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Federal Reserve Bank of Chicago
Levered Returns and Capital
Structure Imbalances
Filippo Ippolito, Roberto Steri,
Claudio Tebaldi, and Alessandro T. Villa
January 8, 2022
WP 2022-42
https://doi.org/10.21033/wp-2022-42
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
*
Levered Returns and Capital Structure Imbalances
Filippo Ippolito1
Universitat Pompeu Fabra and CEPR
Roberto Steri2
University of Luxembourg
Claudio Tebaldi3
Bocconi University
Alessandro T. Villa4
FRB Chicago
Abstract
We revisit the relation between equity returns and financial leverage through the lens of a dynamic trade-off model
with costly capital structure rebalancing. The model predicts that expected equity returns depend on whether a firm’s
leverage is above or below its target leverage. We provide empirical evidence in support of the model predictions.
Controlling for leverage, overlevered (underlevered) firms earn higher (lower) returns. A quantitative version of our
model reproduces key facts about capital structure rebalancing and equity returns for U.S. corporations. Overall,
our results indicate that financial flexibility crucially affects the link between leverage and equity returns.
Keywords: leverage, cross section of returns, target leverage, dynamic capital structure, financial frictions.
JEL Classification Numbers: G12, G32.
? We thank Thomas Geelen, Juan-Pedro Gomez, Minjoo Kim, Christian Laux, Gyongyi Loranth, Stefano Sacchetto, Lukas Schmid,
Gustavo Schwenkler, Victoria Vanasco, Josef Zechner for valuable comments. We also have benefited from comments from seminar
and conference participants at Universitat Pompeu Fabra, the Vienna Graduate School of Finance, the 2017 Financial Intermediation
Research Society Annual Meeting, the 2017 Edinburgh Corporate Finance Conference, the 2017 4nations Cup, and the 2017 Annual
Meeting of the Spanish Finance Association. Disclaimer: The views expressed in this paper do not represent the views of the Federal
Reserve Bank of Chicago or the Federal Reserve System. Declaration of conflicts of interest: none. All errors are our own.
1 Barcelona Graduate School of Economics and CEPR, Universitat Pompeu Fabra. E-mail: filippo.ippolito@upf.edu.
2 Department of Finance at the University of Luxembourg. E-mail: roberto.steri@uni.lu.
3 Department of Finance at Bocconi University. E-mail: claudio.tebaldi@unibocconi.it.
4 Federal Reserve Bank of Chicago. E-mail: alessandro.villa@chi.frb.org.
Preprint submitted to Journal of Monetary Economics
January 8, 2022
1. Introduction
The second proposition of Modigliani and Miller (1958) (MM) is one of the pillars of the theory of corporate
finance. It is widely taught in corporate finance courses and employed by practitioners around the world. The
proposition establishes a positive relation between financial leverage and expected equity returns that is ordinarily
5
applied to compute the cost of equity for different levels of leverage. Over the years, a number of academics have
investigated the relation between equity returns and leverage, and have provided mixed results. In some cases, equity
returns appear to be unrelated to leverage. In other cases, leverage is significantly and negatively related to equity
returns.
In this paper we formalize an intuition for why the fundamental result of MM can be at odds with the data. We
10
suggest that firms’ limited financial flexibility in adjusting their capital structure crucially affects the link between
leverage and returns. We show that, in a dynamic setting, firms tend to partially adjust their capital structure
toward a target leverage ratio. We offer empirical and quantitative evidence that accounting for capital structure
imbalances, defined as deviations from target leverage, helps in coming to terms with the contradictory evidence on
“levered returns”.
15
We begin by illustrating the intuition for our results in a simple two-period model. This stylized setup offers a
“look-alike” MM equation for levered returns. In the model, firms choose their capital structure to maximize their
equity value in the presence of three frictions: taxes, bankruptcy costs, and debt adjustment costs. In the absence
of these three frictions, MM’s second proposition holds in its traditional form. Debt adjustment costs instead create
room for capital structure imbalances. Firms have a motive to raise additional debt as long as the marginal tax
20
shield of an additional dollar of debt exceeds the marginal bankruptcy cost. The optimal financial policy of a firm
is to increase its leverage as long as the margin between tax shields and bankruptcy costs is positive, and reduce
its leverage otherwise. Absent adjustment costs, a firm changes leverage until the two marginal effects are perfectly
equal. This happens when leverage is set equal to target leverage, which is defined as the optimum in the absence of
adjustment costs. With adjustment costs, a firm generally finds it too costly to reach the target, and has to settle
25
for a leverage ratio that is either too low or too high with respect to the target. This is commonly referred to as
“partial adjustment”, and it implies that in the cross section some firms are underlevered (leverage smaller than
target), while others are overlevered (leverage larger than target).
From an asset pricing perspective, a quasi MM result can still be obtained, once a correction factor γ is applied
to the original equation. The resulting relation between market leverage and equity returns is the following:5
RE = RA +
γd
k − γd
RA −
RD
γ
,
(1)
5 For a in-depth discussion of the use of market versus book values of debt in empirical tests of structural models, see Bretscher,
Feldhütter, Kane, and Schmid (2020).
2
where k is the market value of the assets, d is the market value of debt, RA is the return on the unlevered assets,
RD is the market return on debt, and RE is the levered equity return.
30
The correction factor γ reflects differences between average bankruptcy costs and average tax shields in correspondence of firms’ optimal capital structure. As adjustment costs induce capital structure imbalances, γ depends
both on observed leverage and on how far a firm is from its target. The correction factor is equal to one in the
frictionless benchmark, in which case we revert to the standard MM equation. In general, γ can be greater or smaller
than one. More precisely, γ is smaller than one if the average (per-dollar of debt outstanding) tax shield exceeds
35
the average bankruptcy cost, and greater than one otherwise. In other words, for a firm with γ smaller than one,
equity returns respond to leverage as if the firm had a debt load smaller than its market value d. In this case, the
firm obtains higher average tax shields than bankruptcy costs and receives a net benefit from carrying debt. Instead,
when γ is greater than one, equity returns behave as if the firm effectively had more debt than its market value d,
because average bankruptcy costs are higher than average tax shields, and the firm bears a net cost of carrying its
40
outstanding debt stock.6 Observe that γ is different from one in the special case of static tradeoff theory, which does
not include adjustment costs. Optimal capital structure choices only equate the marginal values of bankruptcy costs
and tax shields, while their average values still differ in this case. This is because tax shields and bankruptcy costs,
in general, vary non-linearly with firms’ debt. However, as all firms are be able to instantaneously respond to shocks
by adjusting their capital structure to the target, capital structure imbalances play no role.
45
The key novelty of equation (1) with respect to the second proposition of MM is to show that both the amount
and the unit cost (or benefit) of debt for a firm determine the effect of leverage on its equity returns. As in MM,
capital structure affects how the risk of unlevered assets propagates to equity payouts through the amplification of
both good and bad cash flow outcomes. We show that the amplification effect is also driven by the effective net
cost (or benefit) of debt, which itself is driven by capital structure imbalances. By measuring how far a firm is from
50
its target, relative leverage is a measure of a firm’s capital structure imbalance. Relative leverage is defined as the
difference between leverage and target leverage. It is negative for underlevered firms and positive for overlevered
firms. γ and, consequently, equity returns depend positively on relative leverage. For a given leverage ratio, firms
with higher relative leverage have higher equity returns, while firms with lower relative leverage earn lower equity
returns.
55
The economic intuition for this result is the following. Take first the case of an underlevered firm and suppose
that bankruptcy costs increase more than proportionally with the firm’s debt stock, while tax shields increase
proportionally. As long as the marginal tax shield is greater than the marginal bankruptcy cost, every dollar of debt
raised by the firm brings a net benefit because it generates a positive net margin. As the firm keeps raising more
debt, the margin on an extra dollar of debt shrinks because the bankruptcy costs increase faster than the tax shields.
6 γ captures only the non-diversifiable components of bankruptcy costs and tax shields. Therefore, only the systematic component of
bankruptcy costs and tax shields enters our mechanism. Thus, the effect of capital structure imbalances cannot be simply diversified
away by shareholders through their portfolio allocations.
3
60
However, as long as the margin between tax shields and bankruptcy costs is positive on average, debt generates a net
benefit for the firm. As a result, in comparison to the frictionless environment of MM, the amplification of leverage
on returns is dampened. Vice versa, in the case of a sufficiently overlevered firm, debt brings a net loss to the firm,
and the amplification effect of leverage on returns is stronger than in the original MM framework.
Although this simple illustration is useful to develop intuition for our key results, it is too stylized to serve as
65
a basis for a quantitative investigation of its predictions. We then embed the key economic tradeoffs in a more
general infinite-horizon dynamic environment, which we calibrate to U.S. data. The dynamic model allows for crosssectional firm heterogeneity in the form of productivity shocks. Every period, firms make endogenous investment
and financing decisions. Firms are financially constrained as external equity is costly and firms can endogenously
default on their debt claims. Within the model, we extend the definition of target in the two-period illustration
70
to a dynamic environment. In addition, we adopt a flexible asymmetric functional form for debt adjustment costs.
As debt adjustment costs can arise from several sources, this choice allows us to keep the model tractable while
capturing, albeit in reduced form, the direct and indirect costs of issuing and withdrawing debt.7
Within the calibrated model, we simulate artificial panels of firms and use them as a laboratory to test the
empirical predictions. We complement our quantitative analysis with empirical evidence linking capital structure
75
imbalances and equity returns on a sample from the widely used CRSP and Compustat datasets. Our empirical
evidence is based on standard reduced-form estimates of target leverage in the corporate finance literature. Taken
together, our quantitative and empirical analyses provide a set of consistent empirical facts, which support the key
predictions of our two-period illustration. First, firms partially adjust their capital structure toward their target
leverage. Second, market leverage has very little explanatory power once standard controls, namely firm’s market
80
capitalization and book-to-market equity, are considered. Third, when relative leverage is included alongside leverage
as an explanatory variable for equity returns, the coefficient of relative leverage is positive, statistically significant,
and economically sizable. Instead, leverage has little predictive power. This suggests that heterogeneity in target
leverage ratios confounds the relation between leverage and returns.8 Finally, in the cross section, we find that small
firms adjust their capital structure faster than large firms toward their targets. Accordingly, the premium associated
85
with relative leverage is more pronounced for large firms, which appear to benefit less from adjusting toward their
target capital structure.
The paper is structured as follows. Section 2 develops the two-period illustration for our key results. Section 3
presents empirical evidence in support of the key model predictions. Section 4 lays out our infinite-horizon model,
whose quantitative properties are assessed in Section 5. Section 6 concludes.
7 For example, Fischer, Heinkel, and Zechner (1989) and Strebulaev (2007) discuss underwriting and management fees, Acharya,
Bharath, and Srinivasan (2007) study seniority issues that render large changes in the capital structure costly, and Diamond (1991) and
Myers (1977) consider liquidation and agency costs associated with the use of debt.
8 When target leverage and market leverage are included in the same empirical specification, the coefficient of leverage is positive,
while the coefficient of target leverage is negative. Thus, the traditional positive relation between leverage and returns is restored when
one controls for target leverage. Intuitively, for an underlevered firm, the higher is the target, the larger are the net benefits of debt it
internalizes at any given level of leverage. On the contrary, a higher target implies a smaller net cost of leverage for an overlevered firm.
4
90
Related Literature. This paper mainly relates to the literature that examines how financing frictions introduce deviations from the Modigliani-Miller theorem and ultimately determine cross-sectional spreads in expected equity
returns. Recent papers along these lines include Livdan, Sapriza, and Zhang (2009), Gomes and Schmid (2010),
Caskey, Hughes, and Liu (2012), Ozdagli (2012), Obreja (2013), Ozdagli (2015), Doshi, Jacobs, Kumar, and Rabinovitch (2019), and Friewald, Nagler, and Wagner (2021). Our model and the associated empirical findings provide
95
a number of new contributions. From a theoretical point of view, we reconcile the traditional intuition of MM’s
Proposition 2 with a multi-period environment where leverage changes are costly. We also provide a rationale for the
“non-result” of leverage on returns, i.e. the fact that leverage does not appear to drive equity returns (see Bhandari
1988, Fama and French 1992, Penman, Richardson, and Tuna 1992, George and Hwang 2010, Gomes and Schmid
2010). We show that once the value of debt is adjusted to account for the relevant frictions, the original insight
100
of MM’s proposition carries on through into a more complex set up than the one assumed in their original paper.
Our theoretical findings rely on simple microeconomic trade-offs and have a clear intuition that fundamentally relies
on the benefits and costs of debt finance. Our model shows that MM’s result can be obtained from the production
side of the firm, rather than from a no-arbitrage argument.9 Finally, the model clarifies the interplay of different
measures of leverage (observed leverage, target leverage, and relative leverage) and their individual relation with
105
equity returns.
From an empirical point of view, this work is primarily related to Korteweg (2010), who uses equity prices to
estimate the net benefits to leverage for a large cross section of firms in a dynamic tradeoff setting. Our results are
consistent with Korteweg’s in that we provide additional evidence that the tradeoff between costs and benefits of
leverage is quantitatively a first-order margin reflected in stock prices. We offer a measure of leverage that can be
110
consistently used to predict returns. We show that relative leverage can be computed with a two-step procedure
using historical accounting data and that results are not sensitive to how relative leverage is measured. We provide
empirical evidence that matches well to the predictions of the model, and we document that leverage per se is an
unreliable predictor of returns. Another related empirical paper is Caskey, Hughes, and Liu (2012). Their analysis
primarily relies on the “kink” proxy in Graham (2000) to define overlevered and underlevered firms depending on
115
variation in their marginal tax benefits. Remarkably, Caskey, Hughes, and Liu (2012) find that overlevered firms
earn lower average cross-sectional return than underlevered firms. We entertain a different definition of overlevered
and underlevered firms. Our empirical measures of target leverage, which are widely used in the empirical corporate
finance, account for other dimensions than tax benefits, such as investment opportunities, innovation, and asset
depreciation. In this context, we derive out testable hypotheses from a theoretical setup. Finally, we show that
120
our results are quantitatively relevant in a dynamic model, which is less susceptible to measurement error in target
leverage.
Our work builds on the macroeconomic literature that uses the “gap approach” to describe costly adjustments
of production factors (e.g., Sargent 1978, Caballero and Engel 1993, Cooper and Willis 2004, King and Thomas
9 However,
the absence of arbitrage opportunities in the market is still required.
5
2006, Bayer 2009). We rely on the “gap approach” to a generalize the definition of target leverage in the two-period
125
illustration to an infinite-horizon model. As in these studies, we consider adjustments toward a dynamic target,
which they term “frictionless target”. We apply the gap approach to a firm’s debt stock, rather than to a factor of
production. Reassuringly enough, our quantitative results based on the frictionless target are consistent with the
empirical evidence based on reduced-form measures, which also define a dynamic target for each firm in the sample.
More broadly, this paper pertains to the large and growing literature that links corporate investment and financing
130
decisions to the behavior of asset returns through the lens of production-based asset pricing models. An incomplete
list includes Cochrane (1991), Jermann (1998), Gomes, Kogan, and Zhang (2003), Zhang (2005), Gomes, Yaron, and
Zhang (2006), Liu, Whited, and Zhang (2009), Gomes and Schmid (2010), Belo, Lin, and Bazdresch (2014), Gomes,
Jermann, and Schmid (2016), and Gomes and Schmid (2016).
2. A Simple Illustration
135
In this section we propose a stylized model to illustrate the relation between capital structure imbalances and
expected equity returns. The main advantage of a stylized setup is to provide a transparent illustration of the
underlying economic mechanism that drives our main empirical and quantitative results.
2.1. Economic Environment
Consider an economy with three periods, t0 , t1 and t2 , populated by heterogeneous firms indexed by i. Firms
have a set of physical assets ki that is assumed constant over the three periods. The assets generate after-tax profits
A
A
equal to ri,1
(1 − τ )ki in t1 , where ri,1
is the pre-tax return on assets and τ is the statutory tax rate. Default takes
A
A , where r A is a given threshold. In t , the
place when the return on assets is sufficiently low, such that ri,2
< ri,2
2
i,2
A
assets produce ri,2
(1 − τi,2 )ki , where
τi,2
0
=
τ
A
A ,
if ri,2
< ri,2
A
A .
if ri,2
≥ ri,2
A
A
The firm is currently in period t1 , and ri,1
is known with certainty. Instead, ri,2
is a random variable with conditional
140
A
A
distribution f (ri,2
|ri,1
).
Firms enter t1 with an outstanding amount of debt di,0 . In t1 the firm is assumed not to be in default and pays an
D
interest amount (1 + ρD
i,1 )di,0 , where ρi,1 is a known coupon rate. The interest payments on debt in t2 are stochastic,
D
because they depend on whether default occurs or not. If there is no default, ri,2
(di,1 ) is equal to the coupon rate
ρD
i,2 (di,1 ). In the case of default, the return on debt is equal to the recovery rate
D
ri,2
(di,1 ) =
A
(1 + ri,2
(1 − τi,2 ))ki − βi,2 (di,1 )
− 1,
di,1
6
(2)
where the term βi,2 (di,1 ) ≥ 0 represents the bankruptcy costs per dollar of outstanding debt. Total bankruptcy costs
are βi,2 (di,1 )di,1 . For analytical convenience, we assume that unit bankruptcy costs are proportional to each dollar
of debt outstanding in the case of default, that is βi,2 (di,1 ) = βi,2 di,1 , where
βi,2
β
i
=
0
A
A ,
if ri,2
< ri,2
A
A .
if ri,2
≥ ri,2
This implies that total bankruptcy costs increase more than proportionally (quadratically) with the stock of debt
outstanding in case of default.
We assume that adjusting the debt stock carries a cost Θi (di,0 , di,1 ) =
θi
2
2
(di,1 − di,0 ) . The adjustment cost
is a reduced form of accounting for the frictions associated with the issuance and repurchase of securities in an
145
environment with imperfect capital markets. The convexity of the cost function implies that large debt adjustments
within one period entail larger unit costs than small adjustments, such as in Croce, Kung, Nguyen, and Schmid
(2012) and Belo, Lin, and Yang (2014). Convex costs of adjusting capital structure are consistent with Myers and
Majluf (1984) and Krasker (1986). Although the assumption of quadratic adjustment costs is analytically convenient,
we relax it in the dynamic model of Section 4. The assumption on the existence of a cost of adjusting firms’ capital
150
structure is consistent with the empirical finding that firms do not adjust immediately to their target leverage ratio.10
Firm’s Problem. Tax shields and bankruptcy costs affect the cash flows of the firm and are thus reflected in the
value of equity. In period t1 , firm i chooses di,1 to maximize the market value of the firm Vi,1 , that is:11
A
D
Vi,1 ≡ max ri,1
(1 − τ )ki − (1 + ri,1
(1 − τ ))di,0 + di,1 − Θi (di,0 , di,1 )
di,1
A
D
+ E[M ( (1 + ri,2
(1 − τi,2 ))ki − (1 + ri,2
(1 − τi,2 ))di,1 − βi,2 (di,1 )di,1 )] , (3)
where E[·] denotes the expectation operator with respect to the information set available at time t1 and M represents
the stochastic discount factor between t1 and t2 as given by the market. To minimize notation, from now on we will
refer to firm i more simply as the firm, and drop the subscript i.
155
2.2. Optimal Leverage Policy
The presence of adjustment costs posits an economic tradeoff for all firms in the economy. Each firm optimally
adjusts its capital structure by trading off sure cash flows at time t1 in exchange for expected cash flows at time t2 .
10 Fama and French (2002), Leary and Roberts (2005), Flannery and Rangan (2006), Huang and Ritter (2009), Faulkender, Flannery,
Hankins, and Smith (2010), Flannery and Hankins (2010), and Halling, Yu, and Zechner (2015) provide examples of the estimation of
partial adjustment models of capital structure. The adjustment speed toward the target is estimated to be as low as 7% (Fama and
French (2002)) and as high as 30% ( Flannery and Rangan (2006)).
11 The maximization problem in (3) can be equivalently expressed in terms of change in debt, defined as ∆d
i,1 ≡ di,1 − di,0 . To see
this, replace di,1 with ∆di,1 + di,0 and maximize with respect to ∆di,1 . We can observe immediately that the maximization leads to the
same first-order condition in (32). This observation highlights that, although we do not model debt maturity and seniority explicitly,
what matters is the change in debt from one period to the next.
7
If debt claims are fairly priced in the market, that is E[M (1 + r2D (d1 ))] = 1, the first-order condition for the firm is
"
#
∂ r2D (d1 )d1
∂ (β2 (d1 )d1 )
∂Θ (d0 , d1 )
−E M
= E M τ2
.
∂d1
∂d1
∂d1
(4)
and has an intuitive interpretation. The firm finds the optimal debt level at the point where the change in adjustment
costs resulting from raising one additional dollar of debt at time t1 (on the left-hand side) equalizes its expected
discounted marginal net benefit at time t2 (on the right-hand side). The marginal net benefit is the difference between
marginal tax shields and marginal bankruptcy costs. If there were no adjustment costs, the firm would perfectly
160
trade off the marginal tax shields with the marginal bankruptcy costs. Instead, the presence of adjustment costs
implies that at the optimum, a wedge between marginal tax shields and marginal bankruptcy costs arises. The wedge
can be either positive or negative. When the wedge is positive, the marginal tax shield is larger than the marginal
bankruptcy cost, and the opposite holds otherwise.
The optimality condition (4) can be rearranged to unfold the relation between capital structure imbalances and
165
expected equity returns. The following proposition first characterizes the firm’s optimal leverage policy.
Proposition 1 (Optimal Financing Policy). Firms optimally set their leverage ratio
λ1 of the gap between their initial leverage ratio
d0
k
and their target leverage ratio
d1
d0
−
= λ1
k
k
2β(1−τ )
d0
d∗1
−
k
k
d∗
1
k
d1
k
to close a fraction
, that is
,
(5)
qD
τ (rF +qD )
1
F
and d∗1 ≡ 2β(1−τ
where λ1 ≡ θ+2β(1−τ1+r
qD
)qD . rF denotes the risk-free rate, with 1 + rF = E[M ] . qD ≡
) 1+r
F
R r2A
f (rA |rA )dr2A is the probability of default under the risk neutral measure, in which fQ (r2A |r1A ) ≡ (1+rF )f (r2A |r1A )M (r2A )
−∞ Q 2 1
is the risk-neutral probability density of the realization r2A .
Proof. See Appendix.
170
Equation (5) characterizes the firm’s optimal financing policy as a partial adjustment model, consistent with
the specifications ordinarily estimated in empirical studies, such as Flannery and Rangan (2006). The target
d∗
1
k
is
increasing with the marginal benefit of one dollar of debt τ and decreasing with its marginal bankruptcy cost β. The
target is forward-looking in that firms anticipate future economic conditions at time t2 . Specifically, the numerator
of d∗1 increases with the expected discounted value of the tax shields of debt. The larger the non-idiosyncratic
175
probability of default qD , the larger is the expected marginal tax shield τ (rF + qD ) in solvency states, because the
tax-deductible credit spread that lenders charge over the risk-free rate is also larger.12 Instead, the denominator
shows that the target decreases with the expected discounted value of marginal bankruptcy costs in default states,
the per-dollar value of which is 2β and is realized with probability qD . The term 1 − τ in the denominator instead
12 As standard, the risk-neutral probability of default measures the non-diversifiable risk of bankruptcy (i.e., the part correlated with
the stochastic discount factor), and thus the part priced by the market.
8
accounts for the indirect tax benefit of bankruptcy costs. Higher bankruptcy costs increase the coupon rate that
180
lenders charge to compensate for their lower recovery rate given default. The tax shield in solvency states also
increases as a result of the higher required interest payment. Hence, the firm effectively bears only a fraction 1 − τ
of the marginal bankruptcy cost.
The target is equal to the optimum leverage that the firm would choose in the absence of adjustment costs.13
This definition of the target is consistent with the “gap” approach we use in Section 4 to define target leverage in
185
an infinite-horizon economy. Firms adjust their leverage ratio from
gap
d∗
1
k
−
d0
k
between initial leverage
d0
k
and target leverage
d∗
1
k
d0
k
to
d1
k
by closing a fraction λ1 ∈ (0, 1] of the
. The parameter λ1 can be interpreted as the speed of
adjustment of the firm toward its target capital structure. On the one hand, firms with low values of λ1 are slow in
adjusting their capital structure to changes in their investment and financing conditions. On the other hand, firms
with high speeds of adjustment are fast in closing the gap toward their targets.14 Observe that this illustration offers
190
a benchmark in which λ1 does not depend on k, i.e., the firm’s size does not affect the firm’s speed of adjustment. In
the quantitative analysis of Section 5 we explore how firms’ adjustment toward the target are related to firms’ size.
Proposition 1 shows the existence of capital structure imbalances in the economy. We define relative leverage
RL1 as the difference between observed leverage and target leverage, that is:
RL1 ≡
d∗
d1
− 1.
k
k
(6)
Accordingly, firms with RL1 > 0 are overlevered, i.e, their observed leverage is above the target, while firms with
RL1 < 0 are underlevered. Equation (5) can be rewritten as:
d1
d0
λ1
−
=−
RL1 .
k
k
1 − λ1
(7)
From equation (7) we can see that if λ1 < 1 and in t1 a firm adjusts its leverage up from
d0
k ,
the firm will be
underlevered in t2 . Adjustment frictions (captured by λ < 1) will prevent the firm from reaching its target leverage
d∗
1
k
in t1 . Symmetrically, if the firm deleverages in t1 , it will be overlevered in t2 .
Figure 1 presents an example of the optimal leverage policy of the firm for two different values of θ that correspond
to low adjustment costs (Panel A) and high adjustment costs (Panel B). The picture reports
axis and
d1
k
d0
k
on the horizontal
on the vertical axis. Target leverage is at the intersection of the policy function with the 45-degree line.
Firms that are already at their target do not adjust their capital structure (RL1 = 0). In the figure, the optimal
13 To
see this, rewrite (4) assuming that there are no adjustment costs. This yields
"
#
∂ r2D (d1 )d1
∂ (β2 (d1 )d1 )
E M τ2
=E M
.
∂d1
∂d1
Carrying out the necessary substitutions and solving for the optimal leverage ratio we obtain precisely the definition of target leverage
as in Proposition 1.
14 When
there are no adjustment costs, the speed of adjustment equals one. Equation (5) then states that
9
d1
k
=
d∗
1
.
k
leverage policy is linear in
d0
k ,
and the comparison between the two panels highlights that θ affects the slope of the
optimal leverage policy. The higher θ, the steeper the policy function, and the more sluggish is the firm’s adjustment
toward target leverage. To see this, consider two possible levels of
d0
k ,
O
LU
0 and L0 , located respectively below and
above the target. The policy function in Panel B shows that a firm with initial leverage LU
0 leverages up less than
its counterpart in Panel A and will be more underlevered in t2 . Analogously, all else being equal, a firm with initial
leverage LO
0 lowers its leverage less, and is more overlevered if it faces higher adjustment costs.
[Insert Figure 1 Here]
195
2.3. Levered Returns
Since V1 is the cum-dividend equity value of the firm, its realized gross stock return R2E between t1 and t2 is
defined as
R2E =
P2 + D2
D2
=
,
P1
V1 − D1
(8)
where P1 and P2 are the (ex-dividend) stock prices at time t1 and at time t2 , and D1 and D2 are the dividend
payments at time t1 and at time t2 . Notice that the (ex-dividend) stock price P2 at t2 is zero because there are no
more cash flows after that time.15 The dividend payments of the firm at t1 and t2 are
D1 ≡ (1 + r1A (1 − τ ))k − (1 + r1D (1 − τ ))d0 + d1 − Θ
d0 d1
k , k
D2 ≡ (1 + r2A (1 − τ2 ))k − (1 + r2D (d1 )(1 − τ2 ))d1 − β2 (d1 )d1 ).
Exploiting the condition E[M( (1 + r2A (1 − τ2 ))] = 1 and the definition of D2 , after some manipulation, equation (8)
can be expressed as16
R2E = R2A +
γ(d1 )d1
k − γ(d1 )d1
R2A −
R2D
γ(d1 )
,
(9)
in which we define the after-tax gross return on physical assets as R2A ≡ 1 + r2A (1 − τ2 ), the effective cost of debt for
the firm as R2D ≡ 1 + r2D (d1 )(1 − τ2 ) + β2 d1 , and the correction factor as γ(d1 ) ≡ E[MR2D ]. As E[M(1 + r2D (d1 ))] = 1,
the correction factor is equal to
γ(d1 ) = 1 − E[M (τ2 r2D (d1 ) − β2 d1 )].
(10)
The expression in equation (10) has an intuitive interpretation. In a trade-off economy, the presence of tax shields
τ2 and bankruptcy costs β2 d1 creates a discrepancy between the cost of each dollar of debt for the firm R2D and the
market return on debt 1 + r2D (d1 ) earned by debt-holders. This discrepancy is the net benefit (or cost) to leverage
D2
that, from (3), it follows that V1 − D1 = E[M D2 ]. Therefore, the equity return is given by R2E = E[M
.
D2 ]
A
that the condition E[M( (1 + r2 (1 − τ2 ))] = 1 can be derived under the assumption that all firms have already optimized their
initial level of capital k0 , so that ∀i, ki = k0 . k0 is fixed at t = 1, 2. Note that if firms were allowed to re-optimize their capital stock
at t = 1, firms would optimally choose to stay at k0 according to −1 + E[M( (1 + r2A (1 − τ2 ))] = 0. However, our two-period example
simply offers an illustration based on a snapshot over a firm’s life cycle. The more general dynamic environment of Section 4 allows for
cross-sectional firm heterogeneity, which translates into differences in firms’ target capital structures.
15 Observe
16 Note
10
(Korteweg, 2010) and is given by the term E[M (τ2 r2D (d1 ) − β2 d1 )], which is the present value of the average tax
200
shield τ2 r2D (d1 ) minus the average bankruptcy cost β2 d1 . As in MM, leverage propagates asset risk to equity payouts
through the amplification of both good and bad cash flow outcomes. However, the amplification is not only driven
by the amount of debt d1 , but also by the net benefit γ(d1 ) that shareholders internalize for each dollar of debt
outstanding in correspondence with their optimal leverage and the resulting capital structure imbalance.
The correction factor creates an amplification effect for the burden of debt when γ(d1 ) > 1. This happens when
205
the present value of average bankruptcy costs exceeds the present value of average tax shields. Instead, a reduction
effect for debt occurs when γ(d1 ) < 1, which holds when average tax shields are greater than average bankruptcy
costs. Levered equity returns in (9) are computed as if for each dollar of debt the firm effectively had γ(d1 ) dollars of
debt because of its net benefits/costs to leverage. The effect of the discrepancy generated by τ2 and β2 (d1 ) is twofold.
First, the correction factor γ(d1 ) enters the effective leverage ratio
210
ratio is larger than the observed leverage ratio
R2D
γ(d1 ) ,
d1
k−d1 .
γ(d1 )d1
k−γ(d1 )d1 .
When γ(d1 ) > 1, the effective leverage
Second, the correction factor reduces the effective cost of debt
because the additional leverage embedded in γ(d1 ) does not require any supplementary payment besides R2D .
Importantly, γ(d1 ) captures the systematic component of tax shields and bankruptcy costs through their covariances with the pricing kernel M .17 Thus, the effect of capital structure imbalances cannot be simply diversified away
by shareholders through their portfolio allocations. Intuitively, since bankruptcies are concentrated in bad times
215
while tax benefits are higher in good times, shares of overlevered firms are risky investments because their debt’
increases the present value of non-diversifiable bankruptcy costs and boosts the traditional MM amplification effect.
We formally relate γ(d1 ) to capital structure imbalances in the following section.
2.4. Relative Leverage, Target Leverage, and Equity Returns.
As discussed above, equation (9) offers a number of new insights into the relationship between leverage and
220
returns in the presence of frictions. The following proposition links ex-post realized equity returns and capital
structure imbalances.
Proposition 2 (Imbalances and Equity Returns). Realized equity returns of the firm between t1 and t2 are
related to relative leverage and to target leverage through the following relationship
R2E = R2A +
db1
k − db1
b2D ,
· R2A − R
bD ≡
where the effective debt stock db1 is defined as db1 ≡ γ ( ) d1 , R
2
γ
d∗1 d1
,
k k
≡1+
R2D
γ( )
and the correction factor is defined as
α d1
d∗
τ δD
− α 1 + d1
2 k
k
k
17 Observe
(11)
(12)
that, as in the case of MM, our predictions do not rely on a specific asset pricing model (e.g. multi-factor model), but hold
for any pricing kernel M that ensures no arbitrage in the market.
11
or equivalently as
α d1
τ δD
d1
≡1−
+ αRL1 + d1
γ RL1 ,
k
2 k
k
(13)
qD 18
.
with δD ≡ E[M (1 + r2A )|r2A < r2A ] and α ≡ 2β(1 − τ )k 1+r
F
225
Proof. See Appendix.
Taking expectations of both sides of (11) delivers the relation between ex-ante expected returns and capital
structure imbalances at the heart of our analysis of levered returns, that is
E[R2E ] = E[R2A ] +
db1
k − db1
b2D ] .
· E[R2A ] − E[R
(14)
It is worth reminding the reader that in both equations (11) and (14), leverage is expressed at market values, not
at book values, because d1 and k, respectively, represent the market value of debt and of the assets. Therefore, the
prediction of Proposition 2 is on the relation between leverage measured at market value and equity returns.
2.5. Empirical Predictions
230
On the basis of Proposition 2 and the interpretation of the correction factor, we can draw predictions on the
relation between expected equity returns and our three measures of leverage (relative leverage, observed leverage,
and target leverage).
Consider two firms with the same observed leverage and different relative leverage. According to the definition
of the correction factor provided in equation (13), the firm with higher relative leverage has a higher γ ( ). Plugging
235
equation (13) into equation (14), we then have that the firm with higher relative leverage earns higher expected
returns.
Instead, compare two firms with the same relative leverage and different observed leverage. Leverage enters
equation (14) with a mixed sign because of several contrasting effects. First, the firm with higher dk1 has a lower
γ RL1 , dk1 , as (13) shows. Second, since db1 ≡ γ RL1 , dk1 d1 in (14), it is unclear which of the two firms has a larger
240
amplification term
c
d
1
.
c
k−d
1
bD ≡
Finally, the effect of leverage on R
2
with higher leverage plausibly bears a higher cost of debt
R2D ,
R2D
γ (RL1 ,
is also indeterminate. Although the firm
)
it has a lower correction factor as discussed before.19
d1
k
Overall, the relation between leverage and expected returns is theoretically indeterminate. As a consequence, depending on which effect of leverage, if any, prevails in (14), leverage and returns could be positively, insignificantly,
or negatively related in the data.
18 We use the shorthand E[f (X)|X < K] for E[f (X)1
{X<K} ], where X is random variable, f (·) is a function, 1{·} is an indicator
function, and K is a constant.
19 As discussed in the online appendix, r D (d ) is increasing in d if investors are risk averse and bankruptcy states happen when M is
1
1
2
high (bad states).
12
245
Consider next two firms with the same observed leverage and different target leverage. The firm with higher
target leverage must have a lower relative leverage because RL1 =
d1
k
−
d∗
1
k .
Therefore, the conclusions obtained
above for the relation between relative leverage and returns still hold with the opposite sign, as (12) shows. The firm
with higher target leverage will earn lower average equity returns. Finally, compare two firms with the same target
250
and different leverage, a case that can be constructed by taking two otherwise identical firms that differ only in terms
∗
d
of dk0 . Leverage enters with mixed signs both in the definition of γ k1 , dk1 and into equation (14). Therefore, the
relation between leverage and expected returns is indeterminate even after controlling for target leverage.
3. Supporting Empirical Evidence
This section provides empirical evidence linking capital structure imbalances and expected equity returns. This
evidence is based on standard reduced-form estimates of target leverage in the corporate finance literature and to
255
support the following quantitative analysis.
3.1. Leverage Variables
To compute the empirical counterparts of the key variables of the model, we define the market leverage ratio as
M Li,t =
Di,t
,
Di,t + M Ei,t
(15)
where Di,t denotes the stock of interest-bearing debt of firm i in period t and M Ei,t is the stock market capitalization
of firm i in period t. M Li,t is the empirical counterpart of optimal market leverage in the two-period model. To see
D
this, observe that the market value of debt can be obtained as E[M (1 + rt+1
)]dt = dt , and similarly, that the market
260
A
value of the assets is given by E[M (1 + rt+1
(1 − τt+1 ))]k = k.
Defining the target leverage ratio as T Li,t , relative leverage is obtained as the difference between observed and
target leverage, that is
RLi,t ≡ M Li,t − T Li,t .
(16)
3.2. Estimation of Target Leverage
A natural implementation of the policy function obtained in equation (5) of Proposition 1 is the empirical model
of Flannery and Rangan (2006) (FR), which is widely used in the corporate finance literature.
In the model of FR firms partially adjust their leverage over time toward the desired level M Li,t at a speed of
adjustment λ:
M Li,t − M Li,t−1 = λ(T Li,t − M Li,t−1 ) + i,t ,
13
(17)
with
T Li,t = βXi,t−1 .
(18)
T Li,t is modeled as a linear function of a set of firm-specific characteristics Xi,t−1 , and varies both over time and
265
across firms.
Equations (17) and (18) lead to the following estimable model:
M Li,t = (λβ)Xi,t−1 + (1 − λ)M Li,t−1 + i,t .
(19)
Differently from previous models employed in the literature (e.g. Hovakimian, Opler, and Titman 2001; Korajczyk
and Levy 2003), the specification of FR allows for dynamic partial adjustment in the presence of frictions, and is thus
consistent with the evidence provided by Leary and Roberts (2005) and Strebulaev (2007), according to which the
existence of frictions prevents firms from instantaneously adjusting toward their desired capital structure. Excluding
270
M Li,t−1 from the right-hand-side of (19) is equivalent to assuming that a firm’s target leverage always coincides with
its observed leverage, i.e., that there is full adjustment (λ = 1) in each period.20
The estimation of RLi,t requires three steps. First, we estimate equation (19) to obtain the coefficients for Xi,t−1
and M Li,t−1 , that are respectively given by λβ and (1 − λ). Second, we compute λ and β, and proceed to estimate
T Li,t . Third, we compute RLi,t as the difference between observed M Li,t and predicted T Li,t .
275
For the estimation of equation (19), we use the Compustat annual database over the period 1965-2013 including
all companies listed on AMEX, NYSE, and NASDAQ, and excluding firms that are not incorporated in the United
States, financials (SIC codes 6000-6999) and utilities (SIC codes 4900-4999), because of their special characteristics.
M Lit is defined as the book value of short-term plus long-term interest bearing debt (Compustat items DLTT+DLC)
divided by the market value of assets (DLTT+DLC + PRCC F*CSHO).
280
Following FR, the set of control variables Xi,t−1 comprises the following variables. Profitability: EBIT (EBIT)
over total assets (AT); Market Value over Assets: Book value of liabilities plus market value of equity (DLTT+DLC
+ PRCC F*CSHO) over total assets (AT); Depreciation: Depreciation (DP) over total assets (AT); Size: Logarithm
of total assets (AT)21 ; Tangibility: Property, plant, and equipment (PPENT) over total assets (AT); R&D expenses:
R&D expenses (XRD) over total assets (AT); R&D Dummy: Dummy equal to one for firms with missing values for
285
R&D expenses (XRD); Industry ML: Median industry ML calculated each year for two-digit SIC code industries; a
dummy for each fiscal year; and a firm fixed effect. The control variables capture the importance of tax shields and
bankruptcy costs, as well as possible additional determinants of target leverage identified by the empirical literature.
20 If
λ = 1, equation (17) simplifies to
M Li,t = T L∗i,t + i,t ,
that is
E[M Li,t ] = E[T Li,t ].
21 Assets
are deflated by the consumer price index in 2000 dollars.
14
Importantly, as our estimates of target leverage are included in asset pricing tests, all our estimates need to be
free of look-ahead bias. Thus, future information is never used in the formation of portfolios in the context of our
290
asset pricing tests. As the explanatory variables are lagged, our estimation of T Li,t begins in 1966. We require at
least five observations for the estimation to be included in the sample, that means that the first estimates of T Li,t
included in the sample are in 1970 for the set of firms with no gaps between 1965 and 1969. We then continue on a
rolling basis each year using all the accounting information available until that year.
In their review article on estimation techniques, Flannery and Hankins (2013) run a horse race between different
295
estimation procedures for a dynamic panel model like the one in equation (19).22 The evidence on which estimation
technique performs better is mixed. As the FE approach is more computationally efficient in large datasets, we
employ it for the estimation of equation (19).23
More precisely, we run a regression specification as in equation (19) that contains fixed effects for each individual
firm. To guarantee reliable estimates of the firm fixed effects, we exclude firms for which there are less than five
300
years of data. We then compute the relative leverage as per equation (16).24
3.3. Descriptive Statistics
Table 1 provides summary statistics for observed leverage (ML), target leverage (TL) and relative leverage (RL).
Panel A shows that the estimated mean for the target is close to that of the observed market leverage (0.16). TL
is highly persistent over time, as can be seen from the last column, which reports the autocorrelation of degree one.
RL is less persistent than TL. The findings that RL has approximately a zero mean, that TL and ML have similar
means, and that RL is not very persistent are all consistent with the idea that RL represents only a temporary
deviation from the target. Panel B of Table 1 reports some key statistics for the whole Compustat sample for which
leverage can be measured (first row) and for the subsample for which TL can be estimated. This subsample is fairly
comparable to the full sample, but mildly biased toward larger firms.
[Insert Table 1 Here]
Table 2 displays a breakdown of the sample into quintiles of RL. We observe that RL and ML are positively
related. Thus, firms that are overlevered also tend to have high leverage, and vice versa. Rather differently, RL and
TL are not related monotonically: TL first decreases and then increases with RL, reaching a minimum in the third
305
quintile. This finding shows that over-leverage is not the result of having a low target, nor is under-leverage the
22 The comparison is across the following models: OLS, FE, Arellano and Bond (1991) difference GMM, Blundell and Bond (1998)
system GMM, Huang and Ritter (2009) Four-Period Long Differencing, Hahn, Hausman, and Kuersteiner (2007) Longest Differencing,
and Least Square Dummy Variable Correction.
23 For robustness, in unreported tests we run our estimates using Blundell and Bond (1998) system GMM and obtain qualitatively
similar results.
24 For robustness, in the Online Appendix we consider three alternative measures of target leverage.
15
result of having a high target. Both the most overlevered and the most underlevered firms (fifth and first quintiles)
have high targets. The fourth column reports the change in ML. Notably, overlevered firms tend to lever down in
the next year, while underlevered firms tend to lever up. These findings are consistent with Figure 1 of FR, which
shows that in the next year leverage drops for overlevered firms and it increases for underlevered firms. The largest
310
decrease in leverage is for the firms in the highest quintile of RL. As already noted above, due to the monotonic
relation between RL and ML, these firms are also those with the highest level of ML. The fact that high leverage
firms reduce leverage in subsequent years is consistent with the findings of Figure 2 of FR. Column 4 also shows
that the change in leverage is faster for overlevered firms than for underlevered ones, consistent with the findings of
Warr, Elliott, Koëter-Kant, and Öztekin (2012). The remaining columns report mean values of capital stock (K),
315
investment (I/K), profitability (ROA), market capitalization (SIZE), and Tobin’s Q (Q) for each of the quintiles of
RL. Firms with higher relative leverage tend to have larger capital stocks and lower ROA.25 The relations between
relative leverage and investment, market capitalization, and Tobin’s Q are non-monotonic.
[Insert Table 2 Here]
3.4. Evidence on Returns and Capital Structure Imbalances
In our asset pricing tests, we use monthly stock prices and returns of common shares of firms on NYSE, AMEX,
320
Nasdaq covered by the Center of Research in Security Prices (CRSP) from 1980 to 2013.26 De-listing returns are
included in monthly returns. We drop observations if the trading status is reported to be halted or suspended.
Following the standard procedure of Fama and French (1992), we match these monthly data to annual income
statement and balance sheet data from the CRSP/COMPUSTAT merged database. More precisely, we match
monthly prices and returns from July of calendar year t to June of calendar year t + 1 with data from each company’s
325
latest fiscal year ending in calendar year t − 1. We apply the same matching procedure to the rolling estimates
of TL and RL described in Section 3. The matching procedure ensures a minimum gap of six months between
fiscal year-ends and returns. The gap is meant to conservatively avoid look-ahead biases arising from using future
accounting information to predict returns.
For the purpose of our tests, we compute the natural logarithm of market capitalization (SIZE), and the natural
330
logarithm of BM equity. Market capitalization - defined as the product of a company’s stock price times the number
of outstanding shares - is measured in June of calendar year t for the returns between July of calendar year t and June
of calendar year t + 1. We measure BM equity as the ratio between a firm’s book equity and its market capitalization
at the end of December of calendar year t−1. We compute book equity as the sum of shareholders’ equity and balance
25 Since RL is positively correlated with ML, this evidence is consistent with Gomes and Schmid (2010), who highlight that more levered
firms are also larger and less profitable.
26 As discussed in Section 3 we implement rolling estimates of T L . Years from 1965 to 1980 are used as “burn in” period to obtain
i,t
sensible initial estimates.
16
sheet deferred taxes and investment tax credits if available, minus the book value of preferred stocks. Depending on
335
data availability, we estimate the book value of preferred stocks using, in this order, their redemption, liquidation
and par value. Since we consider the natural logarithm of BM equity in our tests, we eliminate firms with negative
book equity from our analysis.
The six-month gap between predicting variables and returns Ri,t+1 from June/July of year t to June/July of year
t + 1 implies that, in the model, leverage variables in year t are observed at the end of the six-month gap (June/July
340
of year t) and that Ri,t+1 spans from June/July of year t to June/July of year t + 1.
3.5. Portfolio Sorts
In this section we perform a set of univariate portfolio sorts to examine how returns vary with both observed
leverage and capital structure imbalances. To do so, we consider sorts across different quintiles of ML and our
measure of capital structure imbalances RL. Table 3 reports equally-weighted excess returns in the left panel and
345
value-weighted excess returns in the right panel. The top row of both panels reports the realized returns (Re ) in excess
of the risk-free rate for different portfolios sorted according to ML.27 In the panel with equally-weighted returns, the
difference between the returns in the portfolio with the highest and the lowest ML is positive and equal to 6.09%. The
Sharpe ratio of the high-minus-low strategy is 0.46. However, returns do not increase monotonically across columns.
As ML increases across portfolios, returns first decrease and then increase. In the panel with value-weighted returns,
350
a similar pattern emerges. The difference between the returns of the portfolio with the highest and the lowest ML is
positive, but returns do not increase monotonically. In this case, the Sharpe ratio of the high-minus-low strategy is
0.10.
As for RL, across the two panels we report the portfolio sorts for RL and for equally-weighted and value-weighted
excess returns. The difference in returns between the high and the low portfolio is positive, equal to 5.56% for value-
355
weighted portfolios, and 8.04% for equally-weighted portfolios. The growth in returns across portfolios is perfectly
monotonic in some cases, while in others there tends to be a mild reduction in returns in the middle portfolios. The
high-minus-low Sharpe ratios in the left panel are roughly double those obtained for the ML portfolios. Similarly, the
high-minus-low Sharpe ratios for the value-weighted RL portfolios are approximately three times larger than those
obtained for the ML sorts.
[Insert Table 3 Here]
27 We
obtain risk-free rate data from Kenneth French’s website.
17
360
3.6. Cross-Sectional Regressions
Although sorts provide an immediate measure of the trading performance of simple long-short trading strategies
involving RL, they might lead to portfolios with few or even no stocks and become unfeasible when multiple variables
are considered at the same time. Since, as discussed in Section 2.5, our model offers predictions on the joint behavior
of RL, TL, and ML, we use the Fama and MacBeth (FMB) methodology to provide direct estimates of their marginal
365
effects after controlling for the standard predictors of expected equity returns.
Table 4 reports time-series averages of the estimated coefficients of monthly cross-sectional regressions of stock
returns on size, book-to-market (BM), ML, TL and RL. We report FMB tests with a Newey-West correction with
lag-length of 2 to assess which regressors have a coefficient that is significantly different from zero. In their most
comprehensive specification, the FMB regressions of Table 4 take the following form:
Ri,t = β0 + β1 M Li,t−1 + β2 RLi,t−1 + β3 T Li,t−1 + β4 log(Sizei,t−1 ) + β5 log(BM i,t−1 ) + i,t ,
(20)
where Ri,t denotes realized returns, Sizei,t−1 market capitalization, and BMi,t−1 BM of equity. M Li,t−1 , RLi,t−1
and T Li,t−1 are never simultaneously included in the same regression specification, but M Li,t−1 is included together
with RLi,t−1 in columns (4) and (5) and together with T Li,t−1 in column (6) and (7) to test the empirical predictions
of Section 2.5.28
370
The leftmost column shows that leverage has a small positive and insignificant coefficient once we control for
size, and BM. This “no result” for leverage shows that the results obtained in the univariate sorts of Table 3 are not
robust to the controls. These slopes can be interpreted as the average monthly return of a self-financing portfolio
with unit relative leverage, that hedges the effects of the controls in the sample period. In the FMB approach, the
standard error is computed as the standard deviation of monthly returns on this portfolio, divided by the square root
375
of the number of months in the sample (408). Hence, the t-statistics for the coefficients of RL can be approximately
translated into annualized Sharpe ratios of 0.6 (without controls) and 0.3 (with controls).29 Across all columns, size
is negatively related to returns, while B/M is positively related to returns.
It is important to remind the reader that the regressions in columns (1)-(3) are “model free”, in the sense that
our empirical predictions are for regressions that contain at least two of the three leverage measures. We turn to
380
these specifications in columns (4)-(7).
28 Naturally, since RL
i,t−1 = M Li,t−1 − T Li,t−1 , including all three variables at the same time would generate a multicollinearity
problem.
29 The average monthly risk-free rate in our sample is approximately 39 basis points.
18
3.6.1. Relative Leverage and Leverage
Columns (4) and (5) contain one of the main results of our analysis. Column (5) reports an FMB specification in
which we include both ML and RL in addition to the controls. With respect to column (1), the coefficient of ML is
now close to zero, and still not significant. Instead, the coefficient of RL is strongly significant, large, and positive.
385
The results of columns (4) and (5) are consistent with the prediction that RL is positively related to returns once
we control for leverage. Instead, the relation between leverage and returns is undetermined because of contrasting
effects. In the data, according to column (5), the positive effect of leverage on returns appears to dominate, although
it remains not significant.
3.6.2. Leverage and Target Leverage
390
In columns (6) and (7) we report a set of regressions where we include ML and TL. Column (7) also includes
the controls. The model predicts a negative relation between target leverage and returns, controlling for leverage.
Indeed, this is what we observe in the data. The coefficient of TL is negative and statistically significant at the 1
percent level. After controlling for target leverage, observed leverage is positively and significantly related to returns.
While consistent with the model predictions, the last result offers additional insights on the relative importance
395
of the contrasting effects of leverage on returns we described in Section 2.5. In conjunction with the estimates on
columns (4) and (5), the positive coefficient of leverage after controlling for TL suggests that the term
α d1
2 k
in (12)
has an important effect on the correction factor and, eventually, on returns. Indeed, a comparison of (12) and (13)
highlights that all other terms in observed leverage that enter (14) are common to the specifications with RL and ML,
and with TL and ML. Controlling for the target, observed leverage drives
400
c
d
1
c
k−d
1
up through the correction factor, and
increases expected returns as a consequence. Ultimately, accounting for firm heterogeneity in firms’ target capital
structure resurrects the traditional positive relationship between leverage and returns through an amplification effect
a la MM.30
[Insert Table 4 Here]
30 From
a purely empirical perspective, this result follows from the fact that (ignoring the controls and the error term), the specification
Ri,t = β0 + β1 M Li,t−1 + β2 RLi,t−1
implies that
Ri,t = β0 + (β1 + β2 ) M Li,t−1 − β2 T Li,t−1 ,
which is approximately what we observe in the data.
19
4. Dynamic Model
The simple illustration in Section 2 provides intuition for our key results. However, the model is too stylized to
405
serve as a basis for a quantitative investigation of its predictions. In this section, we embed the key economic tradeoffs of our simple illustration in a more general dynamic environment. In the dynamic model, we allow for ex-post
cross-sectional firm heterogeneity in the form of idiosyncratic productivity shocks to study the role of differences
in target leverage across firms. We also introduce aggregate shocks, which affect firms’ profitability and discount
rates. Every period, firms make endogenous investment decisions. This allows for a more realistic analysis of the
410
relationship between firm size, capital structure imbalances, and equity returns. Firms are financially constrained
as external equity is costly and firms can endogenously default on their debt. Endogenous default creates a more
realistic dependence of firm bankruptcy on the cost of borrowing. Finally, we adopt an asymmetric flexible functional
form for debt adjustment costs. This allows for small adjustments to incur possibly large marginal costs while keeping
the model tractable.
Technology and Investment. Time is discrete. We consider the problem of a value-maximizing firm in a perfectly
competitive environment. After-tax operating profits Πi,t for firm i in period t are given by
α
Πi,t = (1 − τ )(eAt eZi,t Ki,t
− F ),
(21)
where τ ∈ (0, 1) is the corporate tax rate, At is an exogenous aggregate shock, Zi,t is a firm-specific shock, Ki,t is firm
i’s capital stock, α ∈ (0, 1) is the capital share in production, and F > 0 is a fixed production cost. The variables At
and Zi,t can be interpreted as shocks to demand, input prices, or productivity. At and Zi,t have bounded support
A = [A, A] and Z = [Z, Z] , respectively. The law of motion of At is described by a Markovian transition function
QA (At , At+1 ). Similarly, Zi,t is Markovian with transition function QZ (Zi,t , Zit +1 ). In our quantitative analysis, we
parameterize At and Ki,t to provide a discrete approximation to continuous AR(1) processes as follows:
log At = µA (1 − ρA ) + ρA log At−1 + σA εA
t ,
log Zi,t = ρZ log Zi,t−1 + σZ εZ
i,t .
415
Z
As εA
t and εi,t are truncated standard normal variables, both At and Zi,t are lognormal, with mean µA ∈ (−∞, ∞)
and 0, persistence ρA ∈ (0, 1) and ρZ ∈ (0, 1), and volatility σA ∈ (0, ∞) and σZ ∈ (0, ∞), respectively.31
At the beginning of each period, firms can scale operations by choosing investment Ii,t . Next period’s capital
stock Ki,t+1 satisfies the standard capital accumulation rule
Ki,t+1 = (1 − δ)Ki,t + Ii,t ,
(22)
where δ ∈ (0, 1) is the depreciation rate of capital.
31 As
common in the literature, we set the mean of firm-specific shocks to zero (e.g., Hennessy and Whited 2007, Gomes and Schmid
2010).
20
Financing. Investment and distributions to shareholders can be financed with either the internal funds generated
by operating profits or new issues. The latter which can take the form of new debt (net of repayments) or external
420
equity.
We denote the firm’s debt stock as Bi,t . Outstanding debt pays a coupon ci,t per unit of time. As we detail
below, the coupon is set in competitive credit markets to compensate expected bankruptcy costs in case of default,
in which lenders recover a fraction ξ ∈ (0, 1) of the firm i’s capital stock. Firms are allowed to refinance their debt
stock by issuing a net amount ∆Bi,t = Bi,t+1 − (1 + ci,t )Bi,t .
Similar to Croce, Kung, Nguyen, and Schmid (2012) and Belo, Lin, and Yang (2014), firms incur costs Λ(∆Bi,t )
of adjusting their debt stock. In our quantitative analysis we choose a flexible parameterization for Λ(∆Bi,t ) to
allow for asymmetries and possibly very large marginal costs for small adjustments. Debt adjustment costs can arise
from different sources, including underwriting and management spreads (e.g., Fischer, Heinkel, and Zechner 1989,
Strebulaev 2007), seniority issues that prevent firms from making large changes in the capital structure (e.g., Acharya,
Bharath, and Srinivasan 2007), costs associated with the liquidation of short-term debt (Diamond 1991), agency costs
associated with long-term debt (e.g., debt overhang and under-investment as in Myers 1977). Specifically, we choose
a linear-exponential (LINEX) functional form (e.g., Varian 1975, Kim and Ruge-Murcia 2009, Aruoba, Bocola, and
Schorfheide 2017), i.e.,
Λ(∆Bi,t ) = λB (eγB ∆Bi,t − γB ∆Bi,t − 1),
425
(23)
with λB ∈ (0, ∞), and γB ∈ (−∞, ∞). In our context, the LINEX functional form is attractive for three main reasons.
First, this flexible reduced-form functional form allows us to keep the model tractable without committing to a specific
mechanism in our quantitative exercise. Second, as previous studies provide limited guidance on functional forms and
economic magnitudes for debt adjustment costs, we can exploit data restrictions to calibrate them to realistic values
in the context of our model. Third, the adjustment cost function is parsimonious. Varying only two parameters, the
430
“scale” λB , and the “asymmetry” γB , Λ(∆Bi,t ) spans a broad spectrum of functional forms.32
Firms can also raise external finance through seasoned equity offerings (SEOs). Let Ei,t denote the equity
issuances. Following the extensive existing literature, we consider equity issuance costs Λ(Ei,t ) that include both a
fixed and a proportional component, i.e.,
Λ(Ei,t ) = (λ0 + λ1 Ei,t )χEi,t >0 ,
(24)
with λ0 ∈ [0, ∞) and λ1 ∈ [0, ∞). We interpret negative values for Ei,t as equity payouts.
Investment financing decisions must satisfy the firm’s budget constraint, which takes the form of the following
accounting identity between uses and sources of funds:
Πi,t + ∆Bi,t + Ei,t + τ δKi,t + τ ci,t = Ii,t + ci,t Bi,t + Λ(∆Bi,t ),
32 The
LINEX functional form nests the quadratic form as an approximation for γB → 0.
21
(25)
where the terms τ δKi,t and τ ci,t reflect the tax deductibility of depreciation and interest expenses, respectively. Net
distributions to shareholders, Di,t , are then defined as equity payout net of issuance costs, i.e.,
Di,t = −Ei,t − Λ(Ei,t ).
(26)
Valuation. Following several studies in cross-sectional production-based asset pricing (e.g., Berk, Green, and Naik
1999, Zhang 2005, Gomes and Schmid 2010), we parameterize the stochastic discount factor of the economy without
explicitly modeling the investor’s problem. As in Zhang (2005), we assume the following stochastic process for the
stochastic discount factor:
logMt = logβ + γt (At − At+1 ),
(27)
where β ∈ (0, 1), γt = γ0 + γ1 (At − µA ), γ0 ∈ (0, ∞), and γ1 ∈ (−∞, 0). This functional form naturally links
to the time-varying risk aversion in Campbell and Cochrane (1999), in which γ0 is interpreted as a “risk aversion”
parameter and γ1 as an “habit formation” parameter. However, as we do not model the household problem explicitly,
435
we remain agnostic about the specific sources of time-varying risk aversion.
Define as Vi,t the equity value of the firm. We assume that shareholders strategically default on their debt
obligations if Vi,t < 0. Thus, interest payments ci,t are determined endogenously as follows:
Bi,t+1 = Et Mt+1 {(1 + ci,t+1 ) Bi,t+1 χ{Vi,t+1 ≥0} + ξKi,t+1 χ{Vi,t+1 <0} } ,
(28)
where ξ ∈ [0, 1) is the fraction of capital the lenders recover in case of default. Equity holders choose investment Ii,t
and debt issuance ∆Bi,t to solve the following recursive problem:
Vi,t =
max {0, Di,t + Et [Mt+1 · Vi,t+1 ]},
Ii,t ,∆Bi,t
(29)
subject to (22), (25), and (28). In the recursive representation (29) the equity value Vi,t = V (Si,t ) is a function of
the state variables Si,t = {Ki,t , Bi,t , ci,t , Zi,t , At }.
Defining a Dynamic Target. To define a dynamic leverage target, we follow the “gap approach”, which has been
extensively used to describe costly adjustments of production factors (e.g., Sargent 1978, Caballero and Engel 1993,
440
Cooper and Willis 2004, King and Thomas 2006, Bayer 2009). These studies consider adjustments toward a dynamic
target, which is termed as the frictionless target. The latter corresponds to the level of a production factor to which
an optimizing agent would eventually adjust to in the absence of changes in the stochastic variables. More formally,
the frictionless target is constructed as the policy function in which adjustment costs are removed for a single period.
∗
∗
In our setup, the frictionless target debt stock Bi,t
= Bi,t + ∆Bi,t
can be obtained from the following optimization
problem:
∗
C
max {0, Di,t
+ Et [Mt+1 · Vi,t+1
]},
∗ ,∆B ∗
Ii,t
i,t
22
(30)
∗
where Di,t
is the equity payout in which current-period debt adjustment costs are removed, i.e., Λ(∆Bi,t ) = 0. The
445
C
is computed using the value function Vi,t in (29), as adjustment costs are set to zero only
continuation value Vi,t+1
for the current period. We then compute the target leverage ratio as
∗
Bi,t
∗ ,
Ki,t
∗
∗
where Ki,t
= Ki,t + Ii,t
. As in the gap
approach, our dynamic target can be interpreted as the capital structure that, conditional on today’s state of the
world, optimally positions the firm to deal with the uncertain funding needs it may have in the future.
Three remarks are in order. First, the frictionless target encompasses the definition of target in the quadratic
450
adjustment model of Section 2 as a special case. More generally, Adda, Cooper, and Cooper (2003) show that the
approximation of the frictionless target as the fixed point where the state-dependent policy function crosses the
45-degree line is exact in the case of a linear-quadratic dynamic programming problem. Second, the frictionless
target generally differs from the static target defined as the debt level that would arise if there were never any
costs of adjustments (e.g., Cooper and Willis 2004, Bayer 2009). The static and the frictionless target coincide in
455
the illustration in Section 2. However, due to its dynamic nature, the frictionless target more closely maps onto the
reduced-form empirical specifications commonly used in the empirical corporate finance literature in which the target
depends on firm-level variables (e.g., Flannery and Rangan 2006, Lemmon, Roberts, and Zender 2008).33 Third, the
structure of our dynamic model does not mechanically impose convergence to the target. Thus, the calibrated model
serves as a lab to quantify to what extent firms adjust their capital structure toward the target.
460
Model Solution. As the model admits no closed-form solution, we resort to numerical dynamic programming in our
quantitative analysis. The model solution is computationally challenging. Equity and debt values are mutually
dependent since the default condition affects the debt pricing equation. In a similar context, Gomes and Schmid
(2010) reduce the dimensionality of the state space using total debt commitments as a state variable. However, their
approach is not viable because of the presence of debt adjustment costs. Thus, we need to solve jointly for both
465
equity values and interest rate schedules and keep track of the dynamics of our five state variables, including the
coupon. The Appendix details our numerical solution method.
Discussion. We lay out a neoclassical dynamic model of investment and financing with endogenous default and a
flexible functional form for debt adjustment costs. As debt adjustment costs can originate from several sources on
which the literature provides limited guidance, we discipline their values by means of calibration. To study firms’
470
adjustments toward target capital structures in a more realistic dynamic environment, we take advantage of the
“gap approach” in the macroeconomic literature. The following quantitative analysis assesses the relevance of the
mechanism we illustrate in Section 2, and for which we provide suggestive empirical evidence in Section 3.4, for
leverage dynamics and equity returns.
33 DeAngelo, DeAngelo, and Whited (2011) consider a static long-run target to which firms would converge after receiving neutral
shocks for many periods in a row.
23
5. Quantitative Analysis
475
5.1. Calibration and Model Fit
Table 5 summarizes our baseline calibration. The calibration frequency is monthly. Details about the computation
of the model-based and data variables are provided in the Appendix. The model features 17 parameters. The first
8 parameters in the table are on the technology side. We set the curvature of the profit function, α, to 0.4, to
roughly match the capital share from the Bureau of Economic Analysis (BEA). This value is similar, for example,
480
to the ones used by Kydland and Prescott (1982), Gomes (2001), and Gomes, Jermann, and Schmid (2016). The
depreciation rate δ is set to be 0.01. This is a fairly common value in the literature, as it implies an annual rate of
roughly 12%. This value is in line with the empirical estimates in Cooper and Haltiwanger (2006) and comparable
to those used by previous studies. For example, Gomes (2001) uses a depreciation rate of 0.145, while Hennessy
and Whited (2005) estimate a value of 0.10. We choose the persistence of the aggregate productivity process, ρA ,
1
485
and its volatility, σA , to be 0.95 3 and 0.007/3, respectively. These monthly values correspond to 0.95 and 0.007 at
the quarterly frequency, consistent with several studies, including Cooley and Prescott (2021), Zhang (2005), and
Gomes, Jermann, and Schmid (2016). We normalize the average aggregate productivity, µA , to -2. µA is purely a
scaling constant that determines the long-run average scale of the economy. As in Zhang (2005), we calibrate the
persistence ρZ and volatility σZ of the idiosyncratic shock process to 0.97 and 0.1, respectively. The fixed cost of
490
operation, F , is chosen to approximately match average profitability in our sample, which leads to a value of 0.18
(or 2.25% of the average capital stock).
The next 3 parameters describe the dynamics of the pricing kernels. We choose β, γ0 , and γ1 to minimize mean
square errors with respect to three aggregate data moments, namely the average Sharpe ratio, the average risk-free
rate, and its volatility (as in Zhang 2005). This procedure yields β = 0.9928, γ0 = 52.71, and γ1 = −50.19.
495
The remaining 6 parameters are on the financing side. We pick the fixed and proportional equity flotation costs,
λ0 and λ1 , to be 0.5 and 0.025. As, for example, Kuehn and Schmid (2012) and Bolton, Wang, and Yang (2021),
we choose λ0 to approximately match the frequency of equity issuance in the data. For the proportional component
λ1 , we pick the same value as in Gomes and Schmid (2010), who also study levered returns. This is also close to
Gomes (2001), who chooses 0.028 based on regressions of flotation costs on amount issued. We choose the recovery
500
rate parameter ξ to be 0.125 of the firm’s capital stock. This implies an average debt recovery rate of 53.9%, which
is close to the 51 % recovery rate for creditors when the firm defaults in Huang and Huang (2012).
We calibrate the debt adjustment cost parameters λB and γB to approximately target the average debt issuance
and the frequency of default in our sample, respectively. The scale parameter λB affects the marginal cost of issuing
debt and, in the model, discourages the use of external debt financing. Instead, positive values of γB imply that
505
issuing debt is more costly than withdrawing debt. Thus, the lower γB (i.e., negative large values), the more likely
firms default due to their inability of reducing leverage following negative shocks. These values imply an average cost
of issuing debt of 0.05 % of the total amount of debt issued, which is at the lower end of the range used by Strebulaev
24
(2007), who chooses values in the range of 0.05 % to 0.35 %. The average costs of withdrawing debt are instead larger,
around 0.1 % of the total amount of debt withdrawn. Overall, the magnitude of debt adjustment costs suggests that,
510
as in Fischer, Heinkel, and Zechner (1989), Goldstein, Ju, and Leland (2001), and Strebulaev (2007), relatively small
adjustment costs significantly affect leverage dynamics. Finally, following Nikolov and Whited (2014), we choose the
tax rate τ to be 0.20. This is as an approximation of the statutory corporate tax rate relative to personal tax rates.
[Insert Table 5 Here]
Table 6 summarizes overall model fit under the parameterization in Table 5. The table compares model-implied
moments, which are tabulated in the first column, with their empirical counterparts, which are tabulated in the
515
second column. Overall, the model does a reasonable job at matching key variables describing the financing policies
of US firms. The model matches quite closely the Sharpe ratios, the annual risk-free rate, and its volatility. The
model produces a sizeable equity premium, which we do not target in the calibration. Both in the model and in the
data, this moment is around 6%. The second set of moments in the table refer to firms’ real and financial policies. On
the real side, the model matches fairly closely average profitability, the volatility and autocorrelation of profitability,
520
and investment ratios. The model does a reasonable job in reproducing average leverage, an untargeted quantity.
Model-implied leverage is 21%, a slight overestimation of its data counterpart of 16 %. Our parameterization also
produces default rates and book-to-market ratios with comparable magnitudes to the data. The third set of moments
in the table describe firms’ capital structure rebalancing. Debt issuance is 19% in the model, and 22% in the data.
Although we target this moment in our calibration, we report model-implied and data moments that describe the
525
relative frequency of positive and negative debt adjustments. The model-implied magnitudes of these additional
non-targeted moments are also close to the data. Finally, the frequency of equity issuance in the model is 5%, fairly
close to its data value of 4%. Overall, the model fit seems reasonable, both for moments that serve as targets in the
calibration, and for untargeted key statistics. In the following sections we use our baseline calibration as a lab to
provide inference about capital structure rebalancing and equity returns.
[Insert Table 6 Here]
530
5.2. Adjustment toward Target Leverage
Table 7 describes capital structure dynamics around target leverage under the baseline calibration of Table 5.
Panel A breaks down the simulated data into quintiles of RL. The top row reports the average values of RL for each
quintile. As RL is defined as the difference between market and target leverage, negative (positive) values refer to
underlevered (overlevered) firms. The bottom row tabulates the corresponding one-period-ahead changes in leverage.
535
Overlevered firms tend to lever down, while underlevered firms tend to lever up. These findings, which are based on
the frictionless target defined as in (30), are in line with the reduced-form evidence in Section 3.4.
25
Panel B reports model-implied and data estimates of adjustment speeds, both in the model and in the data. Panel
B also investigates the role of firm’s size for capital structure adjustments. Data estimates are from the estimation
of the model of Flannery and Rangan (2006) in (19), in which the speed of adjustment λ in (19) now has a fixed
component and a component related to size (total assets AT), i.e., λ = λ0 + λ1 log(AT ). As in the model, target
leverage is observable, and model-implied estimates are from the following specification:
∆M Li,t+1 = λi,t (−RLi,t ) + i,t+1 ,
(31)
where ∆M Li,t+1 = M Li,t+1 − M Li,t and λi,t = λ0 + λ1 log(Ki,t ). Notice that the minus sign in front of RLi,t
captures the fact that adjustments toward the target imply that positive values of RLi,t are associated with negative
values of ∆M Li,t+1 , and vice versa. This is because overlevered firms would tend to reduce their leverage (and vice
540
versa), as the non-parametric evidence in Panel A indicates. Both in the model and in the data, the adjustment
speed in Panel B for the full sample (“all firms”) is computed in correspondence of average size. Adjustment speeds
for large (small) firms are instead evaluated in correspondence of one standard deviation above (below) average size.
The row labeled “All Firms” shows that the average adjustment speed in the model is 0.17, close to its 0.21 data
counterpart. Both in the model and in the data, small firms tend to adjust faster toward target leverage. Estimated
545
adjustment speeds in the “Small Firms” row are 0.22 (model) and 0.27 (data), versus 0.12 (model) and 0.25 (data)
in the “Large Firms” row.
Taken together, the results in Table 7 suggest that firms adjust their capital structure toward a dynamic target
leverage ratio, consistent with the intuition in Proposition 1. In the cross section of size, this pattern is present for
both large and small firms.
[Insert Table 7 Here]
550
5.3. Returns and Capital Structure Imbalances
Table 8 reports average annualized value-weighted stock returns under the baseline parameterization of Table
5. The table shows the results of double sorts on relative leverage (rows) and on leverage (columns). Panel A
tabulates returns for all firms in the simulated economy. Across all leverage portfolios, overlevered firms earn higher
average returns than underlevered firms. The spreads in average returns is more pronounced for firms with high
555
leverage. Leverage is also generally positively related to returns, with the exception of underlevered firms, for which
low-leverage firms earn higher average returns than high-leverage firms.
Panels B and C, instead, report returns for large and small firms, defined as those for which Ki,t is, respectively,
above and below its median. Average returns are positively related to relative leverage, and the spread is more
pronounced for large firms, whose adjustment speeds are lower. Instead, the relationship between leverage and returns
26
560
appears less stable. High-leverage firms earn lower average returns than low-leverage firms among underlevered large
firms. In addition, spreads in returns are economically small for all small firms, for each bin of relative leverage.
[Insert Table 8 Here]
Table 9 reports estimates of cross-sectional regressions of stock returns on relative leverage and leverage on the
same simulated data. Columns (1) to (3) tabulate estimates for the full simulated sample. The results mimic the
patterns in Table 8 and are broadly in line with the non-targeted empirical evidence of Section 3.4. The coefficient
565
on relative leverage is larger in magnitude than the one on leverage. When size and book-to-market are included as
controls in Column (3), the coefficient on leverage becomes negative. This confirms that the relationship between
leverage and returns is ambiguous. Columns (4) to (6) and (7) to (9) refer, respectively, to firms above and below
the median value of size Ki,t . The coefficient on relative leverage is positive for both groups of firms, but bigger for
large firms. Thus, as before, capital structure imbalances appear to be more important for larger firms.
570
All in all, the results in Table 8 and Table 9 corroborate the key intuition of Proposition 2 in a more realistic
environment. Capital structure imbalances are important drivers of the relationship between leverage and crosssectional returns.
[Insert Table 9 Here]
6. Conclusions
In this paper we present a model in which firms maximize the value of equity by choosing an optimal amount of
575
debt finance. There are three frictions: corporate taxes, bankruptcy costs and debt adjustment costs. We show that
the optimal leverage policy for the firm requires a partial adjustment toward a target leverage. Partial adjustment
occurs because there are costs in adjusting debt. We then derive the implications of this optimal policy for the
relation between equity returns and leverage in the presence of capital structure imbalances. We illustrate our key
ideas in a stylized setup, which provides a transparent illustration of the economic forces underlying our empirical
580
results. In this context, we derive a formula that is similar to the celebrated equation of the second proposition of
Modigliani and Miller (1958).
In the presence of the above frictions, we show that one cannot analyze the relation between leverage and returns
without controlling for either target or relative leverage. We show that controlling for leverage, equity returns
are increasing in relative leverage, which is defined as the difference between leverage and target leverage. Instead,
585
controlling for leverage, equity returns are decreasing in target leverage. We find that the sign of the relation between
returns and leverage is theoretically indeterminate (controlling for either relative or target leverage), because there
are different counteracting effects at work.
27
In our setup, both the amount of debt and its unit cost determine the overall effect of leverage on returns. As
in the second proposition of Modigliani and Miller (1958), leverage decisions affect how cash flow risk propagates
590
to equity payouts through the amplification of both good and bad outcomes. Given leverage, capital structure
imbalances influence the unit cost of debt through the non-diversifiable components of bankruptcy costs and tax
shields, which are reflected in the correction factor. Overlevered firms are riskier investments than underlevered
firms because they carry costly debt with high expected (non-diversifiable) bankruptcy costs. As a consequence,
capital structure imbalances either boost or dampen the traditional amplification effect of Modigliani and Miller
595
(1958).
We embed the key economic trade-offs in this simple illustration in a more general dynamic environment, in
which firms make endogenous investment and default decisions. A quantitative version of our model reproduces the
key facts about leverage, capital structure imbalances, and equity returns for U.S. corporations. We also provide
reduced-form empirical evidence, based on standard proxies for target leverage from the corporate finance literature.
600
Overall, our quantitative and empirical analyses corroborate the key intuition from the two-period illustration.
Our work represents a step toward a closer integration between the theory of capital structure, as developed
in the field of corporate finance, and production-based asset pricing theory. Our results indicate that financial
flexibility crucially affects the link between leverage and equity returns. We acknowledge two limitations of our study.
First, for tractability, we are agnostic about the sources of adjustment costs in firms’ capital structure. Although
605
incorporating issues such as seniority and maturity poses conceptual and computational challenges, the literature is
currently missing a quantitative analysis of the frictions that limit firms’ financial flexibility. Second, although it is
reassuring that model-based and reduced-form evidence agree, the mapping between target leverage in the model
and its empirical estimates is not fully understood. Albeit outside of the scope of this paper, a thorough evaluation
of common empirical estimates of target leverage through the lens of structural models would be informative. We
610
leave these tasks to future research.
28
References
Acharya, V. V., S. T. Bharath, and A. Srinivasan, 2007, Does industry-wide distress affect defaulted firms? evidence
from creditor recoveries, Journal of financial economics, 85(3), 787–821.
Adda, J., R. Cooper, and R. W. Cooper, 2003, Dynamic economics: quantitative methods and applications. MIT
615
press.
Arellano, M., and S. Bond, 1991, Some tests of specification for panel data: Monte carlo evidence and an application
to employment equations, The Review of Economic Studies, 58(2), 277–297.
Aruoba, S. B., L. Bocola, and F. Schorfheide, 2017, Assessing dsge model nonlinearities, Journal of Economic
Dynamics and Control, 83, 34–54.
620
Bates, T. W., K. M. Kahle, and R. M. Stulz, 2009, Why do us firms hold so much more cash than they used to?,
The Journal of Finance, 64(5), 1985–2021.
Bayer, C., 2009, A comment on the economics of labor adjustment: Mind the gap: Evidence from a monte carlo
experiment, The American Economic Review, 99(5), 2258–2266.
Belo, F., X. Lin, and S. Bazdresch, 2014, Labor hiring, investment, and stock return predictability in the cross
625
section, Journal of Political Economy, 122(1), 129–177.
Belo, F., X. Lin, and F. Yang, 2014, External equity financing shocks, financial flows, and asset prices, working
paper, National Bureau of Economic Research.
Berk, J. B., R. C. Green, and V. Naik, 1999, Optimal investment, growth options, and security returns, The Journal
of finance, 54(5), 1553–1607.
630
Bhandari, L. C., 1988, Debt/equity ratio and expected common stock returns: Empirical evidence, The Journal of
Finance, 43(2), 507–528.
Blundell, R., and S. Bond, 1998, Initial conditions and moment restrictions in dynamic panel data models, Journal
of Econometrics, 87(1), 115–143.
Bolton, P., N. Wang, and J. Yang, 2021, Leverage dynamics under costly equity issuance, .
635
Bretscher, L., P. Feldhütter, A. Kane, and L. Schmid, 2020, Marking to market corporate debt, Swiss Finance
Institute Research Paper, (21-06).
Caballero, R. J., and E. M. Engel, 1993, Microeconomic adjustment hazards and aggregate dynamics, The Quarterly
Journal of Economics, 108(2), 359–383.
29
Campbell, J. Y., and J. H. Cochrane, 1999, By force of habit: A consumption-based explanation of aggregate stock
640
market behavior, Journal of political Economy, 107(2), 205–251.
Caskey, J., J. Hughes, and J. Liu, 2012, Leverage, excess leverage, and future returns, Review of Accounting Studies,
17(2), 443–471.
Cochrane, J. H., 1991, Production-based asset pricing and the link between stock returns and economic fluctuations,
The Journal of Finance, 46(1), 209–237.
645
Cooley, T. F., and E. C. Prescott, 2021, 1. economic growth and business cycles, in Frontiers of business cycle
research. Princeton University Press, pp. 1–38.
Cooper, R., and J. L. Willis, 2004, A comment on the economics of labor adjustment: Mind the gap, The American
Economic Review, 94(4), 1223–1237.
Cooper, R. W., and J. C. Haltiwanger, 2006, On the nature of capital adjustment costs, The Review of Economic
650
Studies, 73(3), 611–633.
Covas, F., and W. J. Den Haan, 2011, The cyclical behavior of debt and equity finance, American Economic Review,
101(2), 877–99.
Croce, M. M., H. Kung, T. T. Nguyen, and L. Schmid, 2012, Fiscal policies and asset prices, Review of Financial
Studies, 25(9), 2635–2672.
655
DeAngelo, H., L. DeAngelo, and T. M. Whited, 2011, Capital structure dynamics and transitory debt, Journal of
Financial Economics, 99(2), 235–261.
Diamond, D. W., 1991, Debt maturity structure and liquidity risk, the Quarterly Journal of economics, 106(3),
709–737.
Doshi, H., K. Jacobs, P. Kumar, and R. Rabinovitch, 2019, Leverage and the cross-section of equity returns, The
660
Journal of Finance, 74(3), 1431–1471.
Fama, E. F., and K. R. French, 1992, The cross-section of expected stock returns, The Journal of Finance, 47(2),
427–465.
, 2002, Testing trade-off and pecking order predictions about dividends and debt, Review of Financial Studies,
15(1), 1–34.
665
Faulkender, M., M. J. Flannery, K. W. Hankins, and J. M. Smith, 2010, Cash flows and leverage adjustments,
Working Paper.
30
Fischer, E. O., R. Heinkel, and J. Zechner, 1989, Dynamic capital structure choice: Theory and tests, The Journal
of Finance, 44(1), 19–40.
Flannery, M. J., and K. W. Hankins, 2010, Estimating dynamic panel models in corporate finance, Working Paper.
670
Flannery, M. J., and K. W. Hankins, 2013, Estimating dynamic panel models in corporate finance, Journal of
Corporate Finance, 19, 1–19.
Flannery, M. J., and K. P. Rangan, 2006, Partial adjustment toward target capital structures, Journal of Financial
Economics, 79(3), 469–506.
Friewald, N., F. Nagler, and C. Wagner, 2021, Debt refinancing and equity returns, Journal of Finance, Forthcoming.
675
George, T. J., and C.-Y. Hwang, 2010, A resolution of the distress risk and leverage puzzles in the cross section of
stock returns, Journal of Financial Economics, 96(1), 56–79.
Goldstein, R., N. Ju, and H. Leland, 2001, An ebit-based model of dynamic capital structure, The Journal of
Business, 74(4), 483–512.
Gomes, J., L. Kogan, and L. Zhang, 2003, Equilibrium cross section of returns, Journal of Political Economy, 111(4).
680
Gomes, J. F., 2001, Financing investment, American Economic Review, pp. 1263–1285.
Gomes, J. F., U. J. Jermann, and L. Schmid, 2016, Sticky leverage, American Economic Review.
Gomes, J. F., and L. Schmid, 2010, Levered returns, The Journal of Finance, 65(2), 467–494.
, 2016, Equilibrium asset pricing with leverage and default, The Wharton School Research Paper, (92).
Gomes, J. F., A. Yaron, and L. Zhang, 2006, Asset pricing implications of firms financing constraints, Review of
685
Financial Studies, 19(4), 1321–1356.
Graham, J. R., 2000, How big are the tax benefits of debt?, The journal of finance, 55(5), 1901–1941.
Hahn, J., J. Hausman, and G. Kuersteiner, 2007, Long difference instrumental variables estimation for dynamic
panel models with fixed effects, Journal of Econometrics, 140(2), 574–617.
Halling, M., J. Yu, and J. Zechner, 2015, Leverage dynamics over the business cycle, in AFA 2012 Chicago Meetings
690
Paper.
Hennessy, C. A., and T. M. Whited, 2005, Debt dynamics, The Journal of Finance, 60(3), 1129–1165.
, 2007, How costly is external financing? Evidence from a structural estimation, The Journal of Finance,
62(4), 1705–1745.
31
Hou, K., C. Xue, and L. Zhang, 2014, Digesting anomalies: An investment approach, Review of Financial Studies.
695
Hovakimian, A., T. Opler, and S. Titman, 2001, The debt-equity choice: An analysis of issuing firms, Journal of
Financial and Quantitative Analysis, 36(1), 1–24.
Huang, J.-Z., and M. Huang, 2012, How much of the corporate-treasury yield spread is due to credit risk?, The
Review of Asset Pricing Studies, 2(2), 153–202.
Huang, R., and J. R. Ritter, 2009, Testing theories of capital structure and estimating the speed of adjustment,
700
Journal of Financial and Quantitative Analysis, 44(2), 237–271.
Jermann, U. J., 1998, Asset pricing in production economies, Journal of Monetary Economics, 41(2), 257–275.
Kim, J., and F. J. Ruge-Murcia, 2009, How much inflation is necessary to grease the wheels?, Journal of Monetary
Economics, 56(3), 365–377.
King, R. G., and J. K. Thomas, 2006, Partial adjustment without apology, International Economic Review, 47(3),
705
779–809.
Korajczyk, R. A., and A. Levy, 2003, Capital structure choice: Macroeconomic conditions and financial constraints,
Journal of Financial Economics, 68(1), 75–109.
Korteweg, A., 2010, The net benefits to leverage, The Journal of Finance, 65(6), 2137–2170.
Krasker, W. S., 1986, Stock price movements in response to stock issues under asymmetric information, The Journal
710
of Finance, 41(1), 93–105.
Kuehn, L.-A., and L. Schmid, 2012, Investment-based corporate bond pricing, Working Paper.
Kydland, F. E., and E. C. Prescott, 1982, Time to build and aggregate fluctuations, Econometrica: Journal of the
Econometric Society, pp. 1345–1370.
Leary, M. T., and M. R. Roberts, 2005, Do firms rebalance their capital structures?, The Journal of Finance, 60(6),
715
2575–2619.
Lemmon, M. L., M. R. Roberts, and J. F. Zender, 2008, Back to the beginning: persistence and the cross-section of
corporate capital structure, The journal of finance, 63(4), 1575–1608.
Liu, L. X., T. M. Whited, and L. Zhang, 2009, Investment-based expected stock returns, Journal of Political Economy,
117(6), 1105–1139.
720
Livdan, D., H. Sapriza, and L. Zhang, 2009, Financially constrained stock returns, The Journal of Finance, 64(4),
1827–1862.
32
Modigliani, F., and M. H. Miller, 1958, The cost of capital, corporation finance and the theory of investment, The
American economic review, 48(3), 261–297.
Myers, S. C., 1977, Determinants of corporate borrowing, Journal of Financial Economics, 5(2), 147–175.
725
Myers, S. C., and N. S. Majluf, 1984, Corporate financing and investment decisions when firms have information
that investors do not have, Journal of Financial Economics, 13(2), 187–221.
Nikolov, B., and T. M. Whited, 2014, Agency conflicts and cash: Estimates from a dynamic model, The Journal of
Finance, 69(5), 1883–1921.
Novy-Marx, R., 2013, The other side of value: The gross profitability premium, Journal of Financial Economics,
730
108(1), 1–28.
Obreja, I., 2013, Book-to-market equity, financial leverage, and the cross-section of stock returns, Review of Financial
Studies, 26(5), 1146–1189.
Opler, T., L. Pinkowitz, R. Stulz, and R. Williamson, 1999, The determinants and implications of corporate cash
holdings, Journal of Financial Economics, 52(1), 3–46.
735
Ozdagli, A. K., 2012, Financial leverage, corporate investment, and stock returns, Review of Financial Studies, 4,
1033–1069.
Ozdagli, A. K., 2015, An injury to one: Financial frictions and the reaction of stock prices to monetary policy shocks,
Federal Reserve Bank of Boston Working Paper.
Penman, S. H., S. A. Richardson, and I. Tuna, 1992, The book-to-price effect in stock returns: Accounting for
740
leverage, Journal of Accounting Research, 45(2), 427–467.
Sargent, T. J., 1978, Estimation of dynamic labor demand schedules under rational expectations, Journal of political
Economy, 86(6), 1009–1044.
Strebulaev, I. A., 2007, Do tests of capital structure theory mean what they say?, The Journal of Finance, 62(4),
1747–1787.
745
Varian, H. R., 1975, A bayesian approach to real estate assessment, Studies in Bayesian econometric and statistics
in Honor of Leonard J. Savage, pp. 195–208.
Warr, R. S., W. B. Elliott, J. Koëter-Kant, and Ö. Öztekin, 2012, Equity mispricing and leverage adjustment costs,
Journal of Financial and Quantitative Analysis, 47(03), 589–616.
Zhang, L., 2005, The value premium, The Journal of Finance, 60(1), 67–103.
33
Figure 1
Adjustments toward Target Leverage
Figure 1 depicts the optimal leverage policy of a firm for two different values of the adjustment cost parameter θ. Panel A
refers to the case of low adjustment costs (θ = 0.1), while Panel B refers to the case of high adjustment costs (θ = 0.3). Both
panels report initial leverage (d0 /k) on the horizontal axis and optimal leverage (d1 /k) on the vertical axis. Target leverage is
O
at the intersection of the policy function with the 45-degree line. LU
0 and L0 respectively denote two possible levels of initial
leverage below the target and above the target. The model is solved numerically with three possible states (H, M and L) at
time t2 that can occur with probability 0.3, 0.5 and 0.2 respectively. In the low state L firms are insolvent with β2 (L) = 0.5.
In the non-default states, the tax rate is τ = 0.3. The remaining parameters are as follows: M (H) = 0.8, M (M ) = 1.05,
M (L) = 1.1, r2A (H) = 0.9, r2A (M ) = −0.1, R2A (L) = −0.45205, k = 1.
Panel A: Low Adjustment Costs
Panel B: High Adjustment Costs
34
Table 1
Leverage Decomposition: Descriptive Statistics
The table provides summary statistics for our four measures of target leverage and for key firm characteristics. The
sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013. Financial firms
and utilities are excluded. ML is the market debt ratio, TL is the estimated target debt ratio, RL is relative leverage,
defined as the difference between ML and RL. TL is obtained from the estimation of the model of Flannery and
Rangan (2006) with firm fixed effects and at least 5 observations for each firm. The estimation of TL begins in 1970
(using data since 1965) and is done on a rolling basis as described in Section 3, each year using all the accounting
information available until that year. Panel A reports means, standard deviations, and autocorrelation coefficients for
market leverage and for target and relative leverage. Panel B reports average market leverage, target leverage, relative
leverage, capital stock (K), investment (I/K), profitability (ROA), market capitalization (SIZE), Tobin Q (Q) and the
number of observations. All nominal magnitudes are deflated by the consumer price index, to express all nominal values in 2000 dollars. All variables are winsorized at the 1 percent level and are measured as described in the appendix.
Panel A: Leverage Decomposition
Mean
St. Dev.
AC(1)
ML: Observed Leverage
0.16
0.17
0.87
TL: FR with Firm FE (min 5 obs)
0.17
0.15
0.90
RL: ML - TL
0.00
0.10
0.67
Panel B: Firm Characteristics and Sample Selection
LEV
Leverage
TL
RL
K
Investment
I/K
ROA
Valuation
SIZE
Q
N
750
Full Sample
0.16
TL (min 5 obs)
0.17
0.17
0.00
2.49
0.10
0.10
2.64
2.26
81978
3.05
0.08
0.11
3.22
2.10
61683
35
Table 2
Firm Characteristics across Quintiles of Relative Leverage
The table provides summary statistics for key firm characteristics across quintiles of firms sorted by estimated relative
leverage. The sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013.
Financial firms and utilities are excluded. ML is the market debt ratio, TL is the estimated target debt ratio, RL is
relative leverage, defined as the difference between ML and RL and ∆ML is the rate of change of market leverage
in the following year. Our measure of target leverage is obtained from the estimation of the model of Flannery and
Rangan (2006) with firm fixed effects and at least 5 observations for each firm. The estimation of TL begins in 1970
(using data since 1965) and is done on a rolling basis as described in Section 3, each year using all the accounting
information available until that year. The table reports average market leverage, target leverage, relative leverage,
capital stock (K), investment (I/K), profitability (ROA), market capitalization (SIZE), Tobin Q (Q) and the number
of observations. All nominal magnitudes are deflated by the consumer price index, to express all nominal values in
2000 dollars. All variables are winsorized at the 1 percent level and are measured as described in the appendix.
RL - FR with Firm FE (min 5 obs)
Group of RL
ML
Leverage
TL
RL
∆ML
K
Low
0.11
0.24
-0.13
0.04
1.84
0.10
0.14
2.26
2.17
2
0.12
0.16
-0.04
0.00
1.97
0.09
0.13
2.37
2.37
3
0.12
0.12
0.00
-0.02
2.02
0.08
0.11
2.10
2.34
4
0.17
0.13
0.04
-0.08
3.36
0.07
0.10
2.40
2.07
High
0.31
0.18
0.13
-0.13
6.05
0.08
0.09
2.33
1.51
36
Investment
I/K
ROA
Valuation
SIZE
Q
37
6.09
2.03
0.46
8.04
4.07
0.86
Re
[t]
SR
Re
[t]
SR
RL
H-L
ML
Sorting
Variable
12.29
3.05
0.59
15.20
3.41
0.66
L
15.02
3.90
0.76
14.55
3.65
0.73
15.44
4.16
0.82
14.60
3.92
0.77
Equally-Weighted
3
5
15.45
4.10
0.82
15.85
4.09
0.82
7
20.33
4.35
0.94
21.29
4.10
0.91
H
Excess Returns
5.56
1.97
0.35
1.73
0.51
0.10
H-L
5.06
1.28
0.24
7.80
1.99
0.35
L
8.98
2.73
0.50
9.07
3.11
0.54
9.62
3.40
0.60
7.56
2.84
0.49
Value-Weighted
3
5
9.30
3.30
0.58
11.33
3.76
0.67
7
10.62
3.22
0.57
9.53
2.30
0.44
H
In the table stocks are sorted every June in deciles based on their values of market leverage (ML) and relative leverage (RL). RL is obtained from the estimation
of TL from the model of Flannery and Rangan (2006) with firm fixed effects and at least 5 observations for each firm. The estimation of TL begins in 1970 (using
data since 1965) and is done on a rolling basis as described in Section 3, each year using all the accounting information available until that year. The sample
includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013 and covered by the Center of Research in Security Prices (CRSP).
Financial firms and utilities are excluded. The breakpoints for portfolio sorts are computed on the subset of firms traded on the NYSE market. The table reports
returns in excess of the risk-free rate, t-statistics and Sharpe ratios for the bottom decile (L), the top decile (H) and for the third, fifth and seventh decile. We
also report the difference for the excess returns, the t-statistics and the Sharpe ratio between the top decile and the bottom decile (H-L). The left panel reports equally-weighted returns, while the right panel reports value-weighted returns. The sorting variables are matched to monthly returns as described in Section 4.
Table 3
Univariate Portfolio Sorts
Table 4
Levered Returns: Relative, Observed and Target Leverage
For each month between July 1980 and December 2013, we estimate cross-sectional regressions of stock returns on relative leverage (RL), market leverage (ML), target leverage (TL), market capitalization (SIZE), and book-to-market equity (B/M). TL is
obtained from the estimation of the model of Flannery and Rangan (2006) with firm fixed effects and at least 5 observations
for each firm. The estimation of TL begins in 1970 (using data since 1965) and is done on a rolling basis as described in Section
3, each year using all the accounting information available until that year. The sample includes all Compustat firms traded on
NYSE, AMEX and NASDAQ between 1965 and 2013 and covered by the Center of Research in Security Prices (CRSP). Financial firms and utilities are excluded. De-listing returns are included in monthly returns. The table reports Fama-MacBeth coefficient estimates. t-statistics are in parentheses. R2 and N denote the cross-sectional R-squared and the number of observations
respectively. The independent variables are matched to monthly returns as described in Section 3. All variables are described
in the appendix. The symbols (***), (**) and (*) denote statistical significance at the 1, 5 and 10 percent levels respectively.
Observed vs Relative vs Target Leverage
(1)
ML
(2)
(3)
(4)
(5)
(6)
(7)
0.05
(0.19)
0.86∗∗∗
(2.99)
1.77∗∗∗
(4.38)
0.92∗∗∗
(2.82)
1.16∗∗∗
(3.57)
0.61∗
(1.84)
0.91∗∗∗
(3.14)
-1.16∗∗∗
(-3.57)
-0.87∗∗∗
(-2.99)
-0.20∗∗∗
(-4.93)
0.14∗∗
(2.06)
0.01
748724
0.02
727164
0.14
(0.47)
1.69∗∗∗
(4.53)
RL
TL
SIZE
B/M
R2
N
-0.24∗∗∗
(-5.28)
0.17∗∗
(2.46)
0.02
931438
-0.20∗∗∗
(-4.84)
0.14∗
(1.96)
0.01
748724
-0.20∗∗∗
(-4.93)
0.14∗∗
(2.06)
0.02
727164
0.01
748724
38
0.02
727164
Table 5
Parameter Values
The table reports parameter choices for the calibrated model. The frequency of calibration is monthly.
Category
Technology
Description
Capital share
Depreciation
Persistence of aggregate shock
Standard deviation of aggregate shock
Mean of aggregate shock
Persistence of idiosyncratic shock
Standard deviation of idiosyncratic shock
Mean of idiosyncratic shock
Fixed cost of operations
Symbol
α
δ
ρA
σA
µA
ρz
σz
µz
F
Value
0.4
0.01
1
0.95 3
0.007/3
-2
0.97
.1
0
0.18
Pricing Kernel
Time discount
Constant “risk aversion”
Time varying “risk aversion”’
β
γ0
γ1
0.9928
52.71
-50.19
Financing
Fixed equity flotation cost
Proportional equity flotation cost
Debt recovery in bankruptcy
Debt adjustment cost (“scale”)
Debt adjustment cost (“asymmetry”)
Corporate tax rate
λ0
λ1
ξ
λB
γB
τ
0.5
0.025
0.125
.4
−0.2
0.2
39
Table 6
Moments
The table reports model-implied and data moments for the baseline calibration of Table 5. The model-implied
moments are calculated as averages of simulations of 10,000 firms and 2000 time periods. The data source for the
Sharpe Ratio and risk-free rate moments is Zhang (2005). The average annual equity return is computed using data
from Kenneth French’s data library. Default rates are taken from Covas and Den Haan (2011). The remaining data
moments are computed from our sample of nonfinancial, unregulated firms from the annual Compustat dataset. All
moments are annualized. Details on model and data variable definitions are provided in the Appendix.
Model
0.43
0.02
0.04
0.06
Data
0.43
0.02
0.03
0.06
Average profitability
Volatility of profitability
Autocorrelation of profitability
Average investment
Average leverage
Frequency of default
Average book-to-market ratio
0.16
0.07
0.61
0.16
0.21
0.03
0.48
0.15
0.09
0.75
0.14
0.16
0.02
0.57
Average debt issuance
Frequency of positive debt adjustments
Frequency of negative debt adjustments
Frequency of equity issuance
0.19
0.62
0.38
0.05
0.22
0.59
0.41
0.04
Average Sharpe Ratio
Average risk-free rate
Volatility of risk-free rate
Average equity premium
40
Table 7
Adjustments toward Target Leverage - Model
Panel A describes firms’ capital structure adjustments toward target leverage under the baseline calibration of Table
5. RL denotes relative leverage (defined as the difference between market and target leverage) and adjustment
denotes the one-period-ahead change in leverage. Panel B reports model-implied and data estimates of adjustment
speeds for the full sample of firms, for small firms and for large firms. Data figures are obtained with the measure
of target leverage obtained from the estimation of the model of Flannery and Rangan (2006) in Equation 19, with
firm fixed effects and at least 5 observations for each firm. The speed of adjustment λ in (19) has a fixed component
and a component related to size (total assets AT), i.e., λ = λ0 + λ1 log(AT ). Model-implied figures are based on
a regression of leverage adjustments on RL, Ki,t , and their interaction, as described in Equation 31. Both in the
model and in the data, adjustment speeds for all firms are computed in correspondence of average size. Adjustment
speeds for large (small) firms are computed in correspondence of one standard deviation above (below) average size.
All model-implied quantities are based on simulations of 10,000 firms and 2000 time periods.
Panel A: Leverage Adjustments
Group of RL
Low
2
3
RL
Adjustment
-0.029
0.023
-0.002
0.002
Panel B: Adjustment Speeds
Model
All Firms
Small Firms
Large Firms
0.17
0.22
0.12
41
0.000
0.001
4
High
0.015
-0.008
0.065
-0.019
Data
0.21
0.27
0.15
Table 8
Average Returns and Capital Structure Imbalances
The table reports average annualized value-weighted returns of portfolios sorted by terciles of relative leverage
(defined as the difference between market and target leverage) and then by terciles of leverage. All figures are are
based on simulations of 10,000 firms and 2000 time periods under the baseline calibration of Table 5. Panel A refers
to the entire simulated economy. Panels B and C refer, respectively, to firms above and below the median value of
size Ki,t . The cells labeled “OL minus UL” report the difference between the average return of the highest- and the
lowest-tercile portfolio of stocks sorted by relative leverage, for each leverage group. The cells labeled “High minus
Low” report the difference between the average return of the highest- and the lowest-tercile portfolio of stocks sorted
by leverage, for each relative leverage group.
Panel A: All Firms
Leverage
High
Relative Leverage
Low
2
Underlevered (UL)
2
Overlevered (OL)
7.21
7.45
7.48
6.86
6.72
7.53
7.06
8.18
14.89
OL minus UL
0.27
0.67
7.83
High minus Low
-0.15
0.73
7.41
Panel B: Large Firms
Leverage
High
Relative Leverage
Low
2
Underlevered (UL)
2
Overlevered (OL)
8.18
8.26
9.38
7.66
8.39
10.31
5.67
9.35
20.80
OL minus UL
1.20
2.65
15.13
High minus Low
-2.51
1.09
11.42
Panel C: Small Firms
Leverage
High
Relative Leverage
Low
2
Underlevered (UL)
2
Overlevered (OL)
7.09
6.90
7.57
6.36
6.89
7.52
6.28
6.94
7.86
OL minus UL
0.48
1.16
1.58
42
High minus Low
-0.81
0.04
0.29
Table 9
Average Returns and Capital Structure Imbalances: Regressions
The table reports estimated coefficients from cross-sectional regressions of stock returns on relative leverage, market
leverage, target leverage, market capitalization, and book-to-market equity. All figures are are based on simulations
of 10,000 firms and 2000 time periods under the baseline calibration of Table 5. Columns (1) to (3) refer to the entire
simulated economy, columns (4) to (6) and (7) to (9) refer, respectively, to firms above and below the median value
of size Ki,t .
(1)
Relative Leverage
Leverage
0.58
All Firms
(2)
(3)
(4)
Large Firms
(5)
(6)
(7)
Small Firms
(8)
(9)
0.70
0.71
1.12
1.06
1.45
0.37
0.36
0.46
0.13
0.11
-0.45
-0.66
0.18
0.13
Size
-0.03
0.00
-0.05
Book-to-Market
0.02
-0.10
0.01
43
Appendix
Proofs of Propositions
Proposition 1. The FOC for the firm is:
∂ E[M ((1 + r2D (d1 )(1 − τ2 ))d1 + β2 (d1 )d1 )]
∂Θ (d0 , d1 )
=
1−
∂d1
∂d1
.
(32)
From the no arbitrage condition in the debt market, we have that E[M (1 + r2D (d1 ))] = 1. Observe that upon
bankruptcy occurring β2 (d1 ) = βd1 . Then, the following holds
Z Z∞
A
A A
A
M 1 + ρD
2 (d1 ) f (r2 |r1 )dr2 dM +
Z Zr2
M
−∞
r2A
(1 + r2A )k − β2 (d1 )d1
f (r2A |r1A )dr2A dM = 1,
d1
that is
1 + ρD
2 (d1 )
1 − qD
E[M (1 + r2A )|r2A < r2A ]k
qD
+
− βd1
= 1.
1 + rF
d1
1 + rF
Solving for the coupon rate yields
ρD
2 (d1 ) = βd1
E[M (1 + r2A )|r2A < r2A ]k 1 + rF
rF + qD
qD
−
+
.
1 − qD
d1
1 − qD
1 − qD
(33)
Observe that taxes are only paid when there is no default. Thus, the tax shield is
τ ρD
2 (d1 )d1 if no default occurs,
τ2 r2D (d1 )d1 =
0 if default occurs.
Observing that
∂ ρD
rF + qD + 2βd1 qD
2 (d1 )d1
=
∂d1
1 − qD
one obtains
(
D +2βd1 qD
∂ r2D (d1 )d1
. if no default occurs
τ rF +q1−q
D
τ2
=
∂d1
0 if default occurs.
Substituting in equation (4) yields
h
i
rF + qD + 2βd1 qD
d1 − d0 = E M τ
|r2A ≥ r2A − E M (2d1 β) |r2A < r2A ,
1 − qD
which simplifies to
θ (d1 − d0 ) = τ
Collecting all terms with
d1
k
rF + qD + 2βd1 qD
1 − qD
1 − qD
qD
− 2d1 β
.
1 + rF
1 + rF
on the left-hand side and all the rest on the right-hand side one obtains
qD
qD
d1
rF + qD
θk − 2βkτ
+ 2βk
=τ
+ θd0 ,
1 + rF
1 + rF
k
1 + rF
which, after dividing by the coefficient of dk1 on the left hand side and subtracting
rearranged as
!
qD
rF +qD
τ
2β(1 − τ ) 1+r
d1
d0
d
0
1+r
F
F
−
=
.
qD
qD −
k
k
θ + 2β(1 − τ ) 1+r
2β(1 − τ )k 1+r
k
F
F
44
d0
k
from both sides can be
Using the definitions of target leverage and adjustment speed given in the proposition gives the result.
755
Proposition 2. Observe that
∂Θ (d0 , d1 )
λ1
= θ (d1 − d0 ) = θk −
RL1 .
∂d1
1 − λ1
where the first equality follows from the derivative of the adjustment cost, and the second equality follows from
equation (7). Then, equation (4) can be written as
λ1
RL1
kθ −
1 − λ1
"
∂ r2D (d1 )d1
= E M τ2
∂d1
#
∂ (β2 (d1 )d1 )
−E M
.
∂d1
(34)
From the definition of λ1 one obtains
qD
2β(1 − τ ) 1+r
λ1
F
.
=
1 − λ1
θ
(35)
Combining (34) with (35), simplifying the left-hand side of (34), and expanding the derivatives on its right-hand
side, we obtain
qD
∂rD (d1 )
−2β(1 − τ )k
RL1 = E M τ2 r2D (d1 ) − E [M β2 d1 ] + E M τ2 2
d1 − E [M β2 d1 ] ,
1 + rF
∂d1
or, equivalently,
δD k 1 + rF
qD d1
+
+ E [M β2 d1 ] .
β
1 − qD
d1 1 − qD
E M τ2 r2D (d1 ) − E [M β2 d1 ] = −αRL1 − τ
(36)
qD
d1 we obtain
Inserting (36) into the definition of the correction factor (10) and observing that E [M β2 ] d1 = β 1+r
F
γ(d1 ) = 1 + αRL1 + τ
β
qD d1
δD k 1 + rF
+
1 − qD
d1 1 − qD
Therefore, the correction factor can be written as
γ(d1 ) = 1 + αRL1 −
=1+
α d1
2 k
−
d∗
α k1
The proposition follows immediately.
45
α d1
2 k
+
τ δD
+
τ δD
d1
k
d1
k
.
−β
qD
d1 .
1 + rF
760
Variable Definitions
The following table summarizes variable definitions with reference to Compustat and CRSP items.
Variable
Construction
Value of Preferred Stocks (P S)
Book Equity (BE)
Market Leverage (M L)
Market Leverage Growth (∆M L)
Real Assets (K)
Investment/Capital Ratio (I/K)
Return on Assets (ROA)
Tobin’s Q (Q)
Market Capitalization (SIZE)
Book-to-Market Equity (B/M )
If available, in this order: P ST KRV , P ST KL, P ST K.
CEQ + T XDIT C (if available) − P S
DLT T +DLC
AT −BE+P RCC F ·CSHO
F.M L−M L
0.5(F.M L+M L)
AT ·CP I2000
1000
AT −L.AT
0.5(AT +L.AT )
OIBDP
AT
P RCC F ·CSHO+DLT T +DLC
BE+DLT T +DLC
T
log |P RC|·SHROU
(in June)
1000
BE
log |P RC|·SHROU T /1000
σt [Mt+1 ]
In the model, following Zhang (2005), we define the average Sharpe Ratio as St ≡ E
and the risk-free rate as
t [Mt+1 ]
1
Et [Mt+1 − 1. We define the equity premium as the average value-weighted return in the simulated economy, where
V
765
i,t
realized stock returns are computed ex-dividends as Ri,t ≡ Vi,t−1 −D
. Profitability is the ratio of operating profits
i,t−1
Πi,t to capital Ki,t , investment is the ratio of Ii,t to capital Ki,t , leverage is the ratio of Bi,t to Bi,t + Vi,t − Di,t ,
book-to-market is the ratio of Ki,t to Bi,t + Vi,t − Di,t , debt adjustments are changes in debt stock Bi,t − Bi,t−1
divided by capital Ki,t−1 , debt issuance is max{0, Bi,t −Bi,t−1 } divided by Ki,t−1 , and equity issuance is max{0, Ei,t }
scaled by capital Ki,t .
Numerical Solution Method
770
We solve the model using a combination of value function iteration (VFI) and simulation.
Each firm i’s problem is characterized by five state variables, i.e. xi,t ≡ (Ki,t , Bi,t , ci,t , zi,t , At ). We approximate
the value function V (xi,t ) with piece-wise linear interpolation on a grid 7 × 7 × 5 × 5 × 3, respectively. Note that
the projection of V (xi,t ) onto an interpolated structure allows for a precise solution with a relatively parsimonious
number of grid points. We check the robustness of our numerical solution by experimenting with finer grids.
775
780
785
Given xi,t , each firm faces three continuous choices, i.e. (Ki,t+1 , Bi,t+1 , ci,t+1 ), and one discrete choice, i.e.
whether to default or not I(xt ). Choices Ki,t+1 and Bi,t+1 are evaluated on a grid 30 × 30, and we solve numerically
at each step of the VFI for ci,t+1 given each fix evaluation of (Ki,t , Bi,t , ci,t , zi,t , At , Ki,t+1 , Bi,t+1 ). This requires to
solve for 7 × 7 × 5 × 5 × 3 × 30 × 30 non-linear equations at each step of the VFI, given the guess of the value function
and future default. We solve for the coupon using golden search.
The solution with VFI proceed in two steps. First, we find the equilibrium value function Ṽ (i,t ) associated with
an identical model but without endogenous default. Second, we use Ṽ (i,t ) as an initial guess for the model with
endogenous default and we progressively increase the fixed cost F , re-converge the value function and re-initialize.
The convergence criteria on the value function is defined as a max absolute difference between the value functions
in two consecutive iterations of the VFI of 10−4 , or lower.
We then use the four policy functions {K(xi,t ), B( xi,t ), c(xi,t ), I(xi,t )} to perform a long simulation of our economy
(T = 2000) with a panel of i = 1, · · · , 10000 firms, given random draws of idiosyncratic shocks {{zi,t }Tt=0 }10000
and
i=1
aggregate shocks {at }Tt=0 .
46
Online Appendix (Not for Publication)
OA1. Two-Period Example: Additional Content
790
Notable Special Cases: Sticky Leverage and Modigliani-Miller
Equation (9) nests a notable special case. Absent taxes and bankruptcy costs, the first-order condition in (32)
simplifies to
∂ E[M (1 + r2D (d1 )]d1
∂Θ dk0 , dk1
=
.
(OA1)
1−
∂d1
∂d1
If debt is fairly priced, the right hand side of (OA1) is equal to one. Therefore, to maximize firm value the firm
should not change its level of debt and set d1 = d0 . With this policy the firm does not bear any costs of adjustment
and the first derivative of the adjustment cost function is zero. In this case γ(d1 ) = 1 and equation (9) collapses to
the celebrated second proposition of MM, that is
R2E = R2A +
795
d1
R2A − R2D .
k − d1
(OA2)
Equation (OA2) should not be read to mean that firms can costlessly change capital structure and returns (as instead
is the case in MM). Instead, the meaning of equation (OA2) is that if there is cross-sectional heterogeneity in k and
d0 across firms, firms with higher leverage will earn higher returns. Firms are “stuck” at their level of leverage and
returns. A simple subcase of this special case is when there are no adjustment costs, in addition to the exclusion of
taxes and bankruptcy costs. Equation (OA1) holds trivially and equation (OA2) also holds. We have the original
setup of MM, where capital structure is irrelevant and firms are indifferent in their choice of d1 .
Geometric Interpretation
800
Figure OA1 provides a representation of the trade-offs in our model and offers a geometric interpretation of the
correction factor. To start with, it is useful to reinterpret target leverage as the leverage ratio kd that equalizes the net
F +qD ) 34
marginal bankruptcy cost of leverage α kd and the net marginal tax shield of leverage τ (r1+r
. In Panel A, which
F
d
refers to the case of an overlevered firm, the net marginal cost of debt α k is represented by the straight line through
F +qD )
A and C. The straight line through B and C represents the present value of the marginal tax shield τ (r1+r
. The
F
d∗
805
810
two lines intersect at point C, in correspondence of target leverage k1 .
∗
d
Consider now the representations of the correction factor γ k1 , dk1 in (12), as a function of target leverage and
leverage, and γ RL1 , dk1 in (13), as a function of relative leverage and leverage. In (12) and (13), the first term
(“1”) is the present value of one dollar of debt in the absence of frictions.
The second and third term in both (12) and (13) illustrate the economic tradeoffs at work and their effect on the
d∗
correction factor. Notice that their sum α2 dk1 − α k1 in (12) and − α2 dk1 + αRL1 in (13) can both be decomposed as
d∗
− α2 k1 + α2 RL1 by simply exploiting the definition of relative leverage in (6). With reference to Figure OA1 Panel A,
d∗
d∗
the term α2 k1 is equal to the area of the triangle ABC divided by its base k1 , and accordingly captures the average
net surplus from debt that the firm receives when its leverage is equal to target leverage. The surplus arises from the
fact that the marginal tax shield is always greater than the marginal bankruptcy cost up to the target. This term
enters the definition of γ ( ) with a negative sign, because it decreases the effective cost of debt for the firm. The
term α2 RL1 is equal to the area of the triangle CDE divided by the length of its base, RL1 . The term with relative
34 To
d∗
1
k
see this, notice that the first-order condition (4) can be simplified using the definition of α in Proposition 2, and target leverage
can be obtained as the solution of α
d∗
1
k
=
τ (rF +qD )
.
1+rF
Target leverage is the leverage ratio that firms would choose in the absence of
adjustment frictions, in correspondence of which the present value of the marginal tax shield
qD
the net marginal bankruptcy cost α = 2β(1 − τ )k 1+r
.
F
47
τ (rF +qD )
1+rF
is equal to the present value of
815
820
leverage captures the average lost surplus at the optimum, which is different from the target. This term carries a
positive sign in γ ( ) , because if the optimum is different from the target, the firm is not internalizing all the surplus
from its debt and some potential surplus is lost. All else equal, the lost surplus increases the effective cost of debt.
The difference between the areas ABC and CDE (or equivalently CD0 E’) is therefore the net benefit to leverage,
depicted as the trapezoid ABE 0 D0 in Panel A. If the firm is sufficiently overlevered, the area of CDE becomes larger
than that of ABC, resulting in a correction factor that is greater than one. In this case there is a net loss to carrying
so much debt, because the average tax shield is smaller than the average net bankruptcy cost.
825
Finally, the term τ δdD1k in (12) and (13) is the product of the tax rate times the recovery rate per unit of debt in
case of bankruptcy.35 All else equal, the recovery rate of debt in bankruptcy states lowers the coupon that lenders
would otherwise require, and reduces the interest expense and the tax shield in solvency states. This term carries a
positive sign, because it captures a loss of tax shields and thus, all else equal, a higher cost of debt.
830
Panel B of Figure OA1 instead refers to the case of an underlevered firm with the same target leverage of the
firm in Panel A. Following the same logic, the triangle ABC represents the total surplus that the firm gets from debt
when leverage is set equal to the target, and the triangle CDE represents the surplus that the firm loses by choosing
an optimum leverage that is below target leverage. For an underlevered firm, the correction factor is smaller than
one, because the firm always internalizes a net benefit to leverage, which lowers its effective average cost of debt.
Cost of Debt and Leverage
835
The realized return on debt in a given state is ρD
i,2 (di,1 ) in case of solvency, and the recovery rate in case of
default. By the expression of the coupon in (33) it is immediate to conclude that the return on debt is decreasing
with leverage in default states, and increasing in leverage in solvency states. Intuitively, the lender requires a nominal
interest rate on the loan such that it is fairly priced, that is E[M (1 + r2D (d1 ))] = 1. The latter no arbitrage condition
implies that lenders require higher coupons to offset lower recovery rates. Following standard arguments (see, for
example, Cochrane (2001), Chapter 1), the no-arbitrage condition implies that expected return on debt is given by
the following expression:
E[r2D (d1 )] − rF = −(1 + rF )Cov(M, r2D (d1 )),
840
which emphasizes that, in the presence of risk-averse investors, the expected return on debt increases with leverage
if bankruptcy occurs in bad times, in which M takes higher values. In other words, low recovery rates occur in more
valuable states, and must be offset by disproportionately higher coupons in solvency states to preserve no arbitrage
in the market. Accordingly, higher leverage drives the expected return on debt up, as common intuition suggests.
35 This
term and the term (“1”) are not depicted in Figure OA1 because they do not capture marginal effects.
48
OA2. Empirical Evidence: Robustness Checks
Alternative Measures of Target Leverage
845
We implement three additional estimations of the target leverage ratio. The first, which we denote as TL2, is still
based on the partial adjustment model of Flannery and Rangan (2006) (FR) with firm fixed effects. The second and
third alternative measures of target leverage, TL3 and TL4, are non-parametric. We, respectively, construct them
using the rolling median leverage at the firm level and at the industry level. More precisely:
TL2: Flannery-Rangan with convergence in firm fixed effects. As for TL1, we run a regression specification as in equation (19) that contains fixed effects for each individual firm. Differently from TL1, we guarantee
reliable estimates of the fixed effects by excluding firms for which the estimate of the fixed effects does not converge
to a stable value. More specifically, stability of the estimate for firm i in year t is achieved if and only if there exists
a period t∗ such that the fixed effect estimate Fi,t∗ can be computed and satisfies36
|Fi,t∗ − Fi,t∗ −1 | < 0.05 and |Fi,t∗ − Fi,t∗ −2 | < 0.05.
850
Once the stability criterion is satisfied for firm i, the fixed effect estimate Fi,t∗ at t∗ is used to compute target leverage
T Li,t for every year following t∗ and for which equation (19) produces a valid estimate of target leverage for firm i.37
TL3: Firm median leverage with at least five observations. The third measure of target leverage is
computed as the rolling median at the firm level for all firms in the sample with at least five observations.
TL4: 4-digit SIC code industry median leverage. The fourth measure of target leverage is computed as
the four-digit SIC rolling median ML. In this case there is no requirement on the minimum number of observations.
855
860
865
870
A possible concern with the measures of target leverage based on FR is that target and consequently relative
leverage are potentially functions of previously identified determinants of equity returns (indeed some of these variables like profitability, size and market-to-book are related to well known asset pricing anomalies). This may make
it difficult to interpret our results in the empirical section. To address this concern, TL3 and TL4 are based on a
non-parametric approach.
Table OA1 shows that descriptive statistics are consistent across our baseline measure of target TL, TL2, and
TL3 (0.16-0.17). TL4 yields a lower predicted target (0.13). The TL, TL2, and TL3 have a mean that is close to
zero, while RL4 has a slightly larger mean (0.03), which reflects the difference in average between TL4 and ML.
Also, consistency in results across measures is reassuring, because it indicates that our estimates of TL are not
very sensitive to the exact specification of the leverage measure.38 TL4 is the least stringent measure in terms of
data requirements. Accordingly, the sample for which TL4 can be measured is the one that is most similar to the
Compustat universe.
Panels A, B and C of Table OA2 broadly confirm the patterns observed in Table 2, thus providing a robustness
check. Similarly, Table OA3 and Table OA4 are replicas of Table 3 and Table 4, respectively. Overall, the results in
Table OA3 and Table OA4 indicate that the relationship between returns and capital structure imbalances is robust
to different empirical proxies for target leverage.
require that there are no gaps in the firm i’s time series of Fi,s , for t∗ < s < t.
∗ for all years following
due to missing data for one or more regressors in Xi,t−1 , it is not possible to estimate M DRi,t
t∗ . When this occurs, we temporarily remove the firm from the analysis, and start checking again the stability criterion.
38 In unreported results we find that the average of individual firm-level correlations is high (about 0.87) between TL-TL2 and TL3,
and much lower for TL4 (about 0.54). TL and TL2 have a correlation of 1, which stems from the fact that both are estimated with FR.
More precisely, TL is estimated only if there are 5 observations, and TL2 is estimated only if stability is achieved. If both measures can
be estimated, they yield the same value. This does not mean that having two measures is redundant, because the number of observations
in the sample is different across measures.
36 We
37 Occasionally,
49
Book and Net Leverage
Different strands of the literature highlight the importance of distinguishing between leverage measured at market
or at book values, and the importance of accounting for cash holdings in the balance sheet.39 Bates, Kahle, and Stulz
(2009) show that firms in the lower quintiles of size consistently have larger cash-to-assets ratios. Therefore, when
leverage is measured net of cash (both for book and market leverage) smaller firms will appear to have less leverage.
Additionally, book leverage tends to differ significantly from market leverage for firms with high market-to-book
ratios. Therefore, it is possible that by adjusting for cash and by using book values, the distribution of relative
leverage changes unevenly in a way that correlates with size and market-to-book ratios. We define book leverage
ratio as
Di,t
,
(OA3)
BLi,t =
Di,t + BEi,t
where BEi,t is the book value of equity of firm i in period t.
880
In this spirit, Table OA5 replicates the analysis of Tables 4 and OA4 by looking at market leverage measured net
of cash and at book leverage, both gross and net of cash. More precisely, for both debt at market and at book values,
we subtract cash and marketable securities (CHE) from total debt. TL and RL are also re-estimated accordingly.
Table OA5 reports the coefficients of leverage and relative leverage obtained from a set of FMB regressions that
include Size, BM (unreported due to space considerations). Across the three panels, the coefficient of leverage is
volatile, taking both positive and negative signs, and is generally insignificant. Instead, relative leverage is positive
and significant in all (but one) specification. Similarly, when target leverage is included along with leverage, the
results are in line with those in Table 4.
885
From this analysis, we conclude that the measure of leverage that one chooses to employ does not have a first-order
impact on the results. This is an important robustness check, because the previous literature has found contrasting
results on the relation between leverage and returns, depending on the measure of leverage adopted. Our results
suggest that the instability in the results is not driven by the measure of leverage, but by the fact that the relation
between leverage and returns is a priori unclear, as discussed in the model section.
875
Additional Controls
890
895
As a robustness check, we also include post-ranking CAPM betas (β), momentum, profitability, and investment
in our tests. Post-ranking betas are computed following the procedure in Fama and French (1992). We measure
momentum as the continuously compounded return from month m − 12 to month m − 2. Consistent with Novy-Marx
(2013) and Hou, Xue, and Zhang (2014) we measure investment (INV) as the growth rate of total assets (AT) and
profitability (PROF) as the differences between operating revenues (REVT) and the cost of goods sold (COGS),
divided by total assets (AT). Including investment and profitability as controls helps alleviate concerns that the
effect of capital structure imbalances on returns is driven by omitted variables that affect cross-sectional differences
in E[R2A ] and that are not already captured by the control variables in the baseline specification (20).
Table OA6 shows that relative leverage remains largely unaffected by the introduction of the new controls. Both
investment and profitability are significant and with the expected sign. Instead, the little explanatory power of
CAPM betas (β) should on interpreted with caution. As standard40 , our dynamic model has one source of aggregate
risk and predicts a conditional one-factor model, in which leverage variables affect conditional betas. Instead, the
rolling estimates of post-ranking betas are unconditional.
39 See Fama and French (1992), Fama and French (2002) and Flannery and Rangan (2006) for the distinction between market and book
leverage. See Opler, Pinkowitz, Stulz, and Williamson (1999) and Bates, Kahle, and Stulz (2009) for the evidence on cash holdings.
40 See, for example, Zhang (2005) and Gomes and Schmid (2010).
50
Figure OA1
Correction Factor Decomposition
Figure OA1 illustrates the decomposition of the correction factor γ(di,1 ) as discussed in Section 2. The model is solved
numerically with three possible cash-flow states (H, M, and L) at time t2 that can occur with probability 0.3, 0.5 and 0.2
respectively. Panel A and Panel B depict two firms with the same target leverage but different initial leverage ratios, namely
0.75 in Panel A and 0.1 in Panel B. In the low state L firms are insolvent with βi,2 (L) = 0.5. In the non-default states, the
A
tax rate is τ = 0.3. The remaining parameters are as follows: θi = 1, M (H) = 0.8, M (M ) = 1.05, M (L) = 1.1, ri,2
(H) = 0.9,
A
A
ri,2 (M ) = −0.1, Ri,2 (L) = −0.45205, ki = 1.
Panel A: Overlevered Firm
0.14
Marginal bankruptcy cost
Marginal tax shield
D
Benefits and Cost ($ per unit of leverage)
0.11
B
0.07
E’
C
”Lost Surplus”
(CDE)
E
”Surplus at Target”
(ABC)
Net Benefit
to Leverage
(ABE’D’)
D’
A
0.46
Target Leverage
51
0.71
Optimal Leverage
0.9
Panel B: Underlevered Firm
900
0.14
Benefits and Cost ($ per unit of leverage)
Marginal bankruptcy cost
Marginal tax shield
B
0.07
E
C
”Surplus at Target”
(ABC)
”Lost Surplus”
(CDE)
Net Benefit
to Leverage
(ABED)
0.02
A
D
0.15
Optimal Leverage
0.46
Target Leverage
52
0.9
Table OA1
Leverage Decomposition: Descriptive Statistics
The table provides summary statistics for our four measures of target leverage and for key firm characteristics. The
sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013. Financial firms
and utilities are excluded. ML is the market debt ratio, TL is the estimated target debt ratio, RL is relative leverage,
defined as the difference between ML and RL. We consider three alternative measures of target leverage. TL2 is from
the estimation of the model of Flannery and Rangan (2006) with firm fixed effects and at least 5 observations for
each firm. The estimation of TL2 begins in 1970 (using data since 1965) and is done on a rolling basis as described in
Section 3, each year using all the accounting information available until that year. TL2 requires convergence on the estimate of firm fixed effects. TL3 is computed as the rolling median firm leverage for firms with at least 5 observations.
TL4 is computed as the rolling median industry leverage at the 4-digit SIC code level. Reported descriptive statistics
refer to firm-year observations for which, TL2, TL3, and TL4 are not missing. Panel A reports means, standard
deviations, and autocorrelation coefficients for market leverage and for the four measures of target and relative leverage. Panel B reports average market leverage, target leverage, relative leverage, capital stock (K), investment (I/K),
profitability (ROA), market capitalization (SIZE), Tobin Q (Q) and the number of observations for the four measures
of target leverage. All nominal magnitudes are deflated by the consumer price index, to express all nominal values
in 2000 dollars. All variables are winsorized at the 1 percent level and are measured as described in the appendix.
Panel A: Leverage Decomposition
Mean
St. Dev.
AC(1)
ML: Observed Leverage
0.16
0.17
0.87
TL2: FR with Firm FE (convergence)
TL3: Median Firm Leverage (min 5 obs)
TL4: Median Industry Leverage
0.16
0.16
0.13
0.15
0.15
0.12
0.90
0.99
0.91
RL2: ML - TL2
RL3: ML - TL3
RL4: ML - TL4
0.00
0.01
0.03
0.10
0.12
0.14
0.66
0.78
0.83
53
Panel B: Firm Characteristics and Sample Selection
Measure of TL
LEV
Leverage
TL
RL
0.16
K
Investment
I/K
ROA
Valuation
SIZE
Q
N
2.49
0.10
0.10
2.64
2.26
81978
TL2
0.16
0.16
0.00
3.00
0.09
0.11
3.17
2.18
62868
TL3
0.17
0.16
0.01
3.04
0.09
0.11
3.22
2.10
61940
TL4
0.16
0.13
0.03
2.42
0.11
0.10
2.60
2.29
75441
54
Table OA2
Firm Characteristics across Quintiles of Relative Leverage
The table provides summary statistics for key firm characteristics across quintiles of firms sorted by estimated relative
leverage. The sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013.
Financial firms and utilities are excluded. ML is the market debt ratio, TL is the estimated target debt ratio, RL is
relative leverage, defined as the difference between ML and RL and ∆ML is the rate of change of market leverage in
the following year. We consider three alternative measures of relative leverage. Panel A refers to TL2, which is from
the estimation of the model of Flannery and Rangan (2006) with firm fixed effects and at least 5 observations for
each firm. The estimation of TL2 begins in 1970 (using data since 1965) and is done on a rolling basis as described
in Section 3, each year using all the accounting information available until that year. TL2 requires convergence on
the estimate of firm fixed effects. TL3 is computed as the rolling median firm leverage for firms with at least 5
observations. Panel B refers to TL3, which is computed as the rolling median firm leverage for firms with at least 5
observations. Panel C refers to TL4, which is computed as the rolling median industry leverage at the 4-digit SIC
code level. The table reports average market leverage, target leverage, relative leverage, capital stock (K), investment
(I/K), profitability (ROA), market capitalization (SIZE), Tobin Q (Q) and the number of observations for the three
measures of target leverage. All nominal magnitudes are deflated by the consumer price index, to express all nominal
values in 2000 dollars. All variables are winsorized at the 1 percent level and are measured as described in the
appendix.
Panel A: RL2 - FR with Firm FE (convergence)
Group of RL
ML
Leverage
TL
RL
∆ML
K
Low
0.11
0.23
-0.12
0.04
1.92
0.09
0.14
2.33
2.18
2
0.11
0.15
-0.04
0.00
1.94
0.09
0.12
2.33
2.43
3
0.11
0.11
0.00
-0.01
1.85
0.08
0.10
1.96
2.47
4
0.15
0.12
0.03
-0.07
3.19
0.07
0.10
2.32
2.19
High
0.30
0.17
0.13
-0.13
6.09
0.08
0.09
2.40
1.57
55
Investment
I/K
ROA
Valuation
SIZE
Q
Panel B: RL3 - Median Firm Leverage (min 5 obs)
Group of RL
ML
Leverage
TL
RL
∆ML
K
Investment
I/K
ROA
Valuation
SIZE
Q
Low
0.12
0.26
-0.14
0.03
2.59
0.08
0.15
2.24
2.02
2
0.11
0.15
-0.04
-0.01
2.64
0.08
0.13
2.85
2.37
3
0.09
0.09
0.00
0.07
1.44
0.09
0.10
1.73
2.74
4
0.19
0.15
0.04
-0.10
3.76
0.08
0.11
2.78
1.94
High
0.33
0.16
0.17
-0.15
4.91
0.08
0.09
1.88
1.36
Panel C: RL4 - Median Industry Leverage
Group of RL
ML
Leverage
TL
RL
∆ML
K
Low
0.06
0.21
-0.15
0.13
2.37
0.09
0.14
2.11
2.45
2
0.07
0.10
-0.03
0.06
1.89
0.10
0.08
1.91
2.90
3
0.08
0.08
0.00
-0.01
1.75
0.11
0.08
2.03
2.98
4
0.20
0.14
0.06
-0.11
2.97
0.09
0.10
2.28
1.82
High
0.38
0.14
0.24
-0.14
3.13
0.08
0.09
1.11
1.26
56
Investment
I/K
ROA
Valuation
SIZE
Q
57
6.09
2.03
0.46
8.36
4.35
0.89
7.95
3.79
0.87
8.59
4.56
0.93
Re
[t]
SR
Re
[t]
SR
Re
[t]
SR
ML
RL2
RL3
RL4
H-L
Re
[t]
SR
Sorting
Variable
12.74
3.57
0.72
12.60
3.30
0.65
11.56
2.91
0.56
15.20
3.41
0.66
L
13.68
3.57
0.71
14.15
3.88
0.77
14.97
3.89
0.75
14.55
3.65
0.73
15.71
3.48
0.66
14.66
3.80
0.75
15.31
4.02
0.79
14.60
3.92
0.77
Equally-Weighted
3
5
15.44
3.82
0.76
14.74
3.81
0.75
15.72
4.09
0.82
15.85
4.09
0.82
7
21.33
4.31
0.93
20.55
4.23
0.90
19.92
4.32
0.93
21.29
4.10
0.91
H
Excess Returns
2.69
1.15
0.19
5.31
2.14
0.37
5.24
1.85
0.33
1.73
0.51
0.10
H-L
8.52
2.80
0.51
6.34
1.78
0.33
5.11
1.30
0.24
7.80
1.99
0.35
L
7.48
2.62
0.47
8.83
2.77
0.50
8.91
2.68
0.49
9.07
3.11
0.54
8.51
2.69
0.46
7.77
2.28
0.43
9.71
3.46
0.61
7.56
2.84
0.49
Value-Weighted
3
5
9.26
3.13
0.55
9.02
3.23
0.53
7.88
2.50
0.46
11.33
3.76
0.67
7
11.21
2.65
0.49
11.65
3.01
0.55
10.35
3.13
0.56
9.53
2.30
0.44
H
In the table stocks are sorted every June in deciles based on their values of market leverage (ML) and relative leverage (RL). RL is obtained from the
estimation of TL from the model of Flannery and Rangan (2006) with firm fixed effects and at least 5 observations for each firm. The estimation of TL
begins in 1970 (using data since 1965) and is done on a rolling basis as described in Section 3, each year using all the accounting information available
until that year. The sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013 and covered by the Center of
Research in Security Prices (CRSP). Financial firms and utilities are excluded. The breakpoints for portfolio sorts are computed on the subset of firms
traded on the NYSE market. We consider three alternative measures of relative leverage, as described in Table OA1. The table reports returns in excess
of the risk-free rate, t-statistics and Sharpe ratios for the bottom decile (L), the top decile (H) and for the third, fifth and seventh decile. We also report
the difference for the excess returns, the t-statistics and the Sharpe ratio between the top decile and the bottom decile (H-L). The left panel reports
equally-weighted returns, while the right panel reports value-weighted returns. The sorting variables are matched to monthly returns as described in Section 4.
Table OA3
Univariate Portfolio Sorts
Table OA4
Levered Returns: Relative, Observed and Target Leverage
For each month between July 1980 and December 2013, we estimate cross-sectional regressions of stock returns on relative leverage (RL), market leverage (ML), target leverage (TL), market capitalization (SIZE), and book-to-market equity (B/M). We consider three alternative measures of target and relative leverage, as described in Table OA1. The
sample includes all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013 and covered
by the Center of Research in Security Prices (CRSP). Financial firms and utilities are excluded. De-listing returns
are included in monthly returns. The table reports Fama-MacBeth coefficient estimates. t-statistics are in parentheses. R2 and N denote the cross-sectional R-squared and the number of observations respectively. The independent variables are matched to monthly returns as described in Section 4. All variables are described in the appendix. The symbols (***), (**) and (*) denote statistical significance at the 1, 5 and 10 percent levels respectively.
Panel A: Observed and Relative Leverage
(1)
ML
(2)
(3)
0.14
(0.47)
0.78∗∗∗
(2.78)
RL2
0.86∗∗∗
(3.54)
RL3
-0.24∗∗∗
(-5.28)
0.17∗∗
(2.46)
-0.20∗∗∗
(-4.78)
0.16∗∗
(2.11)
-0.20∗∗∗
(-4.86)
0.14∗
(1.90)
0.56∗∗∗
(2.72)
-0.23∗∗∗
(-5.19)
0.17∗∗
(2.09)
0.02
931438
0.02
738975
0.02
730793
0.02
856307
RL4
SIZE
B/M
R2
N
(4)
Panel B: Relative Leverage vs Observed Leverage
ML
RL2
(1)
(2)
(3)
-0.09
(-0.27)
0.85∗∗∗
(2.86)
-0.09
(-0.28)
-0.55
(-1.00)
0.89∗∗∗
(2.98)
RL3
905
-0.20∗∗∗
(-4.90)
0.16∗∗
(2.44)
-0.20∗∗∗
(-4.94)
0.15∗∗
(2.24)
1.03∗∗
(2.03)
-0.23∗∗∗
(-5.30)
0.20∗∗∗
(3.05)
0.02
738975
0.02
730793
0.02
856307
RL4
SIZE
B/M
R2
N
58
Panel C: Observed Leverage vs Target Leverage
ML
TL2
(2)
(3)
(4)
0.77∗∗
(2.37)
-0.85∗∗∗
(-2.87)
0.80∗∗∗
(2.81)
0.49∗∗
(2.22)
-0.89∗∗∗
(-2.98)
TL3
-0.20∗∗∗
(-4.90)
0.16∗∗
(2.44)
-0.20∗∗∗
(-4.94)
0.15∗∗
(2.24)
-1.03∗∗
(-2.03)
-0.23∗∗∗
(-5.30)
0.20∗∗∗
(3.05)
0.02
738975
0.02
730793
0.02
856307
TL4
SIZE
B/M
R2
N
59
60
TL
RL
LEV
TL
RL
LEV
0.02
(0.09)
(LEV)
-0.14
(-0.58)
(LEV)
-0.14
(-0.52)
0.70∗∗∗
(2.66)
0.02
(0.08)
0.52∗∗
(2.03)
(RL2)
0.41∗
(1.81)
0.47∗∗
(2.23)
(RL1)
-0.06
(-0.24)
(RL2)
-0.06
(-0.26)
(RL1)
1.01∗∗
(2.34)
-0.84∗
(-1.66)
(RL4)
0.48∗∗
(2.21)
-0.03
(-0.09)
(RL3)
0.91∗∗
(2.04)
-0.57
(-1.16)
(RL4)
Panel B: Book Leverage
0.63∗∗∗
(2.84)
-0.20
(-0.74)
(RL3)
Panel A: Net Market Leverage
0.56∗∗
(2.03)
-0.70∗∗∗
(-2.66)
-0.52∗∗
(-2.03)
(TL2)
-0.41∗
(-1.81)
0.35
(1.41)
(TL2)
0.54∗
(1.96)
(TL1)
-0.47∗∗
(-2.23)
0.41∗
(1.70)
(TL1)
-0.48∗∗
(-2.21)
0.45∗
(1.96)
(TL3)
-0.63∗∗∗
(-2.84)
0.43∗∗
(2.07)
(TL3)
-0.91∗∗
(-2.03)
0.34∗
(1.91)
(TL4)
-1.02∗∗
(-2.34)
0.18
(1.07)
(TL4)
For each month between July 1980 and December 2013, we estimate cross-sectional regressions of stock returns on relative leverage (RL), market leverage (ML),
target leverage (TL), market capitalization (SIZE), and book-to-market equity (B/M). On top of our baseline measure of TL from the model of Flannery and
Rangan (2006), which we denote as TL1, we consider three alternative measures of target and relative leverage, as described in Table OA1. The sample includes
all Compustat firms traded on NYSE, AMEX and NASDAQ between 1965 and 2013 and covered by the Center of Research in Security Prices (CRSP). Financial
firms and utilities are excluded. De-listing returns are included in monthly returns. The table reports Fama-MacBeth coefficient estimates. t-statistics are in
parentheses. The independent variables are matched to monthly returns as described in Section 4. All variables are described in the appendix. The symbols
(***), (**) and (*) denote statistical significance at the 1, 5 and 10 percent levels respectively.
Table OA5
Book and Net Leverage
61
TL
RL
LEV
-0.13
(-0.63)
(LEV)
-0.17
(-0.86)
0.53∗∗∗
(2.93)
0.54∗∗∗
(3.04)
(RL2)
-0.10
(-0.46)
(RL1)
0.52∗∗∗
(3.21)
-0.12
(-0.53)
(RL3)
0.78∗∗∗
(2.74)
-0.58∗
(-1.68)
(RL4)
Panel C: Net Book Leverage
0.37∗
(1.80)
-0.53∗∗∗
(-2.93)
-0.54∗∗∗
(-3.04)
(TL2)
0.45∗∗
(2.26)
(TL1)
-0.52∗∗∗
(-3.21)
0.41∗∗
(2.50)
(TL3)
-0.78∗∗∗
(-2.74)
0.19
(1.53)
(TL4)
62
R2
N
Momentum
β
B/M
SIZE
PROF
INV
RL4
RL3
RL2
RL1
ML
0.04
925112
-0.26∗∗∗
(-5.26)
0.06∗∗∗
(5.22)
-0.22∗∗∗
(-5.59)
0.20∗∗∗
(3.45)
0.02
(0.09)
0.12
(0.68)
-0.09
(-0.39)
(1)
0.04
726056
-0.26∗∗∗
(-4.24)
0.05∗∗∗
(3.14)
-0.19∗∗∗
(-4.98)
0.16∗∗∗
(2.76)
0.18
(0.18)
0.05
(0.87)
-0.17
(-0.74)
0.95∗∗∗
(3.41)
(2)
0.04
737667
-0.22∗∗∗
(-3.54)
0.06∗∗∗
(4.74)
-0.19∗∗∗
(-4.99)
0.19∗∗∗
(3.35)
0.06
(0.26)
0.11
(0.59)
0.94∗∗∗
(3.32)
-0.33
(-1.29)
(3)
0.04
729644
-0.28∗∗∗
(-4.71)
0.05∗∗∗
(3.19)
-0.19∗∗∗
(-4.99)
0.17∗∗∗
(2.87)
0.02
(0.08)
0.15
(0.83)
1.01∗∗∗
(3.99)
-0.32
(-1.28)
(4)
0.04
850563
1.14∗∗∗
(2.91)
-0.26∗∗∗
(-5.20)
0.06∗∗∗
(5.33)
-0.22∗∗∗
(-5.51)
0.22∗∗∗
(4.05)
-0.05
(-0.22)
0.10
(0.53)
-0.87∗∗
(-2.03)
(5)
For each month between July 1980 and December 2013, we estimate cross-sectional regressions of stock returns on relative leverage (RL), market leverage (ML), target leverage (TL), investment (INV), profitability (PROF), market capitalization (SIZE), book-to-market equity (B/M), post-ranking CAPM
beta (β) and momentum (Momentum). Investment is measured as the growth of firm assets. Profitability in measured as the ratio of operating profits
to book equity. On top of our baseline measure of TL from the model of Flannery and Rangan (2006), which we denote as TL1, we consider three alternative measures of target and relative leverage, as described in Table OA1. The sample includes all Compustat firms traded on NYSE, AMEX and
NASDAQ between 1965 and 2013 and covered by the Center of Research in Security Prices (CRSP). Financial firms and utilities are excluded. De-listing
returns are included in monthly returns. The table reports Fama-MacBeth coefficient estimates. t-statistics are in parentheses. R2 and N denote the
cross-sectional R-squared and the number of observations respectively. The independent variables are matched to monthly returns as described in Section
4. All variables are described in the appendix. The symbols (***), (**) and (*) denote statistical significance at the 1, 5 and 10 percent levels respectively.
Table OA6
Additional Controls
@FedFRASER