Growth options, macroeconomic conditions, and the cross section of credit risk

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Journal of Financial Economics

Journal of Financial Economics 107 (2013) 350–385

0304-40

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Growth options, macroeconomic conditions, and thecross section of credit risk$

Marc Arnold a, Alexander F. Wagner b,c,n, Ramona Westermann d

a University of St. Gallen, Swiss Institute of Banking and Finance, Rosenbergstrasse 52, CH-9000 St. Gallen, Switzerlandb University of Zurich, Swiss Finance Institute, Plattenstrasse 14, CH-8032 Zurich, Switzerlandc Center for Economic Policy Research (CEPR), UKd University of Geneva, Swiss Finance Institute, 42 bd du Pont d’Arve, CH-1205 Geneva, Switzerland

a r t i c l e i n f o

Article history:

Received 12 August 2010

Received in revised form

28 December 2011

Accepted 31 December 2011Available online 28 August 2012

JEL classification:

D92

E44

G12

G32

G33

Keywords:

Capital structure

Credit spread puzzle

Growth options

Macroeconomic risk

Value premium

5X/$ - see front matter & 2012 Elsevier B.V.

x.doi.org/10.1016/j.jfineco.2012.08.017

thank an anonymous referee whose sugge

d the paper. Tony Berrada, Simon Broda, M

ufresne, Rajna Gibson, Michel Habib, Dirk

rwan Morellec, Gabriel Neukomm, Kjell Nyb

Alexandre Ziegler and seminar participants a

the Humboldt University of Berlin, the C.R

ce, and the Research Day of the National Cen

arch Financial Valuation and Risk Manageme

d helpful comments. This research was supp

, the Swiss Finance Institute, and the Researc

e and Financial Markets’’ of the University of

esponding author at: University of Zurich, S

ttenstrasse 14, CH-8032 Zurich, Switzerland

ail addresses: [email protected] (M. Arno

[email protected] (A.F. Wagner),

[email protected] (R. Westermann).

a b s t r a c t

This paper develops a structural equilibrium model with intertemporal macroeconomic

risk, incorporating the fact that firms are heterogeneous in their asset composition.

Compared with firms that are mainly composed of invested assets, firms with growth

options have higher costs of debt because they are more volatile and have a greater

tendency to default during recession when marginal utility is high and recovery rates

are low. Our model matches empirical facts regarding credit spreads, default prob-

abilities, leverage ratios, equity premiums, and investment clustering. Importantly, it

also makes predictions about the cross section of all these features.

& 2012 Elsevier B.V. All rights reserved.

All rights reserved.

stions have greatly

arc Chesney, Pierre

Hackbarth, Mario

org, Tatjana-Xenia

t the University of

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1. Introduction

This paper examines the impact of corporate growthoptions on credit spreads, equity premiums, firm value,and financial policy choices in the presence of time-varying macroeconomic conditions.

The motivation for our study derives from the empiri-cal fact that credit risk, leverage, and equity risk pre-miums exhibit important cross-sectional variation. First,Davydenko and Strebulaev (2007) show that, controllingfor standard credit risk factors, proxies of growth optionsare all positively and significantly related to creditspreads. Similarly, Molina (2005) finds that firms with ahigher ratio of fixed assets to total assets have lower bondyield spreads and higher ratings. Second, firms with more

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M. Arnold et al. / Journal of Financial Economics 107 (2013) 350–385 351

growth options typically have lower leverage (see, e.g.,Smith and Watts, 1992; Fama and French, 2002; Frankand Goyal, 2009). Third, value firms earn higher equityreturns than growth firms (see, e.g., Fama and French,1992). Strikingly, none of these cross-sectional propertiescan be explained by existing structural models of default.The reason is that these models consider firms with onlyinvested assets, but they ignore the fact that growthopportunities constitute an essential element of assetvalues and that firms are heterogeneous in their assetcomposition.1

We provide a model that matches these cross-sectional properties of credit risk, leverage, and equityrisk premiums. In particular, we explicitly incorporateexpansion options of firms into a structural model ofdefault with macroeconomic risk. We show that hetero-geneity in the composition of assets helps explaincross-sectional variation of credit spreads and leverage.Moreover, allowing firms to be heterogeneous withrespect to the importance of growth options in the valuesof their assets explains the aggregate credit spread puzzle,not only qualitatively, but also quantitatively. Importantly,the puzzle is solved while fitting historically reportedasset volatilities and default rates for realistic debt matu-rities. At the same time, the model matches the averageequity premium and explains a significant portion of thecross section of equity risk (the value premium). It alsogenerates a countercyclical value premium, as observed inthe data. Finally, our model is consistent with aggregateand cross-sectional features of default clustering, invest-ment spikes and busts, and recovery rates.

For our analysis, we develop a structural-equilibriumframework in the spirit of Bhamra, Kuehn, and Strebulaev(2010b). Thus, we embed a pure structural model offinancial decisions into a consumption-based asset pricingmodel with a representative agent. Our model simulta-neously incorporates both intertemporal macroeconomicrisk (building on work by Hackbarth, Miao, and Morellec,2006; Bhamra, Kuehn, and Strebulaev, 2010c; Chen,2010), which has been shown to be important for explain-ing credit spreads and leverage, as well as expansionoptions. Macroeconomic shocks to the growth rate andvolatility of earnings, as well as to the growth rate andvolatility of consumption, arise due to switches betweentwo states of the economy: boom and recession. Thechanges in the state of the economy are modeled via aMarkov chain, a standard tool to model regime switches.The representative agent has the continuous time analogof Epstein-Zin-Weil preferences (Epstein and Zin, 1989;Weil, 1990; Duffie and Epstein, 1992b). Therefore, how heprices claims depends on both his risk aversion and hiselasticity of intertemporal substitution. Via the market

1 Recent research focuses on the credit spread puzzle, i.e., the fact

that standard structural models of default significantly underestimate

credit spreads for corporate debt (see, e.g., Elton, Gruber, Agrawal, and

Mann, 2001; Huang and Huang, 2002). Several papers present significant

progress in solving this puzzle (see, e.g., Bhamra, Kuehn, and Strebulaev,

2010a–c; Chen, 2010; Chen, Collin-Dufresne, and Goldstein, 2009;

Gomes and Schmid, 2010a). However, none of these papers addresses

the cross section of credit risk.

price of consumption determined by the agent’s prefer-ences, we are able to link unobservable risk-neutralprobabilities used in the structural model to historicalprobabilities. This modeling approach allows us to studyendogenously the effect of macroeconomic risk on creditspreads and optimal financing decisions.

We allow firms to have expansion options. Theseoptions are converted into invested assets when theunderlying earnings process exceeds the investmentboundary. We pinpoint the isolated effect of a firm’s assetcomposition on credit risk and leverage by assuming, inthe main analysis, that the exercise price of the growthoption is financed through the sale of some assets inplace, i.e., without additional funds being injected into thecompany. We also study equity financing later in thepaper. Default occurs when earnings are below the defaultthreshold in a given regime. Shareholders maximize thevalue of equity by simultaneously choosing the optimaldefault and expansion option exercise policies. The capitalstructure is determined by trading off tax benefits of debtagainst default costs to maximize the ex ante value ofequity, i.e., the value of the firm.

The first result the model yields is that, like in othermacroeconomic models, default boundaries are counter-cyclical, i.e., shareholders default earlier in recession thanin boom. Thus, default is more likely during recession,which, together with countercyclical marginal utilitiesand default costs, raises the costs of debt for all firmscompared with a benchmark model without businesscycle risk.

The central new feature of our model is that the assetcomposition alone matters significantly for the costs ofdebt. Two forces lead to the cross-sectional predictionthat debt is particularly costly for firms with a highportion of expansion options in their assets’ values. First,because options represent levered claims, firms withvaluable growth options are more sensitive to the under-lying earnings process than firms that consist of onlyinvested assets. The volatility of the underlying earningsprocess would, consequently, underestimate the truedefault risk of growth firms. While the literature discussesthis basic idea within equity-financed firms (Berk, Green,and Naik, 1999; Carlson, Fisher, and Giammarino, 2006),little is known about its impact on debt prices. Ourstructural model allows us to jointly analyze a firm’sexpansion policy and financial leverage. We show thatthe combination of these factors is critical for a fullexploration of the quantitative implications of the riski-ness of growth options on credit spreads.

The second driving force is that option values are moresensitive to macroeconomic regime changes than areassets in place. This higher sensitivity is, to some extent,another consequence of the idea that options representlevered claims. Importantly, an additional effect derivesfrom the fact that the optimal exercise boundary ofgrowth options increases in recession and decreases inboom. Intuitively, it is optimal to defer the exercise of anexpansion option when the economy switches to reces-sion, i.e., to wait for better times. Because the moneynessof growth options is regime-dependent, and becauseoptions represent levered claims, the continuation value

M. Arnold et al. / Journal of Financial Economics 107 (2013) 350–385352

of expansion options is more exposed to the macroeco-nomic state than the one of invested assets. Moreover, thechanging moneyness causes expansion options to be lesssensitive to the underlying development of the earningsprocess in recession than in boom, which reduces thevalue of the shareholders’ option to defer default duringbad times. Together, these effects amplify the counter-cyclicality of default thresholds for firms with a highportion of growth options. As marginal utility is highduring bad times, the higher tendency to default inrecession causes larger credit spreads under risk-neutralpricing for firms with expansion options than for thosewith only invested assets.

We then investigate the quantitative performance ofthe model in explaining empirically observed data. Theliterature suggests that an average BBB-rated firm has aten-year credit spread in the range of 74–95 basis points(bps). (This range is obtained by starting from the averagebond yields reported in Davydenko and Strebulaev, 2007;Duffee, 1998 and taking into account that around 35% ofbond yields are due to nondefault components.) With ourmain set of parameters, a model without business cyclerisk produces a mere 29 bps spread for an average firm.A standard macroeconomic model with optimal defaultthresholds in the spirit of Bhamra, Kuehn, and Strebulaev(2010c) or Chen (2010) implies a spread of 56 bps foraverage firms at issue that consist of only invested assets.Our estimate for the average BBB-rated US firm’s assetcomposition is that total firm value is about 60% higherthan the value of invested assets, which corresponds(approximately) to a Tobin’s q of 1.6.2 For such a firm,we obtain a credit spread of about 66 bps when usingoptimal default thresholds, optimal expansion bound-aries, and an earnings volatility such that the averageasset volatility matches the one observed for BBB-ratedfirms. This spread is remarkably higher than the 39 bpsour model implies for a firm with only invested assets.The large difference arises even though leverage is keptconstant; we vary only the characteristics of the assetsthemselves.

As the economy consists of a mix of firms, the resultthat growth firms have higher credit spreads than firmswith only invested assets suggests that our model can alsoexplain the aggregate credit spread puzzle. To evaluatethis conjecture, note that when relating the implicationsof capital structure models for average credit spreads toempirical studies, it is crucial to take into account thatsuch studies use aggregate data over cross sections offirms, not average individual firm-level data (Strebulaev,2007). Following this line of reasoning, Bhamra, Kuehn,and Strebulaev (2010c) investigate how the time evolu-tion of the cross-sectional distribution of firms withdifferent leverage ratios affects credit spreads and defaultprobabilities. Building on their approach, we characterizethe aggregate dynamics by simulating over time a crosssection of individual firms that is structurally similar to

2 Market values can be higher than book values also because of

off-balance sheet assets, so a range exists for the asset composition of

the typical firm.

the empirical distribution of BBB-rated firms not onlywith respect to average leverage ratios but also withrespect to asset composition ratios. The average ten-and 20-year credit spreads of 81 and 100 basis points,respectively, from simulating this true cross section in ourmodel reflect their target credit spreads well. To solve theaggregate credit spread puzzle, a model needs to explainobserved costs of debt while still matching historicaldefault losses (given by the historical default probabilitiesand recovery rates) and asset volatilities. We conse-quently proceed by showing that the model-implieddefault rates and asset volatilities of BBB-rated firms aresimilar to the ones historically reported for realistic debtmaturities.

The nature of assets, thus, has a powerful impact oncosts of debt. Not surprisingly, it also affects the observedfeatures of leverage. At initiation, we find that a firm with anaverage growth option optimally holds about 4–5% lowerleverage than one with only invested assets. In addition, weobtain procyclical optimal leverage decisions of firms, in linewith Covas and DenHaan (2006) and Korteweg (2010). Thereason is that the default risk is higher in recession than inboom. The negative relation between growth options andleverage also maintains when simulating over time ourmodel-implied true cross section of BBB-rated firms. In thissimulation, however, firms deviate from their initiallyoptimal leverage in a way such that the aggregate marketleverage of the whole sample becomes countercyclical,consistent with Korajczyk and Levy (2003) and Bhamra,Kuehn, and Strebulaev (2010c).

We derive additional testable predictions when studyingthe aggregate dynamics of our model economy. Creditspreads and default rates are countercyclical, as reportedin the literature. Next, aggregate investment patterns arestrongly procyclical, with investment spikes often occurringwhen the regime switches from recession to boom, reflect-ing the findings in the empirical investment literature(Barro, 1990; Cooper, Haltiwanger, and Power, 1999). Ourmodel also makes specific cross-sectional predictions. Forexample, realized recovery rates are lower for growth firms.

Finally, we show that the model’s intuition is consis-tent with the literature on the value premium for equity.In the true cross section, our model implies an annualvalue premium, i.e., a difference between the averagevalue-weighted equity premium of the firms in the lowestdecile of the asset composition ratio and the premium ofthose in the highest decile, of 3.47%. Importantly, themodel also explains the empirically reported counter-cyclical pattern of the value premium.

Our paper contributes to several streams of previousresearch. First, the fact that growth options are empiri-cally strongly associated with observed leverage has alsoprompted other explanations. The most prominent ofthese additional explanations, agency, comes in twoprimary forms: a shareholder–bondholder conflict and amanager–shareholder conflict. Appealing to the former,Smith and Watts (1992) and Rajan and Zingales (1995)suggest that debt costs associated with shareholder–bondholder conflicts typically increase with the numberof growth options available to the firm due to under-investment (Myers, 1977) and overinvestment by way of


asset substitution (Jensen, 1986; see also Sundaresan andWang, 2006).3 According to Leland (1998), however, optimalleverage even increases when firms can engage in assetsubstitution. Similarly, Parrino and Weisbach (1999) con-clude that stockholder–bondholder conflicts are too limitedto explain the cross-sectional variation in capital structure.Childs, Mauer, and Ott (2005) show how short-term debtreduces agency costs. Hackbarth and Mauer (2012) demon-strate that the joint choice of debt priority structure andcapital structure can virtually eliminate the suboptimalinvestment incentives of equity-holders. Neither of thepapers incorporates macroeconomic risk.

As for manager–shareholder conflicts, Morellec (2004)shows that agency costs of free cash flow can explain thelow debt levels observed in practice and the negativerelation between debt levels and the number of growthoptions; see also Barclay, Morellec, and Smith (2006).Morellec, Nikolov, and Schurhoff (2012) conclude thateven small costs of control challenges are sufficient toexplain the low-leverage puzzle. It is still a matter ofdebate to what extent conflicts of interest betweenmanagers and stockholders cause the empiricallyobserved patterns. Graham (2000), for example, tests awide set of managerial entrenchment variables and findsonly ‘‘weak evidence that managerial entrenchment per-mits debt conservatism’’ (p. 1931). In any case, our modelis not inconsistent with either of these views. It offers aquantitatively important reason for the cross-sectionalvariation in leverage and credit spreads that derives solelyfrom the nature of assets of firms.4

Second, at the core of our model is the notion thatmacroeconomic (business cycle) risk matters in powerfulways for the costs of corporate debt and financial deci-sions, because firms are more likely to default when doingso is costly (see, e.g., Demchuk and Gibson, 2006; Almeidaand Philippon, 2007; Bhamra, Kuehn, and Strebulaev,2010c; Chen, 2010). What we add to this literature isthe idea that the impact of business cycle risk depends onthe asset base of a firm.

In contemporaneous and independent work, Chen andManso (2010) set up a model similar to ours with expansionoptions. Their focus, however, is on the debt overhangproblem, not on explaining cross-sectional features or thecredit spread puzzle—the central tasks of this paper.

Finally, our structural-equilibrium framework drawson insights from consumption-based asset pricing models(Lucas, 1978; Bansal and Yaron, 2004).

The paper proceeds as follows. In Section 2, we set upour valuation framework. We solve the model in Section3. Section 4 discusses our parameter and firm samplechoices, as well as the optimal default and expansionpolicies. Section 5 outlines qualitative properties of our

3 See Lyandres and Zhdanov (2010) for an explanation for acceler-

ated investment that does not rely on agency.4 An alternative explanation for why low leverage could be optimal

in the high-tech sectors is offered in Miao (2005). In his model, when a

sector experiences technological growth, more competitors enter, lead-

ing to falling prices and possibly to a greater probability of default. Yet

other explanations appeal to the fact that firms have the option to issue

additional debt (Collin-Dufresne and Goldstein, 2001).

model for the aggregate economy. We turn to the quanti-tative implications for BBB-rated firms in Section 6. Thepredictions of our model for the value premium of equityare discussed in Section 7. Section 8 concludes.

2. The model

We build a structural model with intertemporal macro-economic risk, embedded inside a representative agentconsumption-based asset pricing framework. The structuralmodel is based on a standard continuous time model ofcapital structure decisions in the spirit of Mello and Parsons(1992), as extended by Hackbarth, Miao, and Morellec(2006) for business cycle fluctuations. In addition, weexplicitly model growth opportunities. Following Bhamra,Kuehn, and Strebulaev (2010c) and Chen (2010), embed-ding the model of capital structure into a consumption-based asset pricing model allows the valuation of corporatesecurities using the risk-neutral measure implied by thepreferences of the representative agent.

The economy consists of N infinitely lived firms withassets in place and possibly growth options, a largenumber of identical infinitely lived households, and agovernment serving as a tax authority. We assume thatthere are two different macroeconomic states, namely,boom (B) and recession (R). Formally, we define a time-homogeneous Markov chain ItZ0 with state space fB,Rgand generator Q :¼ ½�lB

lR

lB�lR�, in which li 2 ð0,1Þ denotes the

rate of leaving state i. In the main analysis, we considerlBolR, as in Hackbarth, Miao, and Morellec (2006).

The following properties hold. First, the probabilitythat the chain stays in state i longer than some time tZ0is given by e�lit . Second, the probability that the regimeshifts from i to j during an infinitesimal time interval Dt isgiven by liDt. Third, the expected duration of regime i is1=li, and the expected fraction of time spent in thatregime is lj=ðliþljÞ.

Aggregate output Ct follows a regime-switching geo-metric Brownian motion:

dCt

Ct¼ yi dtþsC

i dWCt , i¼ B,R, ð1Þ

in which WtC

is a Brownian motion independent of theMarkov chain, and yi, sC

i are the regime-dependentgrowth rates and volatilities of the aggregate output.In equilibrium, aggregate consumption equals aggregateoutput. Hence, the above specification gives rise touncertainty about the future moments of consumptiongrowth.

To incorporate the impact of the intertemporal dis-tribution of consumption risk on the representativehousehold’s utility, we assume the continuous-time ana-log of Epstein-Zin-Weil preferences (Epstein and Zin,1989; Weil, 1990), which are of stochastic differentialutility type (Duffie and Epstein, 1992a,b). Specifically, theutility index Ut over a consumption process Cs solves

Ut ¼ EPZ 1

t

r1�d

C1�ds �ðð1�gÞUsÞ

ð1�dÞ=ð1�gÞ

ðð1�gÞUsÞð1�dÞ=ð1�gÞ

�1dsjF t

" #, ð2Þ

in which r is the rate of time preference, g determines thecoefficient of relative risk aversion for a timeless gamble,


and C :¼ 1=d is the elasticity of intertemporal substitu-tion for deterministic consumption paths.

As shown by Bhamra, Kuehn, and Strebulaev (2010c)and Chen (2010), the stochastic discount factor mt thenfollows the dynamics:

dmt

mt¼�ri dt�Zi dWC

t þðeki�1Þ dMt , ð3Þ

with Mt being the compensated process associated withthe Markov chain, and

ri ¼ r iþlig�dg�1ðw�ðg�1Þ=ðg�dÞ�1Þ�ðw�1�1Þ

� �, ð4Þ

Zi ¼ gsCi , ð5Þ

ki ¼ ðd�gÞ loghj

hi

� �: ð6Þ

hB, hR solve a non-linear system of equations given inAppendix A.1, Eq. (35). ri are the regime-dependent realrisk-free interest rates, composed of the interest rate if theeconomy stayed in regime i forever, r i, and the adjust-ment for possible regime switches as shown by thesecond term. Zi are the risk prices for systematic Brow-nian shocks affecting aggregate output, and ki is therelative jump size of the discount factor when the Markovchain leaves state i (and, consequently, kj ¼ 1=ki). The no-jump part of the interest rate, r i, is given by

r i ¼ rþdyi�12gð1þdÞðs

Ci Þ

2ð7Þ

and

w :¼ ekR ¼ e�kB ð8Þ

measures the size of the jump in the real-state price densitywhen the economy shifts from recession to boom (seeBhamra, Kuehn, and Strebulaev, 2010c, Proposition 1).

Credit spreads are based on nominal yields and taxesare collected on nominal earnings. To link nominal to realvalues such as the real interest rate, we specify a stochas-tic price index as

dPt

Pt¼ p dtþsP,C dWC

t þsP,id dWP

t , ð9Þ

with WPt being a Brownian motion describing the idiosyn-

cratic price index shock, independent of the consumptionshock Brownian WC

t and the Markov chain. p denotes theexpected inflation rate, and sP,C o0 and sP,id40 are thevolatilities of the stochastic price index associated withthe consumption shock and the idiosyncratic price indexshock, respectively. The nominal interest rate rn

i is thengiven by

rni ¼ riþp�s2

P�sP,CZi, ð10Þ

with sP :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsP,CÞ

2þðsP,idÞ

2q

being the total volatility ofthe stochastic price index.

At any time, the real after-tax earnings process of afirm follows:

dXt,real

Xt,real¼ mi,real dtþsX,C

i,real dWCt þs

X,id dWXt , i¼ B,R, ð11Þ

in which WXt is a standard Brownian motion describing an

idiosyncratic shock, independent of the aggregate output

shock WCt , the consumption price index shock WP

t , and theMarkov chain. mi,real are the real regime-dependent drifts;sX,C

i,real40, the real firm-specific regime-dependent volati-lities associated with the aggregate output process; andsX,id40, the firm-specific volatility associated with theidiosyncratic Brownian shock.

The nominal after-tax earnings process can now bewritten as

dXt

Xt¼ mi dtþsX,C

i dWCt þs

P,id dWPt þs

X,id dWXt , i¼ B,R,

ð12Þ

in which mi ¼ mi,realþpþsP,CsX,Ci,real are the nominal regime-

dependent drifts, and sX,Ci ¼ s

X,Ci,realþs

P,C 40 are the nom-

inal regime-dependent volatilities associated with theaggregate output process. As suggested by the literature,

we posit that sX,CB osX,C

R (Ang and Bekaert, 2004).

Denote the risk-neutral measure by Q. The expectedgrowth rates of the firm’s nominal after-tax earningsunder the risk-neutral measure, ~m i, are given by

~m i :¼ mi�sX,Ci ðZiþs

P,CÞ�ðsP,idÞ2: ð13Þ

Let ~li denote the risk-neutral transition intensities, deter-mined as

~li ¼ ekili: ð14Þ

Following Chen (2010) and Bhamra, Kuehn, andStrebulaev (2010c), the unlevered after-tax asset valuecan be written as

Vt ¼ Xtyi for It ¼ i, ð15Þ

with yi being the price-earnings ratio in state i determined by

y�1i ¼ rn

i � ~m iþðrn

j � ~mjÞ�ðrni � ~miÞ

rnj �

~mjþ ~p~p ~f j: ð16Þ

~p :¼ ~l iþ~lj is the risk-neutral rate of news arrival, and

ð~f B, ~f RÞ ¼ ðlR= ~p,lB= ~pÞ is the long-run risk-neutral distribu-

tion. y�1 can be interpreted as a discount rate, in which thefirst two terms constitute the standard expression if theeconomy stayed in regime i forever, and the last termaccounts for future time spent in regime j. As in Bhamra,Kuehn, and Strebulaev (2010c), the price–earnings ratio inthe main analysis is higher in boom than in recession, i.e.,yB4yR.

Finally, the volatility of the earnings process in regime i is

~s i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsX,C

i Þ2þðsP,idÞ

2þðsX,idÞ

2q

: ð17Þ

The expansion option of the firm is modeled as anAmerican call option on the earnings. Specifically, at anytime t , the firm can pay exercise costs K to achieveadditional future after-tax earnings of sXt for all tZt forsome factor s40. We assume that if a firm exercises itsexpansion option, the option is converted into assets inplace, such that the firm consists of only invested assets.The exercise of the growth option is assumed to beirreversible. At default, bondholders recover not only afraction of the assets in place, but also a fraction of theoption’s value. Intuitively, the option can be exercisedindependently of the considered firm.

6 The scaling property states that, conditional on the current regime

of the economy, the optimal coupon, the optimal default thresholds, the

investment boundaries, as well as the values of debt and equity at

restructuring points are all homogeneous of degree one in earnings.


For the financing of investment, we present twovariants. In the main analysis, we wish to abstract awayfrom the effect of fund injections by debt- or equity-holders to pay the exercise price and instead seek toisolate the effect of growth options in the value of firms’assets on corporate securities. Therefore, we first assumethat, at exercise, the firm pays the exercise costs K of theoption by selling a part of its assets in place.5 In detail,while exercising the option at time t entitles the firm tototal future after-tax earnings of ðsþ1ÞXt for all tZt ,financing the exercise costs requires selling a fractionK=Xt y

iof these earnings, in which i is the realized state of

the economy at the time of exercise. Hence, the totalafter-tax earnings of the firm at any time after exercisecorrespond to ððsþ1ÞÞ�K=ðXt yt ÞÞXt . Second, we also con-sider equity financing of the exercise costs K.

The critical measure to capture the relative importanceof a firm’s expansion option in the value of its assets is theasset composition ratio. We define it as the value of thefirm, divided by the value of invested assets. Certainly,the value of the firm does not only contain the value ofthe invested assets and the expansion option, but also thevalue of the tax shield and bankruptcy costs. Neverthe-less, we use this measure because the direct empiricalanalog of the asset composition ratio is Tobin’s q. Further-more, the impact of the tax shield and bankruptcy costson the ratio is relatively small.

Corporate taxes are paid at a constant rate t, and fulloffsets of corporate losses are allowed. In our framework,firms are leveraged because debt allows it to shield part ofits income from taxation. Once debt has been issued, afirm pays a total coupon c at each moment. Following thestandard in the literature, we assume that firms financecoupons by injecting funds. At any point, shareholdershave the option to default on their debt obligations, aswell as the possibility to exercise an expansion option.Default is triggered when shareholders are no longerwilling to inject additional equity capital to meet net debtservice requirements (Leland, 1998). If default occurs, thefirm is immediately liquidated and bondholders receivethe unlevered asset value less default costs, reflecting theabsolute priority of debt claims. The default costs inregime i are assumed to be a fraction 1�ai of theunlevered asset value at default, with ai 2 ð0,1�. Wesuppose that recovery rates are lower in recession, i.e.,aRoaB (Frye, 2000). The incentive to issue debt is limiteddue to the possibility of costly financial distress.

Equity-holders face the following decisions. First, oncedebt has been issued, they select the default and expan-sion policies that maximize equity value. Hence, bothexpansion and default are chosen endogenously. Second,they determine the optimal capital structure by choosingthe coupon level that maximizes the value of the firm. Themodel does not allow restructuring of debt either whenthe option is exercised or at endogenous restructuring

5 Indirect financing by selling assets often occurs, e.g., when

acquirers divest part of target companies’ assets following takeovers

(Bhagat, Shleifer, and Vishny, 1990; Kaplan and Weisbach, 1992). The

model simplifies in that, in reality, firms have different types of assets.

points. The main reason is that expansion opportunitiespreclude a scaling feature of the model solution.6

The main text presents the model and its solution forinfinite debt maturity. We also solve and use the case offinite debt maturity, in which we consider the stationaryenvironment of Leland (1998): The firm issues debt with aconstant principal p and a constant total coupon c paid ateach moment. A fraction m of the total debt is continu-ously rolled over. In particular, the firm continuouslyretires outstanding debt principal at rate mp and replacesit with new debt vintages of identical coupon and princi-pal. Finite maturity debt is, therefore, characterized by thetuple ðc,m,pÞ. This setup leads to a time-homogeneoussetting. Throughout the paper, it is assumed that debt isissued at par.

3. Model solution

We solve the model by backward induction. First, thevalue of the growth option for given expansion policies isderived. Then, for given corporate policies and capitalstructure, we proceed with the valuation of corporatesecurities for a firm that consists of assets in place andholds an expansion option. Finally, we obtain the expan-sion and default policies that simultaneously maximizethe value of equity, as well as the capital structure thatmaximizes the firm value.

As in Hackbarth, Miao, and Morellec (2006), weassume that the optimal strategies are of regime-dependent threshold type in X (for a formal proof in thecase of expansion thresholds only, see Guo and Zhang,2004). Precisely, suppose that Di and Di are the defaultthresholds in regime i¼ B, R of a firm with only investedassets and of a firm with both invested assets and agrowth option, respectively. Xi denotes the exerciseboundary of the growth option in regime i¼ B, R. In whatfollows, we present the case in which DBoDR, XBoXR,and DBoDR, i.e., the boundaries are lower in boom forboth expansion and default (before and after expansion).7

Finally, we presume that maxfDR,DRgoXB, i.e., we areinterested in firms that exercise their expansion optionwith a positive probability, and we exclude the possibilityof immediate default after expansion. The optimal defaultand expansion policies for relevant parameter regionssatisfy this ordering.

3.1. The value of the growth option

Denote the value functions of the growth option in regimeB and R by GB(X) and GR(X), respectively. Proposition 1

When assuming dynamic capital structure adjustments, the absence of a

scaling property impedes not only closed-form results, but also the

application of numerical solution methods with backward induction.7 We can assume without loss of generality that DB oDR (if not,

interchange the names of the regimes). The case DB oDR , DB oDR , and

XB 4XR (i.e., the exercise boundary in recession is lower than the one in

boom) can be solved by analogous techniques.


states the value of a growth option subject to regimeswitches.

Proposition 1. For any given pair of exercise boundaries

½XB,XR�, the value of the growth option in regime i is given by

GiðXÞ ¼

Ai3Xg3þAi4Xg4 , XoXB, i¼ B,R,

C 1XbR1þC 2XbR

2

þ ~lRsyBX

rnR� ~mRþ

~lR

� ~lRK

rnRþ

~lR

, XBrXoXR, i¼ R,

sXyi�K , XZXi, i¼ B,R,

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð18Þ

in which gk, k¼ 3,4, are the positive roots of the quartic

equation

ð ~mRgþ12~s2

Rgðg�1Þ� ~lR�rnRÞð ~mBgþ1

2~s2

Bgðg�1Þ� ~lB�rnBÞ ¼

~lR~lB,

ð19Þ

and bRk , k¼ 1,2, are given by

bR1,2 ¼

1

2�~mR

~s2R

7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�~mR

~s2R

!2

þ2ðrn

Rþ~lRÞ

~s2R

vuut : ð20Þ

ARk, k¼ 3,4, is a multiple of ABk with the factor

lk :¼1~lB

rnBþ

~lB� ~mBgk�1

2~s2

Bgkðgk�1Þ

� �: ð21Þ

½AB3,AB4,C 1,C 2� solve a linear system given in Appendix A.2.

Proof. See Appendix A.2.

The functional form of the solution (18) is analogous tothe one presented in Guo and Zhang (2004). For eachregime i, the option is exercised immediately wheneverXZXi (option exercise region); otherwise it is optimal towait (option continuation region). Similar to the occur-rence of default, the exercise of the expansion option canbe triggered in two ways: either when the idiosyncraticshock X reaches the exercise boundary Xi in a givenregime or when the regime switches from recession toboom and X lies between XB and XR.

In the option continuation region, the solution (18)reflects the changes in value that occur either when theidiosyncratic shock reaches a boundary or when theregime switches. Proposition 1 shows that in the regionXoXB, i.e., the case in which the option is in thecontinuation region in both boom and recession, thesevalue changes are captured by two terms. When theoption is in the continuation region in recession only,i.e., XBrXoXR, the solution exhibits four terms. Thesefour terms reflect the value changes when leaving thisregion due to hitting a boundary, either XR from below orXB from above or due to a regime-switching inducedexercise of the option. In the option exercise region,XZXi, the firm obtains earnings of sX by investing K.

Proposition 1 determines the value of the growth optionfor any given pair of exercise boundaries XB and XR. In the fullmodel solution, we derive option values for optimal exerciseboundaries of equity-holders in both levered and unleveredfirms. In unlevered firms, the optimal exercise boundaries aredenoted Xunlev

B and XunlevR , respectively. They are determined

by smooth-pasting conditions at the option exercise bound-ary. For ease of notation, we denote the unlevered value ofthe growth option by Gunlev

i , i.e., Gunlevi ðXÞ ¼ GiðX9X

unlevB ,Xunlev

R Þ.Appendix A.2 states the complete set of boundary conditionsfor the unlevered option value and presents the solution.

3.2. Firms with invested assets and expansion options

In this subsection, we derive the value of corporatesecurities of a general firm, as well as the default andexpansion thresholds selected by shareholders.

After exercise, a firm consists of only invested assets,endowed with the initially determined optimal couponlevel. The post-exercise value of corporate securitiesinfluences their pre-exercise value. As the default policyis an ex post policy, the optimal default thresholds afterexercise correspond to the ones of a firm with onlyinvested assets. That is, equity-holders optimally adapttheir default policy upon expansion. Debt-holders antici-pate this change. Let diðXÞ denote the value of corporatedebt of a firm with only invested assets; diðXÞ the value ofdebt of a firm with invested assets and an expansionoption in regime i¼ B, R. The solution for diðXÞ can befound in Appendix A.3, the derivation being analogous toHackbarth, Miao, and Morellec (2006). Proposition 2states the value of infinite maturity debt of a firm withinvested assets and an expansion option.

Proposition 2. For any given set of default and exercise

boundaries ½DB,DR,XB,XR�, the value of infinite maturity debt

in regime i is given by

diðXÞ ¼

aiðXyiþGunlevi ðXÞÞ, XrDi, i¼ B,R,

C1XbB1þC2XbB

2þC5Xg3þC6Xg4

þ ~lBaRyR

rnB� ~mBþ

~lB

X

þc

rnBþ

~lB

, DBoXrDR, i¼ B,

Ai1Xg1þAi2Xg2þAi3Xg3

þAi4Xg4þc

rpi

, DRoXrXB, i¼ B,R,

B1XbR1þB2XbR

2þZðXÞ

þ ~lRc

rPi ðr

nRþ

~lRÞ

þc

rnRþ

~lR

, XBoXrXR, i¼ R,

di sX�K

yi

� �, X4Xi, i¼ B,R:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð22Þ

Gunlevi denotes the unlevered option value in regime i

(see Proposition 1), and

bi1,2 ¼

1

2�~m i

~s2i

7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�~m i

~s2i

!2

þ2ðrn

i þ~liÞ

~s2i

vuut , ð23Þ

C5 ¼ aRl3

l3A

unlev

B3 , ð24Þ

and

C6 ¼ aRl4

l4A

unlev

B4 : ð25Þ

8 Because the exercise of the option is financed by selling assets in

place, the debt value after exercise is not homogeneous in X. The

function ZðXÞ captures this nonhomogeneity after exercise and cannot,

therefore, be simplified to a finite sum.


gk, k¼ 1,2,3,4 are the roots of the quartic equation

ð ~mRgþ12~s2


2~s2


~lR~lB:

ð26Þ

ARk, k¼ 1,2,3,4, is a multiple of ABk with the factor

lk :¼1~lB

rnBþ

~lB� ~mBgk�1

2~s2

Bgkðgk�1Þ

� �, ð27Þ

and rpi is the perpetual risk-free rate given by

rpi ¼ riþ

rj�ri

~pþrj

~p ~f j, ð28Þ

in which ~p ¼ ~l1þ~l2 is the risk-neutral rate of news arrival

and ð~f B, ~f RÞ ¼ ðlR= ~p,lB= ~pÞ is the long-run risk-neutral

distribution. The function ZðXÞ is given by

ZðXÞ ¼ ~lR

Xi,k ¼ 1,2

2ð�1Þiþ1sgk ABk

~s2Rðb

R2�b

R1Þðgk�b

Ri Þ

Xgk2F1

� �gk,bRi ,bR

i �gkþ1;�K

sXyB

� �, ð29Þ

in which 2F1 is Gauss’s hyperbolic function. dið�Þ denotes

the value of debt of a firm with only invested assets.½AB1,AB2,AB3,AB4,C1,C2,B1,B2� solve a linear system given in

Appendix A.4.


In each regime, the firm faces three different regionsdepending on the value of X: Below the default threshold,i.e., XrDi, the firm is in the default region where itdefaults immediately, and debt-holders receive a fractionai of the total asset value. The firm is in the continuationregion if X is between the default threshold and theexercise boundary, i.e., DioXrXi. Finally, the exerciseregion is reached if X4Xi, i.e., X is above the exerciseboundary.

In the continuation region, the value of corporate debtis determined by three components. The first componentis the value of a risk-free claim to the perpetual stream ofcoupon. The second and third components reflect thechanges in the value of debt that occur either due to theidiosyncratic shock reaching a boundary or due to aregime switch. Proposition 2 shows that for the regionDRoXrXB, i.e., when the firm is in the continuationregion in both boom and recession, the solution consistsof five terms. The value of the risk-free claim to thecoupon is given by the last term. The coupon is dis-counted by the perpetual risk-free rate ri

pthat takes into

account the expected future time spent in each regime.The first four terms capture the changes in value due tothe idiosyncratic shock X hitting a region boundary or dueto a change of regime. When DBoXrDR, i.e., the firm is inthe continuation region only in boom, the solution con-sists of six terms. The last term is the value of the risk-freeclaim to the coupon. Here, the discount rate is given bythe nominal interest rate in boom, rn

B, increased by ~lB toreflect the possibility of a regime switch to recession. Thefirst five terms capture the changes in debt value thatoccur when the idiosyncratic shock reaches a boundary orwhen the regime switches to recession. For the regionXBoXrXR, i.e., when the firm is in the continuation

region only in recession, the solution consists of fiveterms. The last term is the value of a risk-free perpetualclaim to the coupon. To account for a possible regimeswitch to boom, the discount rate is here given by theinterest rate in recession, rn

R, increased by ~lR. The remain-ing four terms capture the value changes due to reachinga region boundary, either XB from above or XR from below,or due to a regime switch to boom triggering immediateoption exercise.8

Remark 1 shows how to express the value of finitematurity debt, tax benefits, and bankruptcy costs usingProposition 2. Let tiðXÞ and biðXÞ denote the value of thetax shield and bankruptcy costs of a firm with both assetsin place and an expansion option in regime i¼ B,R,respectively; t iðXÞ and biðXÞ, the corresponding valuefunctions of a firm with only invested assets.

Remark 1.

(i)
The value of finite maturity debt with principal p anda fraction m of debt continuously rolled over is givenby Eq. (22) in Proposition 2, in which c and rn
i arereplaced by cþmp and ri

nþm, respectively, and di isreplaced by the value of finite maturity debt of a firmwith only invested assets.

(ii)
The value of the tax shield is given by Eq. (22) inProposition 2, in which c and ai are replaced by ctand 0, respectively, and di in the last line of Eq. (22) isreplaced by t .
(iii)
The value of bankruptcy costs is given by Eq. (22) inProposition 2, in which c and ai are replaced by 0 and1�ai, respectively, and di in the last line of Eq. (22) isreplaced by b.

Next, the total firm value fi in regime i¼ B,R is given bythe value of assets in place yiX, plus the value of theexpansion option GiðXÞ and the value of tax benefits fromdebt ti(X), less the value of default costs bi(X), i.e.,

f iðXÞ ¼ yiXþGiðXÞþtiðXÞ�biðXÞ: ð30Þ

As the total firm value equals the sum of debt andequity values, the equity value eiðXÞ, i¼ B,R, can, hence, bewritten as

eiðXÞ ¼ f iðXÞ�diðXÞ ¼ yiXþGiðXÞþtiðXÞ�biðXÞ�diðXÞ: ð31Þ

Equity-holders select the default and investment poli-cies that maximize the value of equity ex post. Denotethese policies by Dn

i and Xn

i , respectively. Formally, thedefault policy that maximizes the equity value is deter-mined by postulating that the first derivative of the equityvalue has to be zero at the default boundary in eachregime. Simultaneously, optimality of the option exerciseboundaries is achieved by equating the first derivative ofthe equity value at the exercise boundary with the firstderivative of the equity value of a firm with only invested

Table 1Baseline parameter choice.

This table describes our baseline scenario. Panel A contains the annualized parameters of a typical BBB-rated Standard

& Poor’s 500 firm. Panels B and C show our parameter choice for the expansion option and the macroeconomy,

respectively. The asset composition ratio is the value of the firm divided by the value of the invested assets.

Parameter Parameter value

Boom Recession

Panel A: Firm characteristics

Initial value of after-tax earnings (X) 10 10

Tax advantage of debt (t) 0.15 0.15

Nominal earnings growth rate ðmiÞ 0.0782 �0.0401

Systematic earnings volatility (sX,Ci Þ

0.0834 0.1334

Recovery rate (ai) 0.7 0.5

Panel B: Expansion option parameters of a typical firm (asset composition ratio¼1.6)

Exercise price (K) 310 310

Scale parameter if initiated in boom (s) 1.89

Scale parameter if initiated in recession (s) 2.05

Panel C: Economy

Rate of leaving regime i ðliÞ 0.2718 0.4928

Consumption growth rate ðyiÞ 0.0420 0.0141

Consumption growth volatility ðsCi Þ

0.0094 0.0114

Expected inflation rate ðpÞ 0.0342 0.0342

Systematic price index volatility ðsP,C Þ �0.0035 �0.0035

Idiosyncratic price index volatility ðsP,idÞ 0.0132 0.0132

Rate of time preference ðrÞ 0.015 0.015

Relative risk aversion ðgÞ 10 10

Elasticity of intertemporal substitution ðCÞ 1.5 1.5

9 Our qualitative results do not depend on the ratings of firms.


assets after expansion, evaluated at the correspondingearnings in both regimes. These four optimality condi-tions are smooth-pasting conditions for equity at therespective boundaries:

e0BðDn

BÞ ¼ 0,

e0RðDn

RÞ ¼ 0,

e0BðXn

BÞ ¼ e 0B ðsþ1ÞXn

B�K

yB

� �,

e0RðXn

RÞ ¼ e 0R ðsþ1ÞXn

R�K

yR

� �:

8>>>>>>>><>>>>>>>>:

ð32Þ

We solve this system numerically.For each coupon level c, debt-holders evaluate debt at

issuance anticipating the ex post optimal default andexpansion decisions of shareholders. As debt-issue pro-ceeds accrue to shareholders, shareholders do not careonly about the value of equity, but also about the value ofdebt. Hence, the optimal capital structure is determinedex ante by the coupon level cn that maximizes the value ofequity and debt, i.e., the value of the firm. Denote by f ni ðXÞ

the firm value given optimal ex post default and expan-sion thresholds as determined by the system equations(32). The ex ante optimal coupon of this firm solves

cn

i :¼ argmaxc

f ni ðXÞ: ð33Þ

As indicated in Eq. (33), the optimal initial capital struc-ture depends on the current regime.

4. Results

This section summarizes the model results for indivi-dual firms. Section 4.1 presents the parameter choice. Wedescribe the firm sample in Section 4.2. Next, Section 4.3discusses the properties of the expansion option. Section4.4, finally, analyzes the optimal default policies of indi-vidual firms with different portions of the expansionoptions’ value in the overall value of assets.

4.1. Choice of parameters

Table 1 summarizes our parameter choice. Panel Ashows the firm characteristics that are selected to roughlyreflect a typical BBB-rated Standard & Poor’s (S&P) 500 firm.9

We start with an initial value of the idiosyncratic after-taxearnings X of 10. While this value is arbitrary, neither creditspreads nor optimal leverage ratios depend on this choice. Asis standard in the literature, we set the tax advantage of debtto t¼ 0:15 (Hackbarth, Miao, and Morellec, 2006). Bhamra,Kuehn, and Strebulaev (2010c) estimate growth rates andsystematic volatilities of nominal earnings in a two-regimemodel. Their estimates are similar to those obtained by otherauthors who jointly estimate consumption and dividendswith a state-dependent drift and volatility (e.g., Bonomoand Garcia, 1996). Hence, the real earnings growth ratesðmi,realÞ and volatilities ðsX,C

i,realÞ are chosen such that the

10 In these definitions, we follow, e.g., Baker and Wurgler (2002),

Fama and French (2002), and Daines, Gow, and Larcker (2010). Book

debt is total assets (item 6, AT) minus book equity. Book equity is total

assets minus total liabilities (item 181, LT) minus preferred stock (item

10, PSTKL, replaced by item 56 (PSTKRV) when missing) plus deferred

taxes (item 35, TXDITC) plus convertible debt (item 79, DCVT). The

market value of equity is given by the closing price (item 24, PRCC_F)

times the number of common shares outstanding (item 25, CSHO).11 The regime dependent volatilities and default thresholds also

affect optimal exercise boundaries in boom and recession. We find that

the valuation of earnings is the dominating effect for reasonable

parameter values.


nominal growth rates and systematic volatilities correspondto their empirical counterparts. The relation sX,C

B ¼ 0:0834o0:1334¼ sX,C

R captures the observation in Ang andBekaert (2004) that asset volatilities are lower in boom thanin recession.

Following Acharya, Bharath, and Srinivasan (2007), weassume that recovery rates fall during recession. They reportthat recovery in a distressed state of the industry is lowerthan the recovery in a healthy state of the industry by up to20 cents on a dollar. The reason can be financial constraintsthat industry peers of defaulted firms face as proposed by thefire sales or the industry-equilibrium theory of Shleifer andVishny (1992) or time-varying market frictions such asadverse selection. We choose recovery rates as aB ¼ 0:7and aR ¼ 0:5, respectively. This choice matches the 20 centson a dollar difference in Acharya, Bharath, and Srinivasan(2007) and is close to the standard of 0.6 used in theliterature (Hackbarth, Miao, and Morellec, 2006; Chen,2010). Our qualitative results are insensitive to the choiceof ai as long as aB4aR.

Panel B shows the parameters we use to capturegrowth options. We select an exercise price of K¼310.The choice of a relatively high K is motivated by ourintention to investigate firms that do not exercise theirexpansion option immediately. The scale parameter s for atypical firm is calibrated such that the asset compositionratio at initiation given optimal financing equals theaverage Tobin’s q of 1.6 in our sample of BBB-rated firms.In particular, s is set to s¼1.89 for firms initiated in boomand to s¼2.05 for firms initiated in recession. To analyzegrowth firms with a larger (smaller) portion of optionvalues in the overall value of their assets, we use higher(lower) scale parameters at initiation.

Panel C, finally, lists the variables describing the under-lying economy. The rate of leaving regime i ðliÞ, theconsumption growth rates ðyiÞ, and the consumption growthvolatilities sC

i are estimated in Bhamra, Kuehn, andStrebulaev (2010c). We take the same values for compar-ability. In the described economy, the expected duration ofregime B (R) is 3.68 (2.03) years, and the average fraction oftime spent in regime B (R) is 64% (36%). The inflationparameters are estimated using the price index for personalconsumption expenditures from the Bureau of EconomicAnalysis from 1947 to 2005. We obtain an expected inflationrate ðpÞ of 0.0342, a volatility of the price index of 0.0137,and a correlation between the price index and real nondur-ables plus service consumption expenditures of �0:2575.These parameters imply a systematic price index volatility ofsP,C ¼�0:0035 and an idiosyncratic price index volatility ofsP,id ¼ 0:0132.

The annualized rate of time preference, r, is 0.015; therelative risk aversion, g, is equal to 10; and the elasticityof intertemporal substitution, C, is set to 1.5. This para-meter choice is commonly used in the literature (Bansaland Yaron, 2004; Chen, 2010).

Our choice of parameters implies that real interestrates are rB¼0.0416 and rR¼0.0227 in the baselinespecification. The relative decline in the value of investedassets following a shift from boom to recession is equal to12.61%, which is similar to the one assumed in Hackbarth,Miao, and Morellec (2006).

4.2. Firm sample

Balance sheet and ratings data are collected over theperiod from 1995 to 2008 from Compustat. We use datafor BBB-rated firms. We calculate the quasi-market lever-age of a firm as the ratio of book debt to the sum of bookdebt and market value of equity. Tobin’s q is defined astotal assets plus the market value of equity minus thebook value of equity divided by total assets.10 We deletefinancial and utility firms from the sample. For each firm,we calculate the average of the leverage ratios and Tobin’sq’s over the observation period. Next, we cut extremevalues of both average leverage and Tobin’s q at 1% toavoid having our results driven by outliers. Our samplethen consists of 717 distinct firms. Fig. 1 plots theresulting data points. For the entire sample of BBB-ratedfirms, the mean leverage is 41.83%, and the mean Tobin’sq (asset composition ratio) is 1.59.

4.3. Properties of the expansion option

To understand the implications of our model for creditspreads, it is instructive to first consider some propertiesof the expansion option.

Fig. 2 depicts the equity value maximizing exercisepolicy of the expansion option in a typical firm initiated inboom. Recall that the expansion policy is simultaneouslydetermined with the default policy.

The area above the dashed line is the exercise region inrecession, and the area below the dashed line correspondsto the continuation region. In boom, the regions aredefined analogously with respect to the solid line. Thegraph is drawn for optimal leverage. Exercising the optionat time t entitles the firm to total future after-tax earningsof ðsþ1ÞXt for all tZt . As expected, the endogenousexercise boundaries decrease with s. For example, con-sider initiation in boom: With a scale parameter ofs¼1.89 (that induces an asset composition ratio of 1.6at initiation), the corresponding optimal option exerciseboundaries are Xn

B ¼ 18:26 and Xn

R ¼ 19:55. Setting s to2.73 creates a growth firm with an asset composition ratioof 2.2, and optimal option exercise boundaries ofXn

B ¼ 12:88 and Xn

R ¼ 13:90. Importantly, Fig. 2 also showsthat the expansion option is exercised at lower levels ofthe idiosyncratic earnings X in boom than in recession.Intuitively, the main reason is that the value of the optionof waiting is higher in recession due to the potentialswitch to boom with a higher valuation of earnings.11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

Leverage

Tobi

n’s

q

Fig. 1. Cross section of BBB-rated firms. This scatterplot shows the average leverage and Tobin’s q for each observed BBB-rated firm over the period from

1995 to 2008.

1.4 1.6 1.8 2 2.2 2.4 2.612

14

16

18

20

22

24

26

S

X

Exercise boundary in recessionExercise boundary in boom

Fig. 2. Optimal exercise boundary. The solid line shows the optimal exercise boundary in boom for a range of scale parameters s. The dashed line

represents the corresponding exercise boundary in recession. The graph is drawn for optimal leverage with infinite debt maturity. The baseline

parameter specification from Table 1 is used.


The same qualitative option value properties also hold atnonoptimal leverage levels.

Fig. 3 plots the value of the expansion option as afunction of the after-tax earnings X, using jointly optimalexpansion and default policies.

The option’s value is affected by the current regime.When the asset value jumps due to a regime switch, sodoes the value of the option. Critically, relative valuechanges of expansion options are higher than relativevalue changes of assets in place when the regimeswitches. The reasons are that options represent leveredclaims and that the endogenous exercise boundary ishigher in recession than in boom, as shown in Fig. 2.12

In addition, Fig. 3 shows that both option value functionsare convex, but the value function in boom is steeper than

12 Relative value changes are determined in Appendix A.2. In

untabulated results, we confirm numerically that the relative value

changes are higher for expansion options than for the underlying assets

in place for plausible parameter values.

the one in recession. Therefore, the expansion option’s valueis less sensitive to the underlying earnings in recession thanin boom. Intuitively, the exercise boundary increases and theearnings’ drift decreases in recession, which drives optionsout-of-the money. As a consequence, an expansion optionrepresents a less levered claim in bad times. While inrecession the volatility of X is higher, the sensitivity of agrowth option’s value to changes in the earnings is lower. Asdiscussed in Section 4.4, this lower sensitivity attenuates theincrease in the equity-holder’s default option due to a highervolatility of X during recession.

4.4. Optimal default policy

This subsection explains how the optimal default policy isaffected by the presence of growth options in the value offirms’ assets. To keep the intuition tractable, we do notcomment on the (minor) impact of the exercise boundarieson default thresholds, which arises due to the simultaneousoptimization of the expansion and default policy.

0 2 4 6 8 100

20

40

60

80

100

120

140

X

Opt

ion

valu

e

Expansion option in recessionExpansion option in boom

Fig. 3. Option values. The solid line represents the value of the expansion option in boom for a range of starting earnings between 0 and 10. The dashed

line shows the corresponding values of the same option in recession. The graph is drawn for optimal leverage with infinite debt maturity. The baseline

parameter specification from Table 1 is used.

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

2

2.5

3

3.5

4

Asset composition ratio

Default threshold in boomDefault threshold in recession

X

Fig. 4. Default policy and asset composition. The solid line represents the default threshold in boom for a range of asset composition ratios. The dashed

line shows the default threshold in recession. The graph is drawn for constant leverage (41.83%) at each point. Debt maturity is assumed to be infinite.

The baseline parameter specification from Table 1 is used, with s being varied to generate the desired asset composition ratio.

13 When the scale parameter is changed but the coupon is left

constant, default thresholds are not directly comparable. The reason is

that the total asset value increases with s for every X. Considering


For all firms – those with and those without an expansionoption – the optimal default policy is determined byrecognizing that, at any time, shareholders can either makecoupon payments and retain their claim together with theoption to default or forfeit the firm in exchange for thewaiver of debt obligations. When the economy shifts fromboom to recession, the present value of future after-taxearnings declines mainly because firm earnings have a lowerdrift and because they become both more volatile and morecorrelated with the market. This present value declinereduces the continuation value (the expected value fromkeeping the firm alive) for equity-holders, inducing them todefault earlier (at higher earnings levels) in recession. Werefer to this effect as the value effect. A high earningsvolatility in recession makes the option to default morevaluable, which defers default in bad times. This is thevolatility effect. As in the models for invested assets ofBhamra, Kuehn, and Strebulaev (2010c) and Chen (2010),the value effect usually dominates the volatility effect,generating higher default thresholds in recession, i.e., leading

to countercyclical default thresholds. Countercyclical defaultthresholds together with a high volatility in bad times implycountercyclical default probabilities, consistent with empiri-cal evidence (Chava and Jarrow, 2004; Vassalou and Xing,2004). In addition, default losses are empirically reported tobe higher in recession because many firms experience poorperformances during such times. Combined with highermarginal utilities in bad times, these mechanisms raise thepresent value of expected default losses for bondholders,which leads to higher credit spreads and lower optimalleverage ratios than in standard contingent claim models.

Fig. 4 draws the equity value maximizing defaultpolicy of levered firms initiated in boom. The graph showsdefault thresholds for a range of asset composition ratios.Leverage is held constant at 41.83%.13 The solid line


shows the default threshold in boom; the dashed line theone in recession. For example, for a firm with onlyinvested assets, the optimal default thresholds areDn

B ¼ 2:45 and Dn

R ¼ 2:69. For an average firm with an assetcomposition ratio of 1.6, they are Dn

B ¼ 3:19 and Dn

R ¼ 3:55,and for a growth firm with an asset composition ratio of2.2, they are Dn

B ¼ 3:52 and Dn

R ¼ 3:94. In the no-defaultregion above the line corresponding to a given regime, thecontinuation value for equity-holders exceeds the defaultvalue and it is optimal for shareholders to keep injectingfunds into the firm.

Two points from Fig. 4 are particularly noteworthy.First, the optimal default thresholds increase as the assetcomposition ratio increases, inducing a higher defaultprobability. This finding evolves from the observationthat growth options represent levered claims, which arerelatively more sensitive than invested assets to a givendecrease in X. Second, while all firms are more likely todefault in recession than in boom, the increasing distancebetween Dn

B and Dn

R for larger asset composition ratiosindicates that the countercyclicality of default boundariesis particularly pronounced for growth firms. The reason isthat due to the higher relative value change of growthoptions upon a regime switch, the value effect is strongerfor a firm with a high asset composition ratio. In addition,because options represent less levered claims in recessionthan in boom, the increase in the equity-holders’ defaultoption – due to the higher volatility of X when the regimeswitches to recession – is attenuated for growth firms. Inother words, the volatility effect, which tends to decreasethe distance between the default thresholds, is weaker forfirms with larger expansion options.

5. Aggregate dynamics of leverage, asset composition,investment, and defaults

To validate our structural equilibrium framework withintertemporal macroeconomic risk and investment, weanalyze the dynamic properties of our model-impliedeconomy. In this section, we qualitatively compare theaggregate predictions for the entire economy with empiri-cally reported capital structure, investment, and defaultpatterns.

5.1. Simulation

We generate a dynamic economy of average firmsimplied by our model. We consider 1,000 identical firmswith infinite debt maturity. Initially, each firm’s after-taxearnings are X¼10, and the option scale parameter isassumed to be s¼1.89 if the firm’s initial regime is boomand s¼2.05 otherwise. These choices of s imply an assetcomposition ratio of 1.6 in both states at initiation, givenoptimal leverage. Firms receive the same macroeconomicand inflation shocks but experience different idiosyncraticshocks. Each firm observes its current earnings as well as

(footnote continued)

constant leverage assures that the considered coupon changes consis-

tently with the increase in the total asset value when we alter s.

the current regime on a monthly basis and behavesoptimally. If the current earnings are below the corre-sponding regime-dependent default threshold, the firmdefaults immediately; if the current earnings are abovethe corresponding regime-depending option exerciseboundary, the firm exercises its expansion option; other-wise, the firm takes no action.

In our model, firms have a growth option, which can beexercised only once. To maintain a balanced sample offirms, and to avoid having the average asset compositionratio systematically trending toward the one of a firmwith only invested assets when we simulate the economyover time, we exogenously introduce new firms. Inparticular, we substitute each defaulted or exercised firmby a new firm whose growth option is still intact. Newfirms have initial after-tax earnings of X¼10 and anoption scale parameter s according to the current regimeas described above.

To ensure convergence to the long-run steady state, wefirst simulate the economy for 100 years. The startingperiod for the reported results is the final period of thefirst 100 years of simulation. Next, we simulate the modelfor 200 years and present the aggregate dynamics.

5.2. Results

We start by discussing the cyclicality of leverage.Hackbarth, Miao, and Morellec (2006) generate counter-cyclical optimal leverage ratios in their macroeconomicmodel. As in our framework, the optimal coupon rate ofdebt initiated in boom exceeds the one in recession. At thesame time, the value of assets is greater in boom. Thesecond effect dominates the first, generating the counter-cyclicality in optimal leverage. We also incorporate theempirical fact that asset volatility is regime-dependent.Because asset volatility decreases in boom and increasesin recession, our optimal coupon rate varies morethan in Hackbarth, Miao, and Morellec (2006) when theregime changes. With this extension, the change in thevalue of optimal debt dominates the change in the valueof assets, generating procyclical optimal leverage ratiosfor realistic parameter values, in line with Covas andDenHaan (2006) and Korteweg (2010). Fig. 5 plots thesimulated market leverage in the economy. Shadedareas represent recessions. Even though our optimalinitial leverage ratios are procyclical, the simulated timeseries shows that actual aggregated market leverage iscountercyclical. The reason is that when firms are stuckwith the debt issued at initiation, the equity valuedeclines more than the debt value during recessions,which tends to increase leverage in bad times. Thisprediction conforms to Korajczyk and Levy (2003), whoshow that unconstrained firms’ leverage ratios varycountercyclically.

Fig. 6 shows the time series of the aggregateasset composition ratio in the simulated economy. Asexpansion options are more sensitive to the underlyingstochastic processes than invested assets, the ratiobehaves procyclically, as reported in the literature.

We investigate aggregate default rates in Fig. 7. Simu-lated default rates are countercyclical, consistent with the

Fig. 5. Time series of market leverage. The solid line shows the aggregate market leverage of the simulated economy. The shaded areas represent times

of recession. Standard parameters from Table 1 are used. Debt maturity is assumed to be infinite.

Fig. 6. Time series of asset composition ratio (ACR). The solid line shows the aggregate ACR of the simulated economy. The shaded areas represent times


14 The distance to default in the aggregate economy of firms with

only invested assets is trending over time. The reason is that firms that

default are replaced, but there are no option exercises after which well-

performing firms could be replaced. Consequently, we do not compare

absolute default rates of the two economies, but rather the fraction of

defaults occurring in each regime.


empirical fact that most defaults occur during economicrecessions. In addition, the graph shows several spikes indefault rates that occur right at the time when theeconomy enters into a recession, consistent with theempirical evidence in Duffie, Saita, and Wang (2007)and Das, Duffie, Kapadia, and Saita (2007) (see, e.g.,around years 50 and 90). Defaults can occur because theidiosyncratic earnings reach the default threshold in agiven regime or due to a change of the macroeconomicregime from boom to recession. The clustered defaultwaves occur due to an increase in firms’ default thresh-olds upon such a regime change. All firms between thetwo thresholds default simultaneously when the regimeswitches to recession, even though their earnings do notexhibit instantaneous regime-induced changes. After suchwaves of default, the default frequency tends to remainhigh during recessions.

As a refinement of this general result, we expect thatthe tendency to default during recession should be parti-cularly pronounced for firms with high expansion options.This prediction is suggested by the fact that the degree ofcountercyclicality of default thresholds is positively

related to the initial asset composition ratio. We investi-gate the propensity to default during recession in adynamic, simulation-based setting by counting defaultrates of two separate aggregate economies. The first one isdesigned as above, consisting of firms with both assets inplace and growth options, such that the asset compositionratio at initiation is 1.6. The second setting consists offirms with only invested assets. To construct a number ofcross-sectional distributions of firms, we first simulate 20dynamic economies for ten years. Using each economyobtained at the end of the first ten years, the default ratesin both regimes are observed for 50 subsequent simula-tions of the following 20 years, resulting in a total of 1,000simulations. The average percentage of defaults thatoccurs during recession is then calculated.14 We find that

Fig. 7. Monthly default rates. The solid line shows the percentage of firms that default during a given month in the simulated economy. The shaded areas

represent times of recession. Standard parameters from Table 1 are used. Debt maturity is assumed to be infinite.

Fig. 8. Monthly expansion rates. The solid line shows the percentage of firms that exercise their expansion options during a given month in the simulated

economy. The shaded areas represent times of recession. Standard parameters from Table 1 are used. Debt maturity is assumed to be infinite.


in the first economy, on average, 75.41%, 76.79%, and 77.66%of total defaults of firms with assets in placeand growth opportunities occur during recession over five,ten, and 20 years, respectively. In the economy in whichfirms only have invested assets, the corresponding numbersare considerably smaller at 66.40%, 71.66%, and 73.71%.

This finding is also related to the observation that, onaverage, growth firms have lower recovery rates thanvalue firms (Cantor and Varma, 2005). The standardargument offered by Shleifer and Vishny (1992) is thatgrowth firms as potential buyers of growth assets havelittle cash relative to the value of assets. Hence, they arelikely to be themselves credit-constrained when othergrowth firms sell their assets upon default, which lowersrecovery rates. Our model delivers an alternative expla-nation: Growth options in the value of firms’ assets createa propensity to default during recession, when recoveryrates are low.

A significant literature suggests that business cycleshocks common to all firms play a crucial role in

explaining aggregate investment. In particular, evidenceexists that aggregate investment is characterized by bothepisodes of very intense investment activity and periodsof very low investment activity (Doms and Dunne, 1998;Oivind and Schiantarelli, 2003). Moreover, aggregateinvestment and the probability of investment spikes arestrongly procyclical (Barro, 1990; Cooper, Haltiwanger,and Power, 1999). Our model reflects these features. First,when the regime switches from recession to boom, firmsin the region between the two investment boundariesexercise their expansion option simultaneously by invest-ing K. Fig. 8 shows that investment spikes often occurupon such regime switches (see, for example, around year35 or year 60). After these spikes, simulated investmentrates tend to remain high during boom due to the positivedrift of the earnings. Hence, we observe procyclicalinvestment spikes followed by higher investment activityduring booms. At the other end, investment activity oftendries out when the economy switches from boom torecession, because the optimal exercise boundary jumps

Fig. 9. Time series of credit spread. The solid line shows the average credit spread of the simulated economy. The shaded areas represent times


Table 2Target credit spreads and default probabilities.

This table lists our target credit spreads and default probabilities.

Panel A reports annualized target average credit spreads for various debt

maturities. They are calculated as the BBB-rated bond minus Treasury

yields of Davydenko and Strebulaev (2007) and Duffee (1998), net of a

35.5% nondefault component. Credit spreads are quoted in basis points.

Panel B reports average cumulative issuer-weighted default rates in

percent for BBB-debt over five, ten, and 20 years for US firms (Moody’s,

2010).

Panel A: Target credit spreads (basis points)

Debt maturity

Spread calculation Short Medium Long

Davydenko and Strebulaev (2007) 74 74 92

Duffee (1998) 96 95 128

Panel B: Historical BBB default probabilities (percent)

Years

Time period Five Ten 20

1920–2009 3.136 7.213 13.684

1970–2009 1.926 4.851 12.327


up and the expected earnings’ drift turns negative. Ourmodel also predicts that observed investment wavesshould be mainly driven by firms with high expansionoptions.

Finally, we plot simulated average credit spreadsin Fig. 9. Credit spreads are calculated as ðc=diðXÞÞ�rp

i ,in which rp

i is the perpetual risk-free rate defined in Eq.(28). Consistent with the empirical literature (Fama andFrench, 1989), we find countercyclical credit spreads.When the economy stays in boom, credit spreads tendto decline as distances to default increase due to thepositive expected drift of the earnings and the lowerdefault threshold. Conversely, in recession, credit spreadsrise as distances to default tend to decline and thevolatility increases.

6. Quantitative implications and empirical predictions

In this section, we discuss the quantitative implica-tions and empirical predictions of our model. The atten-tion is restricted to BBB-rated firms because it has beenargued that the pricing of very high-grade investmentfirms is dominated by factors other than credit risk suchas liquidity risk or a tax component (Longstaff, Mithal,and Neis, 2005; De Jong and Joost, 2006). We start bydetermining target observed average credit spreads.Duffee (1998) estimates an average yield spread in theindustrial sector between BBB-rated bonds and Treasuryyields of 198, 148, and 149 bps for bonds with a meanmaturity of 21 years (long), 8.9 years (medium), and 4.7years (short), respectively. Davydenko and Strebulaev(2007) report somewhat lower spreads of 143 bps forbonds with 15–30 years (long), 115 bps for seven to 15years (medium), and 115 bps for one to seven yearsmaturity, respectively.15 From these spreads, we subtract35.5% to reflect the results in Longstaff, Mithal, and Neis

15 The estimates of short and medium maturities in Huang and

Huang (2002) are higher because of the embedded call options in the

corporate bond sample and the inclusion of two recessions with high

spreads.

(2005) and Han and Zhou (2008), who find nondefaultcomponents in BBB bond yields of 29% and 42%, respec-tively. We arrive at a plausible target range of around92–128 bps for long maturities, 74–95 bps for mediummaturities, and 74–96 bps for short maturities.16 Panel Ain Table 2 tabulates these target credit spread ranges.In Panel B, we report empirical default rates of BBB-rateddebt over five, ten, and 20 years from Moody’s (2010).

We discuss the implications of our model for creditspreads and leverage along two dimensions. First, wefollow the traditional way of investigating a typicalindividual firm. Second, we implement an approach

16 We recalculate target ranges by subtracting the absolute non-

default component for BBB firms of 61.8 bps reported in Han and Zhou

(2008) or by subtracting the 29% reported in Longstaff, Mithal, and Neis

(2005) for an earlier sample period. Our model’s performance does not

depend on the exact definition of targets.

Table 3Implications for credit spreads.

This table demonstrates the implications of our model for credit

spreads of BBB-rated firms. The asset composition ratio is defined as

firm value divided by the value of the invested assets. Parameters are

taken from Table 1, and the leverage is set equal to 41.83%. In the one-

regime model, parameters are chosen to match their unconditional

mean. The standard two-regime model is adapted from Bhamra,

Kuehn, and Strebulaev (2010c). Annualized credit spreads for various

debt maturities are calculated as the coupon divided by the debt value,

minus the yield on an otherwise identical risk-free bond. They are

quoted in basis points. Credit spreads of typical firms in Panels B and C

are obtained by weighting the credit spreads in boom and recession by

the average expected time spent in each regime, respectively. Panel D

contains the average credit spreads of our simulated true cross section of

BBB-rated firms.

Credit spreads

Debt maturity (years)

Asset composition Five Ten 20

Panel A: One-regime model

Average firm 18 29 41

Panel B: Standard two-regime model with only invested assets

Average firm 35 56 78

Panel C: Two-regime model with expansion option

Invested assets (asset composition ratio¼1) 24 39 55

Average firm (asset composition ratio¼1.6) 45 66 84

Growth firm (asset composition ratio¼2.2) 47 69 86

Panel D: Two-regime model with true cross section

Average 57 81 100


similar to the one proposed by Bhamra, Kuehn, andStrebulaev (2010c) in which credit spreads and leverageratios are calculated as cross-sectional averages based ona simulation of the empirical distribution of BBB-ratedindividual firms.

6.1. Credit spreads

We first study a typical firm and then turn to the truecross section of firms in the economy.

18 Bhamra, Kuehn, and Strebulaev (2010c) use higher recovery rates

and lower leverage, and they do not model the impact of principal

6.1.1. Typical firm with endogenous default boundary

Credit spreads for various models on newly issuedcorporate debt are calculated in Table 3 for five (short),ten (medium), and 20 (long) years maturity.17 We followthe standard approach in structural models by calibratingthe idiosyncratic earnings volatility such that the totalasset volatility is approximately 25% in each model, theaverage asset volatility of firms with outstanding ratedcorporate debt (Schaefer and Strebulaev, 2008). In addi-tion, we fix leverage at the average ratio of 41.83% in ourBBB-firm sample.

17 The value of a finite maturity risk-free bond is given in Appendix

A.4, Eq. (112), in which c is replaced by cþmp and ri

nis replaced by

rni þm.

Importantly, the default boundaries and expansionthresholds are assumed to be chosen optimally byequity-holders, as we are interested in whether our modelcan generate both realistic prices of corporate claims andrealistic endogenous default and expansion rates. Specify-ing default boundaries exogenously such that a model’sactual default probabilities match the data (as done inChen, Collin-Dufresne, and Goldstein, 2009 or Huang andHuang, 2002) not only substantially dilutes the value ofthe option to default, but also distorts the value of theexpansion option because the latter depends on thedefault policy.

It is well known that structural models of defaulttypically generate credit spreads that are too low com-pared with their empirical counterpart. To illustrate thispoint, we first analyze the model without business cyclerisk in Panel A of Table 3. The expected drifts andsystematic volatilities of earnings and consumption areset equal to their unconditional means. Panel A showscredit spreads for different maturities of the standardstructural model of Leland (1998). The empirical targetcredit spreads in Table 2 are about five times larger forthe short maturity and about three times larger for themedium and long term than those predicted by thestructural model.

Bhamra, Kuehn, and Strebulaev (2010c) and Chen(2010) derive structural multi-regime models for typicalfirms that consist of only invested assets. We closelyreplicate their approach for an average firm within atwo-regime model. To match the asset volatility of 25%,the idiosyncratic earnings volatility is set to sX,id ¼ 0:21.Panel B reports unconditional credit spreads, calculated asa weighted average of the state-dependent credit spreads,in which the weights correspond to the long-run distribu-tion of the Markov chain. For comparability to our settingwith expansion options, the results without debt restruc-turing are presented. While the credit spreads for typicalfirms of 35, 56, and 78 bps for five, ten, and 20 yearsmaturity, respectively, are clearly higher than in the oneregime model, they are still considerably below theirtargets.18

Next, we investigate our model with expansionoptions for a typical BBB-rated firm. For a given idiosyn-cratic earnings volatility, firms with different asset com-position ratios have different total asset volatilities duethe inherent leverage of their expansion option. More-over, a firm’s asset volatility is not constant over time, asits option’s moneyness changes when X moves toward oraway from the exercise boundary. To obtain the averagevolatility for a certain rating class, the standard approach inthe literature is to average the calculated asset volatilitiesover all firms with the same rating (Vassalou and Xing,2004; Duan, 1994; Schaefer and Strebulaev, 2008). We

repayments on default thresholds, which results in marginally lower

credit spreads in their static case. Chen (2010) obtains larger ten-year

credit spreads in a model with nine states and a dynamic capital

structure but uses higher leverage and a cash flow volatility that induces

a much higher asset volatility than empirically observed.


calibrate the idiosyncratic volatility sX,id to the empiricallyreported average asset volatility of 0.25. Given an idiosyn-cratic volatility sX,id, we simulate model-implied samples ofBBB-rated firms over ten years and calculate the resultingaverage asset volatility. (Details on the simulation can befound in Appendix A.5.1.) The calibration yieldssX,id ¼ 0:168, which ensures that the average asset volatilityof our simulated BBB-rated firms with expansion optionscorresponds to its empirical counterpart.19

Panel C of Table 3 shows the resulting credit spreadsfor typical firms. Several aspects are noteworthy aboutthese results. Our model increases the unconditionalcredit spreads of an average firm for five, ten, and 20years from 18 bps to 45 bps (þ150%), from 29 bps to66 bps (þ128%), and from 41 bps to 84 bps (þ105%),respectively, compared with the one regime model inPanel A. To understand this large effect, recall first thatmacroeconomic models generate larger credit spreadsthan one-regime models because recessions are times ofhigh marginal utility, so that default losses that occur duringthese times affect investors more. An important economicimplication is that the average duration of bad times in therisk-neutral world is longer than in the actual world. Becausethe representative agent uses risk-neutral and not actualprobabilities to account for risk and to compute prices, creditspreads are larger and the agent behaves more conserva-tively than historical default losses imply. Second, if firmshave a higher tendency to default in recession, this discre-pancy increases due to the higher risk premium. Our modelshows that because of the strong sensitivity of option valuesto regime switches, and because they are less sensitive to theunderlying earnings during recession, the countercyclicalityof default thresholds is more pronounced for firms withlarger growth options. The resulting stronger countercycli-cality of the default probability of growth firms thus drivesup their credit spreads. As can be seen in row 2 of Panel C,the credit spreads for an average firm, consisting of bothinvested assets and growth options, are 45 bps, 66 bps, and84 bps for debt maturities of five, ten, and 20 years,respectively. This is, respectively, 29%, 18%, and 8% higherthan the credit spreads of an average firm in the standardmacroeconomic model with only invested assets.20

Besides the fact that they generate too low creditspreads, another problem of existing structural modelsis that the implied term structure of credit spreads atinitiation is much steeper than its empirical counterpartfor a typical firm. The reason is that the implied spreads

19 We also repeat this exercise with different specifications, such as

alternative simulation length and debt maturity. The resulting idiosyn-

cratic volatilities are fairly insensitive to these variations. An alternative

approach is to calibrate the idiosyncratic volatility to the cumulative

default probability of BBB-rated firms (Chen, 2010). This procedure,

however, usually leads to asset volatilities that are higher than the ones

empirically observed.20 We cannot directly compare the results for invested assets in

Panel C with the ones for average firms in Panel B, even though the latter

consist of only invested assets. The reason is that, in our model, the

idiosyncratic volatility is calibrated such that the asset volatility of the

entire sample of BBB-rated firms matches 0.25, whereas, in Panel B, sX,id

is chosen such that firms with only invested assets have an asset

volatility of 0.25.

are particularly low at the short end. Most existing studieswith macroeconomic models use the default thresholds ofinfinite maturity debt (that is, debt without principalrepayments) to numerically calculate the risk-neutraldefault probability for each maturity. As the credit riskliterature identifies firms’ debt maturity as an importantdeterminant of credit risk (Gopalan, Song, and Yerramilli,2010; He and Xiong, 2012), we endogenously deriveoptimal default thresholds also for finite debt maturityfollowing the approach of Leland (1998). Due to thecontinuous principal repayments, these thresholds areconsiderably higher for short maturities than for infinitedebt, resulting in larger credit spreads at the short end.The resulting term structure of credit spreads for anaverage firm in Panels A, B, and C is consequently flatterand, hence, closer to the shape observed in target spreadsthan when using default thresholds of infinite maturitydebt.21

The rows in Panel C of Table 3 identify the cross-sectional relation between the asset composition ratioand credit risk. To tease out the effect of growth optionson credit spreads, we vary the asset composition ratio byaltering s. As raising s increases the value of the expansionoption, we simultaneously adapt the coupon to maintaina constant leverage of 41.83%.22 This exercise shows thatthe asset base of the firm is an important driver of creditrisk, implying a positive relation between the portion ofgrowth options in the value of a firm’s assets and the costsof debt. In particular, altering the asset composition ratioof a firm from 1 to 2.2 increases credit spreads by about56–96%, depending on the debt maturity. This effect isremarkable given that we solely vary the assets’ charac-teristics. It arises for two reasons in our model. First,because options are levered, and due to the endogenousinvestment boundary, expansion options are more sensi-tive to the underlying uncertainty and, hence, morevolatile. This higher volatility drives up the default prob-ability of growth firms. Second, a higher portion of theexpansion option’s value in the overall asset value of afirm induces a higher countercyclicality of the defaultprobability, which raises expected default costs. Thehigher default probability and larger default costs bothincrease the costs of debt for growth firms.

While firms with growth options generally have ahigher credit spread than firms with only invested assets(ceteris paribus), credit risk is concave in the assetcomposition ratio. This concavity occurs because firms

21 We use default boundaries for the appropriate debt maturities in

both Panels B and C to highlight the pure effect of expansion options on

credit spreads.22 Alternatively, changing both s and K to alter the asset composi-

tion qualitatively retains the aggregate and cross-sectional predictions.

Holding s constant while only varying K implies large decreases in the

option exercise boundaries for relatively small increases of the asset

composition ratio. In the extreme, a firm with a very low K exercises its

expansion option almost immediately. In essence, credit spreads then

virtually mirror those of a firm with only invested assets, diluting the

model’s cross-sectional predictions. Also, any variation in K changes the

costs of investment. By only varying s, we instead avoid that our results

are driven by different sizes of the expected financing in case of equity-

financed investment costs.

23 Other debt maturities yield virtually identical results for the

matching accuracy.24 The market leverage is matched with an average distance of

0.0248. The average percentage distance of the asset composition ratio

of 0.0549 is larger. This number is driven by a few firms with unusually

high asset composition ratios. As they would optimally exercise their

expansion option immediately in our model, these firms are matched

with model firms with a somewhat lower asset composition ratio. We

expect a minor impact of this limitation on our results, because firms

with unusually high asset composition ratios also have very low

leverages and, hence, are not driving our average credit spreads.25 We follow Bhamra, Kuehn, and Strebulaev (2010c) in measuring

average credit spreads over a five-year period. During longer periods,

many firms could deviate substantially from the initial average distribu-

tion and would, therefore, not be BBB-rated anymore.


with a larger asset composition ratio are closer to theirexercise boundary, where credit spreads also reflect thatthe asset volatility and the countercyclicality of thedefault thresholds decrease when a firm exercises itsexpansion option.

Our model rationalizes empirical properties of thecross section of credit risk. For example, Davydenko andStrebulaev (2007, Table VI) find that market-to-book assetvalues, the ratio of research and development expenses tototal investment expenditure, and one minus the ratio ofnet property, plant, and equipment to total assets are allsignificantly and positively related to credit spreads.Similarly, Molina (2005, Table II) shows that firms witha higher ratio of fixed assets to total assets have lowerbond yield spreads and higher ratings. This evidenceimplies that, empirically, even after controlling for mostfactors relevant to credit risk in standard structuralmodels, credit spreads are higher for growth firms. Hence,while an average firm with valuable growth optionsexhibits, for example, a different tax advantage of debtor payout ratio than a firm that consists of only investedassets, simple variation of such input parameters wouldnot explain these findings. What is needed to address theaggregate puzzle and the cross-sectional evidence is amodel that generates higher explained credit risk thanstandard models for a given level of input parameters. Ourmodel delivers this result.

6.1.2. True cross section

The previous subsection calculates credit spreads of atypical individual firm, which is consistent with thehistorically observed average input parameters of firmsin the same rating class of which the individual firm isrepresentative.

In this subsection, inspired by the work of Bhamra,Kuehn, and Strebulaev (2010c), we employ a simulationapproach to capture the dynamics of the cross-sectionaldistribution of firm characteristics. The central insight ofour approach is that BBB-rated firms are very differentwith respect to both leverage and asset composition ratiosand that credit spreads and default rates are highlynonlinear in these characteristics. Moreover, the previoussubsection considers credit spreads solely at debt issu-ance points, when the principal corresponds to the marketvalue of debt. The majority of empirically reportedspreads are, however, based on observations made attimes when debt is not being issued. To capture theimpact of these issues, it is important to calculate creditspreads and default rates for a simulated sample of firmsthat matches the observed empirical distribution, i.e., thetrue observed cross section of BBB-rated companies. Theresulting average of simulated credit spreads can then becompared with the empirical average credit spread.Simultaneously, the approach allows us to verify whetherthe default probabilities implied by our model correspondto the reported historical default probabilities of BBB-rated firms.

To obtain the implications of the true cross section ofBBB-rated firms, we start by generating a distribution offirms implied by the model. In particular, we set up a gridof optimally leveraged firms with scale parameters s

ranging from zero up to the largest possible value suchthat the option is not exercised immediately. The step sizeis 0.05, and 50 identical firms are considered for eachvalue of the option scale parameter. Earnings paths of allfirms are then simulated forward over ten years, resultingin a model-implied economy populated by more than3,000 firms. This economy has a broad range of leverageratios and asset composition ratios.

In a second step, we match our historical distributionof BBB-rated firms with its model-implied counterpart.For each observation in the average empirical crosssection, we select the firm in our model-implied economywith the minimum distance regarding the percentagedeviation from the target average market leverage andasset composition ratio. The matching is generally veryaccurate. Considering a debt maturity of ten years yieldsan average Euclidean distance of 0.0648, with the 85%quantile being 0.0865.23 That is, on average, only 15% ofthe firms are matched with the root of the sum of thesquared percentage deviations being larger than 8.65%.24

While our initial model-implied economy potentiallycontains firms with different ratings, the describedmatching procedure allows us to construct a cross-sec-tional distribution of model-implied firms that closelyreflects its empirical BBB-rated counterpart.

Next, earnings paths of the 717 matched BBB model-firms are simulated forward for 20 years on a monthlybasis. This simulation is repeated 50 times.

The outcome of both the matching and the forwardsimulation of the matched sample also depends on theparticular realizations of the idiosyncratic shocks and thestates of the economy in the first simulation step. Hence,to explore the distributional properties of our results, theentire procedure is conducted 20 times, which results in atotal of one thousand simulations. Details on the simula-tion are given in the Appendix A.5.2.

Panel D of Table 3 summarizes the results. The averagecredit spreads, calculated during five years after thematching, are 57 bps for five years, 81 bps for ten years,and 100 bps for 20 years.25 Hence, our model closelymatches the historical levels reported in Table 2 for tenand 20 years. Five-year credit spreads are somewhatlower than their target. We also measure the cyclicalityof credit spreads. Average ten-year credit spreads, forexample, are 59 basis points during boom and 115 duringrecession. As expected, they are strongly countercyclical.

Table 4Implications for default rates.

This table shows the simulated cumulative default rates in percent of

our true cross section of BBB-rated firms. Panels A to D vary the

underlying debt maturity used to calculate the default thresholds in

our model.

Default rates

Years

Descriptive statistic Five Ten 20

Panel A: Infinite debt maturity

Average 2.24 6.54 13.76

Median 0.84 3.07 9.34

25th percentile 0.42 1.12 3.35

75th percentile 2.37 9.07 19.80

Panel B: 20 years debt maturity

Average 4.39 10.72 18.67

Median 1.81 6.00 13.81


75th percentile 4.88 14.92 26.57

Panel C: Ten years debt maturity

Average 6.30 13.61 21.92

Median 2.79 8.09 16.95


75th percentile 7.39 19.39 32.50

Panel D: Five years debt maturity

Average 8.05 16.32 25.09

Median 4.04 10.60 20.22


75th percentile 10.32 23.36 36.75

26 The standard deviation of the sojourn times generated by Markov

chains is large. In our model, long sojourn times in recession cause high

default rates for some sample paths. As default rates are nonlinear in the

distance to default, long sojourn times in boom do not counterbalance

the high rates in recession.27 The difference depends on the initial regime and the debt

maturity. For example, with infinite debt maturity, the difference in

optimal initial leverage between a firm with only invested assets and a


Importantly, average credit spreads for the simulatedtrue cross section are considerably higher than the ones ofa typical firm at initiation. There are two reasons for thisresult. First, some firms are near default, and creditspreads are convex in the distance to default. Second,the market value of debt corresponds to the principal atinitiation. In practice, however, firms are not at initiationmost of the time. The actual market value of debt, there-fore, often underestimates the burden from the principalrepayments, and especially so for firms approaching theirdefault boundary. The reason is that the market value canhardly go beyond the principal as it is bounded above bythe value of risk-free debt but can easily reach valuesbelow the principal when earnings deteriorate. Our simu-lation of the true cross section captures these asymmetricdeviations over time, resulting in higher average creditspreads than those of firms observed at initiation. Com-pared with Bhamra, Kuehn, and Strebulaev (2010c), theadditional credit spreads generated from simulating thetrue cross section are lower, because we do not incorpo-rate debt restructuring.

To verify whether our model generates default ratescorresponding to the empirically reported default frequenciesfor realistic debt maturities, we count cumulative defaultrates in the simulated true cross section. The model-impliedaverage and median cumulative default rates over severalyears are reported in each panel of Table 4. Panel A presentsdefault rates over five, ten, and 20 years from simulations

with firms issuing infinite maturity debt. Panels B, C, and Dshow default rates from simulations with firms issuing finitematurity debt. Due to the principal repayments, defaultthresholds of firms with finite maturity debt are considerablyhigher than those of firms with infinite maturity debt.Simulated credit spreads are consistent with a range ofrealized ex post default rates, as observed default rates varydepending on a particular realization of good and bad states.Therefore, we also report the 25% and 75% percentiles of thedistribution.

Empirically, Datta, Iskandar-Datta, and Patel (2000)report a mean maturity of initial public offering bondsof 12 years, Guedes and Opler (1996) obtain an averagematurity of 12.2 years for seasoned debt offers, andDavydenko and Strebulaev (2007) measure a mean timeto maturity of BBB bonds in the industrial sector of 9.51years. Panel C of Table 4 shows that, when assuming thatfirms have a debt maturity of ten years, our model-implied median default rates over five, ten, and 20 yearsare very close to the historical default probabilitiesobserved from 1920 to 2009 reported in Table 2. Hence,for a realistic debt maturity, our median economy isconsistent with historical default frequencies of BBB-rated firms. The average default rates are somewhatlarger than their targets due to a few realizations withlong sojourn times in recession, resulting in high defaultrates.26 Panels A and D show that while the generatedrates tend to be too low in Panel A, but too large in PanelD, historical default frequencies still fall within the25–75% range of model-implied median default rates formost years.

The large difference between Panel A and D in bothaverage and median default rates illustrates that debtmaturities and the associated default thresholds have animportant effect on model-implied default rates. Therefore,incorporating a realistic debt maturity is important whencalibrating models with endogenous default thresholds.

In sum, our results demonstrate that the average creditspreads implied by our model for the true cross section aresimultaneously consistent with historically observed averageasset volatilities and, especially, for typical debt maturities,with default rates reported for BBB-rated firms.

6.2. Leverage

This subsection analyzes the features of leverage ratiosresulting from our model. We first investigate how growthoptions affect the initial choice of optimal leverage in ourmodel. At initiation, a firm consisting of only invested assetshas an optimal leverage that is between 4 and 5 percentagepoints higher than the one of a typical firm with an assetcomposition ratio of 1.6 for all debt maturities.27 The reason

Table 5Implications for leverage.

This table demonstrates the implications of our model for the leverage

features of the true cross section of BBB-rated firms. Leverage ratios

(given in percent) are calculated as the market value of debt divided by

the market value of the firm. The asset composition ratio is defined as

firm value divided by the value of the invested assets. Parameters are

taken from Table 1. The debt maturity is assumed to be ten years.

Descriptive statistic Leverage feature

Panel A: Unconditional leverage

Leverage

Average 40.89

Panel B: Conditional leverage

Leverage

Boom Recession

Average 36.94 46.20

Median 34.36 44.19

25th percentile 22.49 29.88

75th percentile 48.51 60.39

Panel C: Regression of leverage on the asset composition ratio

Coefficient

Average �0.184

Median �0.184

25th percentile �0.268

75th percentile �0.096


is that a higher asset composition ratio increases the defaultprobability, particularly so in recessions in which defaultlosses are larger and harder to bear. Due to the resultinghigher costs of debt, firms with growth options optimallyselect lower initial leverage.

As argued by Bhamra, Kuehn, and Strebulaev (2010b),however, it can be misleading to make quantitativestatements simply based on optimal leverage at issue.Hence, we investigate the leverage ratios of our true crosssection of BBB-rated firms simulated over five years aftermatching. For the main analysis, the debt maturity isassumed to be ten years.

Panel A in Table 5 shows that the average leverage is40.89%, which is, naturally, close to the average of 41.83%of our BBB-rated firm sample used for the matching. (Theaverage leverage is 40.57%, 40.93%, and 41.45% for fiveyears, 20 years, and infinite debt maturity, respectively.)

In Panel B, we compare leverage ratios in boom andrecession. While optimal leverage is procyclical at initia-tion, it is countercyclical over time for the true crosssection of BBB-rated firms. In particular, the averageleverage is 36.94% in boom and 46.20% in recession. Thereason is that the market value of equity is more sensitive

(footnote continued)

firm with an asset composition ratio of 1.6 is 4% if the firms are initiated

in boom. (The optimal leverage ratios in this case are 45.4% and 41.4%,

respectively.) For firms initiated in recession, the difference is 4.4%

(¼44.2% minus 39.8%).

to regime switches than the market value of debt, makingleverage countercyclical. This mechanism dominates theoptimally procyclical leverage choice at initiation for ourtypical firms. The result mirrors the property we pre-viously established for the aggregate economy and con-firms that it holds also when matching to real empiricalsamples.

Finally, Panel C investigates the relation betweengrowth options and market leverage. Regressing theaverage leverage of each firm on its average asset com-position ratio in our empirical BBB-rated firm sampleyields a coefficient of �0.165. We conduct the sameregressions with the averages of asset composition ratiosand leverage ratios from each of the one thousandsimulations of the true cross section. The average coeffi-cient from this regression is �0.184, close to its empiricalcounterpart. Hence, the observed magnitude of the nega-tive relation between growth options and market leverageis preserved during the simulation.

Our qualitative finding for the cross-sectional relationbetween growth options and leverage is widely accepted(Bradley, Jarrell, and Kim, 1984; Barclay, Morellec, andSmith, 2006; Johnson, 2003; Rajan and Zingales, 1995).Consistent with the literature, the coefficient is robustlynegative. Moreover, its quantitative size, implied by the25% and 75% quantiles, is comparable to the one inempirical studies. Fama and French (2002), for example,obtain a coefficient of �0.096 in their regression ofmarket leverage on a similar ratio of asset compositionand standard controls, and Johnson (2003) finds thatincreasing the asset composition ratio by one decreasesleverage by around 7.8 percentage points in a pooledregression.

6.3. Robustness

In this subsection, we discuss the robustness of theresults to variations in the critical input parameters. Inaddition, we show how our predictions are affected if weassume that the expansion is financed by issuing equityinstead of selling assets.

To analyze the impact of preferences on our results, weshow ten-year credit spreads and the simulated averageleverage for g¼ 7:5 in the second column of Table 6, avalue that is also sometimes used in the literature (Bansaland Yaron, 2004; Chen, 2010). All other parameters arekept constant at their baseline levels from Table 1. Thedebt maturity is assumed to be ten years.

Lower risk aversion induces a smaller demand forprecautionary savings, which increases the real risk-freerate. At the same time, it raises the risk-neutral earningsdrift, because risk prices for systematic Brownian shocksðZiÞ decrease. Both mechanisms reduce the default prob-ability, leading to the lower credit spreads and slightlylower leverage.

In Column 3 of Table 6, we investigate the impact ofthe exercise costs on credit spreads and leverage. As weare mainly interested in firms with intact expansionoptions, we present the results for K equal to 350, i.e., ahigher K than in the baseline case. (Lowering K inducesmany growth firms to exercise their expansion option

Table 6Credit spreads and leverage for alternative specifications.

This table shows annualized ten-year credit spreads and simulated average leverage ratios (given in percent) of BBB-rated firms for alternative

specifications of our basic model. The asset composition ratio is defined as firm value divided by the value of the invested assets. Credit spreads are

calculated as the coupon divided by the debt value, minus the yield on an otherwise identical risk-free bond. They are quoted in basis points. The altered

parameter is indicated in the first line, all other parameters are taken from Table 1. Credit spreads in the first three lines of Panel A for typical firms at

issue are obtained by weighting the credit spreads in boom and recession by the expected times spent in each regime, respectively. The leverage is set

equal to 41:83% to generate the credit spreads of typical firms. The last row in Panel A contains average credit spreads of our simulated true cross section

of BBB-rated firms. Panel B shows simulated average leverage ratios for BBB-rated firms. The debt maturity is assumed to be ten years.

Specification

g¼ 7:5 K¼350 Equity financing

Panel A: Ten-year credit spreads

Asset composition

Invested assets (asset composition ratio¼1) 33 39 39

Average firm (asset composition ratio¼1.6) 53 67 65

Growth firm (asset composition ratio¼2.2) 56 72 58

Average in true cross section 68 81 77

Panel B: Unconditional leverage

Descriptive statistic

Average 41.10 41.22 41.14


almost immediately.) Generally, credit spreads and theaverage leverage are very similar to the ones of ourbaseline specification. For high asset composition ratios,such as 2.2, credit spreads at initiation slightly increasebecause a higher K induces a larger distance to theoptimal exercise boundary compared with the baselinespecification. This increase in credit spreads from thelarger distance arises because close to the exerciseboundary, credit spreads also reflect the fact that the firmwill imminently be converted into a firm with onlyinvested assets, and, hence, with lower credit risk. Whensimulating the true cross section, the impact of increasingK from 310 to 350 on the average credit spread is below 1basis point.

Finally, we analyze in Column 4 of Table 6 the case inwhich the exercise price of the expansion option (K) isfinanced by issuing additional equity instead of sellingassets. Appendix A.6 presents the solution for the value ofcorporate debt. New equity decreases the leverage afterexercise and, hence, lowers credit risk. As firms with ahigh asset composition ratio are close to the endogenousexercise boundary where new equity financing occurs,credit spreads are strongly reduced for typical growthfirms compared with the benchmark model. In the simu-lation of the true cross section, however, the effect isrelatively small because most firms have a large distanceto the exercise boundary. Those firms that contribute themost to the average credit spread, i.e., distressed firms,are particularly far away from the exercise boundary.In addition, Panel B shows that the average leverage isonly marginally affected.

The result for typical growth firms in Column 4 showsthat, close to firms’ exercise boundaries, credit spreads aredriven by the expected new financing upon investmentand do not primarily reflect the nature of assets. Thisinsight validates our focus on asset financing instead of onequity financing of growth option exercises to analyze the

isolated impact of the asset composition on credit risk andcorporate policy choices.

We conclude that while alternative specifications andsettings can have an impact on the quantitative results,our qualitative aggregate and cross-sectional predictionsare robust.

7. Equity value premium

In this section, we investigate the value premium forequity implied by our model, i.e., the difference in theequity risk premium between value and growth firms.As in the analysis of credit spreads, we show thatconsidering the true cross section is crucial when explor-ing the quantitative implications of our model.

Proposition 3 presents the instantaneous equity riskpremium, defined as the expected difference between theinstantaneous yield on corporate equity and the yield onthe corresponding risk-free security.

Proposition 3. The nominal instantaneous equity risk

premium epiðXÞ of a firm is given by

epiðXÞ ¼e0iðXÞX

eiðXÞgsX,C

i sCi þ

e0iðXÞX

eiðXÞðsX,C

i sP,CþðsP,idÞ2Þ

�li

ejðXÞ

eiðXÞ�1

� �ðeki�1Þ: ð34Þ


The first term of the equity risk premium comes fromthe compensation for the systematic volatility of stockreturns caused by Brownian shocks. The second termaccounts for the fact that the equity premium is calcu-lated in nominal terms. The last term is the jump riskpremium, in which ðejðXÞ=eiðXÞ�1Þ is the volatility of stockreturns that is caused by Poisson shocks.

Table 7Equity premiums of portfolios formed on the asset composition ratio.

This table shows the yearly equity premiums for the ten different portfolios based on the deciles of the asset composition ratio. The equity premium for

each portfolio is calculated as the equity value-weighted average of the premiums of all firms within the corresponding portfolio. It is reported as the

average yearly premium expressed in percent. The leverage for each portfolio (given in percent) is obtained by averaging over firms’ individual leverage

in the corresponding portfolio, in which the individual leverage is calculated as the ratio of the market value of debt to total firm value. The asset

composition ratio is defined as the portfolio average of the firm values divided by the values of the invested assets. The debt maturity is assumed to be

ten years.

Portfolio

Portfolio feature 1 2 3 4 5 6 7 8 9 10

Panel A: Initial cross section

Equity premium 6.79 6.07 6.29 5.73 5.71 5.37 5.37 5.36 5.25 5.65

Panel B: Simulated true cross section

Equity premium 9.48 6.74 6.36 5.95 5.83 5.94 5.71 5.72 5.73 6.01

Leverage 65.70 56.05 52.17 47.83 44.63 41.50 38.83 36.30 33.78 33.85

Asset composition ratio 0.96 1.07 1.15 1.22 1.30 1.38 1.48 1.58 1.71 1.92


In Panel A of Table 7, we investigate the initial, instanta-neous value premium for our cross section of matched BBB-rated firms. We report yearly equity premiums. At matching,firms are sorted into ten portfolios based on their assetcomposition ratio.28 Portfolio 1 contains all firms in thelowest asset composition ratio decile (value firms); Portfolio10 the ones in the highest decile (growth firms).29 Theaverage equity premium for each portfolio is calculated asthe equity value-weighted average of the instantaneousequity premiums of the matched firms in the correspondingportfolio. The panel reports average equity premiums of 20matchings after the corresponding pre-matching simulations.The pattern across the portfolios at matching is in accordancewith the positive relation between the book-to-market ratioand the equity returns reported in the literature.

Because growth options are levered claims, growthfirms are more volatile, which induces a larger equity riskpremium than for value firms. At the same time, however,growth firms hold lower levels of debt. The averageleverage in portfolio 1, for example, is 61.99%, and theone in portfolio 10 is 28.82%. Consistent with the empiri-cal literature (e.g., Bhandari, 1988; Fama and French,1992; Gomes and Schmid, 2010b), financial leverageincreases the equity risk premium in our model. Theeffect of leverage dominates the impact of the volatilitysuch that value firms have a higher equity risk premiumthan growth firms. The yearly premiums in the lowest andhighest asset composition ratio deciles are 6.79% and5.65%, respectively. The value premium calculated as thedifference between these two premiums is 1.14%.

Panel B analyzes the impact of the dynamics of thetrue cross section of firms on the value premium. To thisend, we simulate future earnings paths for each firm in

28 We do not sort based on the market-to-book equity ratio for two

reasons. First, the asset composition ratio unambiguously identifies

value and growth firms in our model. Second, using the market-to-

book equity ratio requires defining model-implied book asset values and

book values of debt. However, no unique definition of book values exists

in our model, and different definitions influence the sorting.29 The average asset composition ratio is 0.99 in the value portfolio,

and 2.08 in the growth portfolio.

our initial cross section of matched firms over five years.The procedure is analogous to the one in the simulationapproach for credit spreads and leverages. However, wedo not consider firms that have already exercised theirgrowth option. The reason is that, as our model does notincorporate new debt financing, the equity premium ofexercised firms is very small due to the low leverage afteroption exercise. In addition, exercised firms have verylarge equity values. Hence, including them causes a heavydownward bias of the equity premium when applyingequity-weighting in the calculation of the average equitypremium.30 The average value-weighted equity premiumin our entire simulated true cross section of BBB-ratedfirms is 5.69% per year, consistent with the average equitypremium reported in the literature (Campbell, Lo, andMacKinley, 1997; Gomes and Schmid, 2010b). Followingthe sorting procedure proposed in Fama and French(1992), simulated firms are then sorted into ten differentportfolios at the beginning of each simulated year. Wemeasure the average value-weighted equity premium ofeach portfolio during the subsequent year. The first line inPanel B shows that the resulting value premium of 3.47%,given by the equity premium of Portfolio 1 minus theequity premium of Portfolio 10, is much larger than inPanel A. The value premium calculated as the differencebetween the top and bottom portfolio quintiles is 1.45%.The second and third lines of Panel B report the averageleverage and asset composition ratios of each portfolio inthe simulated true cross section.

Empirically, the yearly value premium is between 6.29%and 12.55% when comparing the top and bottom book-to-market deciles (Fama and French, 2002; Patton andTimmermann, 2010; Ang and Kristensen, 2012). Gomesand Schmid (2010b) report 7.19% based on portfolioquintiles. Hence, the results in Panel B show that ourmodel explains about 28–55% of the value premium fordeciles. For quintiles, about 20% are explained.

30 The average ten-year credit spread when omitting exercised firms

is 98 bps, which is even higher than the model predicted credit spread in

the main case (see Panel D of Table 3).


The value premium is higher in Panel B than in Panel Abecause the equity risk premium is an increasing andconvex function of the leverage ratio. Hence, wheneverthe economy switches to recession in the dynamic simu-lation, the equity risk premium of value firms with aninitially larger leverage increases, on average, more thanthe one of growth firms with an initially lower leverage.Empirically, Choi (2010) confirms that a further leverageincrease of already highly leveraged value firms duringtimes with large risk premiums contributes to highervalue premiums. Consistent with our simulation results,he argues that the joint dynamics of asset values andleverage drive, at least partially, the value premium.31

A direct consequence of this dynamic source of thevalue premium is that it is strongly countercyclical. In thesimulation of the true cross section of matched firms overfive years, the yearly value premium based on portfoliossorted by asset composition ratio deciles is, on average,8.74% in recession and 0.78% in boom. Based on quintiles,the value premium is 3.52% in recession and 0.41% inboom. Our result is consistent with the growing body ofliterature that shows that value firms are particularlyrisky in bad times. For example, Petkova and Zhang(2005) and Chen, Petkova, and Zhang (2008) find thatthe value effect is empirically much stronger in bad timesthan in good periods.

In sum, our analysis shows that by simply exploringthe cross-sectional dynamics of firms with endogenousdefault and investment decisions a significant portion ofthe value premium and its countercyclical pattern can beexplained.

8. Conclusion

It is now well accepted that macroeconomic risk is centralfor understanding credit risk and capital structure choices.Specifically, defaults are more likely during recession, whenthey are particularly costly and harder to bear. This counter-cyclicality increases the costs of debt for all firms. But toexplain the cross-sectional variation in apparently excessivecosts of debt, variation inside the firm is needed. This paperformalizes the role of one particularly important aspect ofthis heterogeneity: the asset composition of firms. It is notsurprising that in principle the asset composition can beimportant for optimal capital structure. After all, economistshave devoted much effort to understanding the differencebetween value and growth firms in terms of their financialstructure, starting with Myers (1977) and Jensen (1986).Little was known, however, about the quantitative impor-tance of this factor and its relation with macroeconomic risk.

The present structural equilibrium model allows us tojointly analyze a firm’s expansion policy and financialleverage in the presence of macroeconomic risk. Wedemonstrate that incorporating the combination of thesefactors goes a long way toward explaining the empirically

31 The finding that value stocks have higher returns than growth

stocks has prompted many other explanations. For example, Choi (2010)

shows that fixed operating costs can generate a value premium.

Similarly, Zhang (2005) argues that asymmetric adjustment costs

change the underlying business risk of value firms.

observed cross-sectional variation in costs of debt, lever-age, and equity risk premiums. Our model implies thatcompanies with a high portion of expansion options tendto be riskier in general and, at the same time, particularlysensitive to macroeconomic risk. They are not only morevolatile (because growth options represent leveredclaims), but also have a higher propensity to default inbad times than firms with a low portion of expansionoptions. Thus, the default probability and its counter-cyclicality are higher, the greater the ratio of expansionoptions to total assets. Together with higher marginalutility of the representative agent in recession, this rela-tion (exacerbated by costly liquidation in recession)implies higher costs of debt and more important endo-genous shadow costs of leverage for firms with growthoptions than for those with only invested assets. Thus, ourfindings explain why the credit spread puzzle is empiri-cally more pronounced for growth firms and why growthfirms hold less debt even after controlling for standarddeterminants of credit risk. Moreover, because the econ-omy is made up of a cross-sectional mix of firms, themodel accounts, in quantitatively fairly accurate ways, forthe average credit spread puzzle. The model also yields acountercyclical value premium for equity, consistent withthe data.

We have studied one type of real options of firms,namely, growth options. However, firms have a wide andvarying range of options, including abandonment andshut-down options. A model incorporating these optionscould, therefore, yield further cross-sectional predictions.

While recent research has made important progress inenhancing understanding of average credit risk, the crosssection of credit risk has not received sufficient attention.Analyzing it empirically is, fortunately, feasible. Liquidcredit default swap quotes are now widely available on afirm-by-firm basis, allowing researchers to investigatespecific relations between firm-specific characteristicssuch as growth options and credit spreads. Our paperalso provides a theoretical basis that can guide empiricalresearch in this direction.

Appendix A

A.1. The stochastic discount factor

Solving the Bellman equation associated with theconsumption problem of the representative agent, it canbe shown that the stochastic discount factor mt followsthe dynamics Eq. (3) (see Bhamra, Kuehn, and Strebulaev,2010c; Chen, 2010). The parameters hB, hR solve

0¼ r 1�g1�d

hd�gi þ ð1�gÞyi�

1

2gð1�gÞðsC

i Þ2�r 1�g

1�d

� �h1�g

i

þliðh1�gj �h1�g

i Þ: ð35Þ

One-regime model: To isolate the effect of businesscycle risk, we also consider the model with only oneeconomic regime. The dynamics of the stochastic discountfactor then read

dmt

mt¼�r dt�Z dWC

t : ð36Þ


The real interest rate r and the risk price Z are given by

r¼ r ¼ rþdy�12gð1þdÞðs

Ci Þ

2ð37Þ

and

Z¼ gsC : ð38Þ

The nominal interest rate is calculated as

rn ¼ rþp�s2P�s

P,CZ, ð39Þ

and the expected growth rate is given by

~m ¼ m�sX,CðZþsP,CÞ�ðsP,idÞ2: ð40Þ

The earnings–price ratio simplifies to

y�1 ¼ rn� ~m, ð41Þ

and the total earnings volatility is

~s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsX,CÞ

2þðsP,idÞ

2þðsX,idÞ

2q

: ð42Þ

A.2. The value of the growth option

Proof of Proposition 1. For each regime i, the option isexercised immediately whenever XZXi (option exerciseregion); otherwise, it is optimal to wait (option continua-tion region). This structure results in the following systemof ordinary differential equations (ODEs) for the valuefunction.

For 0rXoXB:

rnBGBðXÞ ¼ ~mBXG0BðXÞþ

12~s2

BX2G00BðXÞþ~lBðGRðXÞ�GBðXÞÞ,

rnRGRðXÞ ¼ ~mRXG0RðXÞþ

12~s2

RX2G00RðXÞþ~lRðGBðXÞ�GRðXÞÞ:

8<:

ð43Þ

For XBrXoXR:

GBðXÞ ¼ sXyB�K ,

rnRGRðXÞ ¼ ~mRXG0RðXÞþ

12~s2

RX2G00RðXÞþ~lRðsXyB�K�GRðXÞÞ:

(

ð44Þ

For XZXR:

GBðXÞ ¼ sXyB�K ,

GRðXÞ ¼ sXyR�K:

(ð45Þ

Whenever the process X is in the option continuationregion, which corresponds to system (43) and the secondequation of (44), the required rate of return rn

i (left-handside) must be equal to the realized rate of return (right-hand side). The latter is obtained by Ito’s lemma forregime switches. Here, the last term accounts for apossible jump in the value of the growth option due to aregime switch. It is calculated as the instantaneous prob-ability of a regime shift, ~lB or ~lR, times the associatedchange in the value of the option. The first equation of(44) and the system (45) state the payoff of the optionat exercise, because the process is in the optionexercise region in these cases. The boundary conditionsare given by

limXr0

GiðXÞ ¼ 0, i¼ B,R, ð46Þ

limXrXB

GRðXÞ ¼ limXsXB

GRðXÞ, ð47Þ

limXrXB

G0RðXÞ ¼ limXsXB

G0RðXÞ, ð48Þ

limXsXR

GRðXÞ ¼ sXRyR�K , ð49Þ

and

limXsXB

GBðXÞ ¼ sXByB�K: ð50Þ

Condition (46) ensures that the option value goes to zeroas earnings approach zero. Conditions (47) and (48) arethe value-matching and smooth-pasting conditions of thevalue function in recession at the exercise boundary inboom. The remaining conditions (49)–(50) are the value-matching conditions at the exercise boundaries in boomand recession, respectively.

The functional form of the solution is given by

GiðXÞ ¼

Ai3Xg3þAi4Xg4 , 0rXoXB, i¼ B,R,

C 1XbR1þC 2XbR

2þC 3XþC 4, XBrXoXR, i¼ R,

sXyi�K , XZXi, i¼ B,R,

8>><>>:

ð51Þ

in which AB3, AB4, AR1, AR2, C 1, C 2, C 3, C 4, g3, g4, bR1, and bR

2

are real-valued parameters to be determined.We first consider the region 0rXoXB and plug the

functional form GiðXÞ ¼ Ai3Xg3þAi4Xg4 into both equations

of (43). Comparison of coefficients yields that ABk is a

multiple of ARk, k¼ 3,4, with the factor lk :¼ ð1=~lBÞðr

nBþ

~lB� ~mBgk�12~s2

Bgkðgk�1ÞÞ, i.e., ARk ¼ lkABk. Using this rela-

tion and comparing coefficients, we find that g3 and g4

correspond to the positive roots of the quartic equation

ð ~mRgþ12~s2


2~s2


~lR~lB:

ð52Þ

The reason for taking the positive roots is given byboundary condition (46).

Next, we consider the region XBrXoXR. Plugging thefunctional form GRðXÞ ¼ C 1Xb1þC 2Xb2þC 3XþC 4 into thesecond equation of (44), we find by comparison ofcoefficients that

bR1,2 ¼

1

2�~mR

~s2R

7


2�~mR

~s2R

!2

þ2ðrn

Rþ~lRÞ

~s2R

vuut ,

C 3 ¼~lR

syB

rnR� ~mRþ

~lR

,

C 4 ¼�~lR

K

rnRþ

~lR

: ð53Þ

The remaining unknown parameters are AB3, AB4, C 1, and C 2.Plugging the functional form (51) into conditions (47)–(50)yields

C 1XbR

1

B þC 2XbR

2

B þC 3XBþC 4 ¼ l3AB3Xg3

B þ l4AB4Xg4

B , ð54Þ

C 1bR1X

bR1

B þC 2bR2X

bR2

B þC 3XB ¼ l3AB3g3Xg3

B þ l4g4AB4Xg4

B , ð55Þ

C 1XbR

1

R þC 2XbR

2

R þC 3XRþC 4 ¼ syRXR�K , ð56Þ

and


AB3Xg3

B þAB4Xg4

B ¼ syBXB�K: ð57Þ

This four-dimensional system is linear in its four unknownsAB3, AB4, C 1 and C 2. We define the matrices

M :¼

l3Xg3

B l4Xg4

B �XbR

1

B �XbR

2

B

l3g3Xg3

B l4g4Xg4

B �bR1X

bR1

B �bR2X

bR2

B

0 0 XbR

1

R XbR

2

R

Xg3

B Xg4

B 0 0

26666664

37777775 ð58Þ

and

b :¼

C 3XBþC 4

C 3XB

�C 3XR�C 4þsyRXR�K

syBXB�K

266664

377775, ð59Þ

such that M ½AB3 AB4 C 1 C 2�T ¼ b. Hence, the solution to the

remaining unknowns is given by

½AB3 AB4 C 1 C 2�T ¼M

�1b: & ð60Þ

Relative price change sensitivity: The relative pricechange sensitivity is

G0iðXÞ

GiðXÞ¼

g3Ai3Xg3�1þAi4g4Xg4�1

Ai3Xg3þAi4Xg4, XoXB, i¼ B,R,

C 1b1Xb1�1þC 2b2Xb2�1

þC 3

C 1Xb1þC 2Xb2þC 3XþC 4

, XBrXoXR, i¼ R,

syi

syiX�K, XZXi, i¼ B,R:

8>>>>>>>>><>>>>>>>>>:

ð61Þ

The unlevered value of the growth option: The unleveredvalue of the growth option can be calculated by imposing thesmooth-pasting boundary conditions at option exercise:

limXsXunlev

R

Gunlev0

R ðXÞ ¼ syR ð62Þ

and

limXsXunlev

B

Gunlev0

B ðXÞ ¼ syB: ð63Þ

The solution method is analogous to the one for the leveredoption value up to and including Eq. (53). Then, system(54)–(57) is augmented by the two equations correspondingto the additional boundary conditions:

Cunlev

1 bR1ðX

unlevR Þ

bR1�1þC

unlev

2 bR2ðX

unlevR Þ

bR2�1þC 3 ¼ syR ð64Þ

and

Aunlev

B3 g3ðXunlevB Þ

g3�1þA

unlev

B4 g4ðXunlevB Þ

g4�1¼ syB: ð65Þ

The full system is six-dimensional with the six unknowns

Aunlev

B3 , Aunlev

B4 , Cunlev

1 , Cunlev

2 , XunlevB , and Xunlev

R , linear in the first

four unknowns and nonlinear in the last two unknowns. It issolved numerically.

One-regime model: Denote the investment boundaryby X1. The system to solve is given by

rnGðXÞ ¼ ~mXG0ðXÞþ~s2

2X2G00ðXÞ, XoX1,

GðXÞ ¼ sXy�K , XZX1: ð66Þ

The boundary conditions are given by a value matchingcondition and the fact that the option must becomeworthless when the earnings approach zero:

limXr0

GðXÞ ¼ 0 ð67Þ

and

limXsX1

GðXÞ ¼ syX1�K ð68Þ

The functional form of the solution is

GðXÞ ¼AXb1 , XoX1,

sXy�K , XZX1,

(ð69Þ

in which A and b1 are real-valued parameters to bedetermined. It is straightforward to show that

b1 ¼1

2�

~m~s2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�

~m~s2

� �2

þ2rn

~s2

sð70Þ

and

A ¼ ðsyX1�KÞX�b1

1 : ð71Þ

The relative price change sensitivity of the option is

G0ðXÞ

GðXÞ¼

b1

X, XoX1,

sy

syX�K, XZX1:

8>><>>: ð72Þ

The unlevered value of the option satisfies the additionalsmooth-pasting condition

limXsXunlev

1

Gunlev0ðXÞ ¼ sy: ð73Þ

A.3. Firms with only invested assets

The solution for the values of corporate securities isbased on Hackbarth, Miao, and Morellec (2006).

The valuation of corporate debt: Without loss of gen-erality, we consider the case in which the default bound-ary in boom is lower than in recession, i.e., DBoDR.An investor holding corporate debt requires an instanta-neous return equal to the risk-free rate rn

i . Once the firmdefaults, debt-holders receive a fraction ai of the assetvalue Xyi. The required rate of return on debt must beequal to the realized rate of return plus the couponproceeds from debt. Therefore, an application of Ito’slemma with regime switches shows that debt satisfiesthe following system of ODEs.

For 0rXrDB:

dBðXÞ ¼ aBXyB,

dRðXÞ ¼ aRXyR:

8<: ð74Þ

For DBoXrDR:

rnBdBðXÞ ¼ cþ ~mBXd 0BðXÞþ

12~s2

BX2d 00BðXÞþ~lBðaRXyR�dBðXÞÞ,

dRðXÞ ¼ aRXyR:

8<:

ð75Þ


For X4DR:

rnBdBðXÞ ¼ cþ ~mBXd 0BðXÞþ

12~s2

BX2d 00BðXÞþ~lBðdRðXÞ�dBðXÞÞ,

rnRdRðXÞ ¼ cþ ~mRXd 0RðXÞþ

12~s2

RX2d 00RðXÞþ~lRðdBðXÞ�dRðXÞÞ:

8<:

ð76Þ

The boundary conditions read

limX-1

diðXÞ

Xo1, i¼ B,R, ð77Þ

limXrDR

dBðXÞ ¼ limXsDR

dBðXÞ, ð78Þ

limXrDR

d 0BðXÞ ¼ limXsDR

d 0BðXÞ, ð79Þ

limXrDB

dBðXÞ ¼ aBDByB, ð80Þ

and

limXrDB

dRðXÞ ¼ aRDRyR: ð81Þ

Condition (77) is the no-bubbles condition. The remainingboundary conditions are the value-matching conditions(78), (80) and (81), and the smooth-pasting condition at

the higher default threshold DR for the debt function in

boom dBð�Þ, Eq. (79). As debt-holders do not choose theoptimal default thresholds, there are no smooth-pastingconditions at default to be considered. The functionalform of the solution is

diðXÞ ¼

aiXyi, XrDi, i¼ B,R,

C1XbB1þ C 2XbB

2þC3XþC4, DBoXrDR, i¼ B,

Ai1Xg1þ Ai2Xg2þAi5, X4DR, i¼ B,R,

8>><>>:

ð82Þ

in which AB1, AB2, AR1, AR2, AB5, AR5, C 1, C 2, C3, C4, g1, g2,

bB1, and bB

2 are real-valued parameters to be determined.

We first consider the region X4DR and use thestandard approach of plugging the functional form

diðXÞ ¼ Ai1Xg1þ Ai2Xg2þAi5 into both equations of (76).Comparing coefficients and solving the resulting two-dimensional system of equations for Ai5, we find that

Ai5 ¼cðrn

j þ~liþ

~ljÞ

rni rn

j þrnj~liþrn

i~l j

¼c

rpi

ð83Þ

and that ABk is always a multiple of ARk, k¼ 1,2, with the

factor lk :¼ ð1=~lBÞðr

nBþ


Bgkðgk�1ÞÞ, i.e., ARk ¼

lkABk. Using these results and comparing coefficientsagain, we obtain that g1 and g2 are the negative roots of

the quartic equation

ð ~mRgþ12~s2


2~s2


~lR~lB:

ð84Þ

The reason for taking the negative roots is the no-bubblescondition for debt stated in Eq. (77).

Next, we consider the region DBrXrDR. Plugging the

functional form dBðXÞ ¼ C 1XbB1þ C 2XbB

2þC3XþC4 into thefirst equation of (75), we find by comparison of coefficients

that

bB1,2 ¼

1

2�~mB

~s2B

7


2�~mB

~s2B

!2

þ2ðrn

Bþ~lBÞ

~s2B

vuut ,

C3 ¼~lBaRyR

rnBþ

~lB� ~mB

,

C4 ¼c

rnBþ

~lB

: ð85Þ

The remaining unknown parameters are AB1, AB2, C 1, and

C 2: We plug the functional form (82) into conditions(78)–(81), and obtain a four-dimensional linear system in

the four unknowns AB1, AB2, C 1, and C 2:

AB1Dg1

R þ AB2Dg2

R þAB5 ¼ C 1DbB

1

R þ C 2DbB

2

R þC3DRþC4,

AB1g1Dg1

R þ AB2g2Dg2

R ¼ C 1bB1D

bB1

R þ C 2bB2D

bB2

R þC3DR,

aBDByB ¼ C 1DbB

1

R þ C 2DbB

2

R þC3DRþC4,

l1AB1Dg1

R þ l2AB2Dg2

R þAR5 ¼ aRDRyR: ð86Þ

We define the matrices

M :¼

Dg1

R Dg2

R �DbB

1

R �DbB

2

R

g1Dg1

R g2Dg2

R �bB1D

bB1

R �bB2D

bB2

R

0 0 DbB

1

R DbB

2

R

l1Dg1

R l2Dg2

R 0 0

266666664

377777775

ð87Þ

and

b :¼

C3DRþC4�AB5

C3DR

aBDByB�C3DR�C4

aRDRyR�AR5

2666664

3777775, ð88Þ

such that M ½AB1 AB2 C 1 C 2�T ¼ b. Hence, the solution of the

unknowns is given by

½AB1 AB2 C 1 C 2�T ¼ M

�1b: ð89Þ

Default policy: The value of equity is calculated as firmvalue minus the value of debt. The firm value consists ofthe value of assets in place plus the value of the optionand the tax shield minus default costs. Once debt hasbeen issued, managers select the ex post default policythat maximizes the value of equity. Formally, the defaultpolicy is determined by equating the first derivativeof the equity value to zero at the corresponding defaultboundary:

e 0BðDn

BÞ ¼ 0,

e 0RðDn

RÞ ¼ 0:

8<: ð90Þ

We solve this problem numerically.Capital structure: Denote by f

n

i ðXÞ the firm value of afirm with only invested assets, given optimal ex postdefault thresholds. The ex ante optimal coupon of a firmsolves

cn:¼ argmax

c

fn

i ðXÞ: ð91Þ


One-regime model: Let D1 be the default threshold. Fora risk-neutral agent, the model corresponds to the one ofLeland (1994). Eqs. (39)–(42), provide the parametersused in the setup and solution of the one-regime model.Postulating that the required return must be equal to theexpected realized return plus the proceeds from debt, wefind the following system:

rndðXÞ ¼ cþ ~mXd0

ðXÞþ~s2

2X2d

00

ðXÞ, X4D,

dðXÞ ¼ aXy, XrD: ð92Þ

The boundary conditions are the no-bubbles condition, aswell as value-matching at default:

limX-1

dðXÞ

Xo1,

limXrD

dðXÞ ¼ ayD: ð93Þ


dðXÞ ¼ayX, XoD,

BXb2þA5, XZD,

(ð94Þ

in which B and b2 are real-valued parameters. It isstraightforward to show that

A5 ¼c

r, ð95Þ

b2 ¼1

2�

~m~s2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�

~m~s2

� �2

þ2rn

~s2

s, ð96Þ

and

B ¼ ayD�c

rn

� �D�b2

: ð97Þ

The default policy and capital structure can be deter-mined analogously to the two-regime model.

A.4. Firms with invested assets and an expansion option

As in the main text, we consider the case DBoDR,DBoDR, and XR4XB. We present a constructive proof forthe valuation of corporate debt.

Proof of Proposition 2. For brevity of notation, defines :¼ sþ1. An investor holding corporate debt requires aninstantaneous return equal to the nominal risk-free ratern

i . Hence, an application of Ito’s lemma with regimeswitches shows that debt satisfies the following systemof ODEs.

For 0rXrDB:

dBðXÞ ¼ aBðXyBþGunlevB ðXÞÞ,

dRðXÞ ¼ aRðXyRþGunlevR ðXÞÞ:

8<: ð98Þ

For DBoXrDR:

rnBdBðXÞ ¼ cþ ~mBXd0BðXÞþ

12~s2

BX2d00BðXÞ

þ ~lBðaRðXyRþGunlevR ðXÞÞ�dBðXÞÞ,


8>><>>: ð99Þ

For DRoXoXB:


12~s2

BX2d00BðXÞþ~lBðdRðXÞ�dBðXÞÞ,

rnRdRðXÞ ¼ cþ ~mRXd0RðXÞþ

12~s2

RX2d00RðXÞþ~lRðdBðXÞ�dRðXÞÞ:

8<:

ð100Þ

For XBrXoXR:

dBðXÞ ¼ dB sX�K

yB

� �,

rndRðXÞ ¼ cþ ~mRXd0RðXÞþ1

2~s2

RX2d00RðXÞþ~lR dB sX�

K

yB

� ��dRðXÞ

� �:

8>>><>>>:

ð101Þ

For XZXR:


yB

� �,

dRðXÞ ¼ dR sX�K

yR

� �:

8>>><>>>: ð102Þ

In system (98), the firm is in the default region in bothboom and recession. In this region, debt-holders receiveaiðXyiþGunlev

i ðXÞÞ at default. As the default boundary inboom is lower than the one in recession, system (99)corresponds to the firm being in the continuation regionin boom and in the default region in recession. For thecontinuation region in boom, the left-hand side of the firstequation is the rate of return required by investors forholding corporate debt for one unit of time. The right-hand side is the realized rate of return, computed by Ito’slemma as the expected change in the value of debt plusthe coupon payment c. The last term captures the possiblejump in the value of debt in case of a regime switch,which triggers immediate default. Similarly, Eqs. (100)describe the case in which the firm is in the continuationregion in both boom and recession. The next system,(101), deals with the case in which the firm is in theexercise region in boom and in the continuation region inrecession. After exercising the option, the firm owns totalassets in place with value XyiþsXyi�K , reflecting thenotion that the exercise costs of the growth option arefinanced by selling assets. The value of debt must then beequal to the value of debt of a firm with only investedassets, i.e., dBðXÞ ¼ dBððsþ1ÞX�K=yBÞ, which is the firstequation in (101). The second equation in this case isobtained by the same approach as in (100), in which thelast term captures the fact that a regime switch fromrecession to boom triggers immediate exercise of theexpansion option. Finally, (102) describes the case inwhich the firm is in the exercise region in both boomand recession. The system is subject to the followingboundary conditions:

limXrDR

dBðXÞ ¼ limXsDR

dBðXÞ, ð103Þ

limXrDR

d0BðXÞ ¼ limXsDR

d0BðXÞ, ð104Þ

limXrDB

dBðXÞ ¼ aBðDByBþGunlevB ðDBÞÞ, ð105Þ

limXrDR

dRðXÞ ¼ aRðDRyRþGunlevR ðDRÞÞ, ð106Þ


limXrXB

dRðXÞ ¼ limXsXB

dRðXÞ, ð107Þ

limXrXB

d0RðXÞ ¼ limXsXB

d0RðXÞ, ð108Þ

limXsXB

dBðXÞ ¼ dB sXB�K

yB

� �, ð109Þ

and

limXsXR

dRðXRÞ ¼ dR sXR�K

yR

� �: ð110Þ

Eqs. (103) and (104) are the value-matching and smooth-pasting conditions for the debt value in boom at thedefault boundary in recession. Similarly, Eqs. (107) and(108) are the corresponding conditions for the debt valuein recession at the option exercise boundary in boom.Eqs. (105) and (106) are the value-matching conditions atthe default thresholds, and Eqs. (109) and (110) are thevalue-matching conditions at the option exercise bound-aries. The default thresholds and option exercise bound-aries are chosen by equity-holders, and, hence, we do nothave the corresponding smooth-pasting conditions fordebt. To solve this system, we start with the functionalform of the solution:

diðXÞ ¼

aiðXyiþGunlevi ðXÞÞ, XrDi, i¼ B,R,

C1XbB1þC2XbB

2þC3XþC4

þC5Xg3þC6Xg4 , DBoXrDR, i¼ B,

Ai1Xg1þAi2Xg2

þAi3Xg3þAi4Xg4þAi5, DRoXrXB, i¼ B,R,

B1XbR1þB2XbR

2þZðXÞþB4, XBoXrXR, i¼ R,

di sX�K

yi

� �, X4Xi, i¼ B,R,

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð111Þ

in which AB1, AB2, AR1, AR2, C1, C2, C3, C4, C5, C6, B1, B2, B4,bB

1, bB2, bR

1, bR2, g1, g2, g3, and g4 are real-valued parameters

to be determined (or to be confirmed). Z(X), as stated inthe sixth line of (111), can be expressed in closed formusing Gauss hypergeometric function.

We first consider the region DRoXrXB: Plugging thefunctional form diðXÞ ¼ Ai1Xg1þAi2Xg2þAi3Xg3þAi4Xg4þAi5

into both equations of (100) and comparing coefficients,we find that

Ai5 ¼cðrn

j þ~liþ

~ljÞ

rni rn

j þrnj~liþrn

i~l j

¼c

rpi

: ð112Þ

As before, ABk is always a multiple of ARk, k¼ 1, . . . ,4, with

the factor lk :¼ ð1=~lBÞðr

nBþ


Bgkðgk�1ÞÞ, i.e.,

ARk ¼ lkABk. Using this relation and comparing coefficients,we find that g1, g2, g3, and g4 correspond to the roots of

the quartic Eq. (84), which is given by

ð ~mRgþ12~s2


2~s2


~lR~lB:

ð113Þ

By arguments of Guo (2001), this quartic equation alwayshas four distinct real roots, two of them being negativeand two positive. The value of debt in both regimes issubject to boundary conditions from both below (default)

and above (exercise of expansion option). To meet allboundary conditions, we need four terms with the corre-sponding factors Aik as well as exponents gk, which

requires usage of all four roots of Eq. (113). The no-bubbles condition is already implemented in the value

function di of a firm with only invested assets and, hence,does not need to be imposed again. The unknown para-

meters for this region are ABk, k¼ 1, . . . ,4.Next, we consider the region DBrXrDR. Plugging the

functional form dBðXÞ ¼ C1XbB1þC2XbB

2þC3XþC4 þC5Xg3þ

C6Xg4 into the second equation of (99), we find bycomparison of coefficients that

bB1,2 ¼

1

2�~mB

~s2B

7


2�~mB

~s2B

!2

þ2ðrn

Bþ~lBÞ

~s2B

vuut , ð114Þ

C3 ¼~lB

aRyR

rnBþ

~lB� ~mB

, ð115Þ

C4 ¼c

rnBþ

~lB

, ð116Þ

C5 ¼ aRl3

l3A

unlev

B3 , ð117Þ

and

C6 ¼ aRl4

l4A

unlev

B4 : ð118Þ

The unknown parameters left for this region are C1 and C2.Finally, consider the region XBoXrXR. The corre-

sponding differential equation for i¼R is [see (101)]:

ðrnRþ

~lRÞdRðXÞ ¼ cþ ~mRXd0RðXÞþ1

2~s2

RX2d00RðXÞþ~lRdB sX�

K

yB

� �:

ð119Þ

To solve this inhomogeneous differential equation, we usea standard approach by first finding a fundamentalsystem of solutions of the homogeneous differentialequation and then calculating the solution of the inho-mogeneous equation as the sum of the solutions of thehomogeneous equation and a particular solution of theinhomogeneous equation (Polyanin and Zaitsev, 2003,pp. 21–23).

Eq. (119) is equivalent to

X2d00RðXÞþ2 ~mR

~s2R

Xd0RðXÞ�2ðrn

Rþ~lRÞ

~s2R

dRðXÞ

¼ �2c

~s2R

�2 ~lR

~s2R

dB sX�K

yB

� �: ð120Þ

Therefore, the corresponding homogeneous differentialequation is

X2d00RðXÞþ2 ~mR

~s2R

Xd0RðXÞ�2ðrn

Rþ~lRÞ

~s2R

dRðXÞ ¼ 0: ð121Þ

A fundamental system of solutions is given by fz1,z2g,with

z1 :¼ XbR1 , ð122Þ

z2 :¼ XbR2 , ð123Þ


and

bR1,2 ¼

1

2�~mR

~s2R

7


2�~mR

~s2R

!2

þ2ðrn

Rþ~lRÞ

~s2R

vuut : ð124Þ

These solutions can be calculated by plugging the func-tional form into the homogeneous ODE (121) and solvingfor bR

1,2.For notational convenience, we now define f 2 :¼ X2,

f 1 :¼ ð2 ~mR= ~s2RÞX, f 0 :¼ �2ðrn

Rþ~lRÞ= ~s2

R, and

gðXÞ :¼ �2c

~s2R

�2 ~lR

~s2R

dB sX�K

yB

� �: ð125Þ

These notations allow to write the ODE (120) as

f 2d00RðXÞþ f 1d0RðXÞþ f 0dRðXÞ ¼ gðXÞ: ð126Þ

The general solution of this inhomogeneous ODE is given by

dRðXÞ ¼ B1z1þB2z2þz2

Zz1

g

f 2

dX

W|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}¼:I1ðXÞ

�z1

Zz2

g

f 2

dX

W|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}¼:I2ðXÞ

, ð127Þ

in which W ¼ z1z02�z2z01 is the Wronskian determinant, andB1 and B2 are coefficients [see, e.g., Polyanin and Zaitsev,2003, p. 22, Eq. (7)]. The first two terms of Eq. (127) are alinear combination of the solutions of the homogeneous ODE,and the last two terms are a particular solution of theinhomogeneous ODE.

We start by calculating the Wronskian determinant

W ¼ z1z02�z2z01

¼ bR2XbR

1 XbR2�1�bR

1XbR1�1XbR

2

¼ ðbR2�b

R1ÞX

bR1þb

R2�1: ð128Þ

Hence, the integral I1ðXÞ is given by

I1ðXÞ ¼

Zz1

g

f 2

dX

W

¼

ZXbR

1 X�2 1

bR2�b

R1

X1�bR1�b

R2 gðXÞ dX

¼1

bR2�b

R1

ZX�1�bR

2 gðXÞ dX ð129Þ

¼1

bR2�b

R1

ZX�1�bR

2 �2c

~s2R

�2 ~lR

~s2R

dB sX�K

yB

� � !dX

¼1

bR2�b

R1

ZX�1�bR

2 �2c

~s2R

�

2 ~lR

~s2R

AB1 sX�K

yB

� �g1

þ AB2 sX�K

yB

� �g2

þAB5

�ÞdX

¼�2 ~lRAB1

ðbR2�b

R1Þ ~s

2R

ZX�1�bR

2 sX�K

yB

� �g1

dX|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼:I11ðXÞ

�2 ~lRAB2

ðbR2�b

R1Þ ~s

2R

ZX�1�bR

2 sX�K

yB

� �g2

dX|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼:I12ðXÞ

þ2AB5ð

~lRþrnRÞ

ðbR2�b

R1Þb

R2 ~s

2R

X�bR2 : ð130Þ

We use the definition of the function gðXÞ [see Eq. (125)],and the solution of the debt value of a firm with onlyinvested assets dRð�Þ [see Eq. (82)].

The integrals I11ðXÞ and I12ðXÞ can be evaluated imme-diately with standard computer algebra packages. Alter-natively, using the integral representation of Gausshypergeometric function 2F1ð�, � ,�; �Þ, we can write theclosed-form solution of the integrals as

I11ðXÞ ¼1

g1�bR2

sg1 Xg1�bR2

2F1 �g1,bR2�g1,bR

2�g1þ1;�K

sXyB

� �ð131Þ

and

I12ðXÞ ¼1

g2�bR2

sg2 Xg2�bR2

2F1 �g2,bR2�g2,bR

2�g2þ1;�K

sXyB

� �:

ð132Þ

Plugging the solutions (131) and (132) into the expressionfor the integral I1, (130) yields

I1ðXÞ ¼�2 ~lRAB1

ðbR2�b

R1Þ ~s

2R

1

g1�bR2

sg1 Xg1�bR2

2F1

� �g1,bR2�g1,bR

2�g1þ1;�K

sXyB

� �

�2 ~lRAB2

ðbR2�b

R1Þ ~s

2R

1

g2�bR2

sg2 Xg2�bR2

2F1


2�g2þ1;�K

sXyB

� �

þ2AB5ð

~lRþrnRÞ

ðbR2�b

R1Þb

R2 ~s

2R

X�bR2 : ð133Þ

Similarly, we find for the second integral I2ðXÞ:

I2ðXÞ ¼�2 ~lRAB1

ðbR2�b

R1Þ ~s

2R

1

g1�bR1

sg1 Xg1�bR1

2F1


2�g1þ1;�K

sXyB

� �

�2 ~lRAB2

ðbR2�b

R1Þ ~s

2R

1

g2�bR1

sg2 Xg2�bR1

2F1


2�g2þ1;�K

sXyB

� �

þ2AB5ð

~lRþrnRÞ

ðbR2�b

R1Þb

R1 ~s

2R

X�bR1 : ð134Þ

Plugging (133) and (134) into (127) and simplifying,we finally obtain the solution

dRðXÞ ¼ B1XbR1þB2XbR

2þZðXÞþB4, ð135Þ

with

ZðXÞ ¼X

i,k ¼ 1,2

2ð�1Þiþ1 ~lRsgk ABk

~s2Rðb

R2�b

R1Þðgk�b

Ri Þ

Xgk2F1

� �gk,bRi ,bR

i �gkþ1;�K

sXyB

� �ð136Þ

and

B4 ¼~lR

c

rpi ðr

nRþ

~lRÞþ

c

rnRþ

~lR

, ð137Þ

for some parameters B1 and B2 determined by the boundaryconditions. The first derivative Z0ðXÞ can be calculated as


Z0ðXÞ ¼d

dXZðXÞ

¼d

dXðXbR

2 I1ðXÞ�XbR2 I2ðXÞÞ

¼ bR2XbR

2�1I1ðXÞþ1

bR2�b

R1

XbR2�1X�1�bR

1 gðXÞ

�bR1XbR

1 I2ðXÞ�1

bR2�b

R1

XbR1 X�1�bR

1 gðXÞ

¼ bR2XbR

2�1I1ðXÞ�bR1XbR

1�1I2ðXÞ

¼X

i,k ¼ 1,2

2ð�1Þiþ1 ~lRsgk ABkbRi

~s2Rðb

R2�b

R1Þðgk�b

Ri Þ

Xgk�12F1

� �gk,bRi ,bR

i �gkþ1;�K

sXyB

� �: ð138Þ

To solve for the unknown parameters AB1, AB2, AB3, AB4, C1,C2, B1, and B2, we plug the functional form (111) into thesystem of boundary conditions (103)–(110):

X4

k ¼ 1

ABkDgk

R þAB5 ¼ C1DbB

1

R þC2DbB

2

R þC3XþC4þC5Xg3þC6Xg4 ,

X4

k ¼ 1

ABkgkDgk

R ¼ C1bB1D

bB1

R þC2bB2D

bB2

R þC3XþC5g3Xg3þC6g4Xg4 ,

aBðDByBþGunlevB ðDBÞÞ ¼ C1D

bB1

B þC2DbB

2

B þC3DBþC4þC5Dg3

B þC6Dg4

B ,

X4

k ¼ 1

lkABkDgk

R þAR5 ¼ aRðDRyRþGunlevR ðDRÞÞ,

X4

k ¼ 1

lkABkXgk

B þAR5 ¼ B1XbR

1

B þB2XbR

2

B þZðXBÞþB4,

X4

k ¼ 1

lkABkgkXgk

B ¼ B1bR1X

bR1

B þB2bR2X

bR2

B þXBZ0ðXBÞ,

X4

k ¼ 1

ABkXgk

B þAB5 ¼ dB sXB�K

yB

� �,

B1XbR

1

R þB2XbR

2

R þZðXRÞþB4 ¼ dR sXR�K

yR

� �: ð139Þ

Using matrix notation, we write

M :¼

Dg1

R Dg2

R Dg3

R Dg4

R �DbB

1

R �DbB

2

R 0 0

g1Dg1

R g2Dg2

R g3Dg3

R g4Dg4

R �bB1D

bB1

R �bB2D

bB2

R 0 0

0 0 0 0 DbB

1

B DbB

2

B 0 0

l1Dg1

R l2Dg2

R l3Dg3

R l4Dg4

R 0 0 0 0

l1Xg1

B l2Xg2

B l3Xg3

B l4Xg4

B 0 0 �XbR

1

B �XbR

2

B

l1g1Xg1

B l2g2Xg2

B l3g3Xg3

B l4g4Xg4

B 0 0 �bR1X

bR1

B �bR2X

bR2

B

Xg1

B Xg2

B Xg3

B Xg4

B 0 0 0 0

0 0 0 0 0 0 XbR

1

R XbR

2

R

26666666666666664

37777777777777775

ð140Þ

and

b :¼

�AB5þC3DRþC4þC5Dg1

R þC6Dg2

R

C3DRþg1C5Dg1

R þg2C6Dg2

R

�C3DB�C4�C5Dg3

B �C6Dg4

B þaBðDByBþGunlevB ðDBÞÞ

�AR5þaRðDRyRþGunlevR ðDRÞÞ

�AR5þZðXBÞþB4

XBZ0ðXBÞ

�AB5þ dB sXB�KyB

� ��ZðXRÞþB4þ dR sXR�

KyR

� �

2666666666666666664

3777777777777777775

:

ð141Þ

Thus, the solution to the remaining unknowns is given by

½AB1 AB2 AB3 AB4 C1 C2 B1 B2�T ¼M�1b: & ð142Þ

Proof of Remark 1. (i) In our framework, debt character-istics ðc,m,pÞ are chosen and fixed at initiation. Thissetting allows us to calculate closed-form solutions forthe values of corporate securities of firms with bothinvested assets and growth options, even with finitematurity debt. For given debt characteristics ðc,m,pÞ, thevalue of finite maturity debt satisfies the following systemof ODEs.

For 0rXrDB:



8<: ð143Þ

For DBoXrDR:

ðrnBþmÞdBðXÞ ¼ cþmpþ ~mBXd0BðXÞþ

12~s2

BX2d00BðXÞ



8>><>>:

ð144Þ

For DRoXoXB:

ðrnBþmÞdBðXÞ ¼ cþmpþ ~mBXd0BðXÞþ

12~s2

BX2d00BðXÞ

þ ~lBðdRðXÞ�dBðXÞÞ,

ðrnRþmÞdRðXÞ ¼ cþmpþ ~mRXd0RðXÞþ

12~s2

RX2d00RðXÞ

þ ~lRðdBðXÞ�dRðXÞÞ:

8>>>>><>>>>>:

ð145Þ

For XBrXoXR:


yB

� �,

ðrnRþmÞdRðXÞ ¼ cþmpþ ~mRXd0RðXÞþ

1

2~s2

RX2d00RðXÞ

þ ~lR dB sX�K

yB

� ��dRðXÞ

� �:

8>>>>>>><>>>>>>>:

ð146Þ

For XZXR:


yB

� �,

dRðXÞ ¼ dR sX�K

yR

� �:

8>>><>>>: ð147Þ


dið�Þ denotes the value of debt of a firm with only investedassets with the same principal, coupon, and debt matur-ity. The solution of di is given in Hackbarth, Miao, andMorellec (2006). It corresponds to the value of infinitematurity debt of a firm with only invested assets with acoupon cþmp and interest rates rn

i þm. The boundaryconditions for system (143)–(147) are the same as in thecase of infinite maturity debt [see (103)]. Comparing thissystem (143)–(147) for finite maturity debt to the corre-sponding system (98)–(102) for infinite maturity debt, weconclude that for given debt characteristics ðc,m,pÞ, thevalue of finite maturity debt corresponds to the value ofinfinite maturity debt with a coupon cþmp and nominalinterest rates rn

i þm. Hence, the value of finite maturitydebt is given by the corresponding formula (22) inProposition 2.

(ii) The value of the tax shield satisfies the followingsystem of ODEs.

For 0rXrDB:

tBðXÞ ¼ 0,

tRðXÞ ¼ 0:

(ð148Þ

For DBoXrDR:

rnBtBðXÞ ¼ ctþ ~mBXt0BðXÞþ

12~s2

BX2t00BðXÞþ~lBð0�tBðXÞÞ,

tRðXÞ ¼ 0:

(

ð149Þ

For DRoXoXB:

rnBtBðXÞ ¼ ctþ ~mBXt0BðXÞþ

12~s2

BX2t00BðXÞþ~lBðtRðXÞ�tBðXÞÞ,

rnRtRðXÞ ¼ ctþ ~mRXt0RðXÞþ

12~s2

RX2t00RðXÞþ~lRðtBðXÞ�tRðXÞÞ:

8<:

ð150Þ

For XBrXoXR:

tBðXÞ ¼ t B sX�K

yB

� �,

rnRtRðXÞ ¼ ctþ ~mRXt0RðXÞþ

1

2~s2

RX2t00RðXÞ

þ ~lR tB sX�K

yB

� ��tRðXÞ

� �:

8>>>>>>><>>>>>>>:

ð151Þ

For XZXR:

tBðXÞ ¼ t B sX�K

yB

� �,

tRðXÞ ¼ t R sX�K

yR

� �:

8>>><>>>: ð152Þ

The boundary conditions write

limXrDR

tBðXÞ ¼ limXsDR

tBðXÞ,

limXrDR

t0BðXÞ ¼ limXsDR

t0BðXÞ,

limXrDB

tBðXÞ ¼ 0,

limXrDR

tRðXÞ ¼ 0,

limXrXB

tRðXÞ ¼ limXsXB

tRðXÞ,

limXrXB

t0RðXÞ ¼ limXsXB

t0RðXÞ,

limXsXB

tBðXÞ ¼ t B sXB�K

yB

� �,

limXsXR

tRðXRÞ ¼ t R sXR�K

yR

� �: ð153Þ

Comparing this system (148)–(152) and its boundaryconditions (153) to the system for infinite maturity debt,(98)–(102), and its boundary conditions (103)–(110)yields that the tax shield corresponds to the value of debtwith a coupon of ct and default costs of zero. The solutionfor the value of the tax shield is, therefore, given by thecorresponding Eq. (22) in Proposition 2.

(iii) The system for bankruptcy costs is given by thefollowing.

For 0rXrDB:

bBðXÞ ¼ ð1�aBÞðXyBþGunlevB ðXÞÞ,

bRðXÞ ¼ ð1�aRÞðXyRþGunlevR ðXÞÞ:

8<: ð154Þ

For DBoXrDR:

rnBbBðXÞ ¼ ~mBXb0BðXÞþ

12~s2

BX2b00BðXÞ

þ ~lBðð1�aRÞðXyRþGunlevR ðXÞÞ�bBðXÞÞ,

bRðXÞ ¼ ð1�aRÞðXyRþGunlevR ðXÞÞ:

8>><>>:

ð155Þ

For DRoXoXB:

rnBbBðXÞ ¼ ~mBXb0BðXÞþ

12~s2

BX2b00BðXÞþ~lBðbRðXÞ�bBðXÞÞ,

rnRbRðXÞ ¼ ~mRXb0RðXÞþ

12~s2

RX2b00RðXÞþ~lRðbBðXÞ�bRðXÞÞ:

8<:

ð156Þ

For XBrXoXR:

bBðXÞ ¼ bB sX�K

yB

� �,

rnRbRðXÞ ¼ ~mRXb0RðXÞþ

1

2~s2

RX2b00RðXÞþ~lR bB sX�

K

yB

� ��bRðXÞ

� �:

8>>><>>>:

ð157Þ

For XZXR:

bBðXÞ ¼ bB sX�K

yB

� �,

bRðXÞ ¼ bR sX�K

yR

� �:

8>>><>>>: ð158Þ

The system is subject to the following boundary condi-tions:

limXrDR

bBðXÞ ¼ limXsDR

bBðXÞ,

limXrDR

b0BðXÞ ¼ limXsDR

b0BðXÞ,

limXrDB

bBðXÞ ¼ ð1�aBÞðDByBþGunlevB ðDBÞÞ,

limXrDR

bRðXÞ ¼ ð1�aRÞðDRyRþGunlevR ðDRÞÞ,

limXrXB

bRðXÞ ¼ limXsXB

bRðXÞ,

limXrXB

b0RðXÞ ¼ limXsXB

b0RðXÞ,

limXsXB

bBðXÞ ¼ bB sXB�K

yB

� �,

limXsXR

bRðXRÞ ¼ bR sXR�K

yR

� �: ð159Þ

This system (154)–(158) and its boundary conditions(159) correspond to the system for infinite maturity debt,

32 An unreported robustness analysis confirms that starting with

finite maturity debt in the pre-matching simulation yields similar

results for the post-matching simulation. Credit spreads are slightly

lower as the initial principals are smaller.


(98)–(102), and its boundary conditions (103)–(110), witha coupon of zero and a recovery rate of 1�ai. The solutionfor bankruptcy costs is, therefore, given by the corre-sponding Eq. (22) in Proposition 2. &

One-regime model: Denote the default boundary by D1,the firm’s investment boundary by X1, and the defaultboundary of a firm with only invested assets by D1. Thesystem to solve is

dðXÞ ¼ aðyXþGunlevðXÞÞ, XrD1,

rndðXÞ ¼ cþ ~mXd0ðXÞþ~s2

2X2d00ðXÞ, D1oXoX1,

dðXÞ ¼ d sX�K

y

� �, XZX1: ð160Þ

This system is analogous to the one of the two-regimemodel, (98)–(102). Similarly, the boundary conditions arethe value-matching conditions at default and exercise

limXrD1

dðXÞ ¼ aðyD1þGunlevðD1ÞÞ, ð161Þ

limXsX1

dðXÞ ¼ d sX1�K

y

� �: ð162Þ


dðXÞ ¼

aðyXþGunlevðXÞÞ, XrD1,

E3Xb1þE4Xb2þA5, D1oXoX1,

d sX�K

y

� �, XZX1,

8>>>><>>>>:

ð163Þ

in which E3, E4, A5, b1, and b2 are real-valued parametersto be determined (or to be confirmed). We need to solvefor the region D1oXoX1. By plugging the functionalform (163) into the differential equation (160) and com-paring coefficients, we find that

A5 ¼c

rð164Þ

and

b1,2 ¼1

2�

~m~s2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�

~m~s2

� �2

þ2r

~s2

s: ð165Þ

B3 and B4 are determined by the following two-dimensional linear system defined by the correspondingboundary conditions:

E3Db1

1 þE4Db2

1 þc

r¼ aðyD1þGunlev

ðD1ÞÞ ð166Þ

and

E3Xb1

1 þE4Xb2

1 þc

r¼ d sX1�

K

y

� �: ð167Þ

Using matrix notation and defining

M1 :¼Db1

1 Db2

1

Xb1

1 Xb2

1

24

35 ð168Þ

and

b1 :¼aðyD1þGunlev

ðD1ÞÞ�cr

d sX1�Ky

� �� c

r

24

35, ð169Þ

we find that

½E3 E4�T ¼M�1

1 b1, ð170Þ

½E3 E4�T ¼

1

Db1

1 Xb2

1 �Db2

1 Xb1

1

Xb2

1 �Db2

1

�Xb1

1 Db1

1

24

35

�

aðyD1þGunlevðD1ÞÞ�

cr

d sX1�Ky

� �� c

r

24

35, ð171Þ

which completes the calculation of the solution.The values of finite maturity debt, the tax shield, and

bankruptcy costs can be found analogously to the two-regime model (cf. Remark 1).

A.5. Details on the simulations

In this section, we explain the calibration of theidiosyncratic volatility used to derive our results, anddescribe the simulation of our model-implied BBB sample.

A.5.1. Calibration of the idiosyncratic volatility

We calibrate the firm-level idiosyncratic volatility ofour BBB sample to the empirically observed total assetvolatility of 0.25. The procedure starts by simulating amodel-implied economy for ten years (pre-matchingsimulation). Next, we match the model-implied distribu-tion after ten years with the empirical cross section ofBBB-rated firms and, finally, simulate the obtainedmatched sample for another ten years (post-matchingsimulation). The average asset volatility of the post-matching simulation is then calculated.

We consider infinite maturity debt in the pre-matching simulation for all debt maturities in the post-matching simulation. We do so to abstract away from theimpact of different initial principals on the results, allow-ing us to analyze the pure effect of debt maturities oncredit spreads in the post-matching simulation. In addi-tion, starting with infinite maturity debt yields initialleverage ratios (principals) close to the ones empiricallyreported.32 The model-implied economy is generated asfollows: Starting with a value firm (s¼0), we generate arange of firms by increasing the option scale parameter s

by steps of 0.05, up to the largest possible value of s suchthat the option is not exercised immediately. At initiation,the capital structure is chosen optimally for all firms. Foreach option scale parameter s, 50 firms are considered,resulting in an initial sample of more than three thousandfirms. During the ten-year pre-matching simulation ofthis initial sample, firms default and expand optimally.Defaulted firms are not replaced, and exercised firmscontinue as firms with only invested assets. At the endof the pre-matching simulation, we calculate the model-implied leverage and asset composition ratio for eachfirm, using the assumed debt maturity and the corre-sponding optimal boundaries. We obtain a model-implied


economy of firms covering a broad range of both assetcomposition ratios and leverage ratios.

In the second step, we match our average historicaldistribution of BBB-rated firms with its model-impliedcounterpart. For each observation in the average historicaldistribution, we select the firm in our model-impliedeconomy at the final period of the pre-matching simula-tion that exhibits the minimum distance regarding thepercentage deviation from the target market leverage andasset composition ratio. That is, the empirical observationof a firm with leverage levemp and asset composition ratioacremp is matched with the model-implied firm withleverage levmi and asset composition ratio acrmi if, giventhe set of all model-implied firms, it minimizes theEuclidean distanceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

levemp�levmi

levemp

� �2

þacremp�acrmi

acremp

� �2s

: ð172Þ

The final step conducts a post-matching simulationwith the obtained sample of model-implied BBB firmsover ten years. For each simulation, we obtain the realizedasset volatility for each firm and calculate the resultingaverage asset volatility over firms. When measuring andaveraging asset volatilities, we incorporate the entireinitially matched BBB sample, including the evolution ofthe assets of firms that default during the ten-year post-matching simulation. This approach avoids a weightingbias when averaging over simulations toward firms withlower leverage and asset volatility, which have a smallertendency to default during the post-matching simulation.

The pre-matching simulation and the subsequentmatching are conducted 20 times. The initial regime ischosen according to the stationary distribution of thestates. This approach also guarantees convergence to thesteady-state distribution of regimes at the time of match-ing. For each matched sample of firms, the post-matchingsimulation is run 50 times. These numbers result in a totalof 1,000 simulations. The procedure is conducted fordifferent post-matching debt maturities.

A.5.2. Simulation of the true cross section

To ensure consistency, the simulation of the true crosssection is implemented analogously to the one performedto calibrate the idiosyncratic volatility. We first simulate amodel-implied distribution of firms for ten years (pre-matching simulation) and then match the model-implieddistribution with the average empirical cross section (fordetails, see above). The final step consists of simulatingthe matched sample for 20 years (post-matching simula-tion). We assume that firms default and exercise opti-mally. Defaulted firms are not recorded after default,whereas exercised firms are maintained in the sampleand continue as firms with only invested assets. Creditspreads and leverage ratios are measured during fiveyears after the matching. For each firm in the sample,we calculate the actual credit spread and leverage everymonth and then report the average over all firms and allsimulations. Default rates are observed for five, ten, and20 years. To assess the impact of the realized regimesat initiation and at the time of matching, we present

quantiles of post-matching average rates. As in the calibra-tion of the volatility, the initial state is chosen according tothe stationary distribution. The pre-matching simulation isrun 20 times, and the post-matching simulation is conducted50 times, resulting in a total of 1,000 simulations.

A.6. Financing the exercise of the growth option by issuing

additional equity

We consider the case in which the exercise price K ofthe growth option is financed by issuing additional equity.The corresponding system of ODEs for corporate debt is asfollows.

For 0rXrDB:



8<: ð173Þ

For DBoXrDR:


12~s2

BX2d00BðXÞ



8>><>>: ð174Þ

For DRoXoXB:


12~s2

BX2d00BðXÞþ~lBðdRðXÞ�dBðXÞÞ,


12~s2

RX2d00RðXÞþ~lRðdBðXÞ�dRðXÞÞ:

8<:

ð175Þ

For XBrXoXR:

dBðXÞ ¼ dBðsXÞ,


12~s2

RX2d00RðXÞþ~lRðdBðsXÞ�dRðXÞÞ:

8<:

ð176Þ

For XZXR:

dBðXÞ ¼ dBðsXÞ,

dRðXÞ ¼ dRðsXÞ:

(ð177Þ

The boundary conditions read

limXrDR

dBðXÞ ¼ limXsDR

dBðXÞ,

limXrDR

d0BðXÞ ¼ limXsDR

d0BðXÞ,

limXrDB

dBðXÞ ¼ aBðDByBþGunlevB ðDBÞÞ,

limXrDR

dRðXÞ ¼ aRðDRyRþGunlevR ðDRÞÞ,

limXrXB

dRðXÞ ¼ limXsXB

dRðXÞ,

limXrXB

d0RðXÞ ¼ limXsXB

d0RðXÞ,

limXsXB

dBðXÞ ¼ dBðsXBÞ,

limXsXR

dRðXRÞ ¼ dRðsXRÞ: ð178Þ

Comparing this system (173)–(177) and its boundaryconditions (178) to system (98)– (102) with boundaryconditions (103)–(110), we conclude that the value ofdebt given that the option exercise is financed by issuingadditional equity corresponds to the value of debt giventhat the option exercise is financed by selling assets in place


with an exercise price K of zero. Hence, the value of debt incase of equity-financed exercise costs can be calculated bythe corresponding formula (22) in Proposition 2. In particular,using the properties of Gauss hyperbolic function 2F1 and thedefinition of bR

1,2 in (23), we find that the function ZðXÞ asstated in line 5 of (22) in Proposition 2 simplifies to

ZðXÞ ¼ ~lRB5Xg1þ ~lRB6Xg2 , ð179Þ

with

B5 ¼sg1 AB1

rnR� ~mRg1�

12~s2

Rg1ðg1�1Þþ ~lR

ð180Þ

and

B6 ¼sg2 AB2

rnR� ~mRg2�

12~s2

Rg2ðg2�1Þþ ~lR

: ð181Þ

A.7. The equity risk premium

Proof of Proposition 3. According to Bhamra, Kuehn, andStrebulaev (2010b), the equity premium epiðXÞ is given by

epiðXÞ ¼ Et½dRt�rni dt� ¼ �Et dRt

dpnomt

pnomt

� �, ð182Þ

with

Rt :¼deiðXÞþð1�tÞðX�cÞ dt

ei�ðXÞ, ð183Þ

and i denotes the left limit of the Markov chain at time t.An application of Ito’s lemma shows that

dRt ¼ mR,i�ðXÞ dtþse,Ci� ðXÞ dWC

t þse,Pi� ðXÞ dWP

t þse,Xi� ðXÞ dWX

t

þeiðXÞ

ei�ðXÞ�1

� �dMt , ð184Þ

with

mR,iðXÞ ¼ mi�

e0i�ðXÞX

ei�ðXÞþ

1

2ððsX,C

i� Þ2þðsP,idÞ

2þðsX,idÞ

2Þe00i�ðXÞX

2

ei�ðXÞ

þð1�tÞðX�cÞ

ei�ðXÞ, ð185Þ

se,Ci� ðXÞ ¼

e0i�ðXÞX

ei�ðXÞsX,C

i� , ð186Þ

sP,Ci� ðXÞ ¼

e0i�ðXÞX

ei�ðXÞsP,id, ð187Þ

and

se,Xi� ðXÞ ¼

e0i�ðXÞX

ei�ðXÞsX,id: ð188Þ

Next, the nominal state price density is linked to the realstate price density by pnom

t ¼ prealt =Pt . Hence, using Ito’s

lemma, the dynamics of the nominal state price densitycan be written as

dpnomt

pnomt

¼�ðmpt þp�ðsP,CÞ

2�ðsP,idÞ

2�gsP,CsC

i Þ dt,

�ðgsCi þs

P,CÞ dWCt �s

P,id dWPt þðe

ki�1Þ dMt :

ð189Þ

Plugging (189) and (182) and taking the expectationyields the equity premium

epiðXÞ ¼e0iðXÞX

eiðXÞsX,C

i gsCi þs

P,C�

þe0iðXÞX

eiðXÞðsP,idÞ

2

�lejðXÞ

eiðXÞ�1

� �ðeki�1Þ: & ð190Þ

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Growth options, macroeconomic conditions, and the cross section of credit risk

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