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Liquidity Risk and Corporate Risk-taking

Jing-zhi Huang Penn State University Email: jxh56@psu.edu

Yuan Wang Concordia University

Email: yuan.wang@concordia.ca

Rui Zhong* Central University of Finance and Economics

Email: ruizhong@outlook.com

This Version: November, 2017

*We thank Shusen Qi, Arben Kita, Stylianos Perrakis, Ruoyan Wang, Weixing Wu, and participants at the International Conference on Corporate Finance and Capital Markets, the 9th NCTU International Finance Conference, the 11th Conference on the Asia-Pacific Financial Market, the Second Annual Shanghai Risk Forum, the 15th International Conference on Financial Systems Engineering and Risk Management, the 14th Chinese Annual Conference in Finance, and the seminar at the University of International Business and Economics in China. Rui Zhong acknowledges support from the National Natural Science Foundation of China (NNSFC, Project No.71501197) and the Institute of Structural Finance and Derivatives (IFSID).

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Liquidity Risk and Corporate Risk-taking

ABSTRACT

We construct a theoretical framework to investigate the impact of liquidity risk, in the secondary corporate debt market, on corporate risk-taking preferences. Using closed-form solutions, our model shows that equity holders choose to adopt high-risk projects upon the arrival of illiquidity shocks. This effect is more pronounced for firms with weaker fundamentals. Empirically, we confirm the positive relationship between liquidity risk and corporate risk-taking. We also document that the impact of liquidity risk on corporate risk-taking preferences is more pronounced for smaller firms and firms with lower profits and higher rollover risk. In addition, we use the introduction of the Trade Reporting and Compliance Engine (TRACE) as a natural exogenous liquidity shock and find a decrease of firms’ risk-taking preferences after the TRACE is implemented. Our findings shed light on the managerial behavior literature, which shows that the frictions of the secondary bond market have a real impact on firms’ risk-taking behaviors.

JEL Classification: D22; G12; G32; G33;

Keywords: liquidity risk; corporate debts; risk-taking preference; rollover risk

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1. Introduction

The equity value of a levered firm, following the framework of Merton (1974), is viewed as

the payoff of longing a call option on the firm’s unlevered assets. The option nature of equity

motivates equity holders to take excess firm risks, especially when a firm is close to default. For

example, Asset substitution (Jensen and Meckling, 1976) describes a situation where adopting

risky projects amplifies a firm’s asset risk resulting in an increase of equity value and a decrease

of debt value.1 Literature on the determinants of corporate risk-taking focuses mainly on the

protection of property rights (John et al., 2008), creditor rights (Acharya et al., 2011), large

shareholder diversification (Faccio et al., 2011), corporate taxes (Djankov et al., 2010) and

executive compensation (Bolton et al., 2015). However, little is known about the impact of the

friction of secondary corporate bond market on firms’ risk-taking preferences.

In this study, we propose a theoretical framework to investigate how corporate debt liquidity

risk affects firms’ risk-taking behaviors. Specifically, following Leland (1994, 1998), we assume

that a firm continuously issues and retires debts at a proportional rate to keep the total value of

outstanding debts time invariant, known as debt rollover. We then generalize Leland’s models by

considering idiosyncratic illiquidity shocks that can impact a firm’s debts in the secondary

market and allow the firm to choose projects at its desired risk level. Upon the arrival of

idiosyncratic illiquidity shocks, the new debts that are issued to rollover the retired debts can

only be sold at a discounted price (Amihud and Mendelson, 1986; He and Xiong, 2012). In this

case, the difference between the market price of newly issued debt and the face value of retired

debt, namely the rollover cost, is absorbed by equity holders. Our model shows that rollover

costs, due to illiquidity shocks in the secondary debt market, increase the sensitivity of equity 1Jensen and Meckling (1976) address two types of agency costs. The first type refers to the conflict of interests between equity holders and a firm’s management team. The second type refers to the conflict of interests between equity holders and debt holders.

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value with respect to underlying asset volatility. Consequently, to maximize their value, equity

holders react to illiquidity shocks by adopting risky projects with high asset volatility.2

In our model, a firm’s risk-taking behaviors are reflected by the firm’s asset volatility. We

assume that there are two regimes for asset volatility: low and high, and that shareholders are

allowed to switch between the two regimes by adopting new projects with various levels of risk.

We use a simulated calibration, and show that equity holders optimally switch to a high-volatility

regime in the face of high liquidity risk in the secondary corporate debt market, suggesting a real

impact of liquidity risk on shareholders’ risk-taking behaviors.

Empirically, we employ two proxies for a firm’s risk-taking preferences: implied asset

volatilities (Merton, 1974) and a firm’s earning volatilities (John et al., 2008). We also construct

several illiquidity measures to quantify liquidity risk in the secondary corporate debt market. We

find a significant and positive relationship between the illiquidity measures and the risk-taking

proxies. Further, we conduct a natural experiment using the introduction of the TRACE that is

believed to be an exogenous liquidity shock to the secondary bond market. We find that firms’

risk-taking behaviors significantly decrease after the dissemination of the TRACE, which

supports our theoretical conjecture that liquidity risk in the secondary bond market has a positive

impact on the risk-taking behavior of equity holders. Moreover, we investigate the interaction

between bond illiquidity and firm characteristics to identify the channels through which bond

illiquidity affects shareholders’ risk-taking behaviors. Consistent with the calibrated results from

our model, we find that the impact of liquidity risk on risk-taking preferences is more

pronounced for the firms with weaker fundamentals, such as smaller size, lower profits and

higher roll-over risk. 2The risk-taking decision relies on the trade off between the benefits of a higher option value for equity and the costs associated with higher default risks caused by an increase of equity riskness.

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This study contributes to the literature in three aspects. First, to our knowledge, this paper is

the first to present the theoretical mechanism through which trading frictions in the secondary

market affect shareholders’ risk-taking preferences. This adds to the recent stream of theoretical

research on the economic consequences of debt liquidity risk. Closely related literature includes

Ericsson and Renault (2005), He and Xiong (2012), and He and Milbradt (2013), all of which

focus on the effect of debt liquidity risk on the default component of credit risk. Our research

differs from these previous studies in that it reveals the impact of debt liquidity risk on

shareholders’ risk-taking preferences.

Second, this study enriches the empirical findings about the economic consequences of

liquidity risk on corporate behavior. There is a large strand of literature that examines the impact

of corporate debt liquidity risk on the cost of debt (Longstaff et al., 2005; Chen et al., 2007; Bao

et al., 2011; Lin et al., 2011; Helwege et al., 2014). We use several rigorous econometric

approaches and also conduct an exogenous test using the dissemination of the TRACE as a

natural shock to bond liquidity to show that the illiquidity of secondary corporate debt markets

exacerbates corporate risk-taking behavior. Our empirical evidence advances the understanding

of the economic consequences of liquidity shocks in the secondary corporate debt market and

highlights its real effect on shareholders’ behavior.

Last but not the least, it is important to understand the determinants of corporate risk-taking

since corporate risk-taking behavior significantly affects firm value and the wealth distribution

between equity and debt holders. There is a large volume of literature on the determinants of

corporate risk-taking behavior. For instance, John et al. (2008) find empirical evidence that the

protection of property rights is positively related to corporate risk-taking. Acharya et al. (2011)

find that creditor rights positively contribute to the risk-taking behavior of equity holders. Faccio

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et al. (2011) show that large shareholder diversification affects corporate risk-taking decisions.

Bolton et al. (2015) show that the structure of executive compensation also has an impact on

firms’ risk-taking decisions. In contrast to these studies, this paper identifies bond liquidity as a

new determinant for corporate risk-taking preference.

The remainder of our paper is organized as follows. Section 2 shows the economic setup for

our model. Section 3 illustrates calibrations and presents our results. In Section 4, we construct

empirical models and report the results. Section 5 concludes. Proofs for the lemmas and

propositions are shown in Appendix A and detailed definitions of the variables are reported in

Appendix B.

2. The Model

2.1. Economic Setup

2.1.1. Unlevered Asset Dynamics.

We consider a firm whose projects are financed by equity and debt with a tax-deductible

coupon. As in most of the related literature, the values of the components of a firm’s balance

sheet are estimated as contingent claims of the state variable V, a firm’s unlevered asset value,

representing its economic activities.3 Similar to Leland (1998), under a risk-neutral assumption,

we assume unlevered asset value V follows the process below:

( ) ,

( ) ,

QtL S

t

QtH S

t

dV r q dt dW for V VVdV r q dt dW for K V VV

σ

σ

⎧ = − + ≤⎪⎪⎨⎪ = − + ≤ ≤⎪⎩

(1.1)

3 Goldstein et al. (2001) and He and Milbradt (2013) use the dynamic of a firm’s cash flow instead of V . the two approaches are strictly equivalent under strong capital market assumptions.

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where r is the risk-free rate, q is a firm’s payout rate, K is the default boundary and QW is a

Brownian Motion under a risk-neutral measure. We assume low- and high-asset-volatility

regimes, denoted by subscript L and H , respectively, where L Hσ σ≤ . We allow a firm to

switch between low- and high-volatility regimes by altering its risk-taking preferences. For

instance, if a firm decides to adopt high-risk projects to replace low-risk projects, the firm can

switch to a high-volatility regime that consequently leads to an increase in asset volatility, and

vice versa. Shareholders’ decisions to switch between high- and low-volatility regimes rely on

the trade off between the default costs and equity valuations. Initially, a healthy firm whose

unlevered asset value is much greater than the default boundary will optimally stay in a low-

volatility regime because, as shown in Figure 1,4 switching from a low- to a high-volatility

regime decreases equity value. The risk-switching boundary is denoted by SV . Once the

unlevered asset value decreases and touches the risk-switching boundary but is still above the

default boundary, a firm will switch to a high-volatility regime Hσ by replacing low-risk

projects with high-risk ones, as discussed above. Due to the option nature of equity, an increase

in asset volatility boosts equity value at the cost of reduced debt values. This is asset substitution

(or agency cost),5 which is more pronounced when asset values are close to the default boundary.

If asset values hit or slip below the default boundary K , this firm will go bankrupt and

liquidation will occur immediately.6

4 Since the marginal change of equity value, with respect to the change of equity volatility, is negative when unlevered assets are much greater than the default boundary, it is optimal for equity holders of a healthy firm to decide to stay in the low-volatility regime. Economically, an increase in asset volatility benefits shareholders because of the equity’s option nature; on the other hand, it generates a cost for shareholders because of the amplified default risk. For a healthy firm, equity is considered a deep-in-the-money option on unlevered assets, which is less sensitive to volatility changes. Therefore, it is optimal for a healthy firm to stay in a low-volatility regime. 5 See Leland and Toft (1996), and Leland (1998) for insights on asset substitution. Jensen and Meckling (1976) were the first to systematically discuss agency costs. 6 It is assumed that, in this situation, the absolute priority rule is applied fully.

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2.1.2. Stationary Debt Structure.

We adopt the exponential stationary debt structure, proposed by Leland (1994), under which a

firm keeps issuing and retiring pieces of corporate debt at a proportional rate g . The face value

is denoted by p and the continuous coupon payments by c , per unit time for a debt. Since all

outstanding debts are identical, the total outstanding principals and coupon payments are

/P p g= and /C c g= , respectively. Debt is initially issued at 0t = with principal P and

coupon payment C . As time goes by, the remaining principal at time t is gte P− and the debt

holders receive a cash flow equal to ( )gte C gP− + , provided the firm remains solvent. Therefore,

the average maturity of this debt is the reciprocal of the proportional retirement rate 1/g T= .7

The exponential stationary debt structure provides debt and equity valuation results that are

similar to the finite-maturity debt structure proposed by Leland and Toft (1996), even in the

presence of liquidity risk in the secondary debt market.8

2.1.3. Liquidity Shocks in the Secondary Debt Market.

When a firm keeps retiring debt at face value and issuing new debt at the market price given a

predetermined stationary debt structure, the deviation between the market price and face value

induces a rollover risk. Rollover risk is taken into consideration in our model through the

incorporation of a secondary debt structure, which is similar to that used in Amihud and

Mendelson (1986). It is assumed that each bond holder is exposed to idiosyncratic liquidity

shocks that arrive randomly. Upon the arrival of a liquidity shock, bond holders can only sell

their shares at a discount in order to exit the market. For computational convenience, we assume

7 See Equation (7) in Leland (1994b). 8 See Perrakis and Zhong (2015; 2017) for a comparison, in terms of debt and equity valuations, of the exponential stationary debt structure (Leland 1994) and the finite-maturity stationary debt structure (Leland and Toft 1996). Exponential stationary debt structure has been used in Mauer and Ott (1995), Leland (1998), He and Milbrat (2014), and Perrakis and Zhong (2015; 2017).

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that the arrival of a liquidity shock follows a Poisson distribution with arrival intensity ξ . The

costs for selling bonds are proportional to the market value of the bonds with rate k . This setup

focuses only on analyzing the effect of external market liquidity that can be enhanced by

improving the internal liquidity of a firm through accumulating cash holdings and enlarging

available credit lines (He and Xiong, 2012).

2.2. Debt and Equity Valuations

In our economic setup, debt value, equity value, the optimal risk-switching boundary and the

endogenous default boundary are interconnected and therefore must be calculated simultaneously.

To reduce the dimension of our optimization problem, we first derive, in the following

subsections, the analytical expression for debt and equity value, respectively. Then, taking

advantage of the analytical expressions for corporate debt and equity value, we use a numerical

optimization algorithm to calibrate the optimal risk-switching boundary and endogenous default

boundary.

2.2.1. Debt Valuations.

Given the asset dynamics presented in Equation (2.1) and the exponential stationary debt

structure, to rule out an arbitrage opportunity the total outstanding debt value should satisfy the

following two differential equations:

For SV V≤ ,

2

2 22

1( , ) ( , ) ( , ) ( )2

L LL L L L

D DrD V g C gP gD V g kD V g r q V VV V

ξ σ∂ ∂

= + − − + − +∂ ∂

(1.2)

and for SK V V≤ ≤ ,

10

2

2 22

1( , ) ( , ) ( , ) ( )2

H HH H H H

D DrD V g C gP gD V g kD V g r q V VV V

ξ σ∂ ∂

= + − − + − +∂ ∂

(1.3)

with four boundary conditions,

( )LC gPD Vr g kξ

+→ +∞ =

+ + (1.4)

( ) ( )1HD K Kα= − (1.5)

( ) ( )H S L SD V D V= (1.6)

( ) ( )| |S S

H LS S

V V V V

D DV VV V= =

∂ ∂=

∂ ∂ (1.7)

The first and second terms on the right-hand side of Equations (1.2) and (1.3) are continuous

coupon payments and retired principles received by debt holders, respectively. The third term is

the new debt that is continuously issued at the market price. The liquidity shocks in the corporate

debt market induce expected costs to debt holders, which is represented by the fourth term. And

the last two terms show a change of debt value with respect to the asset dynamics in each

volatility regime.

The four boundary conditions presented in Equations (2.4 through 2.7) are necessary for the

following reasons. When an asset value approaches positive infinity, the default risk of corporate

debt is close to zero, which is captured by the first boundary condition. On the other hand, when

the asset value hits the default boundary, a default event occurs and the debt holders only recover

( )1 Kα− , as shown in the second boundary condition. In addition, the third and fourth boundary

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conditions maintain the continuity of debt dynamics when a firm’s asset value crosses its risk-

switching boundary.

Proposition 1: When an asset dynamic follows Equation (1.1), the analytical solution for

corporate debt value in Equations (1.2) and (1.3) with boundary conditions (1.4), (1.5), (1.6) and

(1.7) is:

( )( )

( )

1 2

1 2

1 2

1 2

,

,

L L

H H

y yL L L S U

y yH H H B S

C gPD V a V a V V V Vr g k

D VC gPD V a V a V V V Vr g k

ξ

ξ

+⎧ = + + ≤ ≤⎪ + +⎪= ⎨

+⎪ = + + ≤ ≤⎪ + +⎩

(1.8)

where

( )

1

1 2 2 2

2

2 1

1 1

2 2 1 2

2 2

1 2

10, 1

,

1 , 1

H

H L H L

H

H H

L H y

y y y yH L H S H Sy

y ySH L

H L

Aa aK B

Aa a a V a VK B

Vy yC gPA K Br g k y y K

αξ

− −

−

⎡ ⎤= = −⎢ ⎥⎣ ⎦

= = +

−+ ⎛ ⎞= − − = − ⎜ ⎟+ + − ⎝ ⎠

(1.9)

2.2.2. Equity Valuations.

Equity holders have the right to claim residual cash flows in the form of dividends, after

coupon payments and rollover costs, which are induced when the market price of debt deviates

from its face value. Leland (1998) derives equity value by deducting debt values and bankruptcy

costs, and adding tax shields to the unlevered asset values,9 without directly solving the above

differential equations. This method cannot be applied after incorporating the liquidity shocks in

9 See Equation (21) in Leland (1998).

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the secondary debt market as rollover costs themselves are a function of debt value. In our case,

the following two differential equations must be satisfied by equity:

For SV V≤ ,

( ) ( )2

2 22

11 ( )2

L LL L L

E ErE qV w C gD V gP r q V VV V

σ∂ ∂

= − − + − + − +∂ ∂

(1.10)

For SK V V≤ ≤ ,

( ) ( )2

2 22

11 ( )2

H HH H H

E ErE qV w C gD V gP r q V VV V

σ∂ ∂

= − − + − + − +∂ ∂

(1.11)

with boundary conditions,

( ) 0HE K = (1.12)

|

H

V K

E lV =

∂=

∂ (1.13)

( ) ( )L S H SE V E V= (1.14)

| |S S

L H

V V V V

E EV V= =

∂ ∂=

∂ ∂ (1.15)

The first and second terms in Equations (1.10) and (1.11) show the cash flows in the form of

dividends and after tax coupon payments, respectively. The difference between the third and

fourth terms represent the rollover costs to equity holders, which is a function of the debt value

that satisfies Equations (1.2) and (1.3). The rollover costs for equity holders increase drastically

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when a liquidity shock hits the debt market as a surprise.10 The last two terms show a fluctuation

of equity value with respect to asset dynamics.

The four boundary conditions identified in Equations (2.12) through (2.15) are explained as

follows. When a firm’s asset value hits the default boundary, the equity value becomes zero,

composing the first boundary condition. We set the first derivative of equity value with respect to

asset value equal to a free parameter l . When a firm’s asset value approaches positive infinity,

the equity value is linear with respect to the asset value with slope l. Moreover, we add boundary

conditions from Equations (1.14) and (1.15) to maintain continuity when a firm’s asset value

crosses the risk-switching boundary. Since the solution of Equations (1.10) and (1.11) depends

on the solution of Equations (1.2) and (1.3), we have to solve these four differential equations for

equity value simultaneously.

Proposition 2: When an asset dynamic follows Equation (1.1), the solutions for equity value

in Equations (1.10) and (1.11) with boundary conditions (1.12), (1.13), (1.14) and (1.15) are:

( ) ( )

( )

1 2

0 1 2 1 22

1 1 3 3 2 2 1 4 4 22

1 1 2 2

1

1 1 121 1

2 L L

SL L L LL

L L L L L L L L L

y yL S L L L L L S L L L L

L L L L L L L L L L L L L

L LH S

w C gP ga qVE V V

ga V ga Vy y y y

E V

φ φ φ φσ η γ η γ η γ η γ

φ φ φ φ φ φ φ φσ η γ η γ η γ η γ

η φ

⎡ ⎤− + − ⎛ ⎞ ⎛ ⎞− −= + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟+ + − +⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞− − − −− − + −⎢ ⎥⎜ ⎟ ⎜ ⎟+ − + + − +⎝ ⎠ ⎝ ⎠⎣ ⎦

++

( ) ( )( )

( )2 1 2 2 22

2 1 12 2

S

L L L L HL H S L

L L L L L V V

Er q E VV

γ φ φ φσ σ

η γ σ η γ =

⎛ ⎞− ∂⎛ ⎞+ − − +⎜ ⎟⎜ ⎟⎜ ⎟+ + ∂⎝ ⎠⎝ ⎠

(1.16)

10 See He and Xiong (2012).

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( ) ( )

( )( )

1 2

0 1 2 1 22

1 3 3 21 2 1 4 4 22

1 1 2 2

1 2

1 1 121 1

2 H H

H H H HH

H H H H H H H H H

y yH H H HH H H H H H

H H H H H H H H H H H H H

H H

H H

w C gP ga qKE V V

ga K ga Ky y y y

l

φ φ φ φσ η γ η γ η γ η γ

φ φ φ φ φ φ φ φσ η γ η γ η γ η γ

φ φ

η γ

⎡ ⎤− + − ⎛ ⎞ ⎛ ⎞− −= + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟+ + − +⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞− − − −− − + −⎢ ⎥⎜ ⎟ ⎜ ⎟+ − + + − +⎝ ⎠ ⎝ ⎠⎣ ⎦

−+

+

(1.17)

where

( )

( )

( )

1 20 1 2

1 2

2 20 2

2 21 3

11

1 112 2

1 112 2

L Ly yS L S L S

L L L L L L

L L L

L L L

w C gP gaqV ga V ga Vy y

r q H Hl

r q H H

η η η η

σ η σ

σ η σ

⎡ − + − ⎤− + − −⎢ ⎥− − −⎢ ⎥⎢ ⎥⎛ ⎞⎛ ⎞+ − − − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦=

⎛ ⎞− − − − −⎜ ⎟⎝ ⎠

(1.18)

The ancillary parameters are defined in the Appendix section of this paper.

3. Model Calibrations

3.1. Calibration

In this section, we perform a simulated analysis to study the impact of liquidity risk on risk-

taking decisions. First, we construct a base case by combining empirical findings and

calibrations from the literature and show the details in Table 1. Similar to Leland (1998), we

normalize the initial asset value 0 $100V = and choose the risk-free rate 8%r = . We set the

corporate debt tax benefit rate as 27%w = , which is the same as that in He and Xiong (2012).

We use a rounded payout rate 2%q = based on the sample firms described in the research of

Huang and Zhou (2008) who document 2.02% and 2.15% average payout rates for A and BB

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rated debts, respectively.11 According to Chen (2010) who documents 40.1% average recovery

rates and 41.5% with a jump-risk premium, we set the proportional recovery rate at 40%α = .

Custodio et al. (2013) examine debt maturities in the United States during last four decades and

find, in their 2008 sample, that around 45.6% of firms have an average debt maturity of three

years.12 We use average maturity 3T = in our base case and test other maturities, such as 1T =

and 5T = , as robustness checks.

[Please Insert Table 1 about Here]

Following He and Xiong (2012), who were inspired by the empirical findings in Edwards et al.

(2007) and Bao et al. (2011), we separate our calibrations into cases based on rating classes. We

set the proportional transaction costs in the presence of liquidity shocks 1.00%k = for an A

rating and 0.50%k = for a BB rating. We choose low asset volatilities 23%Lσ = for a BB rating

and 21%Lσ = for an A rating, based on the calibrations in Zhang et al. (2009).13 In He and

Xiong (2012), the face value of the aggregated debt and the coupon rate are chosen so that one-

year bonds are issued at par and the credit spreads of these bonds are consistent with the

empirical evidence. For example, they set the face value 61.68P = for BB rated bonds.

Following their work, we set 60P = for our BB rated bonds ex ante. Then, we calculate the

corresponding coupon payment and high asset volatility to make sure that debt is issued at par, in

the absence of liquidity shocks, and to generate a credit spread that equals 330 basis points for a

BB rated bond. According to our calibrations, for a BB rated bond, we find

1.9445* 44.72%H Lσ σ= = . For an A rated bond, assuming the same incremental magnitude of

11 See Table 1 in Huang and Zhou (2008). 12 See Table 2 in Custodio et al. (2013). 13 See Table 7 in Zhang et al. (2009).

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asset volatility indicates 1.9445*21% 40.83%Hσ = = . We calculate the corresponding face

value of all outstanding debts to generate a credit spread that equals 100 basis points and find

52.6P = for an A rated bond.

3.2. Risk-taking Decisions

To model the conflict of interests between equity and debt holders, we adopt a two-stage

decision-making process, ex-ante and ex-post, which is similar to Leland’s (1998) framework.

Ex-ante, all stakeholders work together to set up capital structures with the purpose of

maximizing asset values. This is the situation in which there is no conflict of interest between

equity and debt holders. It is assumed that all of the information is available to all stakeholders

ex ante. The initial capital structure and debt structure of a firm are determined by maximizing

the equation with respect to the coupon payment (C), endogenous default boundary (K) and risk-

switching boundary (VS)14 as follows:

( )0, ,max , , ,

SSC K V

v V C K V (1.19)

We use an iteration approach to numerically solve this optimization problem. In each iteration,

we first choose the face value of all outstanding debts exogenously and select the coupon

payment to make the debts issued at par ex ante. Next, we fix the endogenous default boundary

and find the local optimal risk switching boundary to maximize total asset value. Then, we find

the global optimal risk switching boundary by altering the endogenous default boundary. We

repeat these iterations until the global maximum value of assets is reached and extract the

corresponding coupon payment, endogenous default boundary and risk-switching boundary.

14 Note: debt holders know that equity holders may accept risky projects to increase their equity value.

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Once we have the optimized the coupon payment from the above iteration, we can compute

the corresponding total value of debts by assuming that all debts are issued at par. In the

calculation, we set the average maturity of all outstanding debts equal to the average maturity of

debts, calibrated from the previous literature (e.g., Leland, 1998). These characteristics of debt

structure, once in place, cannot be changed by equity holders. Nonetheless, equity holders can

manipulate the risk-switching boundary to maximize equity value. To analyze when equity

holders choose to change the risk-switching boundary, we need to calculate the sensitivity of

equity value with respect to the risk-switching boundary. Similar to Equation (26) in Leland

(1998), the first derivative of equity value with respect to the risk-switching boundary is:

( ), , , ,S S S

L L LS V V V V V V

S S S

dE E E Kz V K C P TdV V K V= = =

∂ ∂ ∂= = +

∂ ∂ ∂ (1.20)

Equation (2.20) consists of two parts. The first term suggests that a change of equity value is

due to a direct change in the risk-shifting boundary. Meanwhile, a change in the risk-shifting

boundary can indirectly affect equity value by moving the endogenous default boundary, as

shown in the second term.

The selection of a risk-switching boundary by equity holders to maximize their value relies on

the trade off between the anticipated equity value appreciation due to the option nature of equity

and the anticipated costs due to an increase of default probability while taking on risky projects.

When the asset value of a firm is much higher than the default boundary, equity is considered as

a deep in-the-money option for which the anticipated costs, due to an increase of default

probability, dominate the appreciation of equity value due to an increase of volatility, resulting in

a negative value for ( ), , , ,Sz V K C P T . When the asset value is low enough, the anticipated

equity appreciation, because switching to a high volatility regime dominates the anticipated costs

18

of a high default probability, suggests a positive value for ( ), , , ,Sz V K C P T . Therefore,

( ), , , ,Sz V K C P T is a decreasing function of asset value. Since ( ), , , ,Sz V K C P T is the partial

derivative of 𝐸" with respect to 𝑉$ conditional on 𝑉 = 𝑉$ , as shown in Equation (2.20),

( ), , , ,Sz V K C P T is also a decreasing function of risk-switching boundary.

[Insert Figure 1 about Here]

Given the calibrations in the base case, Figure 1 shows that ( ), , , ,Sz V K C P T is a decreasing

function of risk-switching boundary. We also note that ( ), , , ,Sz V K C P T behaves differently for

firms with different credit ratings. For instance, ( ), , , ,Sz V K C P T for A-rated firms is smaller

than that for BB-rated firms. Further, a liquidity shock significantly pushes the optimal risk-

switching boundaries upward for both A and BB rated firms.

[Insert Figure 2 and Figure 3 about Here]

In our model, equity holders are not allowed to change the debt structure but can manipulate

the risk-switching boundary and the endogenous default boundary. We plot the optimal risk-

switching and endogenous default boundaries against the liquidity intensities in Figure 2 and

Figure 3, respectively. Consistent with the findings in He and Xiong (2012), we find that the

endogenous default boundary increases as liquidity intensity increases. In addition, we show that

for a BB-rated firm, equity holders prefer a lower ex-post endogenous default boundary rather

than the ex-ante boundary when liquidity intensity is high, indicating that equity holders will

optimally keep weak-fundamental firms alive longer when shareholders are allowed to adopt

risky projects in the face of decreasing asset values. For the optimal risk-shifting boundary, the

deviation between the ex-ante and ex-post boundaries becomes wider as the liquidity intensity

19

increases. Specifically, the ex-ante risk-shifting boundary is almost invariant with respect to

liquidity intensity, but the ex-post risk-shifting boundary is shifted upward significantly when

liquidity deteriorates, especially for the firms with weak fundamentals. This result suggests that

equity holders are more likely to accept high-risk projects to increase asset volatility when

illiquidity increases in the secondary debt market.

4. Empirical Analysis

Our model shows equity holders’ reactions to the deterioration of bond market liquidity. The

static calibration illustrates that equity holders choose to increase asset volatility by adopting

high-risk projects upon the arrival of illiquidity shocks. Our model also shows that this effect

should be more pronounced for firms with weak fundamentals. In this section, we test these main

implications using empirical data.

4.1. Variables and Sample Descriptions

4.1.1. Illiquidity Measures.

To test the relationship between bond liquidity in the secondary market and the risk-taking

behavior of equity holders, we use corporate bond transaction data from the TRACE dataset to

construct liquidity measures. Following prior literature (e.g., Harris and Piwowar 2006, Dick-

Nielsen et al. 2012), we remove trades that are canceled, corrected and commissioned, and bond

transactions under $100,000 to avoid the effect of retail transactions. We then construct four

illiquidity measures: the Amihud ratio, imputed round-trip trades (IRTs), price dispersion and the

inter-quartile range (IQR). The detailed calculations for the illiquidity measures are presented

below.

20

The Amihud ratio (Amihud) for bond i on day t is defined as the average of the absolute

returns of consecutive transactions divided by the trade size 𝑄',)* (in millions $) within day t

𝐴𝑚𝑖ℎ𝑢𝑑',) =123,4

53,46 753,4

689 /53,4689

;3,46 ∗ 10023,4

*?1 ,

where 𝑁',) denotes the number of returns on day t for bond i, and 𝑄',)* and𝑃',)

* denote the trade

size and price for transaction j, respectively. A larger Amihud ratio indicates lower bond

liquidity since bond prices move more for a given trade size. We multiply the Amihud ratio by

100 to make it comparable, in numerical scale, with other illiquidity measures.

IRTs (Feldhütter 2012) implicitly measure transaction costs by assuming a pre-matched

trading arrangement. We define the IRT for bond i on day t as:

𝐼𝑅𝑇',) =53,4IJK753,4

I3L

53,4IJK ,

where 𝑃',)MNOand𝑃',)M'P denote the maximum and minimum prices for bond i on day t,

respectively. A larger IRT implies higher round-trip transaction costs and lower bond liquidity.

Price dispersion (PD) (Jankowitsch et al. 2011; Friewald et al. 2012) is defined as:

𝑃𝐷',) =;3,46 ∗(53,4

6 75S,4)UV3,49

;3,46V3,4

9

,

where 𝐾',) and 𝑃X,) denote the total number of trades and the average price on day t for bond i,

respectively; 𝑄',)* and𝑃',)

* denote the trade size and price for transaction j, respectively. Price

dispersion measures the volume-weighted price difference, reflecting the potential transaction

costs for a trade.

The IQR, used by Han and Zhou (2008) and Helwege et al. (2014), is defined as:

𝐼𝑄𝑅',) =53,4YZ4[753,4

UZ4[

5S,4,

21

where 𝑃',)\])^, 𝑃',)_])^and𝑃X,)indicate the 75th percentile of prices, the 25th percentile of prices

and the average price for bond i on day t, respectively. The logic behind this measure is that less

liquid bonds tend to have a higher price volatility on a given day. Compared to the maximum and

minimum prices of the IRT, the 75th and 25th price percentiles are less sensitive to outliers.

After calculating the trading volume weighted illiquidity measures for each bond on each day,

we winsorize all of the illiquidity measures at 1% and 99% to remove potential outliers. Then,

we weight the daily illiquidity measures using the daily trading volume to calculate the monthly

bond-level illiquidity measures. Next, we take the average of the monthly measures over a year

to construct the annualized bond-level illiquidity measures. Finally, we follow Helwege et al.

(2014) to construct firm-level illiquidity measures using bond-level measures weighted by the

outstanding amount of bonds. The summary statistics of the illiquidity measures are presented in

Table 2.

[Please Insert Table 2 about Here]

4.1.2. Risk-taking Measures.

We adopt two different measures to proxy for firms’ risk taking. Our first measure is asset

volatility implied from the Merton (1974) model. This measure of asset volatility is a critical

input to calculate Distance to Default and is used in many studies (e.g., Hillegeist et al. 2004;

Bharath and Shumway, 2008; Campell et al., 2008; Mansi et al., 2010). Asset volatility is also a

well-known measure for asset substitution or risk shifting (Jensen and Meckling, 1976; Helwege

et al., 2017).

Following the literature, we simultaneously solve the two equations below to obtain the

implied asset volatility𝜎a and asset value V:

𝐸 = 𝑉𝑁 𝑑1 − 𝑒7de𝐷𝑁 𝑑\ (4.1)

22

𝜎f =af

gfga𝜎a (4.2)

𝑑1 =𝐿𝑛 𝑉

𝐷 + 𝑟 + 0.5𝜎a\ 𝑇

𝜎a 𝑇

𝑑\ = 𝑑1 − 𝜎a 𝑇,

where E is the market value of a firm’s equity, D is the face value of a firm’s total debts that

equal short-term debts plus one-half of long-term debts (Crosbie and Bohn, 2001; Vassalou and

Xing, 2004; Campell et al., 2008), r is the one-year treasury rate, T is the time-to-maturity of a

bond that is set to one year (Mansi et al., 2010; Campell et al., 2008), N(·) is the cumulative

standard normal distribution function and 𝜎f is the three-month rolling sample standard

deviation (Campell et al., 2008).

To solve Equations (4.1) and (4.2), we follow Campell et al. (2008) to set the initial values of

asset volatility and asset value. The iteration continues until we find solutions for both the asset

volatility and asset value. We winsorize asset volatility at 1% and 99% to remove outliers. Table

3 reports the summary statistics of asset volatility.

The second risk-taking measure we adopt is earning volatility (John et al. 2008, Zhang,

2009). We define earning volatility as the standard deviation of quarterly income before

extraordinary items deflated by total assets during the two years preceding a fiscal year end. We

remove firms with missing observations during this two-year period.

4.1.3. Sample Selection.

To obtain accounting information, we match firm-level illiquidity measures with the

Compustat Fundamental Annual dataset. Observations with missing information are excluded.

The final sample contains U.S. public firms with bond trading reported in the standard TRACE

from July 2002 to January 2015 and consists of 9,428 firm-year observations for 1,379 firms.

23

4.2. Empirical Results

4.2.1. OLS Specifications.

To assess the effect of bond illiquidity on a firm’s risk-taking activities, we estimate the

regression below:

𝐴𝑠𝑠𝑒𝑡𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) = 𝑎 + 𝑏 ∗ 𝐵𝑜𝑛𝑑𝐼𝑙𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦',) + 𝑐z ∗

𝑂𝑡ℎ𝑒𝑟𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠',) + 𝑌𝑒𝑎𝑟) + 𝐹𝑖𝑟𝑚𝐷𝑢𝑚𝑚𝑦' + 𝑅𝑎𝑡𝑖𝑛𝑔𝐷𝑢𝑚𝑚𝑖𝑒𝑠',) + 𝑒𝑟𝑟𝑜𝑟',) (4.3)

where i and t index the firm and time, respectively. The main dependent variable is asset

volatility or earning volatility, which measure a firm’s risk-taking activities. We use the Amihud

ratio, IQR, price dispersion and the IRT to proxy for the illiquidity of a firm’s bonds.15 We

control for firm characteristics that may affect risk-taking activities, including firm size, leverage,

sales growth and profit (John et al. 2008).

We include the natural logarithm of firm age in the regression to control for borrowing ability

since younger firms are usually associated with a lower reputation in the bond market making it

more difficult for these firms to borrow in the bond market (Diamond 1991, Carty 1996, Johnson

1997, Datta et al. 1999, Cai et al. 2007). We argue that when illiquidity shocks hit the secondary

corporate bond market unexpectedly, equity holders in younger firms prefer to accept riskier

projects because of the higher costs to rollover their debt.

In the regression, we also control for the rollover risk proxied by the percentage of debt due in

one year to the total debt. In addition, we use firm dummies to control for the possibly omitted

firm characteristics that are constant during our sample period, and year dummies to account for

the intertemporal variations that may bias the relationship between bond illiquidity and risk-

taking. We also employ credit rating dummies to control for credit risk in the regression. We

15We apply four measures of bond illiquidity to confirm the robustness of our empirical results.

24

cluster standard errors by firms to avoid inflated t-statistics caused by the autocorrelation

(Petersen 2009).

[Please Insert Table 3 and Table 4 about Here]

As reported in Table 3, we use asset volatility as the dependent variable and find that the

estimated coefficients of the four bond illiquidity measures are all positive and significant. In

particular, one standard deviation increase in the Amihud ratio, IRTs, price dispersion or the IQR

will increase asset volatility by 3.720*0.01/55.3%=6.7%, 7.521*0.006/55.3%=8.2%,

3.779*0.002/55.3%=1.4% or 3.527*0.006/55.3%=3.8%, respectively. The positive relationship

between asset volatility and bond illiquidity confirms the argument that when illiquidity shocks

hit the secondary bond market, equity holders will expropriate debtholders by accepting risky

projects, which increases the firm’s volatility.

In Table 4, we re-estimate the above regressions using earning volatility as the dependent

variable. As expected, we find that the coefficients for all illiquidity measures remain positive

and significant, confirming the positive relationship between bond illiquidity and a firm’s risk

taking. Numerically, one standard deviation increase in the Amihud ratio, IRTs, price dispersion

or the IQR amplifies profit volatilities by 6.46%, 10.16%, 2.68% and 7.88%,16 respectively.

As discussed in our theoretical model, a higher rollover risk will increase a firm’s risk-taking

behavior. As expected, we document significantly positive coefficients for rollover risk in both

asset-volatility and earning-volatility regressions. Also, we note that firms with a longer history

are associated with relatively lower risk-taking activities, but that this relationship is only

significant for the asset-volatility regressions. Consistent with John et al. (2008), we find that

firm size and profit are negatively related to risk-taking activities, and leverage and sale’s growth

16 0.097*0.01/1.5%=6.46%, 0.254*0.006/1.5%=10.16%, 0.201*0.002/1.5%=2.68% or 0.197*0.006/1.5%=7.88%

25

are positively related to risk-taking activities. The positive coefficient of sale’s growth is not

significant for the earning-volatility regressions.

4.2.2. Causality Test Using the TRACE Dissemination Information.

A potential concern in our analysis is that the relationship between bond liquidity and a firm’s

risk-taking behavior might be spurious. To address this issue, we implement a natural experiment

using the introduction of the TRACE as an exogenous shock to bond-market liquidity. Our

identification strategy relies on the phase-in feature of the TRACE introduction.

On July 1, 2002, the Financial Industry Regulatory Agency (FINRA) began disseminating

trades in investment-grade corporate bonds with issuance the size of $1 billion or greater. Over

time, bond coverage expanded in phases and the TRACE was fully implemented by January 31st,

2006, covering essentially all publicly traded bonds. The introduction of the TRACE reduced

transaction costs and improved transparency leading to an increase in bond liquidity (e.g.,

Bessembinder et al., 2006; Edwards et al., 2007; Goldstein et al., 2007).

Since the TRACE implementation is unrelated to a firms’ decision to undertake risky projects,

it represents an exogenous shock to a firm’s liquidity that allows us to isolate the impact of bond

liquidity on a firm’s risk-taking behavior.

In the spirit of Bertrand and Mullainathan (1999a; 1999b; 2003), we create the POST-TRACE

dummy variable that equals one if a firm’s bonds have already been covered by the TRACE

during year t and zero otherwise. The TRACE dummy captures the impact of an increase in

liquidity in the years following the TRACE introduction. In addition, we control for the

Compustat accounting information for each firm. We then estimate the regression as follows:

𝐴𝑠𝑠𝑒𝑡𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) = 𝑎 + 𝑏 ∗ 𝑃𝑜𝑠𝑡𝑇𝑅𝐴𝐶𝐸',) + 𝑐z ∗ 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠',) +

𝑌𝑒𝑎𝑟) + 𝐹𝑖𝑟𝑚' + 𝑅𝑎𝑡𝑖𝑛𝑔𝑠',) + 𝑒𝑟𝑟𝑜𝑟',) (4.4)

26

where all of the variables and parameters are the same as those in Equation (4.3), except the

Post-TRACE dummy.

[Please Insert Table 5 about Here]

As shown in Table 5, we find a negative and significant coefficient for the Post-TRACE

dummy in both the asset-volatility and earning-volatility regressions. This suggests that a

positive exogenous illiquidity shock in the secondary corporate bond market caused by an

external regulation change significantly reduces a firm’s risk-taking activities, supporting our

theoretical conjecture that illiquidity shocks in the secondary corporate bond market amplify the

asset substitution phenomenon.

4.2.3. Possible Mechanisms.

In our theoretical framework, we show that rollover risk increases the liquidity effect on the

asset substitution phenomenon. To test this conjecture, we introduce an interaction term that

equals the product of the rollover risk proxies and illiquidity proxies, as shown below:

𝐴𝑠𝑠𝑒𝑡𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦',) = 𝑎 + 𝑏 ∗ 𝐴𝑚𝑖ℎ𝑢𝑑 + 𝑐 ∗ 𝐴𝑚𝑖ℎ𝑢𝑑 ∗ 𝑟𝑜𝑙𝑙𝑜𝑣𝑒𝑟𝑟𝑖𝑠𝑘

+𝑑z ∗ 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠',) + 𝑌𝑒𝑎𝑟) + 𝐹𝑖𝑟𝑚' + 𝑅𝑎𝑡𝑖𝑛𝑔𝑠',) + 𝑒𝑟𝑟𝑜𝑟',) (4.5)

The regression results in Table 6 show positive and significant coefficients for the interaction

terms with rollover risk, confirming that bond illiquidity affects risk-taking activities through the

rollover channel.

The results in Table 6 also show negative and significant coefficients for the interaction terms

with firm age, suggesting that the bond illiquidity effect on risk-taking activities is more

pronounced for firms with weaker borrowing powers.

[Please Insert Table 6 about Here]

27

Further, our model also implies that the liquidity effect on equity holders’ risk taking should

be more pronounced in firms with weaker fundamentals. To test this conjecture, we use profits,

size, and leverage to proxy for a firm’s fundamentals and construct the interaction terms between

the illiquidity measure and these variables, respectively. The results in Columns (2) and (5) of

Table 5 show that the interaction term is significantly negative, suggesting that risk-taking

activities are stronger in firms with weaker fundamentals.

5. Conclusion

In this study, we show theoretically and empirically that corporate debt illiquidity has a

positive impact on the risk-taking preferences of equity holders. First, we construct a theoretical

model to show the relationship between liquidity risk in the secondary corporate debt market and

equity holders’ risk-taking preferences. We use a simulated calibration, and show that equity

holders optimally choose to increase asset volatility by adopting risky projects in the face of

illiquidity shocks in the secondary corporate debt market. Second, we perform empirical analyses

and confirm the theoretical prediction. Specifically, we document a significant and positive

relationship between bond illiquidity and the risk-taking behavior of shareholders. In addition,

taking the introduction of the TRACE as an exogenous shock to the liquidity of the corporate

bond market, we find that firms’ risk-taking behaviors significantly decrease after dissemination

of the TRACE. Moreover, consistent with our model’s implications, we find that the impact of

liquidity risk on equity holders’ risk-taking preferences is more pronounced for firms with lower

profits, smaller size and higher rollover risk.

This paper is the first study to uncover the impact of trading friction in the secondary

corporate debt market on shareholders’ risk-taking preferences. Our findings deepen

28

understanding of both the corporate bond liquidity literature and the research on firms’ risk-

taking preferences.

29

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Appendix A: Proofs for Proposition

Proof for Proposition 1:

The debt value LD and

HD satisfy differential equations (1.2) and (1.3), respectively. The

general solutions are,

( )( )

1 2

1 2

0 1 2

0 1 2

L L

H H

y yL L L

y yH H H

D V a a V a V

D V a a V a V

= + +

= + + (A.1)

Where,

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

0

22 2 2

1 2

22 2 2

2 2

22 2 2

1 2

22 2 2

2 2

0.5 0.5 2

0.5 0.5 2

0.5 0.5 2

0.5 0.5 2

L L LL

L

L L LL

L

H H HH

H

H H HH

H

C gPar g k

r q r q r g ky

r q r q r g ky

r q r q r g ky

r q r q r g ky

ξ

σ σ σ ξ

σ

σ σ σ ξ

σ

σ σ σ ξ

σ

σ σ σ ξ

σ

+=

+ +

− − − + − − + + +=

− − − − − − + + +=

− − − + − − + + +=

− − − − − − + + +=

(A.2)

As UV → +∞ and 1 20, 0y y> < , we have 1 0La = in order to satisfy the boundary condition

(1.4). According to the other three boundary conditions (1.5), (1.6) and (1.7), we have,

33

( )1 2

2 1 2

2 1 2

1 2

2 1 2

1 1 12 2 1 1 2 2

1

0

0

H H

L H H

L H H

y yH H

y y yL S H S H S

y y yL L S H H S H H S

C gP a K a K Kr g k

a V a V a V

y a V y a V y a V

αξ

− − −

+⎧ + + = −⎪ + +⎪⎪⎪

− − =⎨⎪⎪

− − =⎪⎪⎩

(A.3)

Solving these linear equations, we have (1.9). QED.

Proof for Proposition 2:

We use the Laplace transform approach to solve the differential equations (1.10) and (1.11) with

four boundary conditions, simultaneously. We define lnHVmK

⎛ ⎞= ⎜ ⎟⎝ ⎠

and replace V with it in

(1.11), we have,

( )2

2 22

1 1 1 ( )2 2

HmH HH H H H H

H H

E ErE r q qKe w C gD m gPm m

σ σ∂ ∂⎛ ⎞= − − + + − − + −⎜ ⎟ ∂ ∂⎝ ⎠

(A.4)

with boundary conditions,

( )0 0HE = and ( )| 0

0

H

H

H m

El

m =

∂=

∂ (A.5)

We define the Laplace transformation of ( )H HE m as,

[ ] ( )0

( ) ( ) HsmH H H H HFH s L E m e E m dm

∞−≡ = ∫ (A.6)

And then apply the Laplace transformation to both sides. It gives

34

( ) [ ]2

2 22

11 1( ) ( )2 2 1

H HL H H

H H

w C gPE E qKrFH s r q L L gL D mm m s s

σ σ− +⎡ ⎤ ⎡ ⎤∂ ∂⎛ ⎞= − − + + − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ∂ ∂ −⎝ ⎠ ⎣ ⎦ ⎣ ⎦

(A.7)

Note that,

( ) ( )

( ) ( ) ( )2

2 22

| 0

(0)

0(0)

H

HH

H

HHH

H H m

EL sFH s E sFH sm

EEL s FH s sE s FH s lm m =

⎡ ⎤∂= − =⎢ ⎥∂⎣ ⎦

∂⎡ ⎤∂= − − = −⎢ ⎥∂ ∂⎣ ⎦

(A.8)

Therefore, we have

( ) [ ]2 2 2 211 1 1( ) ( )2 2 1 2H H H H H

w C gPqKr r q s s FH s gL D m ls s

σ σ σ− +⎡ ⎤⎛ ⎞− − − − = − + −⎜ ⎟⎢ ⎥ −⎝ ⎠⎣ ⎦

(A.9)

We define 0Hη > and 0Hγ− < to be the two roots of the following equation with respect to s ,

2 2 21 1 02 2H Hr r q s sσ σ⎛ ⎞− − − − =⎜ ⎟

⎝ ⎠ (A.10)

Then we have 1H H Hz dη = − > and 0H H Hz dγ = + > where,

2

2

12 H

HH

r qd

σ

σ

− −= and

( )1/22 4 2

2

2H H HH

H

d rz

σ σ

σ

+≡ (A.11)

Therefore,

( )( )

( ) [ ]2 211 1 1( ) ( )2 1 2H H H H

H H

w C gPqKFH s gL D m ls s s s

σ ση γ

− +⎧ ⎫= − − + −⎨ ⎬

− + −⎩ ⎭ (A.12)

According to the solutions for the debt value in (A.1), we have,

35

[ ]1 2

1 2 0 1 20 1 2

1 20

( )H H

H H H

y ysm y y H H

H H H H HH H

a a K a KL D m e a a V a V dms s y s y

∞− ⎡ ⎤= + + = + +⎣ ⎦ − −∫ (A.13)

By inserting (A.13) into (A.12), we have

( ) 1 22 20 1 2

1 2

1 111 1( )

2 1 2

H Hy yH H H H

H HH H H H

w C gP gas s ga K ga KqKFH s ls s s s y s y

η γσ σ

η γ

−− +⎧ ⎫− +

= − − + + + −⎨ ⎬+ − − −⎩ ⎭

(A.14)

By inverting the Laplace transformation, we have the equity value,

( ) [ ]

( )

( ) ( )( )

1 1 1

1

2

02

12

1 1

2

( )

2 1 11 1

12 1 1

2

2

H H H H

H H H HH H H H

H H H H H H H H H

H H

m m

H H H H H

m mm m

H H H H H H H

y m y m y m mH

H H H H H H H

H

E m L FL s

qKV e e

l e ew C gP ga e e

ga K e e e ey y

η γ

η γη γ

η γ

η γ σ η γ

σ η γ η γ η γ

σ η γ η γ

σ

−

−

−−

−

=

⎛ ⎞= − +⎜ ⎟+ − +⎝ ⎠

−⎡ ⎤− + − ⎛ ⎞− −+ − +⎢ ⎥⎜ ⎟+ +⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞− −− −⎢ ⎥⎜ ⎟+ − +⎝ ⎠⎣ ⎦

−2 2 2

2

2 2

H H H H H H H H Hy m y m y m mH

H H H H H H

ga K e e e ey y

η γ

η γ η γ

−⎡ ⎤⎛ ⎞− −−⎢ ⎥⎜ ⎟+ − +⎝ ⎠⎣ ⎦

(A.15)

We denote ( )( ) 0 1H H SE m V H H l= + and ( )( )

2 3

| ln SH

H H s

H VmK

E m VH H l

m⎛ ⎞

= ⎜ ⎟⎝ ⎠

∂= +

∂, where

36

( )

( )

11

0 2

02

12

1

2 1 11 1

12 1 11 1

2 1

H H

H H

H HH

S SS

H H H H H

S S

H H H H H

yyS SH

H H H H H

V VqKH VK K

w C gP ga V VK K

V Vga Ky K K

η γ

η γ

η

η γ σ η γ

σ η γ η γ

σ η γ η

−

−

⎛ ⎞⎛ ⎞ ⎛ ⎞= − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ − +⎝ ⎠ ⎝ ⎠⎝ ⎠

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞− + − ⎛ ⎞ ⎛ ⎞⎢ ⎥+ ⎜ − − − ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦

⎛ ⎞⎛ ⎞ ⎛ ⎞− −⎜⎜ ⎟ ⎜ ⎟⎜+ − ⎝ ⎠ ⎝ ⎠⎝

1

2 22

1

22

2 2

1

2 1 1

H H

H H H HH

yS S

H H

y yyS S S SH

H H H H H H H

V Vy K K

V V V Vga Ky K K y K K

γ

η γ

γ

σ η γ η γ

−

−

⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ − − ⎟⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎠ ⎝ ⎠⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥− ⎜ − − − ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ − +⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦

(A.16)

( )1

1 H H

S S

H H

V VHK K

η γ

η γ

−⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (A.17)

( )

11

2 2

02

112

1 1

21 1

12

2 1 1

H H

H H

H HH

S SH HS

H H H H H

S S

H H H

yyS SH

H HH H H H H H

V VqKH VK K

w C gP ga V VK K

V Vga K yy K K y

η γ

η γ

η

η γσ η γ η γ

σ η γ

ησ η γ η γ

−

−

⎡ ⎤⎛ ⎞−⎛ ⎞ ⎛ ⎞= − +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ − +⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞− + − ⎛ ⎞ ⎛ ⎞+ −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞⎛ ⎞ ⎛ ⎞− − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ − +⎝ ⎠ ⎝ ⎠⎝ ⎠

1

2 22

1

22 22

2 2

2 1 1

H H

H H H HH

yS S

H HH

y yyS S S SH

H H H HH H H H H H H

V VyK K

V V V Vga K y yy K K y K K

γ

η γ

γ

η γσ η γ η γ

−

−

⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ + ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥− ⎜ − − + ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ − +⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦ (A.18)

( )3

1 H H

S SH H

H H

V VHK K

η γ

η γη γ

−⎛ ⎞⎛ ⎞ ⎛ ⎞= +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (A.19)

For SV V≤ , we define lnLS

VmV⎛ ⎞

= ⎜ ⎟⎝ ⎠

and insert into differential equation (2.11),

37

( )2

2 22

1 1 1 ( )2 2

LmL LL L L S L L

L L

E ErE r q qV e w C gD m gPm m

σ σ∂ ∂⎛ ⎞= − − + + − − + −⎜ ⎟ ∂ ∂⎝ ⎠

(A.20)

with the boundary conditions,

( )( )(0)L H H SE E m V= and ( ) ( )( )| 0

| ln

0

L SH

H H SL

L m H VmK

E m VEm m= ⎛ ⎞

= ⎜ ⎟⎝ ⎠

∂∂=

∂ ∂ (A.21)

We define the Laplace transformation of ( )L LE m as,

[ ] ( )0

( ) ( ) LsmL L L L LFL s L E m e E m dm

∞−≡ = ∫ (A.22)

We also have,

( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( )( )22 2

20

| ln

(0)

0(0)

L SH

LL H H S

L

H H sLLL H H S

L Lm H VmK

EL sFL s E sFL s E m Vm

E m VEEL s FL s sE s FL s sE m Vm m m= ⎛ ⎞

= ⎜ ⎟⎝ ⎠

⎡ ⎤∂= − = −⎢ ⎥∂⎣ ⎦

∂∂⎡ ⎤∂= − − = − −⎢ ⎥∂ ∂ ∂⎣ ⎦

(A.23)

Following the similar procedure, we have,

( )

( ) ( )

1 20 1 2

2 1 2

2 20 1 0 1 2 3

11 111 ( )

2 1 12 2

L Ly yS L S L S

L LL LL

L LL

w C gPqV ga ga V ga Vs s s s y s ys sFL sr q H H l s H H l H H l

η γσ

η γσ σ

⎧ − + ⎫− + + +− ⎪ ⎪− − −− + ⎪ ⎪

= − ⎨ ⎬+ ⎛ ⎞⎪ ⎪− − − + − + + +⎡ ⎤⎜ ⎟ ⎣ ⎦⎪ ⎪⎝ ⎠⎩ ⎭

(A.24)

where,

38

( )2 1/22 4 2

2 2

1, 01

22 ,

L L L L L L

L L L LL L

L L

z d z d

r q d rd z

η γ

σ σ σ

σ σ

= − > = + >

− − += ≡

(A.25)

By inverting the Laplace transformation in (A.24), we have,

( ) [ ]

( )( )

1 1 1

2

1

2

02

12

1 1

22

( )

2 1 11 1

12 1 1

2

2

L L L L

L L L L

L L L L L L L L L

L L

L

m mS

L L L L L

m m

L L L L L

y m y m y m mL S

L L L L L L L

yL S

L L L

EL m L FL s

qVV e e

w C gP ga e e

ga V e e e ey y

ga V e

η γ

η γ

η γ

η

η γ σ η γ

σ η γ η γ

σ η γ η γ

σ η γ

−

−

−

−

=

⎛ ⎞= − +⎜ ⎟+ − +⎝ ⎠

⎡ ⎤− + − ⎛ ⎞− −+ −⎢ ⎥⎜ ⎟+ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞− −− −⎢ ⎥⎜ ⎟+ − +⎝ ⎠⎣ ⎦

−+

( ) ( ) ( )

( )( )

2 2

2 2

2 20 1 2 32

0 1

2 1 1 12 2

L L L L L L L

L L L L

L L L L

m y m y m m

L L L L

m mL

L L L

m mL L

L L

e e ey y

r q H H l H H l e e

e eH H l

γ

η γ

η γ

η γ

σ σσ η γ

η γ

η γ

−

−

−

⎡ ⎤⎛ ⎞− −−⎢ ⎥⎜ ⎟− +⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎛ ⎞+ − − + + + −⎢ ⎥⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠⎣ ⎦

++ +

+ (A.26)

Now, we impose the boundary condition at Lm →∞ . The equity value must grow linearly when

V →∞ . Since L

L Lm

S

VeV

η

η ⎛ ⎞= ⎜ ⎟⎝ ⎠

and 1Lη > , to avoid explosion we require the coefficient on L Lmeη

in ( )LEL m to collapse to zero. By collecting the coefficients of L Lmeη , we have Equation (2.18).

QED.

39

Appendix B: Variable Definitions

Panel A: Bond illiquidity measures (bond-level) Amihud ratio (AMI)

The AMI is the average absolute value of a bond return divided by the dollar trading volume (in $millions) within the day and then multiplied by 100.

1 1

1

1 tN j j jt

jt j

P P PAmihud

N Q− −

=

−= ∑

Inter-quartile range (IQR)

Inter-quartile range (IQR) is defined as the difference between the 75th percentile and 25th percentile of prices for one day normalized by the average price on that day (Han and Zhou (2007), Pu (2009)), i.e.,

,75 ,25

100i th i th

i t tt i

t

p pIQRp−

= ×

Imputed Roundtrip cost

Imputed roundtrip cost, proposed by Feldhütter (2012) directly estimates roundtrip transaction costs based on trade prices and is defined as:

𝑅𝑜𝑢𝑛𝑑𝑡𝑟𝑖𝑝 =Pmax − Pmin

Pmax

Price dispersion

Price dispersion, introduced by Jankowitsch, Nashikkar, and Subrahmanyan (2011) measures deviation from the expected market valuation of an asset, and is defined as:

Panel B: Firm risk-taking measures Asset volatility

We back out asset volatility following the research of Merton (1974).

Earning volatility

Is the standard deviation of quarterly income before extraordinary items, deflated by total assets during the two years preceding the fiscal year end.

Panel C: Control variables Firm size The natural log of the book value of total assets measured at each fiscal year end.

Leverage The book value of total debt divided by the book value of total assets measured at each

fiscal year end.

Profit Earnings before interest, tax and depreciation scaled by the book value of total assets at each fiscal year end.

40

Sales growth The two-year average growth rate in sales. If the two-year growth data is not available, we use the one-year growth in sales information.

Rollover risk The percentage of debt due in one year scaled by the book value of total debt measured at each fiscal year end.

Firm age A firm's age is equal to the number of years a firm existed in Compustat prior to each fiscal year end.

41

Table 1: Calibrations

Parameters Calibrations Literatures

Bond Market

Risk free rate 8%r = Leland (1998), He and Xiong (2012), Huang and Huang (2012)

Proportional Liquidity Costs 1.00%k = for A rating 0.50%k = for BB rating He and Xiong (2012)

Liquidity shock intensity 1ξ = for both He and Xiong (2012)

Firm Characteristics

Initial Asset Value 0 $100V = Leland (1998), He and Xiong (2012), etc.

Corporate Payout Rate 2%q = Huang and Zhou (2008) Proportional Bankruptcy Cost 40%α = Chen (2010)

Firm’s Debt Structure

Debt Face Value $52.6P = for A rating $60P = for BB rating He and Xiong (2012) for BB rating

Coupon Payment $4.35C = for A rating $5.30C = for BB rating

Average Maturity of Debts 3T = Custodio, Ferreira and Laureano (2010) Debt Tax Benefit 27%w = He and Xiong (2012)

Firm’s Asset Volatilities

Low Asset Volatilities 21%Lσ = for A rating

23%Lσ = for BB rating Zhang, Zhou and Zhu (2009)

High Asset Volatilities 40.83%Hσ = for A rating

44.72%Hσ = for BB rating

42

Figure 1: Marginal Change of Equity Value and Switching Boundary

This figure depicts the marginal changes of equity value with respect to the switching boundary conditional on the asset value equaling the switching boundaries for A (Panel A) and BB (Panel B) ratings. The solid lines show the situations where liquidity risk in the secondary bond market is absent while the dash lines show the corresponding situations with liquidity risk. The base calibrations are used for both ratings.

60 70 80 90 100-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Panel A: A Rating

VS

@E@V

Sj V

=V S

65 70 75 80 85 90 95 100-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Panel B: BB Rating

VS

@E@V

Sj V

=V S

No liquidity riskWith liquidity risk

No liquidity riskWith liquidity risk

43

Figure 2: Optimal Risk-Shifting Boundaries and Liquidity Intensity

This figure depicts the changes of ex-ante and ex-post risk switching boundaries with respect to the intensity of liquidity shocks for A (Panel A) and BB (Panel B) rating classes. The solid and dash lines show the ex-ante and ex-post risk switching boundaries, respectively. The base calibrations are used for both ratings.

0 0.5 1 1.5 2

60

70

80

90

100

110

Panel A: A Rating

ξ

V S

Ex-AnteEx-Post

0 0.5 1 1.5 2

60

70

80

90

100

110

Panel B: BB Rating

ξ

V S

Ex-AnteEx-Post

44

Figure 3: Liquidity Intensity and Endogenous Default Boundaries

This figure depicts the changes of ex-ante and ex-post endogenous default boundaries with respect to the intensity of liquidity shocks for A (Panel A) and BB (Panel B) rating classes. The solid and dash lines show the ex-ante and ex-post risk switching boundaries, respectively. The base calibrations are used for both ratings.

0 0.5 1 1.5 2

60

70

80

90

100

110

Panel A: A Rating

ξ

V S

Ex-AnteEx-Post

0 0.5 1 1.5 2

60

70

80

90

100

110

Panel B: BB Rating

ξ

V S

Ex-AnteEx-Post

45

Table 2: Summary Statistics

This table reports the summary statistics of liquidity measures, risk-taking measures and firm characteristics based on a sample of 1,379 firms over the period 2002-2015 (9,428 firm-year observations). All of the variables are winsorized at 1% and 99%. The variable definitions are given in Table 1.

N Mean Std Dev 25th Pctl Median 75th Pctl

Illiquidity measures AMI 9428 0.009 0.010 0.004 0.006 0.011

IRT 9428 0.006 0.006 0.003 0.005 0.007 PD 9428 0.002 0.002 0.001 0.002 0.003 IQR 9428 0.004 0.006 0.002 0.003 0.005

Risk-raking measures

Asset volatility 9428 0.553 0.440 0.300 0.441 0.665

Earning volatility 9428 0.015 0.023 0.003 0.007 0.015

Control variables Ln(asset) 9428 8.929 1.583 7.805 8.815 9.940

Leverage 9428 0.669 0.196 0.536 0.647 0.787 Profit 9428 0.119 0.081 0.076 0.116 0.163

Sales growth 9428 0.018 0.140 -0.039 0.003 0.050 Rollover risk 9428 0.028 0.050 0.000 0.007 0.035 Ln(firm age) 9428 3.213 0.594 2.819 3.287 3.774

46

Table 3: OLS regressions of risk-taking measures on bond illiquidity

This table tests the relationship between bond illiquidity and asset volatility. It presents estimates from a pooled OLS regression based on a sample of 1,379 firms over the period 2002 through 2015 (9,428 firm-year observations). The dependent variable is asset volatility. Columns (1) through (4) report the results of regressions with the AMI, IRT, PD and IQR as illiquidity measures, respectively. Other control variables are Ln(asset), Leverage, Profit, Sales growth, Rollover risk and Ln (firm age). All of the variables are winsorized at 1% and 99%. The variable definitions are presented in Table 1. All of the regressions control for year-fixed effects, firm-fixed effects and include rating dummies for each rating category. Standard errors are clustered at the firm level and are presented in parentheses. *, ** and *** indicate statistical significance at the 0.10, 0.05 and 0.01 two-tailed levels, respectively

Dependent variable: Asset volatility (1) (2) (3) (4)

AMI 3.720***

(0.399)

IRT

7.521***

(0.677)

PD

3.779**

(1.678)

IQR

3.527***

(0.634)

Ln(asset) -0.025** -0.032** -0.026** -0.025**

(0.013) (0.013) (0.013) (0.013)

Leverage 0.181*** 0.162*** 0.197*** 0.191***

(0.041) (0.041) (0.041) (0.041)

Profit -0.702*** -0.669*** -0.745*** -0.733***

(0.075) (0.075) (0.075) (0.075)

Sales growth 0.051** 0.051** 0.052** 0.051**

(0.024) (0.024) (0.024) (0.024)

Rollover risk 0.421*** 0.436*** 0.425*** 0.427***

(0.078) (0.078) (0.079) (0.078)

Ln(firm age) -0.323*** -0.294*** -0.309*** -0.306***

(0.044) (0.044) (0.044) (0.044)

Constant 1.393*** 1.328*** 1.391*** 1.383***

(0.21) (0.21) (0.211) (0.211)

Year-fixed effect Yes Yes Yes Yes Firm-fixed effect Yes Yes Yes Yes Rating dummies Yes Yes Yes Yes

N 9428 9428 9428 9428

47

Table 4: OLS Regressions of Risk-taking Measures on Bond Illiquidity

This table tests the relationship between bond illiquidity and earning volatility. It presents estimates from a pooled OLS regression based on a sample of 1,379 firms over the period 2002 to 2015 (9,428 firm-year observations). The dependent variable is earning volatility. Columns (1) through (4) report the results of regressions with the AMI, IRT, PD and IQR as illiquidity measures, respectively. Other control variables are Ln(asset), Leverage, Profit, Sales growth, Rollover risk and Ln(firm age). All of the variables are winsorized at 1% and 99%. The variable definitions are presented in Table 1. All of the regressions control for year-fixed effects, firm-fixed effects and include rating dummies for each rating category. Standard errors are clustered at the firm level and are presented in parentheses. *, ** and *** indicate statistical significance at the 0.10, 0.05 and 0.01 two-tailed levels, respectively. Dependent variable: Earning volatility

(1) (2) (3) (4) AMI 0.097***

(0.023) IRT

0.254***

(0.039) PD

0.201**

(0.096) IQR

0.197***

(0.036)

Ln(asset) -0.009*** -0.009*** -0.009*** -0.009***

(0.001) (0.001) (0.001) (0.001)

Leverage 0.029 0.028*** 0.029*** 0.030***

(0.002) (0.002) (0.002) (0.002)

Profit -0.082 -0.081*** -0.083*** -0.089***

(0.004) (0.004) (0.004) (0.004)

Sales growth 0.001 0.001 0.001 -0.002

(0.001) (0.001) (0.001) (0.001)

Rollover risk 0.010** 0.010** 0.010** 0.010**

(0.004) (0.004) (0.004) (0.005)

Ln(firm age) 0.002 0.003 0.002 0.003

(0.003) (0.003) (0.003) (0.003)

Constant 0.120*** 0.118*** 0.120*** 0.126

(0.012) (0.012) (0.012) (0.245)

Year-fixed effect Yes Yes Yes Yes Firm-fixed effect Yes Yes Yes Yes Rating dummies Yes Yes Yes Yes

N 9428 9428 9428 9428

48

Table 5: The TRACE Dissemination Test

This table tests the relationship between risk-taking measures and dissemination of the TRACE. The dependent variables are asset volatility and earning volatility. The TRACE-dissemination is a dummy variable, which equals one if a firm’s trading is available in Standard TRACE, otherwise zero. Other control variables are Ln(asset), Leverage, Profit, Sales growth, Rollover risk and Ln(firm age). All of the variables are winsorized at 1% and 99%. The variable definitions are presented in Table 1. All of the regressions control for year-fixed effects, firm-fixed effects and include rating dummies for each rating category. Standard errors are clustered at the firm level and are presented in parentheses. *, ** and *** indicate statistical significance at the 0.10, 0.05 and 0.01 levels, respectively.

Dependent variable Asset volatility Earning volatility

(1) (2) TRACE-dissemination -0.190*** -0.010***

(0.039) (0.002) Ln(asset) -0.046 0.004

(0.095) (0.007) Leverage -0.669* 0.001

(0.397) (0.020)

Profit 0.573** 0.098**

(0.265) (0.019)

Sales growth -1.189*** -0.030

(0.442) (0.028)

Rollover risk 0.096 0.002

(0.061) (0.004)

Ln(firm age) 0.319 0.011

(0.389) (0.021)

Constant -0.244 0.024

(0.674) (0.045)

Year-fixed effect Yes Yes Firm-fixed effect Yes Yes Rating dummies Yes Yes

N 1802 1802

49

Table 6: Possible Mechanisms

This table presents the panel regressions, which include the interaction-terms. It presents estimates from a pooled OLS regression, based on a sample of 1,379 firms over the period July 1, 2002 to January 31, 2015 (9,428 firm-year observations). The dependent variable from Columns (1) through (3) is asset volatility and from Columns (4) through (6) it is earning volatility. Amihud* Rollover risk, Amihud* Profit, or Amihud* Ln(firm age) denotes the cross term between the Amihud ratio and Rollover risk, Profit or Ln(firm age), respectively. Other variables are the Amihud ratio, Ln(asset), Leverage, Profit, Sales growth, Rollover risk and Ln(firm age). All of the variables are winsorized at 1% and 99%. The variable definitions are presented in Table 1. All of the regressions control for year-fixed effects and firm-fixed effects, and include rating dummies for each rating category. The standard errors are clustered at the firm level and are presented in parentheses. *, ** and *** indicate statistical significance at the 0.10, 0.05 and 0.01 levels, respectively.

Dependent variables: Asset volatility Earning volatility

(1) (2) (3) (4) (5) (6)

Amihud* Rollover

risk 20.575*** (6.196)

0.939*** (0.354)

Amihud* Profit

-39.923*** (4.076)

-0.516** (0.234)

Amihud* Ln(firm age)

-6.341*** (0.656)

-0.198*** (0.038)

Amihud 3.19*** (0.429)

7.23*** (0.534)

24.646*** (2.201)

0.073*** (0.024)

0.142*** (0.308)

0.751*** (0.126)

Ln(asset) -0.024* (0.013)

-0.018 (0.013)

-0.025* (0.013)

-0.009*** (0.001)

-0.009*** (0.001)

-0.009*** (0.001)

Leverage 0.171*** (0.041)

0.156*** (0.040)

0.173*** (0.041)

0.028*** (0.002)

0.029*** (0.002)

0.029*** (0.002)

Profit -0.694*** (0.075)

-0.215** (0.089)

-0.676*** (0.074)

-0.082*** (0.004)

-0.076*** (0.005)

-0.081*** (0.004)

Sales growth 0.051** (0.024)

0.048** (0.024)

0.053** (0.024)

0.001 0.001 0.001 (0.001) (0.001) (0.001)

Rollover risk 0.225** (0.098)

0.417*** (0.078)

0.406*** (0.078)

0.001 (0.006)

0.010** 0.009** (0.004) (0.004)

Ln(firm age) -0.325*** (0.044)

-0.336*** (0.044)

-0.313*** (0.044)

0.002 (0.003)

0.002 (0.003)

0.002 (0.003)

Constant 1.392*** (0.210)

1.282*** (0.209)

1.300*** (0.209)

0.120*** (0.012)

0.119*** (0.012)

0.118*** (0.012)

Year-fixed effect Yes Yes Yes Yes Yes Yes Firm-fixed effect Yes Yes Yes Yes Yes Yes Rating dummies Yes Yes Yes Yes Yes Yes

N 9428 9428 9428 9428 9428 9428