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Product market competition and credit risk

Hsing-Hua Huang, Han-Hsing Lee

Department of Information Management and Finance, Graduate Institute of Finance, National Chiao Tung University, 1001 University Rd., Hsinchu City 300, Taiwan

a r t i c l e

i n f o

Article history: Received 28 June 2011 Accepted 1 September 2012 Available online 14 September 2012 JEL classification:

G13 G32 G33 Keywords:

Product market competition Credit risk

Structural model Hazard model

a b s t r a c t

This study theoretically and empirically investigates effects of product market competition on credit risk. We first develop a real-options-based structural model in a homogeneous oligopoly and show that credit spreads are positively related to the number of firms in an industry. The disparity of firm size in an indus-try is relevant to both product market competition and credit risk, and we therefore extend the model to an asymmetric duopoly case. In particular, we demonstrate that credit spreads of relatively small (large) firms within an industry are positively (negatively) related to Herfindahl-Hirschman index, and the rela-tive firm size in an industry is an important determinant of credit risk. The models’ implications are empirically scrutinized by a reduced-form hazard model and generally supported. By performing out-of-sample analyses, the results demonstrate that firm size together with the interaction terms between intra-industry firm size dummies and competition intensity can effectively predict default.

Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction

The recent global financial crisis has impacted financial markets around the world, emphasizing the importance of correctly fore-casting credit events. The unprecedented scale of corporate defaults has drawn the attention of both academics and practitioners to examine the prediction of defaults and to explore the causes of de-fault clustering. In prior literature, some researchers have indicated that industry characteristics can affect default probabilities.Jorion and Zhang (2007)andLang and Stulz (1992)documented signifi-cant intra-industry contagion effects of bankruptcies through event studies. Jorion and Zhang (2007) empirically showed that intra-industry credit contagion can be captured in credit default swaps (CDS), and further provided evidence that the change in CDS spreads is significantly related to the industry Herfindahl-Hirsch-man index (HHI). It means that the extent of co-movement in firms’ credit quality within an industry can be determined by the intensity of competition, and this in turn explains part of the correlation of credit risk and the phenomenon of clustered defaults. However, the prevailing credit risk models rarely consider this industry effect. This motivates us to fill the gap in the literature by first building a structural model to theoretically illuminate the relationship between industry competition and credit risk, and then empirically

exploring the effect of product market competition on credit risk and default prediction.

Since the seminal papers ofMerton (1974)andLeland (1994), many structural credit risk models have shown that a firm’s capital structure is an important determinant of credit risk.Mauer and Sarkar (2005)and many others clearly demonstrated that a firm’s financing and investment decisions are interdependent. Moreover, Grenadier (2002)andAguerrevere (2009)built real options models to analyze the effect of product market competition on a firm’s investment and operational decisions. Accordingly, this paper develops a simple structural model to analyze a firm’s optimal operational and financing decisions in a symmetric oligopolistic market and scrutinizes the relationship between product market competition and credit spreads.

Several research works on real options have shown that product market competition has a significant impact on firms’ investment and operational decisions (Grenadier, 2002; Aguerrevere, 2009). Recently,Akdogu and Mackay (2012)theoretically and empirically demonstrated that under- and over-investment can be rational when framed in a strategic competitive setting. Research on the effect of competition on other issues of corporate finance has been relatively sparse, but recently more attention has been paid to this issue. For example,Mackay and Phillips (2005)focused on aggre-gate financial leverage,Grullon and Michaely (2007)investigated payout policy,Giroud and Mueller (2008)explored corporate gov-ernance, and Morellec and Nikolov (2009) and Fresard (2010) looked at firms’ cash holdings.Valta (2010)examined how the

0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.jbankfin.2012.09.001

⇑Corresponding author. Tel.: +886 3 571 2121x57076; fax: +886 3 573 3260. E-mail addresses:hhhuang@mail.nctu.edu.tw(H.-H. Huang),hhlee@mail.nctu. edu.tw(H.-H. Lee).

Contents lists available atSciVerse ScienceDirect

Journal of Banking & Finance

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intensity of competition affects the cost of bank loans and provides evidence that banks rationally take into account the industry struc-ture and competition when pricing financial contracts. As far as we know, however, no study in the literature addresses the linkage be-tween product market competition and credit risk.

Some real options models use regression approach to test the implications of the models. Since the main subject of the paper is to investigate the influence of product market competition on credit risk, instead of regression analysis, we employ the duced-form approach. Different from the regression analysis, re-duced-form models can further provide estimates of default probabilities, and recent empirical research in this field has greatly improved the accuracy of default forecasting. In addition, struc-tural-form models assume that valuation of any corporate security can be modeled as a contingent claim on the underlying value of the firm, implicitly assuming that firm value contains sufficient information about the probability of bankruptcy, butBharath and Shumway (2008)indicated that this is unlikely to be the case. They empirically employed a reduced-form hazard model approach and showed that the implied default probability of the Merton model is not a sufficient statistic for default prediction. Therefore, in addi-tion to proposing a theoretical structural model, we empirically analyze the model’s implications by the well-known reduced-form approach – the hazard model.

The early reduced-form models for default prediction employ approaches like discriminant analysis (Altman, 1968) or binary re-sponse models such as logit and probit regressions (Ohlson, 1980; Zmijewski, 1984).Shumway (2001)argued that these models are inconsistent, because their single-period static features do not ad-just period for risk. The hazard model proposed by Shumway (2001) can incorporate time-varying covariates and was later adopted byChava and Jarrow (2004), Hillegeist et al. (2004), Fig-lewski et al. (2006), Agarwal and Taffler (2008), and many others. However, most of the prior reduced-form models did not consider the industry effect, with only a few exceptions likeChava and Jar-row (2004)that revealed the importance of introducing industry effects in the hazard rate estimation. Nonetheless, they merely consider variables such as industry dummies and their interaction terms with accounting ratios, which only demonstrate industry differences as well as the degrees of importance of accounting vari-ables for different industries. If default intensities are different across industries with otherwise identical firm-specific character-istics, it is of interest to investigate the determinants behind the industry effect through the perspective of product market competition.1

Theoretically, we first build a structural model in a homoge-neously oligopolistic industry. We show that credit spreads are positively related to the number of firms and the effect is signifi-cantly amplified when the firm size is small. The number of firms cannot capture the relative size distribution of the firms in an industry while HHI can. Since the relative firm size in an industry is relevant to both HHI and credit risk, we extend our model to an asymmetric duopolistic industry case, demonstrating that credit spreads of relatively small firms are positively related to HHI, while those of relatively large firms are negatively related to HHI. The ef-fect of HHI on credit spreads is amplified when the firm size is small, and a firm’s relative size in an industry is an important determinant of credit risk. For empirical analysis, we provide evi-dence supporting our theoretical models’ predictions through the reduced-form hazard model. We further perform an out-of-sample default prediction accuracy analysis, incorporating the characteris-tics of product market competition. The results demonstrate that

considering firm size together with interaction terms between the intra-industry firm size dummies and competition intensity can effectively predict default.

The major contributions of our paper are summarized as below. We theoretically and empirically examine the effects of product market competition on credit risk, and further identify that the number of firms and HHI in an industry, measuring different dimensions of market competition, can lead to the opposite im-pacts. This undoubtedly makes contributions to the literature and practice of pricing, measuring and forecasting credit risk with con-sideration of market competition.

The remainder of this paper is organized as follows. Section2 describes our models and hypotheses. Section 3 presents the empirical methodology and data. Section4reports the empirical results of the hazard model and the out-of-sample prediction accu-racy analysis. Finally, Section5draws conclusions.

2. Models and hypotheses

In this section we first develop a structural model that employs the symmetric Cournot-Nash equilibrium in order to model firms’ interactions and propose testable hypotheses that demonstrate how credit spreads are related to the number of firms. We then introduce the asymmetric Cournot-Nash equilibrium in a duopoly and propose hypotheses that particularly show the relationships between the two firms’ credit spreads and the Herfindahl-Hirsch-man index (HHI). Finally, we provide numerical illustrations of our models and develop testable hypotheses.

2.1. Homogeneous oligopoly model

For simplicity, all agents are assumed to be risk-neutral and thus all expected cash flows can be discounted at a constant risk-free rate r.2Consider a homogeneous oligopolistic industry with n

infinitely-lived symmetric firms producing q(t) units of output at to-tal cost TC(q(t)) = a0+ a1q(t), where a0denotes fixed cost and a1q(t) is variable costs. Assume that the produced output cannot be stored, i.e. output always equals demand. The industry inversed demand function is thus given by:

PðXðtÞ; tÞ ¼ XðtÞQðtÞ1=c; ð1Þ

where Q ðtÞ ¼Pn

i¼1qiðtÞ,

c

is elasticity of demand, and X(t) is the industry demand shock governed by dX(t) =

l

X(t)dt +

r

X(t)dW(t).. We further assume X(0) = x0> 0 and r 

l

> 0. Industry production capacity is exogenously given by K, where each symmetric firm owns capacity ki= k = K/n.

Similar to the set-up ofAguerrevere (2009), at time t, any firm i in the industry makes its optimal production decision q

iðtÞ ¼ arg max 06qiðtÞ6k

PðtÞqiðtÞ  TCiðqiðtÞÞ, which leads to the symmet-ric Cournot Nash equilibrium given by:

q iðtÞ ¼ 1 nðXðtÞ=ða1Aðn;

c

ÞÞÞ c; if XðtÞ 6 SW; k; if XðtÞ P SW; ( ð2Þ

where A(n,

c

) = n

c

/(n

c

 1)  A and SW = a1AK1/c. When the indus-try demand is lower than the switching point SW, the firm will pro-duce below its full capacity (k). On the other hand, the firm will produce at its full capacity when the demand is high enough.3

We can now define the firm i’s instantaneous after-tax operat-ing net profits as:

p



iðXðtÞ; KÞ ¼ ð1 

s

ÞðPðtÞqiðtÞ  TCiðqiðtÞÞÞ:

1

For example, among others,Duffie et al. (2007), Figlewski et al. (2006), andDuan (2010)incorporated macroeconomic variables into their reduced-form models.

2

Alternatively, we could assume there is a tradable asset that spans the risks the firms face.

3

We assume that the firm is unable to adjust its capacity, thereby allowing us to focus on the firm’s bankruptcy decision.

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According to the symmetric Cournot Nash equilibrium, we have q

iðx; KÞ ¼ qðx; KÞ and

p

iðx; KÞ ¼

p

ðx; KÞ, where i = 1, 2, . . ., n, and thus suppress notation i hereafter. Bringing back the optimal out-put decision and rearranging terms, we can derive:

p

ðx; KÞ ¼ ð1 

s

Þ a1K n 1 nc1 x SW  c  a0   ; if x 6 SW: ð1 

s

Þ xK11=c n  ða0þ a1K nÞ   ; if x P SW: 8 > < > : ð3Þ

Note that

p

is continuous at the switching point. Eq.(3)shows that when the firm operates at full capacity, its net profit is linear in the industry demand shock, whereas its net profit is non-linear in the shock when it operates below full capacity. This non-linearity dem-onstrates the firm’s operational flexibility, which is affected by other competitors’ flexibilities.

To analyze the firm’s bankruptcy decisions, we first derive its unlevered firm value, which is related to its recovery value when default occurs. Since the switch point is exogenously given and the firm’s abandonment policy is endogenously determined, we consider two possibilities: either switch before abandonment, or abandon before switching. When the abandonment trigger is larger than the switch point xA> SW, our model is exactly the same as the standard Leland style model in which a firm with an unlevered as-set value cannot switch its production in the future and each firm always produces at its full capacity. This paper focuses on the case where the abandonment trigger is less than the switch point and hereafter assumes xA< SW. Therefore, we consider the firm’s oper-ational flexibility and take the non-linear state-dependence feature into consideration.

The firm’s unlevered asset value can be derived as below.4For

xA6x 6 SW, Vðx; KÞ ¼ ð1 

s

Þ a1K n 1 n

c

 1 1 gð

c

Þ x SW  c a0 r    ð1 

s

Þ  a1K n 1 n

c

 1 1 gð

c

Þ xA SW  c a0 r   x xA  g2 þ ð1 

s

Þ  1 

g

2

g

1

g

2   ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x SW  g1  ð1 

s

Þ  1 

g

2

g

1

g

2   ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x A SW  g1 x xA  g2 : ð4Þ For x P SW, Vðx; KÞ ¼ ð1 

s

Þ xK 11=c nðr 

l

Þ a0 r þ a1K nr  !  ð1 

s

Þ  a1K n 1 n

c

 1 1 gð

c

Þ xA SW  c a0 r   x xA  g2 þ ð1 

s

Þ  1 

g

1

g

1

g

2   ðSWÞK11=c nðr 

l

Þ þ

g

1

g

1

g

2  a 1K nr 

c



g

1

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x SW  g2  ð1 

s

Þ 1 

g

2

g

1

g

2  ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x A SW  g1 x xA  g2 : ð5Þ

The firm’s optimal abandonment policy xA is determined by the smooth-pasting condition: lim

x#xA

@Vðx;KÞ

@x ¼ 0, where xA

e

(0, SW). The optimal abandonment trigger is numerically solved and is chosen to maximize the unlevered firm value.

We next assume that each identical firm issues perpetual debt, continuously paying coupon flow C. The debt value can be derived as follows. For x P xD Dðx; KÞ ¼C r C r ð1  bÞViðxD;KÞ   x xD  g2 ; ð6Þ

where b is the proportional bankruptcy cost, and thus (1  b) de-notes the recovery rate of the debt if default occurs.

The corresponding equity value can be derived as below. For xD6x 6 SW, Eðx; KÞ ¼ ð1 

s

Þ a1K n 1 n

c

 1 1 gð

c

Þ x SW  c ða0þ CÞ r    ð1 

s

Þ  a1K n 1 n

c

 1 1 gð

c

Þ xD SW  c ða0þ CÞ r   x xD  g2 þ ð1 

s

Þ  1 

g

2

g

1

g

2   ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x SW  g1  ð1 

s

Þ  1 

g

2

g

1

g

2   ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x D SW  g1 x xD  g2 : ð7Þ For x P SW, Eðx; KÞ ¼ ð1 

s

Þ xK 11=c nðr 

l

Þ a0þ C r þ a1K nr  !  ð1 

s

Þ  a1K n 1 n

c

 1 1 gð

c

Þ xD SW  c ða0þ CÞ r   x xD  g2 þ ð1 

s

Þ  1 

g

1

g

1

g

2  ðSWÞK11=c nðr 

l

Þ þ

g

1

g

1

g

2  a 1K nr 

c



g

1

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x SW  g2  ð1 

s

Þ  1 

g

2

g

1

g

2  ðSWÞK11=c nðr 

l

Þ þ

g

2

g

1

g

2  a 1K nr 

c



g

2

g

1

g

2  a 1K n 1 n

c

 1 1 gð

c

Þ  x D SW  g1 x xD  g2 : ð8Þ

The equity holder’s optimal default decision is determined by the fol-lowing smooth-pasting condition: lim

x#xD

@Eðx;KÞ

@x ¼ 0, where xD

e

(0, SW). The optimal bankruptcy trigger can be numerically solved and is cho-sen to maximize the equity value.

We now define the credit spread of the debt as:

CSðx; n; K; x DðnÞ; x  AðnÞÞ ¼ C Dðx; n; K; x DðnÞ; xAðnÞÞ  r: ð9Þ

Eq.(9)shows that credit spreads of debt are linked to the number of firms in an industry, via the recovery value and a firm’s optimal abandonment and default policies. We can therefore investigate the effect of the number of firms on credit spreads (credit risk), and the result is provided in the later numerical subsection. 2.2. Asymmetric duopoly model

Using the number of firms in an industry to measure the intensity of product market competition is adequate in an industry having sym-metric firms. When employing other measures, such as the

Herfin-4

In an online appendix, we provide the detailed derivations and explanations of this paper’s value functions.

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dahl-Hirschman index (HHI) or the four-firm concentration ratio (CR4), the symmetric oligopolistic assumption is somewhat restricted. We hence extend our model to an asymmetric duopoly model.

We assume that Firm 1 has smaller capacity k1=

j

K while Firm 2 has larger capacity k2= (1 

j

)K, where

j

e

(0, 0.5). This can be explained by stating that Firm 2 was the leader when entering the market, therefore enjoying some first-mover advantages, and builds larger capacity, whereas Firm 1 enters later, is in a disadvan-tageous position, and thus builds smaller capacity.5By doing so, we

theoretically capture the effect of relative size on product market competition and on credit risk, which can be empirically examined by using HHI and CR4as measures of intensity of product market competition. We assume that the total cost functions of Firms 1 and 2 are given respectively as TC1(q1(t)) = a0+ 2(1 

j

)a1q1(t) and TC2(q2(t)) = a0+ a1q2(t). When

j

= 0.5, the two firms’ total costs be-come the same, leading to the case of a symmetric duopoly, which is consistent with the previous section’s model. When

j

e

(0, 0.5), the marginal cost of Firm 1 is larger than that of Firm 2, thereby demonstrating that Firm 1 is in a disadvantageous position.

The asymmetric Cournot Nash equilibrium can be shown as:

q iðx;

j

Þ ¼ gið

j

;

c

Þðx=a1Þc; if x 6 SWi; ki; if x P SWi; ( ð10Þ where SWi= a1(ki/gi(

j

,

c

))1/c, i = 1, 2, g1¼1þh1 ð1  1 cð1þhÞÞ=2ð1jÞ  c ; g2¼ h 1þh 1  1 cð1þhÞ   =2ð1 

j

Þ  c , and h ¼ 1þcð12jÞ 2ð1jÞcð12jÞ. We now define HHIðx;

j

Þ  q1ðx;jÞ

q 1ðx;jÞþq2ðx;jÞ 100  2 þ q 2ðx;jÞ q 1ðx;jÞþq2ðx;jÞ 100  2

, and HHI(x; 0.5) = 2500 for any x > 0. In this case, there are only two symmetric firms in the market, and 2500 is the lowest bound of HHI, representing the highest compe-tition intensity in a duopoly.

The equilibrium profit functions of the two firms are therefore given by:

p

 1ðx; KÞ ¼ ð1 

s

Þ a1k1 c2ð1ð1þhÞ1jÞ   x SW1  c  a0   ; if x 6 SW1: ð1 

s

Þð

j

xK11=c ða0þ 2

j

ð1 

j

Þa1KÞÞ; if x P SW1: 8 < : ð11Þ and

p

 2ðx;KÞ ¼ ð1 

s

Þ a1k2 2ð1jÞccð1þhÞð1þhÞ1cð1þhÞþ1   x SW2  c  a0   ; if x 6 SW2: ð1 

s

Þðð1 

j

ÞxK11=c ða0þ

j

a1KÞÞ; if x P SW2: 8 < : ð12Þ

Following the same procedure and employing some similar bound-ary conditions, we derive all the desired formulae, which are sum-marized as below. For xA16x 6 SW1, the unlevered asset value of Firm 1 is given by V1ðx; KÞ ¼ ð1 

s

Þ a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ x SW1  c a0 r    ð1 

s

Þ  a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ xA1 SW1  c a0 r   x xA1  g2 þ ð1 

s

Þ 1g2 g1g2  j ðSW1ÞK11=c ðrlÞ þ g2 g1g2   2jð1jÞa1K r  

c



g

2

g

1

g

2   a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ  x SW1  g1  ð1 

s

Þ 1g2 g1g2  j ðSW1ÞK11=c ðrlÞ þ g2 g1g2   2jð1jÞa1K r  

c



g

2

g

1

g

2   a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ  x A1 SW1  g1 x xA1  g2 : ð13Þ

For xA26x 6 SW2, the unlevered asset value of Firm 2 is given by

V2ðx;KÞ ¼ ð1 sÞ a1k2 2ð1 jÞcð1 þ hÞ cð1 þ hÞ þ 1 cð1 þ hÞ  1   1 gðcÞ x SW2  c a0 r    ð1 sÞ a1k2 2ð1 jÞcð1 þ hÞ cð1 þ hÞ þ 1 cð1 þ hÞ  1   1 gðcÞ xA2 SW2  c a0 r   x xA2  g2 þ ð1 sÞ 1g2 g1g2   ð1jÞðSW2ÞK11=c ðrlÞ þ g2 g1g2  j a1K r  cg2 g1g2   a1k2   2ð1 jÞcð1 þ hÞ cð1 þ hÞ þ 1 cð1 þ hÞ  1   1 gðcÞ  x SW2  g1 ð1 sÞ 1g2 g1g2   ð1jÞðSW2ÞK11=c ðrlÞ þ g2 g1g2  j a1K r  cg2 g1g2   a1k2   2ð1 jÞcð1 þ hÞ cð1 þ hÞ þ 1 cð1 þ hÞ  1   1 gðcÞ  xA2 SW2  g1 x xA2  g2 : ð14Þ

The two firms’ optimal abandonment policies xAiare determined by

the following smooth-pasting conditions: lim x#x

Ai

@Viðx;KÞ

@x ¼ 0, where xAi2 ð0; SWiÞ; i ¼ 1; 2. The optimal abandonment triggers are numerically solved and chosen to maximize the two unlevered firms’ values separately. For xD16x 6 SW1, the equity value of Firm 1 is given by E1ðx; KÞ ¼ ð1 

s

Þ a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ x SW1  c a0þ C r    ð1 

s

Þ a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ xD1 SW1  c a0þ C r    x xD1  g2 þ ð1 

s

Þ 1g2 g1g2  j ðSW1ÞK11=c ðrlÞ þ g2 g1g2  2j ð1jÞa1K r  

c



g

2

g

1

g

2   a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ  x SW1  g1 ð1 

s

Þ 1g2 g1g2  jðSW 1ÞK11=c ðrlÞ þ g2 g1g2  2jð1jÞa 1K r  

c



g

2

g

1

g

2   a1k1 2ð1 

j

Þ

c

ð1 þ hÞ  1   1

c

Þ  x D1 SW1  g1 x xD1  g2 : ð15Þ

For xD26x 6 SW2, the equity value of Firm 2 is given by

E2ðx;KÞ ¼ ð1 

s

Þ a1k2 2ð1 

j

Þ

c

ð1 þ hÞ 

c

ð1 þ hÞ þ 1

c

ð1 þ hÞ  1     1 gð

c

Þ x SW2  c a0þ C r   ð1 

s

Þ  a1k2 2ð1 

j

Þ

c

ð1 þ hÞ 

c

ð1 þ hÞ þ 1

c

ð1 þ hÞ  1     1 gð

c

Þ xD2 SW2  c a0þ C r  x xD2  g2 þ ð1 

s

Þ  1g2 g1g2   ð1jÞðSW2ÞK11=c ðrlÞ þ g2 g1g2  j a1K r  cg2 g1g2   a1k2   2ð1 

j

Þ

c

ð1 þ hÞ 

c

ð1 þ hÞ þ 1

c

ð1 þ hÞ  1   1

c

Þ  x SW2  g1 ð1 

s

Þ 1g2 g1g2   ð1jÞðSW2ÞK11=c ðrlÞ þ g2 g1g2  j a1K r  cg2 g1g2   a1k2   2ð1 

j

Þ

c

ð1 þ hÞ 

c

ð1 þ hÞ þ 1

c

ð1 þ hÞ  1   1

c

Þ  x D2 SW2  g1 x xD2  g2 : ð16Þ

The two equity holders’ optimal default decisions are similarly determined by the following smooth-pasting conditions: lim

x#xDi @Eiðx;KÞ

@x ¼ 0, where xDi2 ð0; SWiÞ; i ¼ 1; 2. The optimal bankruptcy triggers also can be numerically solved and are chosen to maximize the two equity values separately.

The debt values and credit spreads are finally derived as below. For i = 1, 2

5

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Diðx; KÞ ¼ C r C r ð1  bÞViðxDiÞ   x xDi  g2 ; ð17Þ and CSiðx;

j

;K; xD

j

Þ; x Aið

j

ÞÞ ¼ C Diðx;

j

;K; xD

j

Þ; xA

j

ÞÞ  r: ð18Þ

Eq.(18)shows that the credit spreads of debt are linked to the two firms’ asymmetry

j

, which is relevant to the level of HHI via the recovery value and the two firms’ optimal abandonment and default policies. We can therefore investigate the effects of HHI on the two firms’ credit spreads, with the results provided in the later subsection. 2.3. Numerical analyses and hypotheses

In this section we conduct numerical analyses to illustrate the effects of product market competition on credit risk and build test-able hypotheses along two different dimensions of market compe-tition in our theoretical models: the number of firms and HHI.

With the base-case parameters of r = 0.06,

l

= 0.01,

r

= 0.2,

c

= 1.6, n = 10, K = 1000,

s

= 0.35, b = 0.5, a1= 0.06, a0= 0.1, and C = 0.188,6 we first employ the homogeneous oligopoly model to

illustrate the effects of the number of firms on credit spreads and then use the asymmetric duopoly model to demonstrate how the Herfinddahl-Hirschman index (HHI) generates different impacts on the credit spreads of large and small firms.

Using our homogeneous oligopoly model,Fig. 1 first demon-strates the effect of the number of firms on credit spreads, showing that as the number of firms in an industry (n) increases (market competition intensifies), credit spreads increase. The reason is that when market competition intensifies, other things being equal, each firm’s profits (and thus the unlevered asset value) are diluted by other competitors, thereby increasing the possibility of default as well as credit spreads. Secondly, we observe that credit spreads turn lower as the industry demand increases (firm size becomes larger), which is consistent with the results of standard Leland’s style structural models. When industry demand goes up, ceteris paribus, each firm’s profits and the unlevered asset value increase, thereby lowering bankruptcy probability as well as credit spreads. The increase of the industry demand leads to the increase of firm size, thereby showing that credit risk is negatively related to the firm’s own size.7Thirdly,Fig. 1demonstrates that distances between curves are amplified when industry demand weakens. It clearly shows that the sensitivity of credit spreads to the number of firms in an industry increases as the firm size becomes smaller. In view of Fig. 1, all the above results are robust for changes to various parameters (including industry demand volatility, elasticity, and growth rate), and we respectively summarize the above three results as Hypotheses 1a, 1b and 1c below.

Hypothesis 1a. Credit risk is positively related to the number of firms in an industry.

Hypothesis 1b. Credit risk is negatively related to the firm’s own size.

Hypothesis 1c. The sensitivity of credit risk to the number of firms in an industry is amplified when the firm’s own size becomes smaller.

The above results infer directly that an increase in product mar-ket competition intensity (the number of firms in an industry) amplifies a firm’s credit risk. Using the number of firms in an indus-try to measure the intensity of product market competition is only adequate in an industry with similar-size firms, but when firms in an industry have significantly different sizes, the number of firms cannot capture this characteristic. HHI, a commonly accepted mea-sure of market concentration, takes into account not only the num-ber of firms, but also the relative size distribution of the firms in an industry. HHI decreases (representing that competition intensifies) both when the number of firms in an industry increases and as the disparity in size between those firms decreases.

Employing our asymmetric duopoly model and the above base-case parameters,Fig. 2first demonstrates that the credit spreads of small-capacity Firm 1 decrease, while those of large-capacity Firm 2 increase as HHI decreases (as

j

increases, i.e., the asymmetry be-tween the two firms decreases). Our duopoly model fixes the num-ber of firms in a industry to two, and thus HHI purely captures the effect of the disparity in size between the two firms. When HHI is decreasing, the market position of the relatively small Firm 1 im-proves, whereas that of relatively large Firm 2 gets worse, other things being equal. This is because Firm 1 (2) is in a better (worse) market position in an industry whereby HHI becomes smaller (the asymmetry between the two firms becomes smaller). In particular, we predict that the effect of the concentration ratio on credit spreads for relatively small firms is exactly opposite to that for rel-atively large firms. As a consequence, when HHI becomes smaller (competition intensifies), Firm 1 (2) suffers smaller (greater) de-fault risk, leading to lower (higher) credit spreads. We therefore propose the following hypotheses to characterize the above result. Secondly,Fig. 2illustrates one interesting observation that cred-it spreads of the relatively small Firm 1 are robustly greater than those of the relatively large Firm 2. The existing credit risk literature often shows that a smaller (own) firm size leads to higher credit risk, but as far as we know, no study investigates how the relative firm size in an industry affects credit risk.Fig. 2particularly demon-strates that the relatively small Firm 2 suffers a higher credit risk. We give an example to explain the difference between the own and relative firm size arguments. Assume that there are two indus-tries in a market where the first industry contains two firms (Firms A and B) with firm sizes 80 and 20, while the second industry con-tains two firms (Firms C and D) with firm sizes 8 and 2, respectively. The own size effect suggests that Firm C suffers a higher credit risk than Firm B, other things being equal, whereas the relative size in an industry implies that Firm B suffers greater credit risk than Firm C, other things being equal. As a consequence, the relative firm size in an industry is also an important determinant of credit risk.

Finally,Fig. 2also demonstrates that distances between curves are significantly amplified as the industry demand weakens. It clearly shows that the sensitivity of credit spreads to HHI in an industry enlarges when the firm size becomes smaller. We summa-rize all the three results in the following hypotheses.

Hypothesis 2a. The credit risk of a relatively small (large) firm is positively (negatively) related to HHI.

Hypothesis 2b. Relatively small firms in an industry suffer a higher credit risk than relatively large firms.

Hypothesis 2c. The sensitivity of credit risk of relatively small firms to HHI in an industry is higher than that of relatively large firms to HHI.

6

Most of the parameters are chosen fromAguerrevere (2009)andLeland (1994)

except for a0and C, which are unique to our model. Since a0is fixed costs, related to a

firm’s operating leverage, and C is the coupon payment, related to a firm’s financial leverage, we choose these two parameters to match the medians of operating and financial leverages of our sample firms (2.35 and 0.48, respectively). Our empirical section details the samples. The operating leverage is defined as the ratio of revenues minus variable costs to the revenues minus fixed and variable costs, while financial leverage is defined as the debt to asset ratio.

7

The inference that higher industry demand induces a larger firm size is similar to

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Fig. 1. Effect of the number of firms on credit spreads. This figure shows the effects of the number of firms in an industry on credit spreads (CS) under various settings of return volatilities of demand shocks (r), industry demand price elasticity (c), and expected growth return of demand shocks (l). Here, X is the industry demand shock and n is the number of firms in industry. Other parameters are the same as the base-case parameters.

Fig. 2. Effect of Herfindahl-Hirschman Index on Credit Spreads. This figure shows the effects of Herfindahl-Hirschman Index (HHI) on credit spreads. The left panel denotes the credit spreads of the relatively smaller firm 1 (CS1), while the right panel denotes those of the relatively larger firm 2 (CS2). Here, X is the industry demand shock and HHI is

the sales-based HHI defined as HHIðjÞ  q1ðx;jÞ

q 1ðx;jÞþq2ðx;jÞ 100  2 þ q2ðx;jÞ q 1ðx;jÞþq2ðx;jÞ 100  2

, which is calculated at X = 40 forj= 0.5, 0.4, 0.3, and 0.2. Other parameters are the same as the base-case parameters.

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In a brief summary, we predict that a larger number of firms in an industry (more competitive) leads to a higher credit risk. Never-theless, the relatively small (large) firms suffer from lower (higher) credit risk when HHI is lower in an industry (more competitive). The identification that the number of firms and HHI in an industry, measuring different dimension of market competition generate different impacts on credit risk is essentially important when we explore the relation between competition and credit risk or pre-dicting default.

3. Methodology and data 3.1. Empirical methodology

The empirical performance of credit risk models in pricing risky debts and analyzing credit spreads is generally unsatisfactory, be-cause the illiquid corporate bond market hinders theoretical mod-els from accurately pricing risky debts. However, predicting the credit quality of a corporate security could be a good application, because it is less affected by additional factors such as liquidity, tax differences, and recovery rates. Recently, some researchers have used credit default swap (CDS) spreads as a direct measure of credit quality. Nonetheless, comprehensive and a long history of CDS data is not available, especially for small companies that are crucial for our study.8Therefore, we use equity data for

empir-ical analysis. The advantage of using stock market data is good qual-ity and availabilqual-ity of return data for long periods. However, calibration of our theoretical models using equity data is practically infeasible due to the complexity of our structural models. Accord-ingly, we explore the effect of product market competition on de-fault intensity, which can be estimated through reduced-form models and is closely related to credit spreads in our theoretical models. In addition, reduced-form models by construction can also incorporate other accounting and financial information to control for factors that may affect default intensity.

In the framework of reduced-form modeling, the default of firm i is described by default time

s

i, and the default time can be mod-eled through its stochastic intensity ki. If the firm is alive at time t, then the intensity at time t for firm i satisfies:

kiðtÞ ¼ lim Dt!0

Pðt <

s

i6t þ

D

tj

s

iPt; FtÞ

D

t :

In a word, the default probability within a small time period Dt after t is close to kiðtÞDt, where kiðtÞ depends on information avail-able at time t as represented by Ft. This information contains all intensities of firms and all default histories up to time t. Under the reduced-form setting, modeling the default probability for firm i thus reduces to modeling its default intensity kiðtÞ. As a bond cred-it spread may be affected by factors such as liquidcred-ity unrelated to credit quality, we use default probability as the measure of credit quality in our empirical study.

To analyze the impact of product market competition on default intensity, we adopt the popular Cox proportional hazard model, which has been used byBharath and Shumway (2008), Hillegeist et al. (2004), and many others in empirical studies. Proportional hazard models assume that the probability of default at time t, con-ditional on survival until time t, is kðtÞ ¼

u

ðtÞ½expðxðtÞ0bÞ, where / (t) is the ‘‘baseline’’ hazard rate and the term exp (x(t)’b) allows the

expected time to default to vary across firms, according to covari-ate x(t). The baseline hazard rcovari-ate /(t) is common to all firms and the Cox proportional hazard model does not impose any structure on /(t). Cox’s partial likelihood estimator provides estimates of b, and the details of estimation can be seen inCox and Oakes (1984). 3.2. Data

In the spirit of the broad definition of bankruptcy byBrockman and Turtle (2003)andDichev (1998), our default samples are de-fined as firms that are delisted due to bankruptcy, liquidation, or poor performance. This is because many firms were delisted from the stock exchange for reasons other than liquidation, bankruptcy or merger/acquisition. A significant portion of firms are delisted due to poor performance or failure to meet exchange listing requirements. Specifically, a firm is considered as ‘‘performance delisted’’ byBrockman and Turtle (2003)if it is given a CRSP delist-ing code with the first digit of 4 (liquidation), or between 550 and 591 (poor performance).

We regard these performance-related delistings as companies under financial distress and use these firms as default samples to perform our empirical tests. This is because delisting from a stock exchange can trigger a credit rating downgrade by rating agencies and drastically reduce the value of corporate bonds, leading to a substantial increase in credit spreads. Sometimes when a firm is delisted, its creditors can withdraw lines of credit. Both downgrade and credit line withdrawal make it more expensive for a financially distressed company to raise capital for operations, which further in-creases its default risk. Moreover, a company in bad shape does not necessarily file for bankruptcy around the date of delisting for var-ious reasons. After delisting from the exchange, a firm can be either acquired by another firm or file for bankruptcy several years later. Government intervention, as witnessed during the recent global financial tsunami, has a direct impact in reducing the number of bankruptcy filings (Duan, 2010). The most extreme example can be Fannie Mae and Freddie Mac. They are government agencies and have extremely low probabilities of filing for bankruptcy. Nonetheless, they still were delisted for failing to meet the require-ments set forth by the stock exchange.9Shareholders of these firms

suffered huge losses due to financial distress from delisting. Thus, default risk and hazard rate estimation should not be tied only to events of bankruptcy filings. Overall, the broad definition of ruptcy (or financial distress) can be of equal importance to bank-ruptcy prediction.

Since the sample size of defaults is relatively small, our sam-pling period is from January 1985 to December 2009.Table 1 sum-marizes defaulting and other-exit firms by the major industry categories. Other-exit firms are those delisted due to merger or acquisition. The majority of delisted firms are manufacturing and service companies.Table 2reports the number and the percentage (over active firms) of defaulting and other-exit samples over the years. One can find that the number and proportion of non-finan-cial delisted companies during the subprime-mortgage crisis are less than those during the burst of Dotcom-bubble between 2001 and 2003. This is somewhat unanticipated in terms of severity for the two economic downturns. Government intervention clearly, to some degree, reduces the number of firms forced out of the mar-ket. Therefore, examining the effect of product market competition on performance delisted firms rather than bankruptcy can effec-tively increase the sample size of default firms and reduce the po-tential bias of default events in the hazard model.

8

For example,Ericsson et al. (2009),Das et al. (2009), andTang and Yan (2010)

conducted analyses of credit spreads using CDS data. However, the total number of sample firms in the above-mentioned studies is very small (less than 300) compared with ours (around 15,000). We also perform our analyses using bond yield spreads from TRACE transactions data, but they are still subject to the similar sample selection bias problem. Most of the sample bonds are issued by relatively larger firms in industry.

9

Fannie Mae and Freddie Mac were delisted from NYSE on July 7, 2010 with final trading prices of $0.25 and $0.34, respectively. They were delisted by NYSE, because they could no longer meet the standard for NYSE continued listing – a minimum price of $1 per share.

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An important empirical issue for product market competition research is industry classification. This study, for robustness, con-ducts all the tests for both Fama–French 49-Industry and 3-digit SIC level classifications.10The 4-digit SIC definition is not examined

in our study, because it is too fine and thus too many industries un-der this specification comprise only one or two firms.11Furthermore, some of the four-digit codes may fail to define sound economic mar-kets as pointed out by Clarke (1989) and Kahle and Walking (1996).12

Following literature, we exclude all firms with SIC codes start-ing with 6 (financial firms) and with first two digits bestart-ing 49 (util-ities). In the Fama–French 49-Industry classification, we exclude firms classified as utilities (Fama–French Industry Code 31),

finan-cial companies (Industry Code 45 to 48), and firms that cannot be well-classified by Fama and French (Industry Code 49).Figs. 3 and 4 show the frequency of median monthly number of firms per industry under Fama–French 49 and 3-digit SIC industry classifica-tions, respectively. The median number of firms in an industry with a 3-digit SIC industry classification is generally far fewer than that under the Fama–French classification. One can observe that even under the 3-digit SIC industry classification, there are 85 industries with fewer than 5 firms and 134 industries with less than 10 firms, confirming the difficulty in using 4-digit SIC classi-fication for empirical tests. In sum, excluding financial and utility companies, there are a total of 44 industries in the Fama/French classification and 245 industries in the 3-digit SIC classification.

Equity prices are collected from CRSP, and financial statement information is retrieved from Compustat. Our sampling period is from January 1985 to December 2009. The quarterly accounting information is from 1984 to 2009, because some firms under finan-cial distress stopped filing finanfinan-cial reports a long time before they were delisted from the stock exchanges. Following prior literature, we lag all accounting information by 3 months due to reporting

Table 1

Summary of default firms and other exit firms by industry category. This table reports the number of default firms and other exits over the years from 1985 to 2009. Percent is the percentage of default (other exit) firms in a given industry category over the total number of default (or other exit) firms.

Industry category SIC code Class Number of

default firms

Percent (%) Number of other exit firms

Percent (%)

1 <1000 Agriculture, forestry, and fisheries 30 0.49 26 0.31

2 1000 to less than 1500 Mineral industries 434 7.05 427 5.02

3 1500 to less than 1800 Construction 107 1.74 72 0.85

4 2000 to less than 4000 Manufacturing 2228 36.20 2923 34.38

5 4000 to less than 4900 Transportation and Communications 365 5.93 539 6.34

6 4900 to less than 5000 Utilities 82 1.33 231 2.72

7 5000 to less than 5200 Wholesale trade 304 4.94 262 3.08

8 5200 to less than 6000 Retail trade 457 7.42 501 5.89

9 6000 to less than 6800 Finance, insurance, and real estate 680 11.05 1838 21.62

10 7000 to less than 9000 Services 1257 20.42 1650 19.40

11 9100 to less than 10,000 Public administration 211 3.43 34 0.40

Total 6155 100 8503 100

Table 2

Summary of default firms and other exit firms over the sample period. This table reports the number of default firms and other exit firms for each year during the sample period from 1985 to 2009. Percent is the percentage of default firms and other exit firms over active firms in that year. Financial versus non-financial companies are reported separately.

Year Active non-financial Default non-financial Other exit non-financial Active financial Default financial Other exit financial Num Percent (%) Num Percent (%) Num Percent (%) Num Percent (%) Num Percent (%) Num Percent (%)

1985 4237 91.43 185 3.99 212 4.57 567 91.60 20 3.23 32 5.17 1986 4299 90.09 245 5.13 228 4.78 615 92.48 19 2.86 31 4.66 1987 4564 92.37 166 3.36 211 4.27 722 94.88 13 1.71 26 3.42 1988 4512 89.76 219 4.36 296 5.89 724 91.18 35 4.41 35 4.41 1989 4409 91.10 209 4.32 222 4.59 703 91.78 36 4.70 27 3.52 1990 4404 92.04 235 4.91 146 3.05 711 93.68 25 3.29 23 3.03 1991 4363 92.77 248 5.27 92 1.96 740 93.55 35 4.42 16 2.02 1992 4531 92.24 302 6.15 79 1.61 747 92.34 39 4.82 23 2.84 1993 4836 95.37 134 2.64 101 1.99 827 94.84 16 1.83 29 3.33 1994 5321 94.16 164 2.90 166 2.94 1395 94.51 21 1.42 60 4.07 1995 5467 93.07 194 3.30 213 3.63 1444 90.82 31 1.95 115 7.23 1996 5922 93.26 148 2.33 280 4.41 1427 89.92 19 1.20 141 8.88 1997 6142 91.75 218 3.26 334 4.99 1385 88.90 21 1.35 152 9.76 1998 6048 88.65 357 5.23 417 6.11 1350 87.15 47 3.03 152 9.81 1999 5606 87.35 345 5.38 467 7.28 1359 90.54 27 1.80 115 7.66 2000 5525 88.30 274 4.38 458 7.32 1379 89.95 32 2.09 122 7.96 2001 5136 86.93 426 7.21 346 5.86 1310 90.66 29 2.01 106 7.34 2002 4738 89.51 355 6.71 200 3.78 1269 92.63 35 2.55 66 4.82 2003 4360 90.21 248 5.13 225 4.66 1242 93.03 20 1.50 73 5.47 2004 4281 93.35 100 2.18 205 4.47 1199 90.42 23 1.73 104 7.84 2005 4199 91.84 126 2.76 247 5.40 1230 93.75 23 1.75 59 4.50 2006 4189 92.57 61 1.35 275 6.08 1212 92.03 12 0.91 93 7.06 2007 4063 90.07 49 1.09 399 8.85 1182 91.77 9 0.70 97 7.53 2008 3969 91.62 168 3.88 195 4.50 1142 91.58 42 3.37 63 5.05 2009 3404 84.05 217 5.36 429 10.59 1022 88.18 51 4.40 86 7.42 Total 5393 6443 680 1846 10

We classify all the companies to 49 industries according to Kenneth French’s website. (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/ det_49_ind_port.html)

11

For the 4-digit SIC definition, 145 out of 440 industries have 2 or less companies. A total of 255 industries have fewer than 5 firms.

12

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delay. If the accounting variable is missing, we substitute it with the most recent or closest observation prior to it.

4. Empirical results 4.1. Covariates

4.1.1. Variables of major interests

4.1.1.1. The intensity of competition. We first summarize the most widely used measures of product market competition. Variables for intensity of competition are taken from papers in prior litera-ture, including Morellec and Nikolov (2009), Valta (2010), and Gasper and Masa (2006), among others.

1. Natural logarithm of the number of firms per industry (Ln_N): The number of firms in an industry may affect the ability of a firm to influence price. In a perfectly competitive market, firms are just price takers, while the firm can decide prices in the case of a monopoly. In practice, most industries are somewhere in-between these two cases. Following Morellec and Nikolov (2009), we use the logarithm of the number of firms.

2. Herfindahl-Hirschman Index (HHI): HHI is the sum of the squared fractions of all individual firms’ market share. A higher value of HHI means a more concentrated industry.13

3. Four-firm concentration ratio (CR4): This measure is computed as the combined market share of the four largest firms over the aggregate sales volume in each industry. Similar to HHI, higher values of CR4imply more concentrated industries.

4.1.1.2. Relative size of firms within an industry. Our numerical anal-yses in Section2.3 indicate that the intensity of competition in terms of HHI and the relative firm size (relatively small versus rel-atively large firms) can jointly affect credit spreads, and therefore we introduce the following dummy variables for the relative size of firms in an industry and their interaction with three different competition varaibles.

1. Relative size dummies (Relative_size_S and Relative_size_M): To isolate the effect of the relative size of firms from market capi-talization (firm’s own size in the finance literature), we use dummy variables to divide the samples of each industry into three groups. In accordance with the variables of market com-petition, we use a firm’s sales volume to represent its relative size in a given industry.14 Thus, firms are sorted into three

groups – Small, Medium, and Big – by using 30 percentile and 70 percentile of sales volume within the given industry as the cutoff points. Note that a firm can have a large market capitaliza-tion even as it belongs to the Small relative size group (thus, Relative_size_S = 1) due to its relatively small sales volume as compared to other companies in an industry. That is, firms with large own size can be classified into the Small relative size group.

2. SIC_DN5: This dummy variable is set to 1 for industries with less than five companies. When SIC_DN5 is equal to 1, Rela-tive_size_S and Relative_size_M are set to zero. A plausible cut-off choice is five companies for two reasons. First, the computation of the four-firm concentration ratio loses its meaning for industries with less than five firms. Second, it seems unreasonable to divide firms into three groups for such a small industry.

4.1.1.3. Firm’s own size.

1. Firm’s own size (FirmSize): In the numerical analysis of Sec-tion2.3, it is apparent that firm’s own size can play a crucial role in determining credit spreads. When firm’s own size is large, credit spreads tend to be very small. Following the extant literature, firm’s own size is defined as the logarithm of each firm’s equity value, computed as stock price times shares out-standing, divided by the total market value of NYSE/AMEX to make the size stationary. Note that firm’s own size is not new in the reduced-form literature. Therefore, we have no intention to claim that FirmSize is a measure completely resulting from product market competition; instead, from another perspective it provides economic interpretation of the effect of a firm’s own size on default risk.

0 1 2 3 4 5 6 7 8 9 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Number of Firms within Industry

Frenquency

Fig. 3. Number of Firms within an Industry under the Fama–French 49-Industry Classification. This figure presents the median of the monthly sample number of firms in an industry during the sample period from 1985 to 2009.

0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Number of Firms within Industry

Frequency

Fig. 4. Number of firms within an industry under the 3-digit SIC industry classification. This figure presents the median of the monthly sample number of firms in an industry during the sample period from 1985 to 2009. There are two 3-digit SIC level industries not presented in this frequency plot, because the median number of firms is over 200. The first industry with the first 3-digit SIC code of 283 includes companies related to medicinal chemicals, pharmaceutical, and biological products. The second industry with the first 3-digit SIC code of 737 includes computer programming, data processing, and prepackaged software companies.

13 Note that the recent research byAli et al. (2009)indicates that the Census HHI is

better for capturing actual industry competition than measures that are obtained based on Compustat firms. The Census HHI is based on data from both public and private firms in an industry, and HHI is computed by summing the squares of the sales of individual companies for the 50 largest firms or all the companies in the industry, whichever is lower. However, the U.S. Census Bureau only reports these indices for manufacturing firms every five years. Therefore, due to the very low frequency disadvantage of the Census HHI data, we decide to use HHI computed from public firms in our monthly analysis.

14 We also conduct an analysis by using market equity as the criterion of relative

size dummy. The results are very similar to those using sales as the measure of relative size.

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4.1.2. Control variables

We include the following firm-specific covariates widely used in the prior literature.

1. The firm’s implied default probability from the Merton model (

p

Merton):

p

Merton is derived from the Merton (1974) model, and N(DTD) represents a company’s bankruptcy probability, where N(.) is the standard normal cumulative distribution function and DTD (Distance to Default) is regarded as a vola-tility-adjusted measure of leverage. Merton’s model is widely adopted in both industry (see Crosbie and Bohn, 2003) and academics (seeDuffie et al., 2007; andBharath and Shumway, 2008). Here,

p

Mertonis estimated on a monthly basis using the preceding 1-year daily equity values. Our method for

comput-ing

p

Merton is based on the iterated procedure used by

Vassalou and Xing (2004)andBharath and Shumway (2008). The online Appendix gives a detailed description of its computation.

2. The firm’s trailing 1-year excess stock return (Exret_y): Fol-lowingShumway (2001)and others, we measure each firm’s trailing 1-year excess return in month t as the return of the firm minus the value-weighted market return. Each firm’s trailing 1-year returns are calculated by cumulating monthly returns. Market returns are obtained from Kenneth French’s website.

3. Idiosyncratic risk (Idio_Risk): The 1-year idiosyncratic volatility is calculated by regressing the monthly stock return on the market return over the preceding 12 months.

4. The ratio of net income to total assets (NI/TA). 5. The ratio of total liabilities to total assets (TL/TL).

We note that there are still many other variables that have been used for reduced-form models in the literature, such as trailing 1-year return on the S&P 500 index, 3-month U.S. Treasury rate, real GDP growth, the cash to total assets ratio, and the market-to-book ratio. Control variables included in our study are by no means com-prehensive. Nonetheless, the main purpose of this paper is to investigate how product market competition can affect default risk. To ease any possible estimation difficulty, we tentatively leave those variables for future studies.15

Table 3

Summary statistics of independent variables under the Fama–French 49 Industry Classification. This table reports descriptive statistics of firm-month samples from 1986 to 2009. Ln_N is the natural logarithm of the number of firms per industry; HHI is the Herfindahl-Hirschman Index; CR4is the four-firm concentration ratio. Figures in parentheses indicate

the number of firms (N) in an industry; S, M, and B denote the relative size groups by respectively dividing each industry into 3 subgroups of small, medium, and big firms. Thirty percentile and seventy percentile of sales volume in the given industry are used as cutoffs; FirmSize is defined as the logarithm of each firm’s equity value divided by the total market equity of NYSE/AMEX.pMertonis the firm’s implied default probability calculated from the Merton model; Exret_y is the firm’s trailing 1-year excess stock return; Idio_Risk

is idiosyncratic risk; NI/TA is the ratio of net income to total assets; TL/TL is the ratio of total liabilities to total assets.

Variable Status N_Obs Mean Median Standard deviation

Ln_N (N) Non-default 1,398,995 5.1195 (215.7383) 5.2311 (187) 0.7824 (146.2770) Default 5161 5.1714 (229.9702) 5.2958 (200) 0.7854 (164.7853) CR4 Non-default 1,398,995 0.4592 0.4159 0.1659 Default 5161 0.4712 0.4381 0.1690 HHI Non-default 1,398,995 0.1014 0.0664 0.0982 Default 5161 0.1093 0.0722 0.1076 Ln_N_S (N_S) Non-default 409,855 5.1034 (213.6932) 5.2204 (185) 0.7955 (146.2522) Default 3603 5.1746 (229.2784) 5.2832 (197) 0.7788 (162.3743) Ln_N_M (N_M) Non-default 563,726 5.1198 (215.7018) 5.2364 (188) 0.7820 (146.0622) Default 1221 5.1706 (230.6675) 5.3566 (212) 0.7903 (164.5213) Ln_N_B (N_B) Non-default 425,045 5.1347 (217.7501) 5.2523 (191) 0.7700 (146.5855) Default 337 5.1409 (235.0920) 5.1761 (177) 0.8383 (190.0354) HHI_S Non-default 409,855 0.1020 0.0668 0.0986 Default 3603 0.1121 0.0729 0.1104 HHI_M Non-default 563,726 0.1013 0.0663 0.0984 Default 1221 0.1042 0.0705 0.1022 HHI_B Non-default 425,045 0.1009 0.0663 0.0975 Default 337 0.0978 0.0686 0.0947 CR4_S Non-default 409,855 0.4609 0.4175 0.1668 Default 3603 0.4757 0.4399 0.1715 CR4_M Non-default 563,726 0.4589 0.4154 0.1660 Default 1221 0.4626 0.4293 0.1642 CR4_B Non-default 425,045 0.4580 0.4151 0.1649 Default 337 0.4538 0.4366 0.1576 FirmSize Non-default 1,398,995 10.9495 11.0666 2.0839 Default 5161 14.4030 14.4984 1.4501 pMerton Non-default 1,398,995 0.1417 0.0015 0.2634 Default 5161 0.7263 0.8962 0.3340 Exret_y Non-default 1,398,995 0.0164 0.0202 0.6754 Default 5161 0.7700 0.8090 1.1140 Idio_Risk Non-default 1,398,995 0.1542 0.1248 0.1314 Default 5161 0.3024 0.2599 0.2223 NI/TA Non-default 1,398,794 0.0185 0.0074 0.2885 Default 5161 0.2286 0.0747 2.4553 TL/TA Non-default 1,398,995 0.4899 0.4791 0.4759 Default 5161 0.9325 0.7713 2.7018

15 In an unreported study, we also include two widely used macro variables (Duffie et al., 2007) in our test – the trailing one-year return on the S&P 500 index and the three-month U.S. Treasury rate (CMT rate in percentage terms). The coefficients of these two covariates are both positively significant as in prior studies. We also incorporate two firm-specific variables – the cash to total assets ratio and the market-to-book ratio – in empirical tests. The inclusion of these covariates does not distinctively affect the levels of significance of other variables. However, there is no improvement of out-of-sample prediction capability by including these two macro variables or two firm-specific variables mentioned above. Since they do not make up the major interest of this paper, we do not report the results so as to conserve space.

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Table 3reports descriptive statistics of firm-month variables un-der Fama–French 49-industry classification. As we expect, the num-ber of firms in the industry (Ln_N or N) for default samples is higher than that of non-default samples. However, the default group has higher average concentration ratios, HHI, and CR4, indicating more concentrated and less competitive industries. This may at first look puzzling, but when the effect of the relative size of firms in an indus-try also comes into play, relatively small firms in a more concen-trated industry (with high HHI and CR4) could even be more easily driven out of the market than equally small firms in a less concen-trated industry (with low HHI and CR4). This can make firms in the default group have a higher HHI (CR4) than the non-default sam-ples, which is confirmed by observing the HHI (CR4) of sales-ordered subgroups. In Panel A ofTable 3for the default samples, the average HHIs of small, medium, and big relative size subgroups are 0.1121, 0.1042, and 0.0978, respectively. The decreasing trend of concentra-tion ratios conforms to our conjecture that HHI is indeed negatively associated with the relative size of firms. The results of CR4s are similar and present the same pattern. In addition, we find that the number of default firms in small, medium, and big size groups are 3603, 1221, and 337, respectively. It appears that default firms are relatively small firms in their respective industries. The average FirmSize, measured as a firm’s market equity, of companies in the default group is much smaller than that for the non-default group.16

All control variables are in line with results in the literature –

p

Merton, idiosyncratic risk and TL/TA are larger, while NI/TA and trailing 1-year excess return are smaller for the default samples.

4.2. Empirical results of the hazard model

We present our empirical results under the Fama–French 49-industry classification in Section 4.2and leave the results under

the 3-digit SIC classification to robustness analysis in Section4.3. Before reporting our empirical results, we first briefly discuss our variables constructed in the previous section. Since the firm’s size can be characterized in terms of the firm’s own value or its relative size in an industry, empirically we use a firm’s equity value (FirmSize) as a proxy for firm’s own size and employ the relative or-der of the firm’s market share in an industry (Relative size dum-mies; Relative_size_S and Relative_size_M) to represent the relative size of the firm. Relative size dummies (Relative_size_S and Relative_size_M), computed as the relative order of market shares, measure the degree of ‘‘the relative market power’’ or ‘‘relative competitive position’’ within an industry. Relative size dummies can effectively measure firms’ relative size distribution in a given industry.

We report estimates of the hazard model under the Fama– French 49-Industry classification inTables 4 and 5. Table 4 pre-sents the results for the logarithm of the number of firms per industry (Ln_N), andTable 5reports those for HHI and CR4. From Hypothesis 1a, the number of firms in an industry (Ln_N) is ex-pected to be positively related to the hazard rate since a higher Ln_N indicates higher competition intensity, which implies dis-tressed firms should have higher probabilities of default in more competitive industries. Model 1 ofTable 4shows the coefficient of Ln_N is positive and significantly explains default intensity. This is consistent with Hypothesis 1a that credit risk is are positively re-lated to the number of firms in an industry.

Fig. 1also demonstrates the importance of the effects of a firm’s own size on credit spreads. Credit spreads are much more sensitive to competition intensity when firm’s own size is small. It is only when the firm’s own size is small enough that the credit spread does increase remarkably in terms of economic magnitude. It im-plies that other uncontrolled firm characteristics can hinder us from uncovering the real relationship between product market competition and credit risk. Accordingly, in addition to competition related variables, the five control variables of firm

Table 4

Hazard model estimates under the Fama–French 49-Industry Classification. This table reports estimates of several Cox proportional hazard models with time-varying covariates. The sample period is from 1986 to 2009 and there are 1398,995 firm-month samples and 5161 defaults in the sample. p-Values are in parentheses, below the estimates. Variables of product market competition are defined as follows: Ln_N is the natural logarithm of the number of firms in an industry; Relative_size_S and Relative_size_M are relative size dummies of small and medium firms, respectively. Thirty percentile and seventy percentile of sales volume in the given industry are used as cutoffs; FirmSize is defined as the logarithm of each firm’s equity value divided by the total market equity of NYSE/AMEX. Five firm-specific control variables are as follows:pMertonis the firm’s implied default

probability from the Merton model; Exret_y is the firm’s trailing 1-year excess stock return; Idio_Risk is idiosyncratic risk; NI/TA is the ratio of net income to total assets; TL/TL is the ratio of total liabilities to total assets.

Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Panel A. Hazard model estimates of the number of firms per industry (Ln_N)

Ln_N Coefficient 0.2103 0.0378 0.0157 0.0292 0.0069 p-Value (<.0001) (0.0301) (0.3675) (0.0935) (0.6917) Ln_N Relative_size_S Coefficient 0.2748 0.1044 p-Value (<.0001) (<.0001) Ln_N Relative_size_M Coefficient 0.1002 0.0256 p-Value (<.0001) (0.0126) Relative_size_S Coefficient 1.7620 0.7313 1.7618 0.7306 p-Value (<.0001) (<.0001) (<.0001) (<.0001) Relative_size_M Coefficient 0.8012 0.2876 0.8019 0.2871 p-Value (<.0001) (<.0001) (<.0001) (<.0001) FirmSize Coefficient 0.4928 0.4306 0.4238 0.4241 p-Value (<.0001) (<.0001) (<.0001) (<.0001) pMerton Coefficient 2.7992 1.6610 2.9108 1.8849 2.8830 1.9180 2.8900 1.9161 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Exret_y Coefficient 1.1729 0.9046 1.0099 0.8773 1.0284 0.8788 1.0231 0.8795 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Idio_Risk Coefficient 4.2602 3.3208 3.6360 3.2013 3.6842 3.1978 3.6668 3.2001 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) NI/TA Coefficient 0.0644 0.0417 0.0448 0.0339 0.0550 0.0389 0.0539 0.0392 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) TL/TA Coefficient 0.0163 0.0212 0.0199 0.0224 0.0150 0.0203 0.0155 0.0202 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) 16

Note that the average of firm size is negative, because it is the logarithm of a small fraction of the total NYSE/AMEX market equity value.

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characteristics (

p

Merton, Exret_y, Idio_Risk, NI/TA, and TL/TA) are in-cluded in the following hazard models (Models 2–9).

The results of Models 2 and 3 inTable 4support Hypothesis 1b and confirm the importance of firm’s own size to the hazard rate. The coefficients of FirmSize are significantly negatively related to default intensity, controlling for the five firm characteristics. Although the negative relationship between credit risk (default probabilities) and firms’ equity value (FirmSize) is not the exclu-sive prediction of our theoretical models, we provide theory-based explanations through the effect of product market competition.17

The signs of the five control variables are in line with previous empirical studies and are all statistically significant.

The results of Models 4 and 5 confirm our Hypothesis 1c that the sensitivity of credit risk to the number of firms in an industry is amplified when the firm size becomes smaller. Ln_N Rela-tive_size_S and Ln_NRelative_size_S are both significantly posi-tively related to hazard rates, and the economic significances of the interaction terms between Ln_N and Relative_size_S are larger than those for Relative_size_M. For example, in Model 5, the coef-ficient of Ln_NRelative_size_S is 0.1044 and this is around four times the magnitude of 0.0256, the coefficient of Ln_N Rela-tive_size_M. These results are consistent with Hypothesis 1c, which suggests that credit spreads are much more sensitive to the intensity of competition when firm size is smaller. Note that

Table 5

Hazard model estimates under the Fama–French 49-Industry Classification. This table reports estimates of several Cox proportional hazard models with time-varying covariates. The sample period is from 1986 to 2009 and there are 1398,995 firm-month samples and 5161 defaults in the sample. p-Values are in parentheses, below the estimates. Variables of product market competition are defined as follows: HHI is the Herfindahl-Hirschman Index; CR4is the four-firm concentration ratio; Relative_size_S and Relative_size_M are

relative size dummies of small and medium firms, respectively. Thirty percentile and seventy percentile of sales volume in the given industry are used as cutoffs; FirmSize is defined as the logarithm of each firm’s equity value divided by the total market equity of NYSE/AMEX. Five firm-specific control variables are as follows:pMertonis the firm’s

implied default probability from the Merton model; Exret_y is the firm’s trailing 1-year excess stock return; Idio_Risk is idiosyncratic risk; NI/TA is the ratio of net income to total assets; TL/TL is the ratio of total liabilities to total assets.

Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Panel A. Hazard model estimates of the Herfindahl-Hirschman Index (HHI)

HHI Coefficient 1.2110 1.1534 0.9580 1.1389 0.9807 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) HHI Relative_size_S Coefficient 3.1464 1.7992 p-Value (<.0001) (<.0001) HHI Relative_size_M Coefficient 1.2378 0.3944 p-Value (<.0001) (0.0893) Relative_size_S Coefficient 1.7620 0.7313 1.7616 0.7347 p-Value (<.0001) (<.0001) (<.0001) (<.0001) Relative_size_M Coefficient 0.8012 0.2876 0.7998 0.2876 p-Value (<.0001) (<.0001) (<.0001) (<.0001) FirmSize Coefficient 0.4916 0.4635 0.4238 0.4225 p-Value (<.0001) (<.0001) (<.0001) (<.0001) pMerton Coefficient 2.7864 1.6709 2.8287 1.7675 2.8830 1.9180 2.8824 1.9247 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Exret_y Coefficient 1.1810 0.9014 1.1310 0.8935 1.0284 0.8788 1.0296 0.8780 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Idio_Risk Coefficient 4.2889 3.3118 4.0981 3.2734 3.6842 3.1978 3.6905 3.1964 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) NI/TA Coefficient 0.0660 0.0418 0.0631 0.0412 0.0550 0.0389 0.0557 0.0397 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) TL/TA Coefficient 0.0152 0.0211 0.0140 0.0203 0.0150 0.0203 0.0145 0.0199 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Panel B. Hazard model estimates of the four-firm concentration ratio (CR4)

CR4 Coefficient 0.6963 0.7142 0.5981 0.7473 0.6238 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) CR4Relative_size_S Coefficient 2.2134 1.1143 p-Value (<.0001) (<.0001) CR4Relative_size_M Coefficient 0.4180 0.2340 p-Value (<.0001) (0.0067) Relative_size_S Coefficient 1.7620 0.7313 1.7662 0.7383 p-Value (<.0001) (<.0001) (<.0001) (<.0001) Relative_size_M Coefficient 0.8012 0.2876 0.8031 0.2894 p-Value (<.0001) (<.0001) (<.0001) (<.0001) FirmSize Coefficient 0.4907 0.4313 0.4238 0.4211 p-Value (<.0001) (<.0001) (<.0001) (<.0001) pMerton Coefficient 2.7881 1.6730 2.8908 1.8914 2.8830 1.9180 2.8885 1.9299 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Exret_y Coefficient 1.1816 0.9026 1.0630 0.8829 1.0284 0.8788 1.0292 0.8786 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) Idio_Risk Coefficient 4.2878 3.3132 3.8193 3.2157 3.6842 3.1978 3.6851 3.1953 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) NI/TA Coefficient 0.0651 0.0410 0.0566 0.0392 0.0550 0.0389 0.0547 0.0390 p-Value (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001) TL/TA Coefficient 0.0144 0.0204 0.0108 0.0184 0.0150 0.0203 0.0136 0.0191 p-Value (<.0001) (<.0001) (0.0002) (<.0001) (<.0001) (<.0001) (<.0001) (<.0001)

17 As mentioned previously, we do not intend to claim that FirmSize is a measure

resulting completely from the effect of competition; rather, we wish to provide theory-based explanations and empirical tests in our study.

數據

Fig. 1. Effect of the number of firms on credit spreads. This figure shows the effects of the number of firms in an industry on credit spreads (CS) under various settings of return volatilities of demand shocks ( r ), industry demand price elasticity ( c ), a
Fig. 3. Number of Firms within an Industry under the Fama–French 49-Industry Classification
Table 3 reports descriptive statistics of firm-month variables un- un-der Fama–French 49-industry classification
Table 7 reports accuracy ratios of three different prediction horizons under the Fama–French 49-industry classification

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