Marking to Market Corporate Debt ∗

Lorenzo BretscherLBS†

Andrew KaneDuke‡

Peter FeldhutterCBS§

Lukas SchmidUSC & CEPR¶

November 11, 2020

Abstract

Models of capital structure and credit risk make predictions about market valua-tions of debt, but are routinely tested on the basis of book debt from common datasources. In this paper, we propose to close this gap. We construct a rich data seton firm level debt market valuations by carefully matching data on corporate bondand loan secondary market transactions. We document significant discrepancies be-tween market and book values, especially for distressed firms. We use our dataset toi) provide novel rules of thumb on how to adjust leverage and unlever returns usingstandard datasets, and ii) to revisit a number of prominent empirical patterns involvingcorporate debt. Using a market-based measure of Tobin’s Q, we find little evidence forinvestment cash-flow sensitivity in our data. We find that using market debt valuessignificantly improves default prediction, and do not detect a credit spread puzzle. Inasset pricing tests, we find a leverage premium, but no evidence for a value premiumafter controlling for market leverage. Moreover, a novel measure of financial distress,namely market-to-book debt, predicts stock returns positively in the cross-section, in-consistent with a financial distress puzzle.

∗We thank Christopher Jones, Arthur Korteweg, Wenhao Li, Tatyana Marchuk, Erwan Morellec, ChrisParsons, Thomas Kjaer Poulsen, Norman Schurhoff, Toni Whited and seminar participants at BI Oslo,Erasmus University Rotterdam, Federal Reserve Bank of Atlanta, HEC Lausanne, USC, and the CEPRAdvanced Forum in Financial Economics for valuable comments. We also thank Sudheer Chava for sharingdata on firm bankruptcies.†Department of Finance, Email: lbretscher@london.edu‡Fuqua School of Business, Email: andrew.kane@duke.edu§Department of Finance, Email: pf.fi@cbs.dk¶Marshall School of Business, Email: lukas@marshall.usc.edu

1 Introduction

Much of theoretical and empirical research in finance revolves around the determinants and implica-

tions of corporate debt. While theoretical models often make predictions about market valuations

of debt, empirical tests are often indirect as common datasets such as Compustat only report book

values. This mismatch makes it difficult to evaluate whether challenges to theoretical models stem

from inadequate modelling and predictions or from mismeasurement.

In this paper, we propose to evaluate and close this gap by constructing a rich dataset that

carefully matches data on secondary market transaction prices for corporate bonds and loans.

Our matching procedure leaves us with a monthly dataset that traces out on average more than

eighty percent of the debt structures of around 1200 firms per month between 1998 and 2018. Our

approach thus not only extends standard measurement by exploiting market values, but also by

increasing its frequency. We use our data to revisit a number of empirical puzzles revolving around

corporate debt that have received attention in the literature.

We start out by motivating important differences between market and book debt values in a

simple dynamic model of firm financing and investment. On the basis of our dataset, we then

empirically examine novel market-based measures of key variables in the literature involving corpo-

rate debt, such as leverage, Tobin’s Q, and introduce a novel variable measuring the market value

of debt relative to its book value, namely debt market-to-book. In most empirical work, market

leverage is measured as the ratio of book debt relative to the sum of market equity and book debt,

given data availability. This quantity is sometimes referred to as ’Quasi-Market Leverage’ (see e.g.

Strebulaev (2007)). Given our data, instead we construct market leverage on the basis of market

values of debt as well. Similarly, Tobin’s Q is commonly measured as the sum of market equity

and book debt relative to the replacement value of capital. In our analysis, we construct it on the

basis of market valuations of debt.

In our empirical applications we focus on four puzzles that feature prominently in the literature,

starting with investment regressions, where standard measurement, contrary to the predictions of

Q theory, attributes a significant role to cash flow measures. We then revisit bankruptcy prediction,

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where standard measures of distance to default often exhibit very little predictive power, in contrast

to the predictions of standard structural models of default. Related to default prediction, we revisit

the credit spread puzzle which refers to the notion that standard structural models of corporate

bond pricing appear to underpredict the credit spreads observed in the data. Finally, we consider

the cross-section of asset returns, where there is only weak evidence of a link between common

measures of leverage and risk premia, in contradiction to standard textbook logic.

We find i) substantial differences between measures based on market and book debt values

respectively, especially for distressed firms, and ii) that using market measures significantly tighten

the link between theory and empirics in our applications. First, we find little evidence for investment

cash flow sensitivity when our market based measure of Tobin’s Q is used as a regressor. Second,

market based measures substantially improve on common bankruptcy predictors. Third, we find

no evidence of a credit spread puzzle, as structural models of credit risk price corporate bonds

surprisingly well once calibrated to our data. Fourth, we document strong linkages between the

cross-section of returns and market leverage, and market-to-book debt, respectively. In particular,

we find a leverage premium, but no evidence for a value premium after controlling for market

leverage, and no evidence for a financial distress puzzle on the basis of our market-to-book debt

distress indicator.

Given the documented differences between our market based and traditional measures of debt

related variables, we attempt to provide researchers and practitioners using standard data with

rules of thumb that give guidance on how to adjust book debt measures to marked based variables

accurately. In particular, we provide rules of thumb for adjusting leverage ratios and asset volatili-

ties, variables that are critical in the empirical assessments of models of capital structure, default,

and investment. Similarly, our dataset allows us to compute asset returns and related measures of

business risk in a market based way, which allows us to assess the accuracy of the standard practice

of unlevering equity returns by measures of leverage. We find substantial differences between our

market based asset returns and standard computations. Our rules of thumb provide some guidance

on how to improve such calculations in a simple manner.

Our empirical approach rests on exploiting and merging a variety of datasets in order to con-

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struct debt market valuations. Starting in 1998, we use the Lehman Fixed Income database for

corporate bond transaction data which we extend from 2002 onwards using the Trade Reporting

and Compliance Engine (TRACE) database for corporate bond transaction data. We complement

information on corporate bond prices with secondary market pricing data on corporate loans for the

period from the Loan Syndication and Trading Association (LSTA) and the LPC market-to-market

pricing service between 1998 and 2017. We carefully aggregate the corresponding valuations at the

firm level and complement them with standard data on corporate bond, loan, and firm character-

istics.

To be more specific, regarding investment regressions on Q, we find that accounting for mar-

ket valuations in the construction of Q increases the R2 across standard specifications by up to

15 percent. Once we account measurement error, following Erickson and Whited (2012), the cor-

responding improvement rises to up to 13 percent. When additionally accounting for intangible

capital in our market based Q measure cash flows lose their significance and the corresponding point

estimate is very close to zero. Similarly, regarding default prediction, we show that consistently

using market values improves the empirical performance of well established predictor variables for

firm-level default. In addition, we propose a new variable that contains information orthogonal

to existing predictors. Finally, regarding the cross-section of returns, we confirm, unconditionally

the well-known significant return spread between high and low book-to-market stocks in univariate

sorts. In the same way, we find a similar significant spread between high and low market lever-

age firms. Intriguingly, in bivariate sorts of returns on both book-to-market and market leverage,

the conditional return spreads on book-to-market lose their significance, while those on market

leverage retain them. This finding underlines the relevance of market leverage in explaining the

value premium. Similarly, our market-to-book debt ratio, which is a significant predictor of default,

gives rise to a positive relation between distress risk and stock returns, inconsistent with a financial

distress puzzle.

Literature Our paper belongs to a small, but growing number of papers that exploit enhanced

access to new databases on secondary debt market transactions, such as TRACE, or loan data from

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LPC, to provide better measurement of debt variables. Most of these efforts focus on applications

in asset pricing. Important contributions, starting with Choi (2013), include Choi and Richardson

(2016), Choi and Kim (2018), and Choi, Donangelo and Kim (2020), which examine linkages

market measures of debt instruments and various measures of leverage, as well as risk premia, and

the degree of segmentation between equity and debt markets. Davydenko, Strebulaev and Zhao

(2012) study the dynamics of corporate bond and loan prices after a default event for a select set of

firms. In a contemporaneous contribution, Bartram, Grinblatt and Nozawa (2020) use a measure

of the market value of a corporate bond to its book and link to bond level returns and potential

mispricing. Relative to these important contributions, we are, to the best of our knowledge, the

first to provide a systematic exploration of market relative to book debt valuations, and applying it

to long-standing questions in corporate finance and asset pricing. Moreover, regarding applications

in asset pricing, we provide novel results on the basis of extended datasets. In a recent contribution,

Charles and Korteweg (2020) use options data to improve measurement of debt valuations.

At the broadest level, by providing novel measurement, our paper contributes to the vast

literature on the determinants of corporate capital structure and leverage.

More specifically, in terms of our empirical applications, we build on and contribute to a large

literature on tests of the Q theory of investment, default prediction, corporate bond pricing, and

the cross-section of returns, and especially the value premium, respectively.

Regarding tests of Q theory, beginning with Fazzari, Hubbard and Petersen (1988) a large

empirical literature emerged exploring the links between investment and Q on the basis of firm-

level data. These studies typically obtain small coefficients on Q and positive and significant

coefficients on cash flow. Fazzari et al. (1988) , Gilchrist and Himmelberg (1998) and most of the

subsequent literature interpret these findings as a reflection of the presence of financial frictions.

In a seminal contribution, Gomes (2001), followed by Cooper and Ejarque (2003), challenges this

interpretation by presenting specifications of simulation models with financial frictions in which

the explanatory power of cash flows nevertheless is subsumed by Q. In other words, the presence

of financial frictions is neither necessary nor sufficient to generate cash flow effects. Our empirical

approach is more agnostic by showing that in our sample, cash flow effects are driven out when Q

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is constructed on the basis of market valuations, as theory requires. By emphasizing the role of

mismeasurement, our approach is related and builds on Erickson and Whited (2012), who present

a general methodology that addresses mismeasurement in Q, and Peters and Taylor (2017), who

provide an approach to account for intangible investment and capital in Q.

A long literature has developed methods to predict corporate default. A prominent stream in

that context builds on the seminal work in Merton (1974), in which default probabilities are cap-

tured by the notion of a company’s distance to default. Implementing that concept in practice is

not straightforward, as it depends on variables not observable in common datasets. Among others,

Bharath and Shumway (2008) and Atkeson, Eisfeldt and Weill (2017) propose implementations un-

der various assumptions. Relatedly, in an influential contribution, Campbell, Hilscher and Szilagyi

(2008) propose a flexible default predictor based on firm characteristics. All the implementations

share that they require data on market leverage. We contribute to this literature by showing that

our market based variables significantly improve default prediction.

The observation, most forcefully made in by Fama and French (1992), that there is a value pre-

mium in the cross-section of stock returns, in that companies with high book-to-market equity earn

high expected returns unexplained by market betas, has triggered a vast literature in asset pricing.

Among others, Berk, Green and Naik (1999), Gomes, Kogan and Zhang (2003), Carlson, Fisher

and Giammarino (2004), Zhang (2005), and Kogan and Papanikolaou (2014) provide influential

explanations of this anomaly based on cross-sectional differences in production technologies and

exposures to sources of uncertainty abstracting from financial frictions. On the other hand, Gomes

and Schmid (2010), Ozdagli (2012), and Doshi, Jacobs, Kumar and Rabinovitch (2019), observe

a tight empirical correlation between standard measures of leverage and book-to-market equity,

in that value firms exhibit significantly higher leverage than growth firms, and suspect that this

may the mechanism underlying the value premium. Their evidence is based on restrictive model

assumptions. We contribute to this literature by providing direct evidence on the links between

leverage and book-to-market equity based on our novel market based measures. In our asset pricing

tests, we find a leverage premium, but no evidence for a value premium after controlling for market

leverage.

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Outline This paper is structured as follows. In the next section, we use a simple continuous-time

dynamic capital structure model with analytical solution to motivate relevant differences between

market and book values of debt in the cross-section and in the time series. In section 3, we detail

our data sources and variable construction, and provide a first look at our novel measures. Section

4 describes empirical applications based on our dataset, and section 5 provides a few concluding

remarks.

2 Market versus Book Debt in a Simple Model

To set the stage for our empirical analysis and to motivate some of the relevant questions, in this

section we present a simple continuous-time dynamic capital structure model in a real options

setting that delivers closed-form expressions linking market and book values of debt and corporate

decisions on financing and investment. This setup allows us to formalize our basic argument and

illustrate its implications for the cross-section and time series of debt measures in a tractable model.

For the purpose of illustration, we use a simple specification along the lines of Gomes and Schmid

(2010), or Sundaresan, Wang and Yang (2015).

2.1 Environment

We consider the problem of value-maximizing firms, indexed by the subscript i, that operate in

a perfectly competitive environment. Time is continuous. The instantaneous flow of (after-tax)

operating profits , Πi, for each firm i is specified by the expression

Πi = (1− τ)XtKαi , 0 < α < 1,

where Ki is the productive capacity of the firm, τ is the corporate tax rate, and X is an exogenous

state variable that captures the state of aggregate demand or productivity. We assume that Xt

follows the stochastic process dXt/Xt = µdt + σdεt and that εt is a standard Brownian motion

under a risk-neutral measure.

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Financing and Investment In our simple environment, a typical firm is endowed with an

initial capacity K0 and a single option to expand it to K1 by purchasing additional capital in the

amount I = K1−K0 > 0. We assume that the relative price of capital goods is one and that there

are no adjustment costs to this investment. In what follows we will say that the firm is “young” if

it has not yet exercised this growth option and “mature” if this option has already been exercised.

Firms are financed with both debt and equity issues. For tractability, we make three simplifying

assumptions regarding the nature of debt financing available to firms. First, we assume that debt

takes the form of a consol bond that pays a fixed coupon ci per period for each firm i. Second, we

restrict new debt to be issued only to finance investment spending so that a firm is simultaneously

choosing optimal investment and financing policies. Third, we assume that debt is restructured

at the time of new issues. Existing debt is retired at its current market value and new debt is

issued. Given these assumptions we now denote ci and B(X, ci) as, respectively, the flow of interest

payments and the market value of debt for a firm with productive capacity equal to Ki.

Given our assumptions it follows that the instantaneous dividends for the equity holders of a

mature firm, with capital K1, are equal to

(1− τ) (XtKα1 − c1) .

Given debt and its associated coupon payment, c1, the equity value of a mature firm, V (X; c1),

satisfies the following Bellman equation:

V (X; c1) = (1− τ) (XtKα1 − c1) dt+ (1 + rdt)−1E[V (X + dX; c1)]. (1)

Equation (1) holds only as long as the firm meets its obligations to the debt holders. However,

equity holders have the option to close the firm and default on their debt repayments if the prospects

for the firm are sufficiently bad. If equity holders have no outside options this (optimal) default

occurs whenever V (X; c1) reaches zero. Formally, default occurs as soon as the value of X reaches

some (endogenous) default threshold, XD1 . This threshold is determined by imposing the usual

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value matching and smooth pasting conditions, requiring that at XD1 equity value satisfies

V (XD1 ; c1) = 0 and V ′(XD1 ; c1) = 0.

Young firms are similar to mature firms but, in addition, they possess an option to invest

and expand their productive capacity to K1. For young firms the flow of operating profits (and

dividends) per unit of time is thus given by the expression

(1− τ) (XtKα0 − c0) .

This yields the following Bellman equation for equity value, V (X; c0):

V (X; c0) = (1− τ) (XtKα0 − c0) dt+ (1 + rdt)−1E[V (X + dX; c0)]. (2)

As before equation (2) holds only as long as the firm meets its obligations to the debt holders.

Letting XD0 denote the default threshold for a young firm we require that V (X; c0) satisfies the

following boundary conditions:

V (XD0 ; c0) = 0 and V ′(XD0 ; c0) = 0.

Moreover, the equity value for a young firm also reflects the existence of a growth opportunity.

Intuitively, if demand grows sufficiently so that X is above an investment threshold, say XI , the

firm will choose to expand its productive capacity to K1. Thus, at this investment threshold firm

value must obey the additional boundary conditions

V (XI ; c1) + (B(XI ; c1)−B(XI ; c0))− I = V (XI ; c0)

V ′(XI ; c1) + (B′(XI ; c1)−B′(XI ; c0)) = V ′(XI ; c0),

where (B(XI ; c1)−B(XI ; c0)) denotes the value of net new debt issues at the time of investment.

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2.2 Valuation

To compute the value of a mature firm, given a pre-determined coupon payment, c1, we use Ito’s

Lemma in equation (1) and impose default when X = XD1 to solve the associated differential

equation. This procedure implies that the value of equity for a mature firm satisfies the expression

V (X; c1) =(1− τ)XKα

1

r − µ− (1− τ)c1

r+A11X

v1 , (3)

where v1 < 0, and the value for the constant A1 > 0 can be obtained using the relevant boundary

conditions at the default threshold.1

The first term in equation (3) is the present value of the future cash flows generated by existing

assets, K1. From this value we must then deduct the present value of all future debt obligations,

which is captured by the term (1−τ)c1r . Finally, the last term shows the impact of the option to

default on the value of the firm to its shareholders.

In the case of a young firm we apply Ito’s Lemma to the Bellman equation in (2) and solve the

associated differential equation to obtain the expression

V (X; c0) =(1− τ)XKα

0

r − µ− (1− τ)c0

r+A10X

v1 +A0Xv0 , (4)

where v0 > 1 and v1 < 0 are the roots of the quadratic equation r = vµ+0.5v(v−1)σ2 and A10 > 0

and A0 > 0 are determined by imposing the boundary conditions at the investment and default

threshold.

The first three terms in equation (4) for the equity value of young firms seem identical to those

of mature firms and capture, respectively, the present value of the future cash flows generated by

existing assets, K0, and future debt obligations, as well as the present value of the option to default

on these obligations.

1 Standard computations that

A11 = −(

(1− τ)XDKα1

r − µ− (1− τ)c1

r

)(1

Xv1D1

),

while v1 is the negative root of the quadratic equation r = vµ+ 0.5v(v − 1)σ2.

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Despite these similarities the value of young firms, V0, differs from that of mature firms, V1,

in two important ways. First, the equity value of young firms will depend on the (positive) value

of future growth options, captured here by the term A0Xv0 . Second, mature firms are larger

(K1 > K0) which implies that they are also more levered so that c1 > c0.

Debt Value and Coupon Payments Before computing the value of each firm we need to

determine both the market value of debt outstanding and the value of the instantaneous coupon

payments, since both of these values are linked to the firm’s decision to invest. It is the possibility

of default that naturally induces a deviation between the market and the book value of debt at any

point in time. As in Leland (1994), as long as the firm does not default the market value of debt

paying a per-period coupon of ci, satisfies the Bellman equation

B(X; ci) = cidt+ (1 + rdt)−1E[B(X + dX; ci)]

Upon default debt holders are able to recover a fraction, ξ > 0, of the value of the firm upon

default. Formally, this leads to the following boundary condition on debt:

B(XDi ; ci) = ξ(1− τ)XDiK

αi

r − µ,

where XDi denotes the threshold level of demand upon which firm i optimally chooses to default.

Effectively this boundary condition assumes that, after accounting for transaction costs, debt hold-

ers will take over the firm and be entitled to the entirety of its future cash flows.2 Given this

boundary condition at default we can easily construct the expression for the market value of debt

for firm i, B(X; ci). This is given by

B(X; ci) =cir

+(B(XDi ; ci)−

cir

)( X

XDi

),v1 (5)

where v1 < 0 is defined as above. Note that this expression implies that the market value of debt

2 Note that there is no boundary condition at the restructuring threshold since we are assuming that theyoung firm’s debt is callable at market value.

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converges to ci/r as X approaches infinity.

The exact value of the optimal periodic coupon payment, ci, can now be determined by maxi-

mizing the joint value to equity and bond holders as follows

c1 = arg maxcV (XI ; c) +B(XI , c) and c0 = arg max

cV (X0; c) +B(X0, c) (6)

where X0 is the (arbitrary) value of demand process X at the birth of the firm when initial

leverage is decided.

Note that since both V (·) and B(·) are increasing in X it follows immediately that c1 > c0 if

XI > X0. Since, by definition, the young firm invests immediately at XI it must be the case that

young firms with unexercised growth options have less debt than large mature firms.

2.3 Some Numerical Examples

While our stylized environment is not sufficiently rich for a detailed quantitative analysis, it nev-

ertheless allows to provide some illuminating numerical examples in a straightforward way. Our

objective is to illustrate that a common dynamic capital structure with standard parameter choices

predicts substantial differences between market and book based measures of leverage and Tobin’s

Q.

In the spirit of the exercise, rather than fully calibrating the model, we pick the parameters

in line with the literature, along the lines of Gomes and Schmid (2010). Specifically, we set the

baseline parameter values in our examples to r = 0.05, µ = 0.0, τ = 0.2, σ = 0.25, α = 0.65, and

K0 = 1. By varying K1 and the recovery rate on debt ξ = 0.0, we can explore the effects of

differential growth and investment opportunities, as well as financial constraints and distress, on

discrepancies between market and book based measurement of debt across firms.

Figure 1 provides some initial results by means of simulated time series plots. We show simula-

tions over 120 months, which is close to the average maturity of corporate long-term debt. In our

baseline specification, parameters are picked such that firms issue debt so as to exhibit a market

leverage ratio of around 0.3, close to the empirical evidence. Importantly, as the upper left panel

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shows, ’true’ market leverage and ’quasi market leverage’ where debt is measured at the book value

coincide at the beginning of the simulation. That is because book debt is market debt at issuance.

However, over time, as the economic environment changes with the evolution of the state variable,

market and book debt begin to diverge, sometimes substantially. This divergence is also reflected in

the evolution of Tobin’s Q and its empirical counterpart, where debt values are inferred from book

values, as the top right panel shows. In a bond market context, this observation of divergence can

be captured by a notion whose equity market analogue has attracted a lot of interest. Specifically,

we can define the market-to-book ratio of corporate bonds. The lower left panel shows the evolution

of this quantity over time. The divergence of market and book values of debt is reflected in the

long and volatile swings that this quantity exhibits. Finally, we compare market and book debt

implied leverage ratios more directly by looking at their ratios. Anticipating the next section3 , we

label this quantity market value ratio, i.e., MVR. Once again, our simple model predicts that MVR

exhibits long and volatile swings around one, which represents the benchmark case in which book

and market values coincide.

While we cast the initial results in terms of time series, the same approach gives a first sense

of cross-sectional predictions, by varying across firms, i) bankruptcy costs, and ii) the moneyness

of the growth option by adjusting K2. While the first adjustment allows us to capture firms closer

to distress with higher leverage ratios, the second provides a representation of firms that have

exhausted their growth potential. Figures 2 and 3 illustrate the results and reinforce the notion

that market and book values of debt can persistently diverge even in a simple model as ours. This

presents a major challenge to measurement and testing of models, to which we turn next.

3 Measurement and Data Construction

A main contribution of our paper is the compilation of a rich dataset on market values of corporate

debt. In this section, we describe our data sources and discuss in detail the construction of our

final dataset.

3 See section 3 regarding the empirical challenges surrounding this approach

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3.1 Data Sources

In order to construct our main data set, we rely on the following data sources: Lehman Fixed

Income database for corporate bond quote data between 1998 and 2002; the Trade Reporting

and Compliance Engine (TRACE) database for corporate bond transactions data from July 2002

onwards; the Mergent Fixed Income Securities Database (FISD) for information on corporate bonds;

the WRDS-Reuters DealScan data base for information on loans; the Loan Syndication and Trading

Association (LSTA) and the LPC market-to-market pricing service for secondary loan pricing data

for the period between May 1998 February 2017; CRSP/Compustat Fundamentals Annual and

Quarterly, respectively, to obtain company balance sheet information; CRSP/Compustat Security

Monthly for monthly equity stock price information.

3.2 Corporate Loan Data

We start our data construction by matching monthly facility pricing data to gvkey company iden-

tifiers. To do so, we use the linking table from Chava and Roberts (2008). We restrict our analysis

to loans that are either denominated in US Dollars, United Kingdom Pounds, or Euros. Following

Beyhaghi and Ehsani (2017) and Choi (2013) we further restrict our analysis on term loans and

revolvers. On average, we have loan pricing information on three different loans (the median is two)

available for our sample companies. Table 1 reports how this procedure affects the total number

of observations. We compute monthly returns for individual term loans based on the procedure

described in the S&P/LSTA U.S. Leveraged Loan 100 Index Methodology note.4 Effectively, at

time t, the total return on a loan consists of price, principal repayment, and interest returns:

rLt =Part × (Pt − Pt−1) + (Part − Part−1) + (AIt −AIt−1) + C

Part−1 × Pt−1 +AIt−1(7)

where Part is the par value (remaining face value) of the loan at time t, adjusted for any repayments

that already took place. Pt is the market price of the loan at time t. (Part − Part−1) is the principal

4 This methodology is also applied by Beyhaghi and Ehsani (2017). More details can be found onhttp://us.spindices.com/indices/fixed-income/sp-lsta-us-leveraged-loan-100-index.

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repayment at time t and is assumed to be made at par. AIt is accrued interest at time t based

on a 360-day basis. Finally, C is the coupon payment paid. In our sample, 62.37% of the loans

repay principle on a quarterly basis, 10.01% semi-annually, 4.74% monthly, 4.32% annually. The

remaining 18.56% of the loans belong to ”Bullet“ loans that do not pre-pay any principal before

expiration.

For the returns on revolvers we follow Choi (2013) and abstract from principal repayments. In

addition, we assume that 20% of revolving loans are withdrawn. Compared to Sufi (2009) who

reports that on average 30% of revolving loans are withdrawn this is a conservative assumption

which prevents us from overstating the importance of revolving loans.

Finally, for firms with multiple loans we replace missing price data with the value-weighted

price of otherwise similar loans (i.e. term loans or revolvers) with non-missing price data of the

same company in a given month.

3.3 Corporate Bond Data

For the period January 1997 to June 2002, we use month-end quotes provided by Merrill Lynch

(ML). These data are used by, among others, Schaefer and Strebulaev (2008), Acharya, Amihud

and Bharath (2013), and Feldhutter and Schaefer (2018a). From July 2002 and onwards, we

use the TRACE database. For fixed-coupon bonds, we rely on the WRDS bond returns database

containing monthly returns and month-end prices. The database provides access to cleaned datasets

of corporate bond transactions, sourced from TRACE Standard and TRACE Enhanced datasets,

and we use their prices based on the last transaction in the month. The WRDS bond returns

database does not provide prices and returns for floating-rate bonds, and for those we source data

directly from TRACE Standard and TRACE Enhanced. We filter the data according to Dick-

Nielsen (2009) and Dick-Nielsen (2014) and use the last transaction in the month.

For the Merrill Lynch dataset and the floating-rate bonds in TRACE, we compute monthly

14

returns as:

rt =(Pt +AIt + Coupont)− (Pt−1 +AIt−1)

Pt−1 +AIt−1(8)

where rt is bond’s month-t return, Pt is its clean price at month-end t, AIt is its accrued interest

at month-end t, and Coupont is any coupon paid between month-ends t − 1 and t. For floating-

rate bonds we use Mergent FISD’s database FISD COUPON FORMULAINDEX to calculate the

current floating rate in month t and assume that this floating rate is accrued in the same month.5

To calculate the potentially time-varying amount outstanding for each bond, we use Mergent

FISD’s database FISD AMT OUT HIST.

Bond information We obtain bond information from the Mergent Fixed Income Securities

Database (FISD) and limit the sample to senior unsecured fixed rate or zero coupon bonds. We

exclude bonds that are callable, convertible, putable, perpetual, foreign denominated, Yankee, or

have sinking fund provisions.6 We use only bonds issued by industrial firms and restrict our sample

to bonds with a maturity of less than 20 years to be consistent with the maturities of the default

rates we use as part of the estimation. After we merge the bond data with firm variables, the

number of observations is 69,809. We winsorize spreads at the 1% and 99% level.

Corporate bond yield spreads As in Feldhutter and Schaefer (2018b), Bai, Goldstein and

Yang (2020), and others we calculate corporate bond yield spreads relative to the swap rate and

use on a given date the available rates among the 1-week, 1-month, 2-month, and 3-month LIBOR

and 1, 2, 3 ,4 ,5 ,6, 7, 8, 9, 10, 12, 15, 20-year swap rates and linearly interpolate to obtain a swap

rate at the exact maturity of the bond.

5 If interest rate frequency is 0, 99, 13, 14, or missing, we assume that it is quarterly for floating-ratebonds and semi-annually for fixed-rate bonds.

6 For bond rating, we use the lower of Moody’s rating and S&P’s rating. If only one of the two ratingagencies have rated the bond, we use that rating. We track rating changes on a bond, so the same bond canappear in several rating categories over time.

15

3.4 Corporate Debt Data

Next, we combine the corporate loan and bond data. To do so, we calculate total book and

market values of both loans and bonds at the company-month level. In addition, we compute

value-weighted monthly loan and bond returns for each company. For a total of 828 companies we

have continuous price and return data for both loans and bonds. We use this data to estimate the

following regressions:

P loani,t = α+ βP bondi,t + γP bond2

i,t + δcpnbondi,t × TBillt + ηcpnbondi,t + θt + εi,t (9)

where P loani,t is the loan price for a company i at time t, P bondi,t is the corresponding bond price,

cpnbondi,t is the coupon rate on bonds, TBillt is the 3-month Treasury Bill rate at time t (collected

from FRED), θt are time fixed effects and α is a constant. In comparison to the similar regression in

Davydenko et al. (2012) we also include the coupons of bonds and the interaction term of coupons

with the Treasury Bill. This helps us to better capture price dynamics during the recent period

of the zero lower bound. Moreover, rather than running this regression on the pooled sample, we

run separate regressions for four subsamples that are based on the S&P issuer credit rating. In

particular, we form the following four groups: (1) high rating - companies with ratings of at least

BBB+; (2) medium rating - companies with ratings BBB or BBB−; (3) medium to low rating

- companies with ratings BB+, BB, or BB−; (4) low rating - companies with ratings of B+ or

lower. Our definition of the subsamples is led by the relative frequency of the individual ratings

and the attempt to form four groups that are not too different in sample size. The regressions

produce an average adjusted R2 of 72%. The explanatory power does not improve when we control

for additional characteristics of corporate bonds such as duration or time-to-maturity or additional

macroeconomic controls. We then use the regression coefficients to estimate loan prices for sample

firms with missing values.

To extrapolate loan returns, we estimate the following regressions:

rxloani,t = α+ βrxbondi,t + γrxbond2

i,t + δcpnbondi,t × TBillt + ηcpnbondi,t + θt + εi,t (10)

16

where rxloani,t and rxbondi,t are monthly excess returns on corporate loans and bonds. Choi (2013) ap-

plies a similar regression model to estimate loan returns. We additionally include squared monthly

bond excess returns to capture non-linearities as well as information on the coupons and the interac-

tion between coupons and the prevailing interest rate to account for the fact that our sample spans

the zero lower bound period. Importantly, similar to the price regressions above, we determine

rating group specific coefficients to improve the estimated excess loan returns.

In a final step, we merge our corporate debt data with information about quarterly company

fundamentals and monthly stock prices. Moreover, in order to calculate firm-level loan and bond

returns we value-weight corresponding security specific returns for a company in a given month.

3.5 Sample Summary Statistics

Our final data set spans the period from March 1998 until March 2018 and consists of 286,876 firm-

month observations. That is, on average we observe 1,223 firms. We restrict our empirical analysis

to non-financial (exclude sic codes between 6000 and 6999), non-regulated companies (exclude

sic codes between 4900 and 4999) and companies that are traded on either NYSE, AMEX, and

NASDAQ exchanges. Bonds account on average for 59% (65%) of the book value of debt taken

from quarterly (annual) balance sheet data.7 The face value of loans, on the other hand, account

for 30% (32%) of book value of debt from quarterly (annual) balance sheets.8 That is, our data

covers on average 82% (84%) of the quarterly (annual) book value of total debt. Table 2 reports the

sample frequencies of loan- and bond-month observations for specific security characteristics. For

example, term loans (fixed coupon bonds) account for roughly three quarters of our loan (bond)

data. Moreover, the vast majority of loans and bonds is senior debt.

Table 3 reports summary statistics of loan and bond prices as well as monthly loan, bond, and

equity excess returns. The level of observation is firm-months. That is, loan and bond returns are

simply value-weighted averages of the individual loan and bond returns for a given company in a

7 The book value of the debt is defined as sum of total long-term debt and debt in current liabilities. Thatis, DLTTQ + DLCQ for quarterly and DLTT + DLC for annual balance sheet data.

8 The loan percentages exclude companies that have loans but no bonds outstanding as including thesefirms would result in artificially high percentages for loans.

17

given month. Loan prices rarely trade above par due to their repayment option. In comparison,

bonds trade frequently above par as witnessed by an above par average bond price. This is largely

driven by the second half of our sample which includes the zero lower bound period. As expected,

loan returns are less volatile and smaller in magnitude than bond returns.

As discussed above, our sample contains on average 1,223 firms for each sample year. This

makes up for a representative cross-section compared to the CRSP/Compustat universe. In fact,

comparing the sum of the book values of total liabilities of all our sample firms with the corre-

sponding sum for the universe of CRSP/Compustat firms results in an average coverage of 77.1%

as reported in Table A1.

4 A First Glance at the Data

In this section, we introduce a number of variables that our dataset allows us to compute and that

we find instructive. With those variables at hand, we present a number of useful rules of thumb

that allow to approximate marked based debt variables using common datasets.

4.1 Market vs Book Values: Capital Structure Implications

With the rich data set at hand, we are able to calculate monthly company-specific time series

of book and market leverage and compare them to each other. Doing so allows us to get a first

impression on whether market values of corporate debt are indeed different from book values. That

is, we start by calculating the following four measures of book leverage (BL) and market leverage

18

(ML):

BLQit =DebtBV,Qit

DebtBV,Qit + EMV,Mit

BLYit =DebtBV,Yit

DebtBV,Yit + EMV,Mit

BLMit =DebtBV,Mit

DebtBV,Mit + EMV,Mit

MLMit =DebtMV,M

it

DebtMV,Mit + EMV,M

it

where BV (MV ) stands for book (market) value of debt and E is the monthly measured market

value of equity (monthly common shares outstanding times monthly closing share price). The

market value of debt is measured as the sum of market values of loans and bonds. The appendix

provides more details on the construction of MV Debt. Moreover, Q (Y , M) indicate quarterly

(annual, monthly) frequency of company balance sheet data. In comparison to existing studies,

our data set allows us to improve the measurement of leverage along two dimensions: (1) data

frequency; we have monthly rather than annual or quarterly data, and (2) market prices; we use

market values instead of book values.

To study which of the two dimensions, frequency or market values, contributes most to differ-

ences between corporate book and market leverage, we calculate the following ratios:

BV RQit =1−BLMit1−BLQit

BV RYit =1−BLMit1−BLYit

MVRit =1−MLMit1−BLMit

MMRQit =1−MLMit

1−BLQit

MMRYit =1−MLMit1−BLYit

The ratios BV RQt and BV RYt measure the importance of the increase in the data frequency from

19

quarterly to monthly and from annual to monthly, respectively.9 MVRt measures the importance

of using market rather than book values. Finally, MMRQt and MMRYt report to what extent our

marking to market approach is different from existing (quarterly and annual) measures of leverage.

Figure 4 plots the distribution of BV RQt , BV RYt , and MVRt across time. We conclude that it is

predominantly the change from book to market values in debt that leads to large cross-sectional

differences between measures of book and market leverage. Figure 5 corroborates this conclusion as

5a and 5b closely mirror Figure 4c. We further investigate the distribution of MMRQt by splitting

the sample into companies with a low (high) S&P issuer credit rating where low rating companies

are defined as companies with an S&P issuer credit rating of B+ or worse. In contrast, high rating

companies are firms with a rating of at least BBB+. Figure 6 plots the distributions of MMRQt,L

and MMRQt,H , respectively. As expected, the cross-sectional differences between book and market

leverages are much more severe for low compared to high rating firms. That is, MMRQt,L ranges

between 0.75 and 2.6 while MMRQt,H ranges between 0.93 and 1.1.

4.2 Proxies for Market Leverage

Next, we consider a number of variables that are traditionally relevant in calibrations of structural

models of credit risk. Specifically, these models are typically calibrated to quasi-leverage, i.e.

calibrated to

LBV =DBV

EMV +DBV.

In a recent influential paper, Bai et al. (2020) [BGY henceforth] point out that theoretically the

correct leverage the models should be calibrated to is

LMV =DBV

EMV +DMV,

9 Note that we choose to calculate ratios1−BLM

it

1−BLQit

rather thanBLM

it

BLQit

because the second ratio is problematic

for firms with very low levels of debt.

20

because the denominator reflects the market value of the firm and therefore the market value of debt

should be used.10 They approximate the market value of debt as follows. For each major rating

category, they compute the average ratio priceface value in their corporate bond sample and apply this

ratio to the face value of all debt outstanding of firms with that rating. The first row in Table 15

shows their calculated ratios. Importantly, they find that the market value of debt is substantially

below book value for the most risky firms as the ratio of 0.72 for the C rating shows. The next line

in the table shows the ratios calculated using our sample of corporate bonds. Consistent with their

results we find that C-rated corporate bonds trade substantially below face value at an average

ratio of 0.82.

The market value of debt is a weighted average of the market value of bonds and bank debt

and the BGY estimates are potentially biased because they do not include bank debt. To examine

the severity of the bias, we calculate in the third row in Table 15 the average market-to-book debt

ratio of the issuing firms. We see that the correct ratios are substantially closer to one than the

ratios calculated using the BGY approach. In particular, the correct ratio is 0.95 for C-rated firms.

There are at least three reasons for this. First, bank debt has substantially higher average recovery

rates than bond debt. Second, bank debt on average has lower duration than bonds and bank

debt prices are likely less sensitive to changes in credit risk than bonds. Third, there is a negative

correlation (-0.42 in our data set) between the market-to-book ratio of individual bonds and the

ratio of market value of bank debt to the market value of total debt of the issuing firm.11 This

implies that bonds with low market prices on average have a smaller weight in the calculation of

total debt value.

In absence of detailed data on bank debt prices, an approximate adjustment of the leverage to

account for market values is potentially important. As an alternative to BGY’s rating-specific ad-

justment of the leverage ratio, we propose to approximate market leverage as a n-degree polynomial

10 In contrast to later in the paper, we use book value of debt in the numerator. The reason is that instructural models, the firm defaults when firm value is sufficiently low compared to the face value promisedto debt holders.11 A likely explanation for the negative correlation is that if a larger fraction of a firm’s debt is bank debt,

bondholders – being de facto junior to bank debt holders – expect to receive a smaller recovery in bankruptcy.

21

in book leverage. Specifically, we estimate the regression

DBVj

EMVj +DMV

j

=N∑i=1

βi(LBVj )i + εj , j = 1, ..., T

for N = 1, 2, 3, and 4 and where LBVj =DBV

j

EMVj +DBV

j. Table 16 Panel A shows the regression results.

The vast majority of market leverage is explained by a first-order polynomial and we see a modest

incremental benefit of including higher-order terms. The results do not clearly point to an optimal

order of the polynomial and we use the ”out-of-sample” pricing evidence in Section 4.3 to inform

the choice of order.

To compare the proxies for market leverage, Table 16 Panel B shows the prediction error variance

(PEV) 1T

∑Tj=1

(DBV

j

EMVj +DMV

j− PMj

)2where PMj is the predicted market leverage of observation j

by model M . The BGY adjustment leads to a PDE of 0.00376, which is higher than that of making

no adjustment, 0.00362. This implies that that the leverage adjustment suggested in BGY leads

to worse market leverage estimates than using book leverage as is common in the literature. Panel

B also shows that a polynomial in book leverage leads to better market leverage estimates than

using book leverage or the BGY adjustment. Thus, in absence of bank debt prices the estimated

polynomials in Table 16 can be used as an estimate for market leverage.

4.3 Proxies for Asset Volatility

Asset volatility is a another key input in structural models and the standard approach is estimate

asset volatility by unlevering equity volatility. One approach is using a model linking asset and

equity volatility. The risk is that the model is misspecified leading to biased asset volatility es-

timates. An alternative approach is to rely on direct measures of asset volatility to estimate a

function that calculates asset volatility with equity volatility and leverage as inputs. Feldhutter

and Schaefer (2018b) (FS) take this approach and approximate the ratio between asset volatility

22

and equity volatility, σAσE

as

(1− LBV )(1.00 ∗ 1{LBV ≤0.25} + 1.05 ∗ 1{0.25<LBV ≤0.35} + 1.10 ∗ 1{0.35<LBV ≤0.45} (11)

1.20 ∗ 1{0.45<LBV ≤0.55} + 1.40 ∗ 1{0.55<LBV ≤0.75} + 1.80 ∗ 1{0.75<LBV }

](12)

where σE is equity volatility and σA is asset volatility. Their estimate (based on results in ?)

ignores the contribution of bank debt when calculating asset volatility and is therefore potentially

biased.

Our data set allows us to calculate asset volatilities that take bank debt into account. To provide

a potentially more acccurate alternative to the function in equation (11), we do as follows. For

each firm-year in the sample where we observe 12 monthly equity and asset returns, we calculate

the standard deviation of asset and equity returns and compute the ratioσA,j

σE,j. We then estimate

the regression

σA,jσE,j

− 1 =N∑i=1

βi(LBVj )i + εj , j = 1, ..., T (13)

for N=1, 2, 3, and 4 and where LBVj =DBV

j

EMVj +DBV

j.

Table 17 Panel A shows regression results. The vast majority of the variation inσA,j

σE,jis explained

by a first-order polynomial as evidenced by an R2 of 0.9897. A second-order polynomial improves

predictions slightly as the (small) increase in R2 and significant coefficient on the second-order

term shows. Higher-order terms are neither significant nor improve the R2. Overall the regressions

suggest a second-order polynomial as the best proxy for the volatility ratio.

Table 17 Panel A compares the accuracy of the volatility estimator of FS with that of the

polynomial estimates by calculating the prediction error variance 1T

∑Tj=1

(σAj

σE,j− PMj

)2where

PMj is the predicted volatility ratio of observation j by model M . The PDEs of the polynomial

estimators are in the range 0.00210-0.00231 and much small than the PDE of 0.00727 of the FS

estimator. This implies that in absence of bank debt data, asset volatilites are more accurately

estimated using the second-order polynomial compared to using the FS estimator.

23

4.4 Asset Returns vs Unlevered Equity Returns

Our data set allows us to directly calculate company-specific asset returns. We simply calculate

asset returns based on corresponding debt and equity returns and market value based capital

structure weights,

rAt+1 =LoansMV

t

LoansMVt + BondsMV

t + EquityMVt

rLt+1 +BondsMV

t

Loansmvt + BondsMV

t + EquityMVt

rBt+1+

EquityMVt

LoansMVt + BondsMV

t + EquityMVt

rEt+1.

(14)

Having ”true” firm-specific asset returns further enables us to evaluate traditional ways of unlevering

equity returns. First, we use lagged quasi-market leverage to unlever equity returns as follows

rUi,t = rEi,t(1− Li,t−1) (15)

where Li,t−1 = DBVi,t−1/(D

BVi,t−1 + EMV

i,t−1) (see, among others, ?). Second, we use a standard Merton

model to calculate the firm-specific market value of debt. Third, we make use of our data set

and use actual market leverage to unlever equity returns as in equation 15. Finally, we rely on the

suggested rules of thumb from section 4.2 which allow us to calculate the ratio DBVi,t /(D

MVi,t +EMV

i,t =

0.92LBVi,t + 0.16(LBVi,t )2 and, hence, market leverage. Figure 8 reports the results graphically. The

four panels show the distributions for the corresponding differences between asset and unlevered

equity returns expressed in percent (i.e., 100(rAi,t − rUi,t)). Moreover, we differentiate between seven

rating groups based on S&P ratings (AAA, AA, A, BBB, BB, B, and C). The distributions of

differences are represented by boxplots which report the median (red line), the interquartile range

(IQR, the ”box”) and the minimum and the maximum (first quartile - 1.5*IQR and third quartile

+ 1.5*IQR, respectively). As expected, unlevered equity returns are quite close to asset returns

for high rating firms. However, for firms with lower credit ratings unlevered equity returns are

very different from asset returns. Moreover, unlevering using quasi-market leverage is far from

ideal. In comparison, unlevering using the Merton model seems preferable as it decreases the

dispersion in the rating specific-differences drastically. Interestingly, using actual market leverage

24

improves the results even more. While the differences for non-investment grade debt is still sizeable,

there is virtually no difference for investment grade debt. Importantly, applying the rule of thumb

from section 4.2 yields very similar results. Overall, there emerge two main take aways from this

analysis: 1) relying on market values or appropriate rules of thumb is preferred to other alternatives

of unlevering equity returns and 2) ignoring the cost of debt when unlevering equity returns is costly

(in terms of measurement) for firms which have costs of debt that are significantly different from

risk-free rates. In other words, it is difficult to accurately approximate asset returns for firms with

low credit ratings without residing to the corresponding cost of debt. As a consequence, results

that are based on unlevered equity returns and focus on low rated companies should be taken with

a grain of salt.

5 Further Empirical Applications

We now revisit a number of empirical patterns involving corporate debt that have received attention

in the recent literature. Our novel dataset provides a perspective on whether these patterns are

puzzling from the viewpoint of theory or of measurement.

5.1 Market Values and the Investment-q Relation

This section explores how making use of market values of debt affects the empirical investment-q

relation. To this end, we compare the empirical performance of q measures that are either based

on book or market values of debt. Moreover, we introduce a novel measure of firm-level bond-q.

5.1.1 Variables Definitions

Tobin’s q: We follow Peters and Taylor (2017) and measure total q by scaling firm value by the

sum of physical and intangible capital:

qtotit =Vit

Kphyit +Kint

it

. (16)

25

We measure replacement cost of physical capital, Kphy, as the book value of property, plant and

equipment (Compustat item PPEGT). Kint is measured as in Peters and Taylor (2017). Finally,

the firm’s market value V is measured either, following the literature, using the book value of debt

or, alternatively, using the market value of debt. That is, we calculate two variants of V defined as

follows 12

Vit = EquityMV +DebtBV

VMVit = EquityMV +DebtMV ,

which allows us to calculate in addition to the traditional (book value) q defined in equation (16)

a market value based version of q

qtot,MVit =

VMVit

Kphyit +Kint

it

.

In addition to the relatively recent suggested total q, we also examine the literature’s standard

Tobin’s q measure used by Fazzari et al. (1988), Erickson and Whited (2012), and many others.

Again, we calculate a book and market value based version of this measure,

qit =Vit

Kphyit

qMVit =

VMVit

Kphyit

.

Erickson and Whited (2006) and Erickson and Whited (2012) compare several alternative Tobin’s

q measures, including the market-to-book-assets ratio, and they find that q works best.

Investment and Cash Flow Measures: We follow the literature and measure standard physical

12 Note that the market value of equity is calculated as the product of share price and number of sharesoutstanding (PRCC F x CSHO), the book value of debt is the sum of current and long-term liabilities(DLTT + DLC). In addition, we subtract the firm’s current assets (ACT) which include cash, inventory, andmarketable securities. This definition of firm value follows Peters and Taylor (2017). Finally, the marketvalue of debt is calculated as discussed in section ??.

26

investment (i∗), firm’s physical (iphy), intangible (iint), and total investment rates (itot) as

i∗it =Iphyit

Kphyi,t−1

iphyit =Iphyit

Ktoti,t−1

iintit =Iintit

Ktoti,t−1

itotit = iphyit + iintit

We measure physical investment, Iphy, as capital expenditures (CAPX), and we measure intangible

investment, Iint, as in Peters and Taylor (2017). Moreover, in line with their paper, we define total

cash flows that recognize R&D and part of SG&A as investments

ctotit =IBit +DPit + Iintit (1− κ)

Kphyi,t−1 +Kint

i,t−1(17)

5.1.2 Empirical Evidence on the Investment-q Relation

Table 4 presents summary statistics for the variables discussed above. As can be seen from com-

paring sample moments across rows using market values rather than book values changes the

distributional properties of the q measures. Market value based q measures have a similar mean,

median, but the standard deviation and skewness are lower.

OLS Results

Table 5 report results from OLS regressions of investment on lagged q, firm, and year fixed effects.

The columns compare different investment measures. In what follows, we focus on R2 values of

the various regressions as the regression coefficients suffer from measurement-error bias (see, for

example, Erickson and Whited (2006) and Abel (2014)).

Theoretically, the R2 of this regression should be 100%. Consistent with the literature, we find

R2 well below 100% both for book value as well as market value based measures. One explanation

for the low R2 is that we measure q with error. One component of of this error is due to the fact

27

that the existing literature uses book values of debt to approximate for market values. Consistent

with this argument Table 5 shows that the within R2 are always higher for market value based q

measures compared with their book value based counterparts. That is, using market values leads

to an increase in R2 between 7% and 14%. The magnitude of this improvement is comparable to

the increase in within R2 when using qtot rather than q in our sample period (R2 increases between

3% and 27% as reported in Table A2 in the Online Appendix). This conclusion is robust to using

total q from Peters and Taylor (2017) or the standard q which abstracts from intangible assets.

Importantly, the increases in within R2 when replacing qtotit with qtot,MVit (or qit with qMV

it ) are

highly statistically significant for all four investment measures.

It is tempting to run horse races by including book value and market values based q measures

in the same regression. Since both variables proxy for actual q with error, their resulting slopes

might be biased in an unknown direction, making the results difficult to interpret (Klepper and

Learner (1984)). Nevertheless, we note that regressing the four investment measures on both book

and market based q leads to positive and highly statistically significant coefficients on our market

value based q’s and insignificant slopes for book value based q’s. Corresponding results are reported

in Table A3 in the Online Appendix.

To sum up, our empirical evidence suggests that using market rather than book values of debt

improves the performance of q in classical investment regressions.

Bias-corrected results

Even though using market values of debt helps to improve the measurement of the true, unobserved

q, our measure is still only a noisy approximation thereof. As a result, standard OLS coefficients

suffer from measurement-error bias. To address this issue, we now estimate the previous models

while correcting this bias using Erickson, Jiang and Whited (2014) higher-order cumulant estimator.

Compared to the Erickson and Whited (2002) estimator, the cumulant estimator has better finite-

sample properties and a closed-form solution, which makes the numerical implementation easier

and more reliable. We use the third-order cumulant estimator. The cumulant estimator provides

28

unbiased estimates of β in the following classical errors-in-variables model:

iit = ai + qitβ + zitα+ uit (18)

pit = γ + qit + εit (19)

where p is a noisy proxy for the true, unobservable q, and z is a vector of perfectly measured

control variables. The cumulant estimator’s main identifying assumptions are that p has non-zero

skewness, β 6= 0, and that u and ε are independent of q, z, and each other.

Estimation results are in Table 6. First, we discuss the slopes on q. Comparing the q-slopes

of book and market value based measures reveals that the coefficients are weakly larger for market

value based measures. The same conclusion holds when we additionally control for cash flows

(see Panel B of Table 6). In theory, the q-slopes measure the inverse of the capital adjusment-

cost parameter. Hence, a low coefficient on q implies high adjustment costs. The literature since

Summers (1981) have argued that investment-q regressions imply implausibly large adjustment

costs. Hence, the fact that the slopes on market value based q measures are weakly larger compared

to book value measures helps to alleviated this concern.

In addition to delivering unbiased q-slopes, the cumulant estimator produces two useful test

statistics. The first, ρ2, is the hypothetical R2 from (18). Loosely speaking, ρ2 tells us how well

the true, unobservable q explains investment. The second statistic, τ2, is the hypothetical R2 from

(19). It tells us how well our q proxy explains the true q.

Comparing the market value based q regressions with book value based q regressions, we see

that market value measures produce a stronger investment-q relation. That is, ρ2 increases from

0.26 to 0.30 (a 15% increase) and from 0.10 to 0.13 (a 30% increase), respectively. Moreover, using

market values also leads to a better proxy for Tobin’s q, τ2 increases from 0.62 to 0.66 (a 7%

increase) and from 0.71 to 0.82 (a 15% increase), respectively. While the model fit is still far from

perfect, using market values clearly brings the data evidence closer to the relationship implied by

theory. Finally, Panel B additionally corrects for total cash flow as defined in (17). Two main

results arise. First, the results discussed above are robust to controlling for cash flow. That is,

29

both ρ2 and τ2 are always higher for market value compared book value based q measures. Second,

consistent with the literature, we find positive and significant cash flow slopes for qtot and q. This

result contradicts the classical q-theory which implies a zero cash flow slope. Importantly, however,

when using market value based q measures the cash flow coefficients estimates shrink towards zero

and reduce in terms of statistical significance. In fact, when using our benchmark q measure, qtot,

cash flows are no longer significant and the point estimate is very close to zero. These results,

reinforce our conclusions so far: relying on market values leads to an improved proxy for true q

which brings the empirical evidence closer to the implications of q-theory.

Firm-level bond-q

Philippon (2009) introduces an aggregate bond-q measure, which he obtains by applying a structural

model to data on bond maturities and yields. At an aggregate level, bond-q successfully explains

more of the aggregate variation in physical investment than standard q. Importantly, bond-q also

explains changes in aggregate investment reasonably well. In contrast, standard q has very limited

explanatory power for changes in investment. Our data on both firm-level book and market values of

debt allows us to calculate a proxy for firm-level bond-q, namely MBDebtit = MV Debtit/BV Debtit.

First, we document how MBDebtit varies over time in the cross section of firms. To this end,

Figure 7a plots the time series of the 5%, 25%, 50%, 75%, and 95%-percentiles. While the median

experiences fairly limited variation there is ample cross-sectional variation as evidence by the tails

of the distribution. Decomposing the sample into firms with S&P issuer credit ratings below BB

and firms with higher ratings in Figures 7b and 7c makes clear that MBDebtit potentially contains

a lot of information, particularly for low rating firms.

To document the performance of our proxy for firm-level bond-q, we replicate the simple stan-

dard investment regressions in Panel A of Table 7. We find that MBDebt positively and significantly

explains all four measures of investment. That is, firm-level bond-q seems to have explanatory power

for firm-level investment in line with the evidence in Philippon (2009) at the aggregate level. More-

over, the within R2 are of comparable magnitude with those for q in Panel B of Table 5. Motivated

by the graphical evidence in Figures 7b and 7c, we redo the total investment regression from Panel

30

A of Table 7 for investment grade firms and low rating firms (see Panels B and C in Table 7). In

particular, we test whether MBDebt contains information about total investment over and above

the market value based q measures, qMV and qtot,MV . Our results suggest that bond-q only has

explanatory power in excess of qMV and qtot,MV for firms with low credit ratings. That is, for

low rating firms adding MBDebt to q measures leads to highly statistically significant increases in

within R2. This novel firm-level results adds to the evidence at the aggregate level from Philippon

(2009). That said, and in contrast to the aggregate measure of Philippon (2009), MBDebt does not

rely on a structural model and, hence, is free from any modelling assumptions; it is simply a ratio

of two values.

Next, we further explore the informational content of firm-level bond-q for total investment by

regressing changes in total investment on MBDebt. These regressions are motivated by the evidence

at the aggregate level in Philippon (2009) and Peters and Taylor (2017). Both studies show that

aggregate q measures do not explain 4-quarter changes in investment but aggregate bond-q does

to a certain extent. Given our proxy for firm-level bond-q, we can further explore this aggregate

results using the cross section of firms. Table 8 reports regression results for investment grade firms

in Panel A and low rating firms in Panel B. We find that, again, MBDebt is a valuable predictor of

annual changes in investment at the firm-level only for low rating firms. That is, adding MBDebt to

book and market value based measures total q leads to increases in within R2 that are statistically

and economically (between 30% and 35%) significant.

To sum up, our proxy for firm-level bond-q has substantial explanatory power for different

notions of investment. In fact, bond-q contains information for the level and changes of total

investment in low rating firms that is different from the information contained in proxies of q.

5.2 Default Risk and the Value of Debt

This section explores how market values of debt affect the empirical performance of firm default

prediction models. To this end, we compare measures of distance to default that are either based

on book or market values of debt. In addition, we show that MBDebt contains information on

31

firm-level default risk that is not captured by standard measures of distance to default.

5.2.1 Construction of Distance to Default Measures

We study the properties of five distance to default measures. In particular, we follow Bharath and

Shumway (2008) and construct their naive KMV measure which is based on the Merton (1974)

model. Moreover, we replicate the measure developed by Atkeson et al. (2017). In addition to

these two measures which are based on book values of debt, we construct their market value based

counterparts which leaves us with four measures in total. The fifth measure is the ratio of the

market and book value of debt at the firm-level, i.e., MBDebtit .

KMV: Bharath and Shumway (2008) approximate debt volatility with equity return volatility.

Specifically, σDit = 0.05 + 0.25σEit , where σEit is the average daily equity return volatility over the

past month. They then compute overall asset volatility as a weighted average of debt volatility and

equity volatility as follows:

σVit =Eit

Eit +DBVit

σEit +DBVit

Eit +DBVit

σDit ,

where Eit is the firm’s market capitalization, DBVit is the book value of the firm’s debt. Using this

proxy of asset volatility, Bharath and Shumway (2008) calculate distance to default as

KMVit =ln[(Eit +DBV

it )/DBVit ] + (ri,t−1 − 0.5σVit

2)

σVit,

where ri,t−1 is the firm’s past stock return. We make use of our data by substituting the market

value of debt with the book value of debt where appropriate. Moreover, rather than approximating

debt return volatility to calculate asset volatility, we can directly calculate asset return volatility

from our firm-level data. That is, we use capital structure weights (in market values), equity,

and debt returns to deduce asset returns as discussed in section ?? below which gives us a direct

32

measure of asset return volatility, σAit . Hence, KMVMVit is defined as

KMVMVit =

ln[(Eit +DMVit )/DBV

it ] + (ri,t−1 − 0.5σAit2)

σAit

AEW: AEW denotes the distance to default measure from Atkeson et al. (2017). In their study,

these authors study distance to insolvency which is defined as

DIit =

(AMVit −DMV

it

DMVit

)1

σAit.

However, due to data limitations - they do not observe market values of assets and debt as well as

asset volatility directly - they approximate distance to insolvency with the inverse of the average

of daily equity return volatility over the past month,

AEWit =1

σEit.

In contrast, we use our data to directly calculate the distance to insolvency proxy as follows,

AEWMVit =

(AMVit −DMV

it

DMVit

)1

σAit.

Finally, our last variable of interest, MBDebtit , is defined as discussed above.

5.2.2 Empirical Evidence

Table 9 presents summary statistics for the various measures discussed in section 5.2.1. Panel A

shows that the market value based default measures have a lower mean and standard deviation

compared to their book value based counterparts. Interestingly, the pairwise correlations of market

and book value based measures are far from perfect (0.74 and 0.88, respectively). This suggests that

using market values leads to non-trivial changes in the cross section. Moreover, the pairwise corre-

lations of KMVMV and AEWMV with MBDebt are very low (0.23 and 0.33, respectively). That is,

MBDebt contains very different and potentially interesting information compared to KMVMV and

33

AEWMV . In particular, MBDebt seems to be a natural candidate to contain information about

firm-level default risk.

Predicting Firm-level Default

We start our analysis with standard univariate predictive regressions in Table 10. To this end, we

run logit regressions where the dependent variable is a dummy which equals one in case a firm is in

default within one quarter, one year, or two years. Firm-level instances of default are from Chava

and Purnanandam (2010).13 The number of defaults within a year during our sample period are

tabulated in Table A1 in the Online Appendix. The results in Table 10 are in line with the evidence

documented in the literature. That is, both KMV and AEW negatively predict firm-level default

events. This is true for the one quarter, one year, and two years horizon (see Panels A, B, and

C of Table 10). Further, we note that using their market value based counterparts KMVMV and

AEWMV does not impair the negative and statistically highly significant relationship with default.

In fact, the pseudo R2 is significantly higher for models with KMVMV and AEWMV compared to

models with KMV and AEW at all prediction horizons. This resonates well with the findings from

section 5.1 and suggests once more that consistently using market values improves the empirical

performance of variables motivated by theoretical models.

Finally, the fifth column of Table 10 reports results for MBDebt. The estimated coefficients are

negative and highly statistically significant in all specifications. Hence, a market value of debt that

is low relative to the book value is indicative of elevated default probability. The pseudo R2 for

models with MBDebt is slightly lower compared to models with KMVMV and AEWMV . However,

given the low correlation of MBDebt with other predictors, MBDebt might still be an important

variable in default prediction models as it likely contains information that is different from existing

predictor variables. To further test this conjecture, Table 11 reports results from multivariate

default prediction models. That is, we test six different combinations of the various predictor

variables, all of them including MBDebt. As before, we study three different prediction horizons

(see Panels A, B, and C of Table 11). The results confirm that MBDebt contains information

13 We thank Sudheer Chava for sharing the updated default data.

34

about firm-level default probabilities that is orthogonal to the information from KMV, KMVMV ,

AEW, AEWMV (see columns 1 to 4 across Panels A, B, and C); KMV and AEW, KMVMV and

AEWMV (see columns 5 and 6 across Panels A, B, and C). For all six model specifications across

all prediction horizons, adding MBDebt to the set of regressors significantly improves the model fit

as witnessed by the very low p-values from corresponding likelihood ratio tests where the restricted

model corresponds to the model employing the same set of regressors without MBDebt.

Table 12 further estimates univariate prediction models as in column five of Table 10. To further

dissect the negative relationship of MBDebt and firm-level default, we re-estimate the model on

subsamples of small and large firms. Small (large) firms are defined as either have market value of

assets or market value of equity which is below (above) the median. We find that the predictive

power of MBDebt is stronger for small firms compared to large firms. That is, pseudo R2 is

statistically significantly higher for small compared to large firms. This finding is consistent with

section 5.1 where we show that MBDebt predominantly contains information for firms with low

credit ratings as credit ratings are strongly negatively correlated with firm size.

Firm-level Market Values of Debt and large scale Default Prediction Models

So far, we have shown that consistently using market values improves the empirical performance of

well established predictor variables for firm-level default. In addition, we proposed a new variable

that contains information orthogonal to existing predictors. In what follows, we consider large scale

default prediction models and test whether using market values both for equity and debt leads to

an improvement in empirical performance. To this end, we study two reference models from the

finance literature, the models of Bharath and Shumway (2008) and Campbell et al. (2008). Tables

13 and 14 report the results. That is, in both tables we report regression results for the existing

models in columns one, three, and five for prediction horizons one quarter, one year and two years,

respectively. In addition, columns two, four, and six report similar regressions where all book value

based regressors are substituted with their market value counterparts. The conclusion from both

tables is the same: using market rather than book values improves the empirical performance of

the two models at all prediction horizons. Even though the resulting increases in pseudo R2 seem

35

relatively small in magnitude, they are statistically significant. Hence, these results suggest that

a larger set of regressors combined stands a better chance at spanning the information contained

in market values. Hence, unconditionally, large scale models do not seem to suffer much from

abstracting from market values. Nevertheless, even in large scale regression models using market

rather than book values is the dominant empirical strategy.

5.3 Corporate Bond Pricing and the Credit Spread Puzzle

Structural models of credit risk are widely used in financial economics. There is a large literature

studying the impact of different assumptions on asset value dynamics and risk premiums on the

models’ ability to price debt. In contrast, there is little research on the impact of different ways

to calculate the key model inputs. However, a recent paper by Bai et al. (2020) argues that how

leverage is calculated is crucial for assessing the models’ ability to price debt. Feldhutter and

Schaefer (2018b) find, using quasi-leverage as input, that the Black-Cox models matches credit

spreads well on average and thus there is hno so-called credit spread puzzle. In contrast, Bai et al.

(2020) find that once market leverage is used, the Black-Cox model underpredicts investment grade

spreads, reviving the puzzle. Bai et al. (2020) [BGY[ do not include bank debt in their calculation

of market leverage. Our data set allows us to provide a more accurate estimation of market leverage

and advance our understanding of structural models’ ability to price debt. Furthermore, we improve

on existing methods of calculating market leverage and asset volatility.

Pricing of Corporate Bonds

To next study the implications of using better measures of market laverage and asset volatility for

bond prices. We follow the recent literature and calibrate the Black-Cox model to historical default

rates. We use the approach in FS and extract a default boundary – d in equation XX – common

for all firms, that provides the best fit to the cross-section of historical default rates.

Specifically, we find d by the following procedure. For each observed spread in the data sample

on bond i with a time-to-maturity T issued by firm j on date t, we calculate the firm’s T -year

36

default probability πP (dLjt, σA,j , δjt, θ, T ) where Ljt is the time-t estimate of the firm’s leverage,

σA,j is firm j’s asset volatility and θ is a constant Sharpe ratio such that the drift of the firm value

is µjt = θσj + rTt − δjt. As in FS we calculate a constant asset volatility for each firm by calculating

the average of time 1,...,T asset volatility. We use the T -year riskfree rate as rTt and use Chen,

Collin-Dufresne and Goldstein (2009)’s estimate of the Sharpe ratio of 0.22. We follow FS and refer

to their paper for further details (such as the calculation of δjt and equity volatility).

For a given rating a and maturity T - rounded up to the nearest integer year - we find all bond

observations in the sample with the corresponding rating and maturity. For a given calendar year

y we calculate the average default probability πPy,aT (d) and we then calculate the overall average

default probability for rating a and maturity T , πPaT (d), by computing the mean across the N years,

πPaT (d) = 1N

∑Ny=1 π

Py,aT (d). We denote by πPaT the corresponding historical default frequency for the

period 1970-2015 reported by Moody’s (2015). For rating categories AAA, AA, A, BBB, BB, and

B and horizons of 1-20 years we find the value of d that minimizes the sum of absolute differences

between the annualized historical and model-implied default rates by solving

min{d}

B∑a=AAA

20∑T=1

1

T

∣∣∣πPaT (d)− πPaT∣∣∣. (20)

With the estimated default boundary d we calculate model spreads implied by the Black-Cox

model. For a given rating r, we then find all bonds at the end of a given month t that have this

rating, calculate the average actual bond yield spread (to the swap rate) sart, and do this for all

months in the sample where we have observations available. For the Black-Cox model, we likewise

calculate a time series of monthly average model credit spread sMr1 , ..., sMrT . Table 18 shows the

pseudo-R2 calculated as 1− 1/T∑

t(sart−sMrt )2

1/T∑

t(sart)

2 .

Table 18 shows that the BGY leverage adjustment leads to worse pricing (average R2 = 0.544)

than no adjustment (average R2 = 0.604) consistent with our finding that book leverage is a

better predictor of market leverage than the BGY adjusted leverage ratios. We also see that our

suggested polynomium approach to estimating asset volatility improves pricing (average R2 in the

range 0.621-0.629) and there is almost no impact on the R2 of including terms higher than n = 2.

37

When we use our suggested market leverage polynomium estimates, Table 18 shows that the best

pricing performance is when the polynomial is second-order (n = 2) as the average R2 is 0.632.

Once we include higher-order terms average R2’s drop which could be due to overfitting in the

polynomial.

Overall, the results in Table 18 shows that by using only book leverage and our proposed market

leverage and asset volatility estimators pricing is improved. For our suggested specification using

a second-order polynomial for both market leverage and asset volatility, the average R2 is 0.632

compared to 0.601 when using book leverage and 0.544 when using BGY adjusted leverage. Finally,

when we use the correct market leverage, the average R2 increases to 0.659, so using bank debt

information increases pricing accuracy further.

The Credit Spread Puzzle

The credit spread puzzle is the long-standing puzzle first documented in Huang and Huang (2012)

that on average investment-grade credit spreads are higher than those implied by standard struc-

tural models when benchmarked to historical default rates, recovery rates and the equity premium.

Feldhutter and Schaefer (2018b) find that the traditional approach of calibrating the models to a

single default rate is statistically very imprecise and propose to calibrate the models to a cross-

section of default rates, Once they apply the approach they do not find evidence of a puzzle. Bai

et al. (2020) make the important point that FS use book leverage in their calibration and find

that when they apply the FS approach along with their previously discussed debt adjustments the

puzzle re-appears.

Our data set allows us to calculate market leverage directly and therefore we can examine Bai

et al. (2020)’s argument without relying on a biased proxy for the market value of debt as they

do. Table 19 shows average actual and model-implied credit spreads. Bonds with a BBB rating

are most common and therefore these bonds have been mostly in focus when examining the puzzle.

The actual average BBB-spread is 157bps while the average model-implied spread is 169bps using

book leverage as in FS. In contrast, when using the BGY leverage adjustment, the average model-

implied BBB-spread is only 90bps and statistically significantly different from the actual spread.

38

These results confirm the findings in FS and BGY. The remaining 10 rows in the table calculate

credit spreads using either the correct market leverage or polynomials of different order as proposed

in the previous sections. The BBB-spreads range from 139 to 176 and none of the estimates are

significantly different from the actual spread of 157bps. This shows that when using more detailed

data, the conclusion of BGY is reversed and there is no evidence of a puzzle.

5.4 Risk Premia and Market Values

In this section, we explore how equity, debt, and asset returns are related to measures involving

market values of debt. That is, we first document a strong relationship between market leverage

and returns. Second, we document that MBDebt is not only related to debt but also to equity

returns.

Table 20 reports equity, overall debt, loan, bond, and asset returns for sorts on market leverage

(Panel A) and on MBDebt (Panel B). Asset returns are defined as in 14 and return on overall debt

are defined as follows:

rDt+1 =LoansMV

t

LoansMVt + BondsMV

t

rLt+1 +BondsMV

t

LoansMVt + BondsMV

t

rBt+1. (21)

Our portfolios contain stocks in percentiles 0 to 5, 5 to 20, 20 to 40, 40 to 60, 60 to 80, 80 to 95,

and 95 to 100 for the two characteristics. This portfolio construction procedure follows Campbell

et al. (2008) and pays greater attention to the tails of the distribution. In addition to the individual

portfolio returns, we also report the long-short strategy returns that are long the 0005 (0520) and

short the 9500 (8095) portfolios. We denote these strategies LS0595 and LS2080, respectively.

Sorting on market leverage leads to statistically significant returns for LS0595 for all but loan

returns. This also means that the significance of the return spread in debt returns is entirely due

to the spread in bond returns. Moreover, the portfolio returns are largely monotonically increasing

with the level of market leverage with results being strongest for equity and bond returns.

Next, we explore double sorts of market leverage and book-to-market equity. This is related ?

who investigate the relationship between corporate leverage and book-to-market equity by unlev-

39

ering equity returns. Interestingly, in double sorts of market leverage and book-to-market equity,

we find that the value premium is largely absent. In fact, Table 21 shows that the market leverage

dimension (LMH portfolios) still results in statistically significant spreads while the book-to-market

equity dimension (HML portfolios) does not. Abstracting from the long-short portfolios, sorting

on market leverage leads to almost exclusively strongly monotonic portfolio sorts which is clearly

not the case for book-to-market equity. To sum up, market leverage is strongly related to the

cross-section of returns.

Further, we study return sorts on MBDebt. That is, we sort on firm-level MBDebt as we are

interested in firm-level risk premia. This is very different from Bartram et al. (2020) who study the

cross-section of individual corporate bond returns and sort on market-to-book ratios of individual

securities. Moreover, while their study abstracts from corporate loans, we incorporate quantities

and prices of loans in our analysis. Panel B of Table 20 shows that sorting on firm-level MBDebt

leads to significant cross-section return spreads for equity, debt, bond, and asset returns. That is,

firms with low values of MBDebt, arguably firms facing substantial levels of distress risk, experience

higher future returns. Moreover, conceptually, sorting on MBDebt seems closely related to the well

established sorting characteristic book-to-market equity from ?. However, in practice, sorting on

MBDebt is very different from sorting on book-to-market equity. In fact, the average of annual

correlations of MBDebt and book-to-market equity is even slightly negative (-0.14) and far from

being significantly different from zero. Moreover, conditional double sorts on MBDebt and book-

to-market equity reveal that MBDebt is related to equity risk premia even in this case (see Table

A5 of the Online Appendix).

6 Conclusion

The vast literature on corporate debt faces a challenge as theoretical predictions predominantly

concern market values of debt, while empirical work using common data sources is based on book

values. In this paper, we construct and exploit a novel dataset that traces out on average around

eighty percent of the market values of the debt structure of a broad set of companies based on

40

secondary market corporate bond and loan transactions. Our approach not only extends standard

measurement by exploiting market values, but also by providing capital structure data at monthly

frequency.

We document significant discrepancies between market and book values, especially for distressed

firms, and use our data to revisit a number of empirical patterns involving corporate debt that

have received attention in the literature. Using a market-based measure of Tobin’s Q we find little

evidence for investment cash-flow sensitivity in our data, and find that using market debt values

significantly improves default prediction. In asset pricing tests, we find a leverage premium, but

no evidence for a value premium after controlling for market leverage.

41

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7 Tables

Table 1: Data Construction and Sample Size

The column sample refers to loan-month and bond-month observations, respectively. Sampleperiod is from January 1997 until March 2018.

Panel A: Loan DataFilter Sample Size

- 360,079Focus on USD, GBP, EUR loans 350,991Focus on term loans and revolvers 338,443Restrict to loans with non-missing price data 328,021

Panel B: Bond DataFilter Sample Size

- 2,173,565Focus on bonds of listed companies 1,934,824Delete observations with missing gvkey 1,824,148Restrict to bonds with non-missing price data 1,823,386

47

Table 2: Loan and Bond Characteristics

This Table reports in Panel A the various different types of corporate loans and bonds in oursample. Panel B reports the percentages of loans and bonds for a certain seniority.

A. Type Loans Bonds

Term Loans 76.5% -Revolvers 23.5% -Fixed Coupon Bond - 75.5%Variable Coupon Bond - 13.1%Zero Coupon Bond - 11.5%

B. Seniority Loans Bonds

Senior 99.2% -Senior Secured - 4.5%Senior Unsecured - 89.9%Senior Subordinated 0.2% 3.4%Mezzanine 0.5% -Junior - 0.0%Junior Subordinated 0.1% 2.1%

Table 3: Price and Return Summary Statistics

This Table reports summary statistics of loan and bond prices (per 100 USD face value) as well asmonthly loan, bond, and equity excess returns (expressed in percentages).

Variable p1 p50 p99 mean std

loan prices 75.28 98.87 101.45 97.37 5.09Bond prices 57.80 103.40 122.51 102.07 10.86

Loan returns -3.14 0.25 3.10 0.24 1.02Bond returns -8.53 0.40 10.21 0.50 2.62Equity returns -32.12 0.77 39.70 1.07 12.89

48

Table 4: Q: Summary Statistics

The statistics are based on the sample of firms from 1998 to 2018. The investment and q measuresare defined as discussed in section 5.1.1.

Variable Mean Median Std Skewness

A. Investment MeasuresCAPX/PPE (i∗) 0.10 0.08 0.07 2.60Physical Investment (iphy) 0.05 0.04 0.05 3.28Intangible Investment (iint) 0.06 0.05 0.05 1.57Total Investment (itot) 0.12 0.10 0.07 2.04

B. Book Value based Q MeasuresStandard q (q) 3.66 1.60 6.45 4.73Total q (qtot) 1.06 0.81 1.07 4.40

C. Market Value based Q MeasuresStandard q (qMV ) 3.51 1.58 5.59 3.72Total q (qtot,MV ) 1.02 0.80 0.88 2.39

49

Table 5: Q: OLS Results

Results are from annual OLS panel regressions of investment on lagged Tobin’s q and firm and yearfixed effects. Each column uses a different investment measure noted in the top rows. Standardinvestment i∗ equals CAPX scaled by PPE. Physical investment (iphy) equals CAPX scaled bytotal capital (Ktot = Kphy + Kint). Intangible investment (iint) equals R&D + 0.3 × SG&A,scaled by Ktot. Total investment equals iphy + iint. Panels A and B show regressions on total qbased on book and market values, denoted qtot and qtot,MV . Panels C and D show regressions onstandard q based on book and market values, denoted q and qMV . Standard errors clustered byfirm are in parentheses. We report within-firm R2. Standard errors for differences in R2 values arecomputed via bootstrap and take into account the correlation across regressions and clustering byfirm. Data is from 1998 - 2018.

Investment scaled by capital

i∗ iphy iint itot

A. Regressions with total q

qtot 0.020∗∗∗ 0.012∗∗∗ 0.007∗∗∗ 0.019∗∗∗

(0.001) (0.000) (0.000) (0.000)Within R2 0.173 0.144 0.191 0.239

qtot,MV 0.024∗∗∗ 0.014∗∗∗ 0.009∗∗∗ 0.023∗∗∗

(0.001) (0.000) (0.000) (0.001)Within R2 0.191 0.163 0.205 0.272

N 11,821 11,821 10,925 10,925Difference in R2 0.018∗∗∗ 0.019∗∗∗ 0.014∗∗∗ 0.032∗∗∗

(0.002) (0.001) (0.003) (0.002)

B. Regressions with standard q

q 0.004∗∗∗ 0.001∗∗∗ 0.001∗∗∗ 0.002∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.142 0.083 0.132 0.138

qMV 0.005∗∗∗ 0.001∗∗∗ 0.001∗∗∗ 0.002∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.172 0.087 0.140 0.152

N 11,821 11,821 10,925 10,925Difference in R2 0.030∗∗∗ 0.004∗∗ 0.008∗∗∗ 0.014∗∗∗

(0.002) (0.002) (0.002) (0.001)

50

Table 6: Bias-Corrected Results

Results are from annual OLS panel regressions of total and standard investment (itot and i∗)on lagged Tobin’s q, firm fixed effects, and (in Panel B) contemporaneous cash flow. ρ2 is thewithin-firm R2 from a hypothetical regression of investment on true q, and τ 2 is the within-firmR2 from a hypothetical regression of our q proxy on true q. Standard errors clustered by firm arein parentheses. Data is from 1998 - 2018.

Investment scaled by capital

itot itot i∗ i∗

A. Standard and total q

qtot 0.033∗∗∗

(0.002)qtot,MV 0.038∗∗∗

(0.003)q 0.006∗∗∗

(0.001)qMV 0.006∗∗∗

(0.001)ρ2 0.265 0.300 0.103 0.128τ 2 0.625 0.661 0.712 0.819N 11,821 11,821 10,925 10,925

B. Correcting for total cash flows

qtot 0.036∗∗∗

(0.002)qtot,MV 0.045∗∗∗

(0.004)ctot 0.028∗∗ -0.003 0.045∗∗∗ 0.027∗∗

(0.014) (0.015) (0.014) (0.015)q 0.007∗∗∗

(0.001)qMV 0.006∗∗∗

(0.001)ρ2 0.301 0.356 0.130 0.144τ 2 0.554 0.558 0.576 0.825N 11,821 11,821 10,925 10,925

51

Table 7: Q: OLS Results with MBDebt

Results are from annual OLS panel regressions of investment on lagged Tobin’s q and firm and year fixed effects.In Panel A, each column uses a different investment measure noted in the top rows. Standard investment i∗ equalsCAPX scaled by PPE. Physical investment (iphy) equals CAPX scaled by total capital (Ktot = Kphy + Kint).Intangible investment (iint) equals R&D + 0.3 × SG&A, scaled by Ktot. Total investment equals iphy + iint.Panel A shows regressions on market-to-book ratio for debt, denoted MBDebt. Panel B (C) shows regressions fortotal investment on MBDebt, Qtot, and qtot,MV for investment grade (low rating) firms only. The ratings are thethe S&P issuer credit ratings. Standard errors clustered by firm are in parentheses. We report within-firm R2.Standard errors for differences in R2 values are computed via bootstrap and take into account the correlation acrossregressions and clustering by firm. Data is from 1998 - 2018.

Investment scaled by capital

i∗ iphy iint itot

A. Full Sample - MBDebt and total q

MBDebt 0.113∗∗∗ 0.075∗∗∗ 0.007∗∗ 0.080∗∗∗

(0.009) (0.006) (0.003) (0.007)Within R2 0.099 0.097 0.094 0.133N 12,777 12,777 11,838 11,838

Total investment scaled by capital

itot itot itot itot

B. Investment Grade Firms Only (> BBB)

MBDebt -0.015 -0.026(0.011) (0.018)

qtot 0.015∗∗∗ 0.015∗∗∗

(0.001) (0.001)qtot,MV 0.018∗∗∗ 0.018∗∗∗

(0.001) (0.001)Within R2 0.378 0.378 0.383 0.385Difference in R2 0.000 0.002

(0.001) (0.002)N 3,195 3,195 2,925 2,925

C. Low Rating Firms Only (< BB)

MBDebt 0.126∗∗∗ 0.095∗∗∗

(0.018) (0.017)qtot 0.022∗∗∗ 0.021∗∗∗

(0.001) (0.001)qtot,MV 0.035∗∗∗ 0.033∗∗∗

(0.002) (0.002)Within R2 0.259 0.286 0.337 0.354Difference in R2 0.027∗∗∗ 0.017∗∗

(0.006) (0.007)N 1,927 1,927 1,680 1,680

52

Table 8: Changes in Investment: OLS Results with MBDebt

Results are from annual OLS panel regressions of annual changes in total investment on laggedTobin’s q and firm and year fixed effects. Panel A (B) shows regressions on MBDebt, Qtot, andqtot,MV for investment grade (low rating) firms only. The ratings are the the S&P issuer creditratings. Standard errors clustered by firm are in parentheses. We report within-firm R2. Standarderrors for differences in R2 values are computed via bootstrap and take into account the correlationacross regressions and clustering by firm. Data is from 1998 - 2018.

Changes in total investment scaled by capital

∆itot ∆itot ∆itot ∆itot ∆itot

A. Investment Grade Firms Only (> BBB)

MBDebt 0.018 0.019 0.018(0.012) (0.012) (0.012)

qtot -0.002∗∗ -0.002∗∗

(0.001) (0.001)qtot,MV -0.001∗ -0.001∗∗

(0.001) (0.001)Within R2 0.057 0.060 0.060 0.059 0.058Difference in R2 0.000 -0.001

(0.002) (0.001)N 2,886 2,886 2,886 2,886 2,886

B. Low Rating Firms Only (< BB)

MBDebt 0.120∗∗∗ 0.114∗∗∗ 0.124∗∗∗

(0.021) (0.022) (0.022)qtot -0.001 -0.003

(0.002) (0.002)qtot,MV 0.001 -0.002

(0.002) (0.002)Within R2 0.093 0.069 0.090 0.069 0.093Difference in R2 0.021∗∗∗ 0.024∗∗∗

(0.002) (0.002)N 1,635 1,635 1,635 1,635 1,635

53

Table 9: Summary Statistics

Panel A reports the mean, median, standard deviation, and the number of observations for thebankruptcy predictors used in the empirical analysis. The definition and calculation of the variablesis discussed in section B of the online appendix. Panel B reports pairwise correlation coefficients.

A. Summary Statistics and CorrelationsMean Median Std N

KMV 7.09 6.59 4.79 227, 037KMVMV 6.61 5.94 4.51 227, 037AEW 3.96 3.61 1.99 227, 037AEWMV 2.56 2.36 1.64 227, 037MBDebt 0.99 1.01 0.12 227, 037

B. Pairwise CorrelationsKMV KMVMV AEW AEWMV

KMV 1KMVMV 0.743 1AEW 0.778 0.610 1AEWMV 0.896 0.740 0.885 1MBDebt 0.246 0.234 0.242 0.325

54

Table 10: Predicting Default: Univariate Regressions

This Table reports results from univariate predictive regressions. In particular, firm bankruptciesare measured as in Chava and Purnanadam (2010) and predictor variables include book and marketvalue versions of KMV and AEW as defined in section 5.2 as well as MBDebt. A reports resultsfor logit regressions where the dependent variable is a dummy that equals one if a firm defaultswithin one quarter. Panels B and C report corresponding results for default in one year and twoyears, respectively. BIC denotes the Bayesian information criterion. The standard error of thedifference in pseudo R2 of book and market value based predictors (in parentheses) is bootstrapped.Significance levels are denoted by * = 10%, ** = 5%, and *** = 1%.

(1) (2) (3) (4) (5)

A. Default within 1 Quarter

KMV −0.417∗∗∗

(0.063)KMVMV −0.570∗∗∗

(0.096)AEW −1.111∗∗∗

(0.200)AEWMV −1.560∗∗∗

(0.273)MBDebt −7.914∗∗∗

(0.770)

Observations 225,295 225,295 225,295 225,295 225,295BIC 1596.5 1573.3 1620.0 1578.7 1650.7Pseudo R2 0.107 0.120 0.093 0.117 0.076Difference in R2 0.013∗∗∗ 0.024∗∗∗

(0.001) (0.000)

55

Table 10: Predicting Default: Univariate Regressions (continued)

(1) (2) (3) (4) (5)

B. Default within 1 Year

KMV −0.312∗∗∗

(0.044)KMVMV −0.507∗∗∗

(0.063)AEW −0.775∗∗∗

(0.126)AEWMV −1.464∗∗∗

(0.235)MBDebt −6.324∗∗∗

(0.605)

Observations 218,305 218,305 218,305 218,305 218,305BIC 1748.8 1683.9 1767.6 1677.5 1797.0Pseudo R2 0.073 0.108 0.063 0.111 0.047Difference in R2 0.035∗∗∗ 0.048∗∗∗

(0.001) (0.002)

C. Default within 2 Years

KMV −0.230∗∗∗

(0.034)KMVMV −0.383∗∗∗

(0.051)AEW −0.664∗∗∗

(0.111)AEWMV −1.172∗∗∗

(0.178)MBDebt −5.050∗∗∗

(0.682)

Observations 203,384 203,384 203,384 203,384 203,384BIC 1697.2 1640.1 1687.4 1624.0 1728.4Pseudo R2 0.046 0.079 0.052 0.088 0.028Difference in R2 0.033∗∗∗ 0.036∗∗∗

(0.003) (0.004)

56

Table 11: Predicting Default: Multivariate Regressions

This Table reports results from multivariate predictive regressions. In particular, firm bankruptciesare measured as in Chava and Purnanadam (2010) and predictor variables include book and marketvalue versions of KMV and AEW as defined in section 5.2 as well as MBDebt. A reports results forlogit regressions where the dependent variable is a dummy that equals one if a firm defaults withinone quarter. Panels B and C report corresponding results for default in one year and two years,respectively. BIC denotes the Bayesian information criterion. The log-likelihood test statistic(LR Test, p-values in parentheses) tests whether adding MBDebt to existing variables leads to animprovement of the model fit. Significance levels are denoted by * = 10%, ** = 5%, and *** =1%.

(1) (2) (3) (4) (5) (6)

A. Default within 1 Quarter

KMV −0.308∗∗∗ −0.234∗∗∗

(0.038) (0.051)KMVMV −0.414∗∗∗ −0.282∗∗∗

(0.053) (0.072)AEW −0.796∗∗∗ −0.298∗∗

(0.112) (0.143)AEWMV −1.146∗∗∗ −0.470∗∗

(0.152) (0.204)MBDebt −5.553∗∗∗ −5.181∗∗∗ −5.639∗∗∗ −4.691∗∗∗ −5.152∗∗∗ −4.628∗∗∗

(0.769) (0.789) (0.751) (0.792) (0.789) (0.822)

Observations 225,295 225,295 225,295 225,295 225,295 225,295BIC 1556.5 1542.2 1574.7 1556.0 1563.9 1548.5Pseudo R2 0.136 0.144 0.126 0.137 0.139 0.148LR Test (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗

57

Table 11: Predicting Default: Multivariate Regressions (continued)

(1) (2) (3) (4) (5) (6)

B. Default within 1 Year

KMV −0.241∗∗∗ −0.183∗∗∗

(0.038) (0.061)KMVMV −0.414∗∗∗ −0.231∗∗∗

(0.055) (0.064)AEW −0.588∗∗∗ −0.215

(0.104) (0.151)AEWMV −1.236∗∗∗ −0.681∗∗

(0.222) (0.268)MBDebt −4.361∗∗∗ −3.423∗∗∗ −4.455∗∗∗ −2.699∗∗∗ −4.078∗∗∗ −2.586∗∗∗

(0.638) (0.710) (0.620) (0.740) (0.641) (0.775)

Observations 218,305 218,305 218,305 218,305 218,305 218,305BIC 1726.6 1676.1 1741.5 1677.7 1735.2 1674.4Pseudo R2 0.092 0.119 0.084 0.118 0.094 0.126LR Test (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗

C. Default within 2 Years

KMV −0.184∗∗∗ −0.080(0.029) (0.053)

KMVMV −0.329∗∗∗ −0.149∗∗

(0.046) (0.060)AEW −0.545∗∗∗ −0.378∗∗

(0.099) (0.159)AEWMV −1.055∗∗∗ −0.694∗∗∗

(0.173) (0.232)MBDebt −3.494∗∗∗ −2.601∗∗∗ −3.231∗∗∗ −1.697∗∗ −3.054∗∗∗ −1.643∗∗

(0.720) (0.765) (0.729) (0.794) (0.725) (0.816)

Observations 203,384 203,384 203,384 203,384 203,384 203,384BIC 1688.8 1641.4 1681.1 1631.8 1689.4 1635.9Pseudo R2 0.058 0.085 0.062 0.091 0.065 0.095LR Test (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗ (0.00)∗∗∗

58

Table 12: Size and MBDebt

This Table reports results from univariate predictive regressions of default within one quarter(1Q), one year (1Y), and two years (2Y) on MBDebt. Moreover, the sample is split along thesize dimension. Panel A reports results for small (below median) and large (above median) firmswhen size is approximated with the market value of assets. Panel B reports corresponding resultswhen size is approximated with the market value of equity. The standard error of the differencein pseudo R2 (in parentheses) is bootstrapped. Significance levels are denoted by * = 10%, ** =5%, and *** = 1%.

small large

1Q 1Y 2Y 1Q 1Y 2Y

A. Sample split based on Market Value of Assets

MBDebt −6.917∗∗∗ −5.231∗∗∗ −4.017∗∗∗ −7.004∗∗∗ −5.725∗∗ −3.815(0.715) (0.680) (0.716) (2.301) (2.392) (2.453)

Observations 111,997 106,958 98,237 112,635 110,797 105,112Pseudo R2 0.063 0.036 0.020 0.042 0.026 0.011Difference in R2 0.021∗∗∗ 0.010∗∗ 0.009∗∗∗

(0.003) (0.003) (0.002)

B. Sample split based on Market Value of Equity

MBDebt −6.719∗∗∗ −5.247∗∗∗ −4.009∗∗∗ −8.484∗∗∗ 0.388 −2.192(0.706) (0.669) (0.714) (2.733) (4.567) (2.938)

Observations 112,407 107,427 98,617 112,974 110,954 104,829Pseudo R2 0.059 0.036 0.020 0.064 0.000 0.003Difference in R2 -0.005 0.036∗∗∗ 0.017∗∗∗

(0.005) (0.003) (0.002)

59

Table 13: Replacing Book with Market Values - Shumway (2001)

This Table reports results for the Shumway (2001) model for default predictions over one quarter,one year, and two years in columns (1), (3), and (5). The results in columns (2), (4), and (6),correspond to similar model regressions, however, book values are replaced with market values.That is, book assets are replaced with market assets and equity volatility is replaced with assetvolatility. BIC denotes the Bayesian information criterion. The standard error of the differencein pseudo R2 (in parentheses) is bootstrapped. Significance levels are denoted by * = 10%, ** =5%, and *** = 1%.

Default within 1 Quarter Default within 1 Year Default within 2 Years

(1) (2) (3) (4) (5) (6)

NITA −6.472∗∗∗ −4.991∗∗ −5.494∗∗

(2.402) (2.468) (2.409)TLTA 3.539∗∗∗ 3.454∗∗∗ 2.357∗∗∗

(0.943) (0.774) (0.707)IncomeAssets

MV −1.184∗∗∗ −0.583∗ −0.738∗

(0.279) (0.298) (0.401)DebtAssets

MV4.092∗∗∗ 4.169∗∗∗ 3.136∗∗∗

(0.743) (0.609) (0.563)EXRET −1.482∗∗ −1.290∗ −0.765 −0.310 0.915 1.276∗

(0.597) (0.674) (0.588) (0.607) (0.737) (0.739)SIGMA 1.660∗∗∗ 0.977∗∗∗ 0.638∗

(0.326) (0.354) (0.336)SIGMAMV 1.534∗∗∗ 1.833∗∗∗ 1.563∗∗∗

(0.340) (0.384) (0.394)

RSIZE −0.610∗∗∗ −0.597∗∗∗ −0.578∗∗∗ −0.462∗∗∗ −0.604∗∗∗ −0.492∗∗∗

(0.066) (0.069) (0.062) (0.066) (0.060) (0.067)

Observations 225,295 225,295 218,305 218,305 203,384 203,384BIC 1385.6 1399.1 1636.4 1621.1 1615.8 1600.5Pseudo R2 0.240 0.247 0.160 0.168 0.121 0.129Difference in R2(%) 0.007∗∗ 0.008∗∗ 0.008∗∗

(0.002) (0.002) (0.003)

60

Table 14: Replacing Book with Market Values - Campbell et al (2008)

This Table reports results for the Campbell et al. (2008) model for default predictions over onequarter, one year, and two years in columns (1), (3), and (5). The results in columns (2), (4),and (6), correspond to similar model regressions, however, book values are replaced with marketvalues. That is, book assets are replaced with market assets, market-to-book equity is replacedwith market-to-book debt, and equity volatility is replaced with asset volatility. BIC denotes theBayesian information criterion. The standard error of the difference in pseudo R2 (in parentheses)is bootstrapped. Significance levels are denoted by * = 10%, ** = 5%, and *** = 1%.

Default within 1 Quarter Default within 1 Year Default within 2 Years

(1) (2) (3) (4) (5) (6)

NIMTAAVG −4.883∗∗∗ −5.039∗∗ −5.687∗∗

(1.786) (2.026) (2.365)TLMTA 1.422∗ 2.665∗∗∗ 2.031∗∗∗

(0.815) (0.883) (0.777)IncomeAssets

MV −0.928∗∗∗ −0.099 −0.479(0.283) (0.397) (0.465)

DebtAssets

MV2.902∗∗∗ 3.283∗∗∗ 2.954∗∗∗

(0.734) (0.725) (0.680)EXRETAVG −4.450∗∗∗ −3.923∗∗∗ −3.264∗∗ −1.812 1.441 2.736∗

(1.124) (1.127) (1.353) (1.287) (1.559) (1.562)SIGMA 1.187∗∗∗ 0.162 0.234

(0.343) (0.382) (0.398)SIGMAMV 1.210∗∗ 1.383∗∗∗ 1.474∗∗∗

(0.504) (0.518) (0.454)RSIZE −0.505∗∗∗ −0.481∗∗∗ −0.405∗∗∗ −0.361∗∗∗ −0.501∗∗∗ −0.452∗∗∗

(0.091) (0.090) (0.088) (0.085) (0.086) (0.084)CASHMTA −0.521 0.302 1.016

(1.520) (1.210) (1.291)MB 0.016 −0.004 0.005

(0.018) (0.019) (0.019)CashAssets

MV −0.134 −0.117 −0.430(0.378) (0.358) (0.390)

MBDebt −1.211 −1.521∗∗ −1.692∗∗

(0.900) (0.669) (0.759)PRICE −0.209 −0.246 −0.314∗ −0.207 −0.162 −0.065

(0.175) (0.171) (0.179) (0.182) (0.182) (0.180)

Observations 225,295 225,295 218,305 218,305 203,384 203,384BIC 1427.6 1421.7 1658.4 1647.0 1645.4 1633.5Pseudo R2(%) 0.252 0.255 0.168 0.174 0.125 0.131Difference in R2(%) 0.003∗ 0.006∗∗ 0.006∗∗

(0.001) (0.002) (0.002)

61

Rating AAA AA A BBB BB B C

Bai, Goldstein, Yang(2020) [BGY] 1.10 1.07 1.03 1.02 1.01 1.00 0.72BGY estimates in our sample 1.10 1.09 1.10 1.10 1.02 0.97 0.82

All bond and bank debt information 1.01 1.02 1.04 1.05 1.01 0.99 0.95Bond-month obs. 2136 7062 22343 22655 8521 4427 2665Firm-month obs. 656 1574 6686 8355 3941 2252 1070

Table 15: Average ratio of market debt to book debt. For each credit rating, this table showsestimates of the average ratio of market debt to book debt. The first set of estimates are fromBai, Goldstein, and Yang(2020) [BGY]. ’BGY estimates in our sample’ applies the calculation

methodology of BGY to our sample. In particular, the estimate is 1Nit

∑i,t

Pijt

Fiwhere Pijt is the

price in month t of bond i issued by firm j, Nit is the number of bond-month observations and Fi isthe par value of bond i. ’All bond and bank debt information’ estimates the market-to-book ratioas 1

Njt

∑j,tMBDebt

jt where MBDebtjt is the market value of all bank debt and bonds outstanding

divided by the book value of all bank debt and bonds outstanding of the issuer firm of bond jin month t. ’Bond-month obs.’ is the number of bond-month observations on which the BGYestimates in our sample are based on. ’Firm-month obs.’ is the number of observations the lasttwo sets of estimates are based on. The sample period is 1997:04-2018:03.

62

A. Regression, dependent variable is DBV

EMV +DMV

(1) (2) (3) (4)(LBV )1 1.00∗∗

(0.001)0.92∗∗(0.002)

1.12∗∗(0.003)

0.90∗∗(0.007)

(LBV )2 0.16∗∗(0.003)

−0.71∗∗(0.013)

0.88∗∗(0.043)

(LBV )3 0.77∗∗(0.011)

−2.45∗∗(0.084)

(LBV )4 1.95∗∗(0.050)

R2 0.9740 0.9752 0.9768 0.9773Obs 69809 69809 69809 69809

B. Prediction error variance of DBV

EMV +DMV

Bai, Goldstein, Yang(2020) [BGY] 0.00376No debt adjustment 0.00362

Regression predictor∑N

i=1 βi(LBV )i

N = 1 0.00361N = 2 0.00344N = 3 0.00323N = 4 0.00316

Table 16: Predictions of DBV

EMV +DMV . Panel A shows results from the regressionsDBV

j

EMVj +DMV

j=∑N

i=1 βi(LBVj )i + εj, j = 1, ..., T for N=1, 2, 3, and 4 and where LBVj =

DBVj

EMVj +DBV

jand T

is the number of firm leverage observations. Panel B shows the prediction error variance1T

∑Tj=1

(DBV

j

EMVj +DMV

j− PM

j

)2given in Panel A and PM

j is the predictedDBV

j

EMVj +DMV

jfrom model

M . The ’BGY’ adjustment calculates market value of debt by using the rating-specific multipliersgiven in Table 15. ’No debt adjustment’ assumes that market value of debt is equal to book valueof debt.

63

A. Regression, dependent variable is σAσE− 1

(1) (2) (3) (4)(LBV )1 −0.95∗∗

(0.002)−1.04∗∗(0.007)

−1.00∗∗(0.017)

−1.01∗∗(0.035)

(LBV )2 0.15∗∗(0.010)

0.00(0.059)

0.06(0.203)

(LBV )3 0.12∗(0.047)

0.02(0.362)

(LBV )4 0.06(0.203)

R2 0.9897 0.9907 0.9907 0.9907Obs 2204 2204 2204 2204

B. Prediction error variance of σAσE− 1

Feldhutter and Schaefer(2018) [FS] 0.00727

Regression predictor∑N

i=1 βi(LBV )i − 1

N = 1 0.00231N = 2 0.00211N = 3 0.00210N = 4 0.00210

Table 17: Predictions of asset volatility. For each bond-year in the sample where we have 12monthly observations of asset and equity returns, we calculate the standard deviation of as-set and equity returns, σA,j and σE,j. Panel A shows results from the regressions

σA,j

σE,j− 1 =∑N

i=1 βi(LBVj )i + εj for N=1, 2, 3, and 4 and where LBVj =

DBVj

EMVj +DBV

j. Panel B shows the predic-

tion error variance 1T

∑Tj=1

(σAj

σE,j− PM

j

)2where T is the number of observations given in Panel A

and PMj is the predicted

σA,j

σE,jfrom model M . The ’FS’ adjustment calculates PM

j using the function

(1−LBVj )(1.00∗1{LBVj ≤0.25}+1.05∗1{0.25<LBV

j ≤0.35}+1.10∗1{0.35<LBVj ≤0.45}+1.20∗1{0.45<LBV

j ≤0.55}+1.40 ∗ 1{0.55<LBV

j ≤0.75} + 1.80 ∗ 1{0.75<LBVj }.

64

Average Inv BB+ BB BB- B+ B B- CCC+ CCC CCC- CC C

No leverage adjustment, FS asset volatility [d = 0.901] 0.604 0.842 0.873 0.773 0.868 0.771 0.677 0.416 0.660 0.582 0.231 0.343 0.209

BGY leverage adjustment, FS asset volatility [d = 0.687] 0.544 0.768 0.631 0.679 0.732 0.701 0.599 0.409 0.655 0.526 0.227 0.276 0.327

No leverage adj., asset vol. adj., Poly. n=1 [d = 0.963] 0.621 0.842 0.886 0.813 0.872 0.778 0.701 0.435 0.705 0.517 0.251 0.410 0.240

No leverage adj., asset vol. adj., Poly. n=2 [d = 0.957] 0.627 0.852 0.871 0.797 0.873 0.790 0.705 0.428 0.704 0.518 0.285 0.429 0.271

No leverage adj., asset vol. adj., Poly. n=3 [d = 0.956] 0.629 0.852 0.872 0.798 0.873 0.791 0.707 0.429 0.705 0.517 0.289 0.434 0.277

No leverage adj., asset vol. adj., Poly. n=4 [d = 0.957] 0.629 0.852 0.872 0.798 0.873 0.791 0.707 0.429 0.705 0.516 0.289 0.434 0.277

Leverage adj, Poly. n=1, asset vol. adj., Poly. n=2 [d = 0.953] 0.627 0.852 0.871 0.797 0.873 0.790 0.705 0.428 0.704 0.518 0.285 0.428 0.271

Leverage adj, Poly. n=2, asset vol. adj., Poly. n=2 [d = 0.907] 0.632 0.861 0.828 0.772 0.862 0.798 0.710 0.426 0.715 0.521 0.353 0.462 0.276

Leverage adj, Poly. n=3, asset vol. adj., Poly. n=2 [d = 0.913] 0.613 0.854 0.824 0.748 0.851 0.788 0.721 0.415 0.718 0.382 0.349 0.431 0.278

Leverage adj, Poly. n=4, asset vol. adj., Poly. n=2 [d = 0.924] 0.619 0.853 0.833 0.760 0.857 0.797 0.729 0.425 0.719 0.375 0.361 0.440 0.277

Correct leverage, asset vol. adj., Poly. n=2 [d = 0.905] 0.659 0.919 0.817 0.824 0.890 0.743 0.793 0.456 0.683 0.658 0.315 0.508 0.298Number of monthly observations 158 240 208 185 198 177 172 204 179 140 100 84 5

Table 18: Comparing actual and model-implied average monthly time series of credit spreads: R2’s. This table shows how theBlack-Cox structural model matches average monthly spreads when leverage and asset volatility are calculated in different ways.For a given rating r, we find all bonds at the end of a given month t that have this rating, calculate the average actual bondyield spread (to the swap rate) sart, and do this for all months in the sample where we have observations available. For eachmodel, we likewise calculate a time series of monthly average model credit spread sMr1 , ..., s

MrT . This table shows the pseudo-R2

calculated as 1 − 1/T∑

t(sart−sMrt )2

1/T∑

t(sart)

2 . ’No leverage adjustment’ assumes that the market value of debt is equal to the book value

of debt. ’BGY debt adjustment’ uses rating-specific market-to-book adjustment factors using only bond prices estimated byBai, Goldstein, and Yang(2020) and given in Table 15. ’Correct rating-adjusted leverage’ uses rating-specific market-to-bookadjustment factors for debt using bond and bank debt prices calculated using our data set and given in Table 15. ’Correctleverage’ uses firm-specific debt values calculated in our data sample using bond and bank debt prices. ’Leverage adj. Poly.’uses an n-degree polynomium in book leverage - given in Table 16 - to approximate market leverage. ’FS asset volatility’calculates asset volatility by multiplying equity volatility with (1 − LBV )(1.00 ∗ 1{LBV ≤0.25} + 1.05 ∗ 1{0.25<LBV ≤0.35} + 1.10 ∗1{0.35<LBV ≤0.45} + 1.20 ∗ 1{0.45<LBV ≤0.55} + 1.40 ∗ 1{0.55<LBV ≤0.75} + 1.80 ∗ 1{0.75<LBV }. ’Asset vol. adj. Poly.’ uses an n-degreepolynomium in book leverage - given in Table 17 - to approximate σA

σE. In the Black-Cox model, the firm defaults the first

time firm value hits the default boundary d × F where F is the total face value of debt. d is estimated such that the averagemodel-implied default probabilities have the closest fit to historical default rates and the table shows the estimate d. The sampleincludes bonds with a bond maturity between one and 20 years and the sample period is 1998:04-2018:03.

65

d AAA AA A BBB BB B CActual spread 25 30 67 157 336 468 794

No leverage adjustment, FS asset volatility 0.901 17∗(9;25)

17∗∗(8;25)

82(55;104)

169(109;221)

311(202;398)

551(340;728)

687(453;885)

BGY leverage adjustment, FS asset volatility 0.687 5∗∗(2;7)

5∗∗(3;7)

45∗∗(32;56)

90∗∗(62;112)

168∗∗(112;212)

289∗∗(187;373)

742(514;994)

Correct rating-adjusted leverage, FS asset volatility 0.875 15∗∗(7;22)

14∗∗(7;21)

72(49;91)

143(94;185)

285(185;365)

523(324;688)

716(475;941)

No leverage adj., asset vol. adj., Poly. n=1 0.963 17(8;25)

18∗(9;28)

85(56;109)

176(109;236)

305(188;405)

533(300;757)

671(402;990)

No leverage adj., asset vol. adj., Poly. n=2 0.957 17∗(8;24)

17∗(8;27)

80(53;103)

168(105;226)

293(179;389)

521(292;742)

676(408;994)

No leverage adj., asset vol. adj., Poly. n=3 0.956 17∗(8;24)

17∗(8;27)

81(53;104)

168(105;226)

292(179;389)

520(291;740)

676(408;1003)

No leverage adj., asset vol. adj., Poly. n=4 0.957 17∗(8;24)

17∗(8;27)

81(53;104)

169(105;226)

292(179;389)

520(291;740)

676(408;1004)

Leverage adj, Poly. n=1, asset vol. adj., Poly. n=2 0.953 17∗(8;24)

17∗(8;27)

80(53;103)

168(105;226)

293(179;389)

521(292;742)

676(408;994)

Leverage adj, Poly. n=2, asset vol. adj., Poly. n=2 0.907 14∗∗(7;21)

14∗∗(6;22)

68(45;87)

148(93;199)

259(159;345)

479(267;680)

645(386;944)

Leverage adj, Poly. n=3, asset vol. adj., Poly. n=2 0.913 15∗∗(7;21)

15∗∗(7;23)

70(47;89)

154(95;208)

255(156;347)

497(266;693)

760(397;967)

Leverage adj, Poly. n=4, asset vol. adj., Poly. n=2 0.924 14∗∗(7;21)

14∗∗(7;22)

71(48;91)

161(99;216)

261(159;352)

503(270;718)

756(403;964)

Correct leverage, asset vol. adj., Poly. n=2 0.905 20(12;27)

15∗∗(7;23)

67(46;84)

139(90;182)

287(174;370)

505(303;692)

756(463;970)

Table 19: Average actual and model-implied spreads. This table shows the average actual and model-implied spreads for bondswith a maturity between 1–20 years. ’Actual spread’ is the actual spread to the swap rate. ’No leverage adjustment’ assumesthat the market value of debt is equal to the book value of debt. ’BGY debt adjustment’ uses rating-specific market-to-bookadjustment factors using only bond prices estimated by Bai, Goldstein, and Yang(2020) and given in Table 15. ’Correct rating-adjusted leverage’ uses rating-specific market-to-book adjustment factors for debt using bond and bank debt prices calculatedusing our data set and given in Table 15. ’Correct leverage’ uses firm-specific debt values calculated in our data sample using bondand bank debt prices. ’Leverage adj. Poly.’ uses an n-degree polynomium in book leverage - given in Table 16 - to approximatemarket leverage. ’FS asset volatility’ calculates asset volatility by multiplying equity volatility with (1−LBV )(1.00∗1{LBV ≤0.25}+1.05 ∗ 1{0.25<LBV ≤0.35}+ 1.10 ∗ 1{0.35<LBV ≤0.45}+ 1.20 ∗ 1{0.45<LBV ≤0.55}+ 1.40 ∗ 1{0.55<LBV ≤0.75}+ 1.80 ∗ 1{0.75<LBV }. ’Asset vol. adj.Poly.’ uses an n-degree polynomium in book leverage - given in Table 17 - to approximate σA

σE. In the Black-Cox model, the

firm defaults the first time firm value hits the default boundary d × F where F is the total face value of debt. d is estimatedsuch that the average model-implied default probabilities have the closest fit to historical default rates and the table shows theestimate d. Confidence bands are simulation-based following Feldhutter and Schaefer (2018) and are at the 95% level.* impliessignificance at the 5% level and ** at the 1% level. The sample includes bonds with a bond maturity between one and 20 yearsand the sample period is 1998:04-2018:03.

66

Table 20: Equity, Debt, and Asset Excess Returns

This table reports monthly excess returns for equity, overall debt, loans, bonds, and assets. Themean returns and standard deviations are reported for annually rebalanced portfolios sorted onMBDebt (Panel A) and market leverage (Panel B). In June of each year, firms are sorted intoportfolios on percentile cutoffs, for example, 0 to 5th percentile (0005) and 95th to 100th percentile(9500). Moreover, LS0595 (LS2080) correspond to long-short strategies that go long the 0005(0520) and short the 9500 (8095) portfolios. The table reports results for equal-weighted portfoliosof equity (rxE), debt (rxD), loan (rxL), bond (rxB), and asset (rxA) excess returns. Standarderrors reported in parentheses are adjusted for heteroscedasticity and autocorrelation (Newey-Westwith 12 lags). Significance levels are denoted by * = 10%, ** = 5%, and *** = 1%. Sample periodfrom April 1998 to March 2018.

A: Market Leverage Sorts

Portfolios 0005 0520 2040 4060 6080 8095 9500 LS2080 LS0595

rxEt 0.606 0.615∗ 0.694∗∗ 0.834∗∗ 0.946∗∗ 1.366∗∗ 2.486∗∗∗ -0.751∗∗ -1.881∗∗∗

(0.369) (0.326) (0.346) (0.356) (0.448) (0.550) (0.790) (0.324) (0.633)

rxDt 0.371∗∗∗ 0.357∗∗∗ 0.381∗∗∗ 0.378∗∗∗ 0.414∗∗∗ 0.467∗∗∗ 0.641∗∗∗ -0.109 -0.265∗∗

(0.086) (0.073) (0.079) (0.086) (0.105) (0.119) (0.164) (0.074) (0.115)

rxLt 0.259∗∗∗ 0.267∗∗∗ 0.235∗∗∗ 0.217∗∗∗ 0.206∗∗∗ 0.220∗∗∗ 0.283∗∗∗ 0.047 -0.017(0.058) (0.060) (0.059) (0.061) (0.069) (0.081) (0.109) (0.032) (0.058)

rxBt 0.403∗∗∗ 0.391∗∗∗ 0.442∗∗∗ 0.465∗∗∗ 0.546∗∗∗ 0.670∗∗∗ 0.986∗∗∗ -0.279∗∗ -0.582∗∗∗

(0.103) (0.094) (0.100) (0.111) (0.136) (0.171) (0.231) (0.112) (0.176)

rxAt 0.487 0.624∗∗ 0.697∗∗∗ 0.636∗∗ 0.704∗∗ 0.730∗∗∗ 1.105∗∗∗ -0.106 -0.569∗

(0.342) (0.290) (0.268) (0.261) (0.279) (0.278) (0.298) (0.177) (0.301)

B: MBDebt Sorts

Portfolios 0005 0520 2040 4060 6080 8095 9500 LS2080 LS0595

rxEt 2.310∗∗∗ 1.297∗∗∗ 0.927∗∗ 0.878∗∗ 0.816∗∗ 0.609 0.496 0.689∗∗ 1.813∗∗∗

(0.653) (0.489) (0.398) (0.380) (0.358) (0.392) (0.341) (0.322) (0.506)

rxDt 0.822∗∗∗ 0.451∗∗∗ 0.363∗∗∗ 0.354∗∗∗ 0.326∗∗∗ 0.317∗∗∗ 0.313∗∗∗ 0.134∗ 0.510∗∗∗

(0.214) (0.129) (0.091) (0.080) (0.074) (0.075) (0.088) (0.074) (0.166)

rxLt 0.309∗∗ 0.238∗∗∗ 0.216∗∗∗ 0.201∗∗∗ 0.201∗∗∗ 0.220∗∗∗ 0.207∗∗∗ 0.018 0.101(0.123) (0.082) (0.065) (0.059) (0.053) (0.053) (0.062) (0.032) (0.071)

rxBt 1.175∗∗∗ 0.657∗∗∗ 0.478∗∗∗ 0.443∗∗∗ 0.384∗∗∗ 0.348∗∗∗ 0.328∗∗∗ 0.309∗∗∗ 0.846∗∗∗

(0.287) (0.184) (0.124) (0.106) (0.096) (0.090) (0.102) (0.114) (0.228)

rxAt 1.322∗∗∗ 0.785∗∗∗ 0.591∗∗ 0.637∗∗ 0.682∗∗∗ 0.625∗∗ 0.465∗ 0.151 0.839∗∗∗

(0.361) (0.287) (0.265) (0.246) (0.262) (0.289) (0.251) (0.166) (0.261)

67

Table 21: Equity, Debt, and Asset Returns

This table reports conditional double sorts of monthly excess equity returns on Book-to-marketequity (BMEquity) and market leverage. The mean returns and standard deviations are reportedfor equal-weighted annually rebalanced portfolios. LMH is the portfolio that goes long firms withlow and short firms with high market leverage. In contrast, HML correspond to the usual strategythat is long firms with high and short firms with low Boot-to-market equity. Standard errorsreported in parentheses are adjusted for heteroscedasticity and autocorrelation (Newey-West with12 lags). Significance levels are denoted by * = 10%, ** = 5%, and *** = 1%. Sample periodfrom April 1998 to March 2018.

Market Leverage Quintiles

BMEquity Low 2 3 4 High LMHQuintiles

Low 0.541 0.548 0.695∗∗ 0.722∗ 1.155∗∗ -0.614∗

(0.398) (0.316) (0.342) (0.372) (0.514) (0.351)2 0.568∗ 0.759∗∗ 0.618 0.699∗∗ 0.828∗ -0.260

(0.325) (0.345) (0.385) (0.344) (0.500) (0.315)3 0.744∗∗ 0.828∗∗ 0.963∗∗∗ 1.146∗∗∗ 1.365∗∗ -0.621∗∗

(0.339) (0.343) (0.341) (0.408) (0.561) (0.314)4 0.635∗ 0.797∗∗ 0.814∗ 0.801∗ 1.310∗∗ -0.675∗∗

(0.348) (0.387) (0.435) (0.458) (0.526) (0.339)High 1.152∗∗∗ 1.131∗∗ 1.232∗∗ 1.393∗∗ 1.998∗∗ -0.846∗

(0.431) (0.481) (0.544) (0.698) (0.801) (0.502)

HML 0.611 0.583 0.537 0.671 0.842(0.378) (0.397) (0.330) (0.481) (0.561)

68

8 Figures

0 20 40 60 80 100 120

months

0.2

0.3

0.4

0.5

0.6ML vs Quasi-ML

0 20 40 60 80 100 120

months

0.4

0.5

0.6

0.7

0.8

0.9

1Q vs Quasi-Q

0 20 40 60 80 100 120

months

0.8

0.85

0.9

0.95

1Market-to-Book Debt

0 20 40 60 80 100 120

months

0.95

1

1.05

1.1

1.15

1.2

1.25MVR

Figure 1: Model SimulationsThe Figure plots illustrative time series from a model simulation for ten years at monthly frequency.The top right panel plots market (ML, solid line) versus book (Quasi-ML, dashed line) basedmeasures of market leverage, the top right panel plots market (Q, solid line) versus book (Quasi-Q, dashed line) based measures of Tobin’s Q, the lower left panel plots the bond’s market-to-bookratio, and the lower right panel plots the ratio of market-based to book-based market leverage.The plots refer to an average firm.

69

0 20 40 60 80 100 120

months

0.55

0.6

0.65

0.7

0.75ML vs Quasi-ML

0 20 40 60 80 100 120

months

0.8

0.85

0.9

0.95

1

1.05

1.1Q vs Quasi-Q

0 20 40 60 80 100 120

months

0.92

0.94

0.96

0.98

1

1.02

1.04Market-to-Book Debt

0 20 40 60 80 100 120

months

0.95

1

1.05

1.1

1.15

1.2

1.25MVR

Figure 2: Model Simulations - Bankruptcy CostsThe Figure plots illustrative time series from a model simulation for ten years at monthly frequency.The top right panel plots market (ML, solid line) versus book (Quasi-ML, dashed line) basedmeasures of market leverage, the top right panel plots market (Q, solid line) versus book (Quasi-Q, dashed line) based measures of Tobin’s Q, the lower left panel plots the bond’s market-to-bookratio, and the lower right panel plots the ratio of market-based to book-based market leverage.The plots refer to a distressed firm.

70

0 20 40 60 80 100 120

months

0.25

0.3

0.35

0.4

0.45

0.5ML vs Quasi-ML

0 20 40 60 80 100 120

months

0.6

0.7

0.8

0.9

1

1.1Q vs Quasi-Q

0 20 40 60 80 100 120

months

0.85

0.9

0.95

1

Market-to-Book Debt

0 20 40 60 80 100 120

months

1

1.05

1.1

1.15MVR

Figure 3: Model Simulations - Moneyness of Growth OptionThe Figure plots illustrative time series from a model simulation for ten years at monthly frequency.The top right panel plots market (ML, solid line) versus book (Quasi-ML, dashed line) basedmeasures of market leverage, the top right panel plots market (Q, solid line) versus book (Quasi-Q, dashed line) based measures of Tobin’s Q, the lower left panel plots the bond’s market-to-bookratio, and the lower right panel plots the ratio of market-based to book-based market leverage.The plots refer to a firm far from the investment boundary.

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

BV

RQ

median[25%, 75%] CI[5%, 95%] CI

(a) BV RQ

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

BV

RY

median[25%, 75%] CI[5%, 95%] CI

(b) BV RY

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

MV

R

median[25%, 75%] CI[5%, 95%] CI

(c) MVR

Figure 4: Importance of Sampling Frequency and Market ValuesThis Figure plots the the book value ratios BV RQ and BV RY which are informative about the roleof of changing sampling frequency. On the left, the market value ratio MVR reveals differencesbetween market and book values.

71

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

MM

RQ

median[25%, 75%] CI[5%, 95%] CI

(a) MMRQ

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

MM

RY

median[25%, 75%] CI[5%, 95%] CI

(b) MMRY

Figure 5: Marking to Market vs Book Values of Corporate DebtThis Figure plots the marking to market ratios MMRQ and MMRY . The first (second) ratiorelates market leverage to quarterly (annual) quasi-market leverage.

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

MM

RQ L

median[25%, 75%] CI[5%, 95%] CI

(a) MMRQL - Low Rating

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan150.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

MM

RQ H

median[25%, 75%] CI[5%, 95%] CI

(b) MMRQH - High Rating

Figure 6: Market-to-Market vs Book Values across RatingsLow rating firms are defined as companies with an S&P issuer credit rating below BB. On the otherhand, high rating companies include companies with an issuer credit rating of at least BBB+.Sample period from April 1998 to February 2017 due to the discontinued S&P Capital IQ issuercredit ratings.

72

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan15 Jul170.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

MB

Deb

t

median[25%, 75%] CI[5%, 95%] CI

(a) MBDebt

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan150.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

MB

Deb

tL

median[25%, 75%] CI[5%, 95%] CI

(b) MBDebtL - Low Rating

Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 Jan150.7

0.8

0.9

1

1.1

1.2

1.3

MB

Deb

tH

median[25%, 75%] CI[5%, 95%] CI

(c) MBDebtH - High Rating

Figure 7: Market-to-Book Debt across RatingsLow rating firms are defined as companies with an S&P issuer credit rating below BB. On the otherhand, high rating companies include companies with an issuer credit rating of at least BBB+.Sample period from April 1998 to February 2017 due to the discontinued S&P Capital IQ issuercredit ratings.

73

AAA AA A BBB BB B CRating Groups

-10

-5

0

5

10

100*

(rA i,t -

rU i,t

)

(a) BV of Debt

AAA AA A BBB BB B CRating Groups

-10

-5

0

5

10

100*

(rA i,t -

rU i,t

)(b) MV of Debt from Merton Model - Total Debt

AAA AA A BBB BB B CRating Groups

-10

-5

0

5

10

100*

(rA i,t -

rU i,t

)

(c) Actual Market Value of Debt

AAA AA A BBB BB B CRating Groups

-10

-5

0

5

10

100*

(rA i,t -

rU i,t

)

(d) Rule of Thumb

Figure 8: Unlevered Equity Returns vs Asset ReturnsThis Figure provides information on the distribution of the differences between unlevered excessequity returns and excess asset returns for various rating groups. Unlevered excess equity returnsare calculated as rUi,t = rEi,t(1−Li,t−1) where lagged leverage Li,t−1 is either calculated using the bookvalue of total liabilities as in Doshi et al. (2019) (panel a) or using the market of debt resultingfrom a Merton model where the face value of debt is approximated with total debt (dlttq+dlcq,panel b). Panel c makes use of the actual market leverage, MLMi,t . Finally, panel d uses the ruleof thumb on how to approximate market leverage suggested in section ??.

74

Appendix

A Measurement of MV Debt

When we measure the market value of debt, we distinguish between two cases: (1) market prices are

available for the total debt outstanding and (2) this is not the case. In the second case, we take two

routes, a conservative approach and an extended approach that allows us to maximise our sample. The

detailed construction of the data is as follows:

• Case I - pricing data on total debt outstanding for a given company is available: the market value

of debt can be measured unambiguously as the sum of market value of loans and bonds.

• Case II (Conservative Approach) - pricing data on total debt outstanding for a given company is

not available: we approximate the market prices of the missing debt with its book value equivalent.

• Case II (Extended Approach) - pricing data on total debt outstanding for a given company is not

available: we derive the market prices on the total debt outstanding as follows

– Scenario A - sum of face values of term loans, maximum amount of revolvers, and bonds is

larger than the total debt outstanding: we relax our assumption already conservative assump-

tion that only 20% of the revolver are withdrawn such that the total debt outstanding is met.

On average, this means an increase of the percentage withdrawn to 28% which is still below

the 30% reported by Sufi (2009).

– Scenario B - no pricing data on revolvers are available: we assume that the fraction of debt

that is missing is priced at the value-weighted average price of all outstanding loans from the

same company in that month.

B Data and Variables Construction

Variables we create based on market values of debt are as follows:

• Market to book debt (MBDebt) – Market debt over book debt

• Market net income to assets (NIMTAM) –Income (NIQ) over market value of assets

• Market leverage (TLMTAM) –Market debt over market value of assets

• Market cash to assets (CASHMTAM) –Cash (CHEQ) over market value of assets

• Volatility of assets (SIGMAM)–Volatility of monthly market asset returns over past 12 months.

Variables from Campbell et al. which do not incorporate market values of debt are as follows:

• Net income to assets (NITA)–Net Income (NIQ) over assets (ATQ).

• Modified net income to assets (NIMTA)–Net income (NIQ) over market capitalization plus total

liabilities (PRCCQ*CSHOQ + LTQ).

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Table A1: Coverage of the Sample

This Table reports the coverage of the sample in terms of the number of firms and total liabilitiescompared with the CRSP/Compustat universe of firms. That is, Firms reports the average numberof firms in the sample for each year. % Coverage is the percentage ratio of total book value ofliabilities of the sample to that of the entire CRSP/Compustat universe. Moreover, the columnBankruptcies reports the number of bankruptcies/defaults occurring in our sample in each year.

Year Firms % Coverage Bankruptcies Year Firms % Coverage Bankruptcies

1998 761 57.9% 0 2009 1,200 77.4% 21999 951 62.6% 0 2010 1,248 79.4% 42000 1,021 71.2% 5 2011 1,266 86.6% 62001 1,109 69.3% 6 2012 1,329 86.9% 52002 1,331 73.3% 8 2013 1,391 85.9% 42003 1,329 72.5% 10 2014 1,450 85.6% 92004 1,408 77.4% 4 2015 1,446 84.9% 82005 1,426 79.7% 6 2016 1,380 87.9% 72006 1,426 79.9% 13 2017 1,283 86.2% 02007 1,370 80.2% 6 2018 967 83.1% 02008 1,282 80.3% 5 Average 1,223 77.1% 5

• Leverage (TLTA)–Total liabilities (LTQ) over assets (ATQ).

• Modified leverage (TLMTA)–Total liabilities (LTQ) over market capitalization plus total liabilities

(PRCCQ*CSHOQ + LTQ).

• Excess Return (EXRET)– Log equity return (log(1+RET)) minus log S&P return.

• Market Capitalization Ratio (RSIZE)–Log market capitalization over S&P

log(PRC*SHROUT/SPclose)

• Return Volatility (SIGMA)–Annualized average daily volatility over the past 3 months

(√

252 ∗ 1N−1

∑r2k where N is the number of days in the past 3 months)

• Cash to Assets (CASHMTA)– Cash and short-term investments (CHEQ) over market capitalization

plus total liabilities (PRCCQ*CSHOQ + LTQ))

• Market to Book (MB)–Market capitalization ((PRCCQ*CSHOQ) to book equity (BE)

• Book equity (BE)–stockholder equity (SEQQ) + deferred taxes and investment (TXDITCQ) -

preferred stock (PSTKQ)

• Stock price (PRICE)–Log stock price (log(PRC)).

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Table A2: Q-Theory: OLS Results

This Table replicates the results of Peters and Taylor (2017). Results are from annual OLS panelregressions of investment on lagged Tobin’s q and firm and year fixed effects. Each column uses adifferent investment measure noted in the top rows. Standard investment i∗ equals CAPX scaledby PPE. Physical investment (iphy) equals CAPX scaled by total capital (Ktot = Kphy + Kint).Intangible investment (iint) equals R&D + 0.3× SG&A, scaled by Ktot. Total investment equalsiphy + iint. Panels A and C show regressions on standard q for the full (Panel A) and 1998-2018(Panel B) sample. Panels B and D show regressions on total q for the full (Panel B) and 1998-2018(Panel D) sample. Standard errors clustered by firm are in parentheses. We report within-firmR2.

Investment scaled by capital

i∗ iphy iint itot

A. Regressions with standard q

q 0.006∗∗∗ 0.002∗∗∗ 0.002∗∗∗ 0.004∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.204 0.135 0.195 0.242N 122,053 122,053 113,641 112,682

B. Regressions with total q

qtot 0.025∗∗∗ 0.012∗∗∗ 0.008∗∗∗ 0.022∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.233 0.193 0.212 0.313N 122,053 122,053 113,641 112,682

A. 1998 - 2018: Regressions with standard q

q 0.005∗∗∗ 0.001∗∗∗ 0.002∗∗∗ 0.003∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.230 0.120 0.200 0.239N 62,692 62,692 58,259 57,925

A. 1998 - 2018: Regressions with total q

qtot 0.022∗∗∗ 0.009∗∗∗ 0.008∗∗∗ 0.017∗∗∗

(0.000) (0.000) (0.000) (0.000)Within R2 0.238 0.169 0.221 0.304N 62,692 62,692 58,259 57,925

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Table A3: Q: Horse Race

This Table reports results from annual OLS panel regressions of investment on lagged book valueand market value based measures of Tobin’s q and firm and year fixed effects. Each column uses adifferent investment measure noted in the top rows. Standard investment i∗ equals CAPX scaledby PPE. Physical investment (iphy) equals CAPX scaled by total capital (Ktot = Kphy + Kint).Intangible investment (iint) equals R&D + 0.3× SG&A, scaled by Ktot. Total investment equalsiphy + iint. Panel A shows regressions on book and market value versions of total q, denoted qtot

and qtot,MV . Panel B reports regressions on standard q based on book and market values, denotedq and qMV . Standard errors clustered by firm are in parentheses. We report within-firm R2. Datais from 1998 - 2018.

Investment scaled by capital

i∗ iphy iint itot

A. Regressions with total q

qtot 0.004∗∗ -0.004∗∗∗ 0.001 -0.002(0.002) (0.001) (0.001) (0.001)

qtot,MV 0.018∗∗∗ 0.017∗∗∗ 0.007∗∗∗ 0.025∗∗∗

(0.002) (0.001) (0.001) (0.001)Within R2 0.182 0.162 0.201 0.263

B. Regressions with standard q

q -0.005∗∗∗ -0.000 -0.000 -0.000(0.001) (0.000) (0.000) (0.001)

qMV 0.009∗∗∗ 0.001∗∗ 0.001∗∗∗ 0.003∗∗∗

(0.001) (0.000) (0.000) (0.001)Within R2 0.160 0.084 0.134 0.140

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Table A4: Summary Statistics

This Table reports the mean, median, standard deviation, minimum, maximum, and the numberof observations for variables used in the empirical analysis. The definition and calculation of thevariables is discussed in section B of the online appendix.

Mean Median Std. Dev. Min Max Observations

Debt to Assets 0.350 0.333 0.169 0.067 0.756 227, 037NITA 0.006 0.009 0.022 -0.127 0.041 227, 037NIMTA 0.004 0.007 0.015 -0.076 0.028 227, 037NIMTAM 0.003 0.009 0.057 -4.290 4.576 227, 037TLTA 0.641 0.645 0.154 0.080 0.886 227, 037TLMTA 0.511 0.515 0.198 0.031 0.847 227, 037TLMTAM 0.367 0.337 0.217 0 0.999 227, 037EXRET -0.007 0.001 0.097 -0.316 0.206 227, 037RSIZE -8.392 -8.107 1.429 -14.025 -6.985 227, 037SIGMA 0.368 0.297 0.236 0.167 1.416 227, 037CASHMTA 0.049 0.030 0.057 0.002 0.374 227, 037CASHMTAM 0.082 0.039 0.144 -0.002 6.172 227, 037MB 3.052 1.734 3.562 0.475 14.606 227, 037PRICE 3.241 3.473 0.852 -0.288 4.065 227, 037

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Table A5: Equity, Debt, and Asset Excess Returns

This table reports conditional double sorts of monthly excess equity returns on Book-to-marketequity (BMEquity) and MBDebt. The mean returns and standard deviations are reported for equal-weighted annually rebalanced portfolios. LMH is the portfolio that goes long firms with low andshort firms with high market leverage. In contrast, HML correspond to the usual strategy thatis long firms with high and short firms with low Boot-to-market equity. Standard errors reportedin parentheses are adjusted for heteroscedasticity and autocorrelation (Newey-West with 12 lags).Significance levels are denoted by * = 10%, ** = 5%, and *** = 1%. Sample period from April1998 to March 2018.

MBDebt Quintiles

BMEquity Low 2 3 4 High LMHQuintiles

Low 0.910∗ 0.659∗ 1.024∗∗∗ 0.757∗∗ 0.586∗ 0.324(0.538) (0.375) (0.317) (0.332) (0.319) (0.346)

2 0.795∗∗ 0.785∗ 0.694∗ 0.699∗ 0.545 0.250(0.358) (0.403) (0.394) (0.383) (0.363) (0.271)

3 1.506∗∗∗ 0.901∗∗ 1.011∗∗∗ 0.898∗∗∗ 0.780∗∗ 0.724∗∗∗

(0.448) (0.440) (0.346) (0.341) (0.369) (0.271)4 1.191∗∗ 0.712 0.857∗ 0.884∗∗ 0.574 0.721∗∗

(0.502) (0.455) (0.461) (0.370) (0.404) (0.342)High 1.941∗∗ 1.547∗∗∗ 1.381∗∗ 0.997∗ 0.733 1.208∗∗

(0.777) (0.517) (0.557) (0.520) (0.481) (0.556)

HML 1.031 0.888∗∗ 0.357 0.240 0.148(0.660) (0.396) (0.374) (0.362) (0.337)

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