as defined in Table II. The bottom-right graph shows the net benefits for a firmwith median characteristics in an expansion period versus in a recession (peak-to-trough as definedby the NBER). Optimal leverage ratios, as determined by maximum net benefits, are marked withan “X.”

capture any benefits or suffer any costs of debt, and Bit would be zero. This re-sult therefore suggests that the market expects low-leverage firms to lever upin the future to capture some benefits from debt financing, but not immediatelyor completely.12 Similarly, highly levered firms experience net benefits as lowas −10%, implying that these firms are worth less than their unlevered coun-terparts. This can only be the case if firms cannot immediately and costlesslyunlever, that is, when there are friction costs to refinancing. I take a closer lookat financial policy after discussing firms’ optimal leverage ratios.

12 My thanks go to an anonymous referee for making me aware of this point.


B. Optimal Capital Structure

The model makes strong predictions regarding firms’ optimal leverage ratios.In this section, I look at how optimal leverage varies with firm characteristicsand whether companies are optimally levered.

B.1. Optimal Capital Structure and Firm Characteristics

A company’s optimal capital structure, L∗, is determined by the leverage ratioat which the net benefits are maximized. In a dynamic capital structure world,this is the leverage ratio that firms choose when they refinance, taking intoaccount that they will incur friction costs to change leverage in the future. Putdifferently, if at a particular point in time firms are allowed to choose leveragewithout incurring friction costs at that point in time, then they choose L∗.

The optimal leverage ratio is marked with an “x” in Figure 2. From the plotsit is clear that L∗ varies systematically with firm characteristics. To get a morecomplete picture of this relation, Figure 3 plots optimal leverage against firmcharacteristics. Consistent with previous findings (see Harris and Raviv (1991)for a comprehensive overview), optimal leverage increases with tangible assetsand is a decreasing function of the market-to-book ratio, depreciation, and thevolatility of operating profits.

The effects of profitability and size on optimal capital structure are of par-ticular interest. Prior studies tend to find a negative relation between prof-itability and leverage, and a positive relation between firm size and leverage(Graham (2000), Myers (2001), Fama and French (2002), Korajczyk and Levy(2003), Frank and Goyal (2004), and studies cited in Harris and Raviv (1991)).In contrast, Figure 3 shows a strong positive relation between profitabilityand optimal capital structure and a negative relation between size and optimalleverage. Since the net benefits model captures optimal leverage, the estimatesare not affected by firms’ temporary deviations from the optimum, unlike ex-isting studies that use cross-sectional or panel regressions. Strebulaev (2007)and Kurshev and Strebulaev (2006) show that in a dynamic capital structuremodel with transaction costs, cross-sectional regressions will produce mislead-ing results on leverage and profitability. Highly profitable firms tend to haveperformed well in the past, which mechanically lowers observed leverage ra-tios. Small firms have lower observed leverage ratios, on average, because theywait longer between refinancings.

In recessions, optimal leverage is lower than in expansions, whereasKorajczyk and Levy (2003) find the opposite result for observed leverage ratios.Bhamra, Kuehn, and Strebulaev (2010) show that it is possible to have a pro-cyclical optimal leverage ratio while observed leverage moves countercyclicallywhen there are transaction costs, due to changes in equity market capitaliza-tion as in Welch (2004). Chen (2010) also predicts procyclical optimal leverageratios in a dynamic capital structure model with macro-economic shocks. How-ever, since only one short recession occurred during the sample period (Marchto November 2001), this result should be taken with some caution.


Figure 3. Optimal capital structure. This figure shows how optimal leverage (L∗) varies withfirm characteristics (on the horizontal axis), based on parameter estimates of the full model (Spec-ification II) and the FF sample, as reported in Table III. Firm characteristics are as defined inTable II. Each plot allows one firm characteristic to range between the 5th to 95th percentile of theobserved data, while keeping the other characteristics at the median of the sample distribution.

B.2. Are Firms Underlevered?

Past research suggests that many firms leave substantial tax benefits onthe table and should therefore take on more debt than we observe (Miller(1977) and Graham (2000)). However, this statement is difficult to evaluatewithout knowledge of the entire net benefits to leverage curve. Armed with thenet benefits estimates in this paper, I look at whether firms are on averageunderlevered, and identify drivers of under- or overleverage.

Table IV compares model-implied optimal leverage with observed leverageratios. For both the SIC2 and FF models, the median firm is underlevered:the median firm in the FF (SIC2) model has a leverage ratio of 0.179 (0.196)compared to an optimal leverage of 0.269 (0.221). Note that this does not implythat firms are necessarily suboptimally levered, as friction costs may preventfirms from refinancing to the optimal leverage ratio (as in Leary and Roberts


Table IVOptimal versus Observed Leverage

Panels A and B show summary statistics of observed and model-implied optimal leverage for thetwo-digit SIC (SIC2) and Fama–French (FF) samples, respectively, where leverage is as defined inTable II. Optimal leverage is calculated for each firm-month from Specification II in Table III. Eachsample is split into subsamples of interest-paying firms, non–interest-paying firms, and firms thatpay neither interest nor dividends. N is the number of firm-months in each subsample.

Panel A: SIC2 Sample

Percentile

Mean 10 50 90 SD

Full sample (N = 24,277)Observed 0.240 0 0.196 0.563 0.226Optimal 0.258 0.128 0.221 0.432 0.152Interest-paying firms (N = 22,595)Observed 0.256 0 0.227 0.575 0.225Optimal 0.264 0.133 0.224 0.441 0.154Non–interest-paying firms (N = 1,682)Observed 0 0 0 0 0Optimal 0.180 0.042 0.196 0.258 0.088Non–interest, non–dividend-paying firms (N = 1,060)Observed 0 0 0 0 0Optimal 0.150 0 0.163 0.247 0.091

Panel B: FF Sample

Full sample (N = 29,753)Observed 0.234 0 0.179 0.573 0.233Optimal 0.316 0.142 0.269 0.575 0.176Interest-paying firms (N = 26,868)Observed 0.259 0 0.225 0.591 0.232Optimal 0.328 0.155 0.279 0.591 0.176Non–interest-paying firms (N = 2,885)Observed 0 0 0 0 0Optimal 0.205 0.065 0.189 0.379 0.127Non–interest, non–dividend-paying firms (N = 1,972)Observed 0 0 0 0 0Optimal 0.163 0 0.159 0.289 0.100

(2005)). Still, it is surprising that firms are on average underlevered over thesample period.

Almeida and Philippon (2007) find that firms are on average correctly levered,in contrast to the above findings. Since they do not consider nontax benefits,such as reductions in agency problems between management and sharehold-ers, when calculating the benefits to debt financing, Almeida and Philipponunderestimate optimal leverage. On the other hand, they use the relativelylow 10% to 23% ex post cost of financial distress estimates based on the LBOsample in Andrade and Kaplan (1998) to calculate the ex ante cost of debt. Inthe next section, I show that my results imply costs of distress in the 15% to


30% range. The existence of nontax benefits coupled with higher distress costssuggests that net benefits as a function of leverage are more “spiked” thanAlmeida and Philippon suggest, that is, net benefits rise and fall faster withleverage. As a result, net benefits at the optimal leverage ratio are higher, andthe cost of being under- or overlevered is greater than their study implies.

Restricting the sample to firms that have interest-bearing debt, the under-leverage becomes less severe but does not disappear altogether. Table IV showsthat the average (median) interest-paying firm in the FF model has a leverageratio of 0.259 (0.225) compared to an optimal leverage of 0.328 (0.279).13 Theunderleverage disappears entirely for firms with at least 5% leverage, definedas net debt relative to firm value: average (median) observed leverage in theFF model is 0.343 (0.313) versus optimal leverage of 0.341 (0.299).

The importance of the low-debt firms in the underleverage result warrantsfurther investigation. Many public firms have a zero debt policy and some ofthese firms refrain from issuing debt for years (Minton and Wruck (2001) andLemmon and Zender (2003)). Firms that do not pay interest (true zero leveragefirms) have optimal leverage ratios around 0.2, and are therefore severelyunderlevered. Table IV shows that zero leverage firms that do not pay dividendsare less underlevered than the zero leverage firms in general. Regressionsof the degree of overleverage (observed minus optimal leverage) in Table Vconfirm these results, and show that they are robust to other variables thatmay capture firms’ deviations from optimal capital structures. When addingfirm fixed effects, the coefficients on zero leverage firms largely disappear. Thisimplies that zero leverage firms tend to remain unlevered but does not answerthe question whether they do so optimally or whether they face friction coststhat are too high relative to the benefits of debt financing. In the next section, Idiscuss financial policy in more detail, and look at whether the market expectszero leverage firms to remain unlevered.

What other factors make firms deviate from optimal leverage? Intuitively,firms in financial distress and in economic hardship are likely to be overlev-ered. Table V shows that, indeed, the degree of overleverage is higher for non-investment grade firms and during economic recessions. Overleverage is alsohigher for large, low-profit firms, which is simply a restatement of the oppositerelation between leverage and profitability, or size, when using observed oroptimal leverage ratios (as discussed above). The coefficient on market-to-bookvalue has conflicting signs and is therefore difficult to interpret.

As a final exercise, I look at a sample of LBO and management buy-out firmsfrom Andrade and Kaplan (1998). With data for 25 of the 32 firms in theirsample, Table VI shows that these firms are typically profitable, with a highproportion of tangible assets and low market-to-book ratios. In the year beforethe buy-out, the median leverage ratio is 0.198, compared to a median optimalleverage in the FF (SIC2) model of 0.336 (0.449). Consistent with popular belief,LBO firms were underlevered before the buy-out. The gain to levering up is

13 Some interest-paying firms still have zero effective leverage as measured by net debt relativeto total value because they have cash in excess of the amount of debt outstanding.


Tab

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Tab

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III

III

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rcep

t−0

.006

−0.1

200.

026

−0.0

69−0

.085

−0.1

27(0

.001

)∗∗∗

(0.0

07)∗

∗∗(0

.016

)∗∗∗

(0.0

01)∗

∗∗(0

.006

)∗∗∗

(0.0

16)∗

∗∗

Fir

m-F

EN

NY

NN

YYe

ar-F

EN

NY

NN

Y

Adj

ust

ed-R

20.

044

0.25

30.

698

0.04

00.

312

0.77

3F

559.

743

1175

.170

227.

476

617.

633

1929

.622

330.

715

p0.

000

0.00

00.

000

0.00

00.

000

0.00

0N

24,2

7724

,277

24,2

7729

,753

29,7

5329

,753


Tab

leV

IL

ever

aged

Bu

y-O

ut

Fir

ms

Th

ista

ble

pres

ents

firm

char

acte

rist

ics

and

mod

el-i

mpl

ied

opti

mal

leve

rage

rati

osfo

r25

leve

rage

dbu

y-ou

tfi

rms

from

An

drad

ean

dK

apla

n(1

998)

,u

sin

gac

cou

nti

ng

data

from

Com

pust

atin

the

fisc

alye

arbe

fore

the

buy-

out.

HL

TD

ate

isth

eda

teof

the

buyo

ut.

Lev

erag

e(L

)is

the

book

valu

eof

debt

(net

ofca

sh)

rela

tive

ton

etde

btpl

us

mar

ket

equ

ity,

calc

ula

ted

from

An

drad

ean

dK

apla

n.F

irm

char

acte

rist

ics

are

defi

ned

inT

able

II.I

repo

rtle

vera

gebo

thin

the

year

befo

re(P

re)

and

afte

r(P

ost)

the

buy-

out.

Opt

imal

leve

rage

(Opt

)is

calc

ula

ted

from

Spe

cifi

cati

onII

inT

able

III

for

both

the

SIC

2an

dF

Fm

odel

s.T

he

gain

tore

fin

anci

ng

(GA

IN)

isca

lcu

late

das

Bit/V

L itat

the

opti

mal

leve

rage

min

us

Bit/V

L itat

the

pre-

buy-

out

leve

rage

.

LS

IC2

FF

Nam

eH

LT

Dat

eP

RO

FD

EP

RV

OL

PP

EM

BL

N(T

A)

Pre

Pos

tO

ptG

AIN

Opt

GA

IN

Bu

rlin

gton

Inds

9/87

0.09

20.

063

0.20

90.

496

0.52

37.

688

0.10

71.

007

0.53

20.

046

0.34

50.

017

Ch

erok

ee5/

890.

143

0.01

40.

183

0.13

01.

055

4.52

6−0

.033

0.41

70.

277

0.04

00.

233

0.03

3F

lori

daS

teel

11/8

80.

123

0.04

20.

365

0.54

90.

619

5.63

40.

198

1.05

80.

546

0.03

70.

387

0.01

3F

ort

How

ard

10/8

80.

253

0.05

70.

144

0.58

21.

059

7.69

40.

113

0.90

30.

632

0.08

40.

487

0.04

9H

arco

urt

Bra

ce7/

870.

178

0.03

10.

147

0.32

50.

625

7.44

40.

235

0.96

50.

432

0.01

30.

336

0.00

4Jo

van

ovic

hH

ills

Sto

res

12/8

50.

087

0.03

70.

200

0.35

00.

162

6.64

00.

169

0.83

90.

442

0.02

20.

310

0.00

7In

terc

o12

/88

0.09

60.

032

0.10

00.

241

0.58

87.

594

0.16

10.

891

0.34

80.

012

0.25

40.

003

KD

I12

/88

0.10

00.

036

0.17

60.

326

0.58

15.

212

0.20

51.

045

0.40

50.

013

0.30

10.

003

Lea

sew

ayT

ran

sp6/

870.

097

0.09

40.

166

0.55

80.

507

6.82

00.

245

0.47

50.

649

0.03

10.

391

0.00

5R

HM

acy

7/86

0.12

20.

044

0.08

40.

464

1.35

67.

765

0.26

50.

855

0.47

70.

015

0.35

10.

003

May

flow

er12

/86

0.07

00.

056

0.21

00.

467

0.66

35.

547

0.30

70.

917

0.49

00.

009

0.33

10.

000

Nat

ion

alG

ypsu

m4/

860.

198

0.04

00.

374

0.45

30.

953

6.95

70.

030

0.85

00.

449

0.06

30.

337

0.04

0P

aper

craf

t10

/85

0.17

70.

021

0.14

20.

217

1.21

84.

968

-0.0

350.

795

0.33

50.

056

0.28

20.

045

Pay

less

Cas

hw

ays

10/8

80.

072

0.04

30.

105

0.52

20.

475

6.70

30.

221

0.90

10.

612

0.04

10.

429

0.01

3P

ay’N

’Pak

3/88

0.05

70.

024

0.08

00.

455

0.36

15.

847

0.24

80.

968

0.56

50.

030

0.41

50.

009

Pla

ntr

onic

s4/

890.

187

0.03

90.

138

0.17

11.

160

4.77

30.

106

0.97

60.

292

0.01

30.

246

0.00

8R

epu

blic

Hea

lth

8/86

0.21

00.

032

0.19

00.

634

0.22

96.

724

0.55

40.

924

0.76

40.

013

0.58

50.

000

Rev

co12

/86

0.05

80.

035

0.23

70.

305

0.83

76.

895

0.17

11.

041

0.33

10.

009

0.22

80.

001

RJR

Nab

isco

3/89

0.20

50.

035

0.05

50.

346

1.15

89.

784

0.13

30.

923

0.43

10.

033

0.33

60.

017

(con

tin

ued

)


Tab

leV

I—C

onti

nu

ed

LS

IC2

FF

Nam

eH

LT

Dat

eP

RO

FD

EP

RV

OL

PP

EM

BL

N(T

A)

Pre

Pos

tO

ptG

AIN

Opt

GA

IN

Sea

man

Fu

rnit

ure

2/88

0.14

50.

020

0.20

00.

379

2.80

54.

625

0.01

30.

943

0.34

00.

052

0.27

90.

037

Spe

cial

tyE

qpm

t9/

880.

174

0.04

40.

200

0.16

10.

464

5.39

30.

231

0.96

10.

292

0.00

10.

234

0.00

0S

outh

lan

d12

/87

0.05

20.

054

0.12

20.

664

0.66

48.

138

0.26

70.

897

0.73

90.

054

0.48

50.

013

Su

perm

arke

tsG

nrl

10/8

70.

041

0.06

60.

040

0.54

00.

765

7.06

70.

081

0.91

10.

612

0.06

50.

398

0.02

6U

SG

7/88

0.18

40.

046

0.38

50.

568

0.71

87.

647

0.26

51.

025

0.55

40.

027

0.39

60.

007

Jim

Wal

ter

1/88

0.18

20.

036

0.15

30.

313

0.81

87.

983

0.21

31.

013

0.40

20.

013

0.31

10.

004

Mea

n0.

132

0.04

20.

176

0.40

90.

815

6.64

30.

179

0.90

00.

478

0.03

20.

347

0.01

4M

edia

n0.

123

0.03

90.

166

0.45

30.

664

6.82

00.

198

0.92

30.

449

0.03

00.

336

0.00

8S

D0.

060

0.01

70.

090

0.15

30.

517

1.31

70.

124

0.15

30.

139

0.02

20.

089

0.01

5


less than the 10% increase in firm value due to tax benefits reported in Graham(2000) because of the compensating increase in the cost of debt. The mediangain to levering up to the optimal leverage ratio, calculated as the change innet benefits, is 0.8% (3%) of firm value for the FF (SIC2) estimates, but is ashigh as 8.4% for the buy-out of Fort Howard in 1988. Immediately after thebuy-out, the median leverage is 0.923, far above the optimum. This result is tobe expected since the intention of an LBO is to pay down debt quickly in thepost-LBO years, ultimately settling on the optimal leverage ratio.14

C. Financial Policy

The results in the previous section show that, most of the time, companieshave too little or too much debt. Is this optimal behavior given that firms facefriction costs to changing capital structure, or are firms leaving money on thetable by not adjusting leverage? In other words, do companies adjust theircapital structure when the gains of doing so are large, or are they persistentlyunder- or overlevered?

Figure 4 plots the gains to refinancing versus the degree of overleverage,defined as observed minus optimal leverage. The gains to refinancing are cal-culated as the change in Bit from adjusting leverage from its current levelto the model-implied optimum, and represent the increase in firm value ifthe firm could change its capital structure without incurring friction costs atthat particular time. In a dynamic capital structure world, this gain is thena measure of the friction costs to refinancing. Figure 4 shows separate plotsfor interest-paying investment grade firms, non–investment grade firms, zeroleverage firms (with no interest-bearing debt), and zero leverage firms that paydividends. Friction costs can generate substantial variation in leverage ratios,consistent with the theoretical results in Fischer et al. (1989). Even with costsof rebalancing their capital structure of only 1% to 2% of firm value, Figure 4shows that we could easily see firms’ leverage ratios vary by 0.2 above or belowthe optimal capital structure.

Figure 4 shows that the potential gains of rebalancing are very large foroverlevered companies, up to 30% of firm value. These gains clearly outweighany issuance costs. Most of the highly overlevered firms are non–investmentgrade, financially distressed firms. As Gilson (1997) argues, the costs of un-levering financially distressed firms are much higher than the mere cost ofissuing equity and buying back debt. Firms cannot force a settlement on allcreditors, giving rise to hold-out problems. In addition, there are regulationsthat discourage institutional lenders from writing down debt or exchangingdebt for equity. To the extent that the extremely overlevered firms are likelyeconomically distressed, they have few (if any) earnings to shield from taxes,and agency benefits like the disciplinary effect of reduced free cash flows aresmall. If these firms tend to remain highly levered due to market frictions, asGilson suggests, their future benefits to leverage are arguably very low. The

14 Since these firms went private, it is not clear what leverage ratio they ultimately settled on.


Figure 4. Gain to refinancing. This figure shows plots of the gain to refinancing from observedleverage to the model-implied optimal leverage versus the degree of overleverage, for the 29,753firm-month observations in the FF model. Overleverage is defined as observed leverage minusoptimal leverage. Observed leverage (L) is the net debt value (market value of debt net of cash)divided by the sum of net debt and market value of equity. Optimal leverage is calculated from thefull model (Specification II) using the FF model, as reported in Table III. The gain to refinancing(GAIN) is calculated as the difference in net benefits (B/V L) at the optimal leverage minus B/V L

at the observed leverage ratio. The sample is split into firm-years in which firms were rated in-vestment grade and paid interest (top left), firm-years in which firms were rated below investmentgrade (top right), firm-years in which firms paid no interest (bottom left), and firm-years in whichfirms paid no interest but did pay dividends (bottom right).

15% to 30% potential gain to unlevering for the most extremely overleveredfirms then consists mostly of reductions in distress costs, and we can roughlyestimate costs of financial distress at 15% to 30% of firm value. This estimateis higher than the 10% to 23% distress costs in Andrade and Kaplan (1998),which is not surprising given that their calculations are based on LBO firmsthat likely face low costs of distress and can therefore bear the extremely highleverage ratios of a buy-out.


For the most severely underlevered firms, levering up to the optimal capitalstructure increases market value by as much as 5% to 8%. The issuance costs ofadjusting the capital structure back to the optimum are much lower: increasingleverage by 0.5 costs 0.5% of firm value.15 Unless these highly underleveredfirms face other friction costs, they should refinance.

A large fraction of the highly underlevered firms are zero leverage firms.The present value of net benefits, Bit, is informative about market expectationsregarding the financial policy of these companies. If they are expected to remainunlevered, net benefits are zero. The zero leverage firms that pay dividends, andare therefore not likely to be financially constrained, have mean (median) netbenefits of 2.72% (2.28%) of firm value in the FF model. The market thereforeexpects these firms to lever up and capture some benefits to debt financing inthe future. The mean (median) gain to levering up is 4.37% (3.32%) of firmvalue, so these firms certainly benefit from refinancing.

The zero debt firms that do not pay dividends are not expected to lever upin the future: the mean (median) net benefits for this subset of companiesare −0.84% (−0.73%).16 Their mean (median) potential gain to relevering is1.81% (1.29%). Given that these firms are on average underlevered by 0.163(see Table IV), the costs of relevering are about 0.16%. The zero leverage firmsthat do not pay dividends act as if they face friction costs that are substantiallylarger than issuance costs alone, such as overvalued equity or managerialaversion to taking on debt.

Pushing the question of financial policy further, do firms actively rebalancecapital structure when the potential gains of doing so are high? To addressthis question, I estimate two logit models of the relation between companies’financing activities and the gains to adjusting capital structure. The basic ideais the same as that in Leary and Roberts (2005), the difference being that Ihave direct estimates of the gain to refinancing. The first model estimates theprobability of levering up as a function of the gain to refinancing, using thesubsample of underlevered firms:

ln(

pi,t+1

1 − pi,t+1

)= β0 + β1 ∗ GAINit + β2 ∗ GAINit

2 + β3 ∗ GAINit3 + ηit,

(10)

where pi,t+1 is the probability of firm i issuing debt or buying back equity worthmore than 20% of its book assets in year t + 1, and GAINit is the change inB/V L from rebalancing leverage from its current level to the optimum in yeart. Figure 5 plots pi,t+1 against GAIN. In the SIC2 model, the probability thatan underlevered firm increases its leverage modestly rises with the potential

15 Altinkilic and Hansen (2000) document bond offering costs of about 1% of issue size. Equity re-purchase costs are probably quite small, consisting primarily of trading costs and SEC restrictionson the timing and amount of share repurchases (rule 10-18b).

16 The results for the SIC2 model are quantitatively similar: firms that pay no interest but dopay dividends have mean (median) net benefits of 1.68% (1.29%). The mean (median) net benefitsfor firms that pay no interest nor dividends are −1.2% (−0.88%).


Figure 5. Probability of refinancing. This figure shows plots of the probability of rebalancingthe capital structure to the optimum as a function of the gain to refinancing. Underlevered firmsare defined as firms with a leverage ratio below (above) the model-implied optimum in a particularyear, where optimal leverage is calculated from Specification II in Table III. The probability oflevering up for underlevered firms is calculated from a logistic regression:

ln(

pi,t+1

1 − pi,t+1

)= β0 + β1 ∗ GAINit + β2 ∗ GAINit

2 + β3 ∗ GAINit3 + ηit,

where pi,t+1 is the probability of firm i issuing debt or buying back equity worth more than 20%of its book assets in year t + 1. The gain to refinancing (GAIN) is calculated as in Figure 4. Thereare 1,213 underlevered firm-years in the SIC2 sample and 1,748 in the FF model. The probabilityof levering down is calculated similarly for overlevered firm-years, with the probability of firm iissuing equity or buying back debt worth more than 20% of its book value in year t + 1 as the depen-dent variable. There are 743 overlevered firm-years in the SIC2 model, and 656 in the FF model.

gain of doing so. In the FF model the probability drops to zero for gains over0.15 (i.e., 15% of firm value), but this region of the support is an extrapolationfrom the data, as the maximum gain to underlevered firms is 0.1 (see Figure 4).

The second model takes the same functional form as (10), but estimates theprobability of levering down for overlevered firms. The dependent variable is


Figure 6. Time series of leverage. This figure shows the observed and optimal leverage forhigh- and low-leverage portfolios. I sort the 132 firms that exist over the entire 1994 to 2004sample period into four leverage portfolios, based on initial leverage in January 1994. The plotgraphs the average observed leverage ratio of the highest and lowest leverage portfolios, andthe associated model-implied optimal leverage from the FF model. Leverage (L) is as defined inTable II.

the probability of firm i issuing equity or buying back debt worth more than20% of its book value in year t + 1. The results in Figure 5 are quite striking, asboth the SIC2 and FF models predict that the refinancing probability first riseswith the potential gain, but then drops down to zero as the gain becomes large.This drop in refinancing probability is consistent with market frictions pre-venting these highly overlevered firms from immediately unlevering, causingsignificant friction costs due to financial distress.

A different way of looking at financial policy is to track firms’ leverage ratiosover time. I sort firms into leverage portfolios in January 1994, including onlyfirms that exist over the entire sample period, and plot the portfolios’ averageleverage over time in Figure 6 . Optimal leverage ratios are quite stable overtime, consistent with Lemmon, Roberts, and Zender (2008). The high leverage


firms slowly lever down towards their optimal leverage ratio, even though the1998 debt crisis and the 2000 to 2001 downturn partially negate their efforts.

The low-leverage portfolio slowly drifts up over time, but never reaches theoptimal leverage ratio. This result is due to firms that persistently keep zeroleverage. The subset of companies that do issue debt lever up substantially to-wards the optimum. This result again highlights the puzzle of the zero leveragefirms.

VI. Conclusions

The net benefits to debt financing are identified from the market valuesand betas of corporate debt and equity. Two identification assumptions arenecessary: (i) firms within an industry have the same (unlevered) asset beta and(ii) the ex ante net benefits to leverage are a function of observable variables.

Using a panel data set of firms over the period 1994 to 2004, I estimate thepresent value of net benefits to debt financing as a function of firm-specific vari-ables. Net benefits are increasing in leverage for low-debt firms but decreaseas leverage becomes very high, implying the existence of an optimal capitalstructure. For the median firm in the sample, net benefits can be as high as5.5% of firm value.

Unlike the cross-sectional regressions used in prior research, the results onoptimal capital structure in this paper are not affected by firms deviating fromtheir optimal capital structure. Contrary to prior empirical evidence but con-sistent with theoretical predictions, I find that optimal leverage is increasingin profitability and decreasing in company size.

An important feature of the identification employed here is that it does notrequire firms to be optimally levered, not even on average. I find that firmsare generally slightly underlevered, but not as much as suggested by priorresearch. The underleverage result is mainly due to the firms that have nointerest-bearing debt (true zero leverage firms). Firms that pay neither interestnor dividends are substantially underlevered, and there is no indication thatthese companies will lever up in the future. On the other hand, the firms thatpay no interest but do pay dividends are expected to lever up in the future,suggesting that they are only temporarily underlevered. Still, it is puzzlingthat zero leverage firms behave as if they face friction costs to relevering thatare substantially larger than issuance costs alone, and the nature of these costsremains an open question.

A sample containing 25 leveraged buy-out firms shows that these firms weresubstantially underlevered in the year before the buy-out. The median gain toincreasing leverage to the model-implied optimum for these firms is 0.8% to3% of firm value, depending on the model specification used.

Many companies in the sample are overlevered. The most overlevered firmsare distressed firms that are rated below investment grade. Assuming a lowpresent value of the benefits to debt for these severely distressed companies,I estimate the present value of the costs of financial distress at 15% to 30%,higher than prior estimates of the costs of financial distress that are based on


a small sample of LBO firms. I find that overlevered firms are more likely torebalance their capital structures when gains to refinancing are moderatelyhigh, but this likelihood decreases as the potential gains rise. This result in-dicates that the friction costs of refinancing distressed companies are large.Refinancing severely distressed firms is generally difficult, time-consuming,and costly to achieve due to debt overhang, bargaining issues, and conflictsamong creditors.

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