Optimal Debt and Profitability in the Trade‐Off Theory

• DOI: http://doi.org/10.1111/jofi.12590
• Published date: 01 February 2018
• Date: 01 February 2018

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 1 •FEBRUARY 2018

Optimal Debt and Profitability in

the Trade-Off Theory

ANDREW B. ABEL∗

ABSTRACT

I develop a dynamic model of leverage with tax deductible interest and an endoge-

nous cost of default. The interest rate includes a premium to compensate lenders for

expected losses in default. A borrowing constraint is generated by lenders’ unwilling-

ness to lend an amount that would trigger immediate default. When the borrowing

constraint is not binding, the trade-off theory of debt holds: optimal debt equates

the marginal interest tax shield and the marginal expected cost of default. Contrary

to conventional interpretation, but consistent with empirical findings, increases in

current or future profitability reduce the optimal leverage ratio when the trade-off

theory holds.

THE TRADE-OFF THEORY OF capital structure is the longest standing theory of

capital structure1and underlies much of the large body of empirical work on

capital structure. According to the trade-off theory, the optimal amount of debt

equates the marginal benefit of a dollar of debt arising from the tax deductibil-

ity of interest payments with the marginal cost of a dollar of debt arising from

increased exposure to default. This framework implies that changes in leverage

over time, or variation in leverage across firms, can be attributed to differences

in the marginal interest tax shield and/or differences in the marginal cost of

default. The trade-off theory has been conventionally interpreted2as implying

that more profitable firms should have higher leverage ratios—a prediction

that runs counter to the empirical fact3that more profitable firms tend to

∗Andrew B. Abel is with the Department of Finance of the Wharton School of the University

of Pennsylvania and the National Bureau of Economic Research. I thank Jeff Campbell, Marco

Giometti, Vincent Glode, Joao Gomes, Christian Goulding, Richard Kihlstrom, Bob McDonald,

Adriano Rampini, Scott Richard, Michael Roberts, Ali Shourideh, Toni Whited, and seminar par-

ticipants at the Federal Reserve Bank of Chicago, Kellogg Finance, MIT Finance, Shanghai Ad-

vanced Institute of Finance, Wharton Finance, the Tepper/LAEF Conference on Macro-Finance,

and the Tel Aviv University Finance Conference for helpful comments and discussion. I also thank

the Editor, Associate Editor, and two referees for helpful comments. Part of this research was

conducted while I was a Visiting Scholar at Chicago Booth and I thank colleagues there for their

hospitality and support. I have read the Journal of Finance’s disclosure policy and have no conflicts

of interest to disclose.

1See Robichek and Myers (1966), Kraus and Litzenberger (1973), and Scott (1976).

2See Scott (1976), Fama and French (2002), and Frank and Goyal (2008).

3See Fama and French (2002).

DOI: 10.1111/jofi.12590

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96 The Journal of Finance R

have lower leverage ratios.4Notable exceptions to this interpretation are pre-

sented in quantitative models by Hennessy and Whited (2005) and Strebulaev

(2007). In the stochastic dynamic model presented in this paper, I analytically

demonstrate an alternative explanation of the negative relationship between

profitability and leverage that is found empirically.

In this paper, I develop and analyze a model of capital structure that in-

corporates an interest tax shield as well as the possibility of default. In some

situations, the optimal amount of debt is determined by the equality of the

marginal benefit of debt arising from the interest tax shield and the marginal

cost of debt associated with increased probability of default—that is, optimal

debt will be characterized by the trade-off theory. However, in other situa-

tions within the model, optimal debt is not characterized by the equality of the

marginal benefit and the marginal cost that epitomizes the trade-off theory.

Because the model allows for situations in which the trade-off theory holds

and situations in which it does not hold, it has the potential to guide empirical

tests by including both a null hypothesis in terms of the trade-off theory and

an alternative hypothesis that offers an explanation of leverage other than the

trade-off theory. In particular, I show that the model developed here accom-

modates situations in which higher profitability (either current profitability or

expected future profitability) is associated with lower leverage, specifically, a

lower value of market leverage, which is the ratio of debt to the market value of

the firm. Furthermore, if the probability of default is nonzero, these situations

arise when and only when the trade-off theory is operative. Thus, the empirical

finding that more profitable firms tend to have lower leverage ratios, which has

been interpreted by others as evidence against the trade-off theory, is viewed

as evidence in favor of the trade-off theory when seen through the lens of the

model presented here.

As the trade-off theory has been developed over the past half century, it

has become increasingly complex, especially in empirical structural models of

the firm designed to capture realistic features of a firm’s environment.5The

model I develop here is stripped of these complexities so that I can focus on

its new features and implications in a framework that admits analytic results

without relying on numerical solution. The model’s biggest departure from

standard models of debt concerns the maturity of debt. Many standard models

of debt6assume that debt has infinite maturity and pays a fixed coupon over the

infinite future, or until the firm defaults. The assumption of infinite maturity

is clearly extreme, but it has been used productively over the years. I also make

an extreme assumption about maturity, but in the opposite direction: I assume

that debt must be repaid an instant after it is issued. The standard specification

with infinite maturity can be viewed as the limiting case of long-term debt,

4For instance, Myers (1993, p. 6) states that “The most telling evidence against the static trade-

off theory is thestrong inverse correlation between profitability and financial leverage. . . . Yet the

static trade-off story would predict just the opposite relationship. Higher profits mean more dollars

for debt service and more taxable income to shield. They should mean higher target debt ratios.”

5See Hennessy and Whited (2007).

6See Modigliani and Miller (1958), Leland (1994), and Gorbenko and Strebulaev (2010).

Optimal Debt and Profitability in the Trade-off Theory 97

while my specification with zero maturity can be viewed as the limiting case of

short-term debt, such as commercial paper, much of which has a maturity of

only one to four days,7as well as overnight repurchase agreements.

The specification of zero-maturity debt is motivated by two considerations.

First, it makes salient the recurrent nature of the financing decision, in contrast

to the once-and-for-all financing decision in many models of debt.8At each

instant, the firm decides whether to repay its debt or to default; if it decides to

repay its debt, it also chooses the amount of debt to issue anew.Because I do not

include any flotation, issuance, or adjustment costs, the amount of debt issued

responds immediately and completely to changes in the firm’s environment,

and hence does not have the rich dynamics documented and analyzed by Leary

and Roberts (2005). Second, zero-maturity debt is always valued at par, which

alleviates the need to calculate the value of debt that would arise with long-

term debt. Therefore, the firm’s decision about whether to default on its debt,

which depends on a comparison of the total value of the firm and the value of

the firm’s debt, becomes transparent.

Because firms can default on their debt, rational lenders need to account

for the probability of default, as well as their losses in the event of default,

to determine the appropriate interest rate on loans to the firm. In this paper,

risk-neutral lenders have the same information as the firm’s shareholders, and

they require a premium above the riskless rate to compensate for their expected

losses in the event of default. In addition, if the amount of debt is sufficiently

large, it will trigger immediate default. In the current framework with zero-

maturity debt, lenders avoid being subject to immediate default by refusing

to lend an amount greater than the contemporaneous value of the firm, which

itself depends on the amount of debt issued.

An important component of the firm’s financing decision is the stochastic

process for the firm’s pretax preinterest cash flow, that is, earnings before inter-

est and taxes (EBIT). I specify a continuous-time continuous-state process for

EBIT.Instead of a diffusion process, as in Leland (1994), for example, I specify a

Markov process in which EBIT remains unchanged for a random length of time

and then a new value of EBIT arrives at a date governed by a Poisson process

with arrival intensity λ. On these dates, the value of EBIT changes by a discrete

amount, and a discrete decrease in the value of EBIT can lead the firm to default

on its debt. In the event of default, a fraction α>0 of the firm’svalue disappears

as a deadweight loss and creditors take ownership of the remaining fraction

1−αof the firm. I do not consider renegotiation between shareholders and

creditors. Shareholders and creditors have common knowledge. If they were to

renegotiate when a new realization of EBIT leads to default, the result of that

renegotiation would simply be a function of the amount of debt and the level of

7In March 2016, the average daily issuance of nonfinancial AA commercial paper was 210

issues, which amounted to $5.3 billion. Of these issues, 53%, which accounted for 55% of the dollar

volume, had a maturity of one to four days. Source: Board of Governors of the Federal Reserve

System, Volume Statistics for Commercial Paper Issuance, Data as of April 12, 2016.

8See Merton (1974), Leland (1994), and Gorbenko and Strebulaev (2010).