Investment-based financing constraints and debt renegotiation

Accepted Manuscript Investment-based financing constraints and debt renegotiation Takashi Shibata, Michi Nishihara PII: DOI: Reference:

S0378-4266(14)00352-5 http://dx.doi.org/10.1016/j.jbankfin.2014.11.005 JBF 4597

To appear in:

Journal of Banking & Finance

Received Date: Accepted Date:

4 March 2014 8 November 2014

Please cite this article as: Shibata, T., Nishihara, M., Investment-based financing constraints and debt renegotiation, Journal of Banking & Finance (2014), doi: http://dx.doi.org/10.1016/j.jbankfin.2014.11.005

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Investment-based financing constraints and debt renegotiation Takashi Shibataa∗ and Michi Nishiharab,c a

Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-osawa,

Hachioji, Tokyo 192-0397, Japan.

b

Graduate School of Economics, Osaka University, ´ 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. c Swiss Finance Institute, Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland. First Version:

March 4, 2014

Second Version: October 16, 2014 This Version:

November 15, 2014

Abstract We consider how equity holders’ bargaining power during financial distress influences the interactions between financing and investment decisions when the firm faces the upper limit of debt issuance. We obtain four results. First, weaker equity holders’ bargaining power is more likely that the firm is financially constrained. Second, the investment quantity is independent of equity holders’ bargaining power. Third, the constrained credit spreads are increasing with equity holders’ bargaining power, contrary to the unconstrained ones. Fourth, higher volatility and weaker equity holders’ bargaining power are likely that the firm prefers to issue debt with renegotiation, compared with debt without renegotiation.

JEL classification: G32; G33, G21. Keywords: Real option; credit constraints; strategic debt service; capital structure. ∗ Corresponding author. Tel.: +81 42 677 2310; fax: +81 42 677 2298. Email addresses: [email protected] (T. Shibata), [email protected] (M. Nishihara)

1

Introduction

Contingent claim models in corporate finance have become a standard framework for the valuation of debt and equity since the seminal work pioneered by Merton (1974), Black and Cox (1976), and Leland (1994). In most of the contingent claim models, it has been assumed that bankruptcy is identical to liquidation (i.e., Chapter 7 of the US bankruptcy code). Once the firm goes into a period of financial distress, the firm is immediately liquidated (i.e., the firm stops operations). In practice, however, most companies in financial distress can try to restructure outstanding debt into a form that is more affordable (i.e., Chapter 11 of the US bankruptcy code). The debt reorganization strategy is an alternative procedure to liquidation. According to Brown (1989) and Gilson et al. (1990), reorganization is preferred because it prevents losses from liquidation during financial distress. There are several studies that incorporate the notion of debt reorganization as an alternative bankruptcy procedure to liquidation. Mella-Barral and Perraudin (1997), Fan and Sundaresan (2000), Francois and Morellec (2004), and Broadie et al. (2007) allow ex ante financing (capital structure) decisions in the presence of debt reorganization during periods of financal distress. However, in these papers, investment decisions are not considered. Because financing decisions correspond to investment decisions, both financing and investment decisions are simultaneously considered. Sundaresan and Wang (2007) consider the interaction between financing and investment strategies, by incorporating investment strategies into previous papers such as Fan and Sundaresan (2000) and Francois and Morellec (2004). The most important parameters are equity and debt holders’ bargaining power parameters, η ∈ [0, 1] and 1 − η, respectively, during periods of financial distress. They show that ex post stronger equity holders’ bargaining power (i.e., higher η) lowers the ex ante amount of debt issuance and reduces the ex ante equity option value before investment. However, Sundaresan and Wang (2007) can be extended in at least two ways, which we undertake in this analysis. The first extension in our paper is to incorporate an upper limit of debt issuance as in Sundaresan and Wang (2007). Let E a (Xt , c, δ) and Da (Xt , c, δ) denote the equity and debt values after investment where δXt is the cash inflow of the firm (Xt ≥ 0 represents the stochastic price and δ ≥ 1 represents the quantity) and c ≥ 0 is the coupon payment (cash outflow of the firm). These values E a (Xt , c, δ) and Da (Xt , c, δ) depend on ex post equity and debt holders’ negotiation powers during the periods of financial distress. The ex ante optimal financing and investment problem is formulated by maxT,c,δ E[e−rT {E a (XT , c, δ)− (I(δ) − Da (XT , c, δ))}], where E represents the expectation operator and I(δ) > 0 is the

1

cost expenditure as a function of δ. Here, we determine endogenously the investment timing T , the coupon payment c (i.e., the optimal amount of debt Da (XT , c, δ)), and the investment quantity δ. In this problem, it is possible for the firm to issue the amount of debt Da (XT , c, δ), which is larger than I(δ) > 0, to maximize its own option value. Under the same parameters as in Sundaresan and Wang (2007), we calculate numerically the optimal amount of debt at the optimal time of investment in Table 1.1 Table 1 reports the optimal amount of debt issuance for η ∈ {0, 0.2, 0.4, 0.6, 0.8, 1}. We see that the optimal amount of debt is larger than the investment cost for η ∈ {0, 0.2, 0.4, 0.6}. In this model, the excess Da (XT , c, δ) − I(δ) > 0 is distributed to equity holders. The excess distribution is illegal in practice. Thus, we need at least the mathematical condition Da (XT , c, δ) ≤ I(δ) based on the practical problem. In addition, debt issuance encourages risk shifting from equity holders to debt holders. The upper limits of debt issuance are typically needed to rule out corporate default by mitigating risk shifting (see Jensen and Meckling (1976)). As a result, we incorporate the condition of Da (XT , c, δ) ≤ qI(δ), where q ∈ [0, 1] is an exogenous parameter for the firm. [Insert Table 1 about here] The second extension of our paper is that the firm endogenously determines the investment quantity δ. Suppose that δ = 1 is exogenous as in Sundaresan and Wang (2007). Then the upper limit constraint is Da (XT , c, 1) ≤ qI(1). Then, the amount of debt issuance is constrained by the two exogenous variables q and I(1). In practice, the firm has various types of investment opportunities and decides the financing (capital structure) and investment (quantity) decisions mutually. Thus, it is reasonable to assume that investment quantity δ is endogenous, which implies that we can regard the upper limit constraint to be endogenous. In particular, an increase in δ increases E a (XT , c, δ), Da (XT , c, δ), and I(δ). Importantly, I(δ) and Da (XT , c, δ) depend on the endogenous investment quantity δ, which implies that both terms of upper limit constraints are endogenous. Thus, we can consider how the equity holders’ bargaining power influences the upper limit constraints and how they affects the financing and investment (timing and quantity) strategies. To summarize, our extensions to Sundaresan and Wang (2007) introduce upper limit constraints of debt issuance and endogenous investment quantity. The important parameters are η (equity holders’ bargaining power), q (financial friction), and σ (cash flow volatility). In this paper, we show how these three parameters influence the interaction 1

They assume the basic parameters: α = 35%, r = 6%, τ = 25%, µ = 0, σ = 15%, I(δ) = 1, and

δ = 1 (I and δ are assumed to be exogenous, not endogenous).

2

between the financing and investment strategies. In addition, in order to make clear the effects of the debt reorganization strategies, we derive the financing and investment strategies in the absence of debt reorganization during the periods of financial distress (i.e., Wong (2010)) and compare the financing and investment strategies in the presence and absence of debt reorganization. Our model provides the following four results. First, we show that whether the firm is financially constrained by the upper limit of debt issuance depends on the three key parameters: q (financial friction), η (equity holders’ bargaining power), and σ (volatility). By increasing η, the firm is not likely to be constrained. The reason is that an increase in η decreases coupon payments c, which leads to a decrease in the amount of debt issuance. By increasing σ, the firm is more likely to be constrained. The reason is that an increase in σ increases coupon payments (i.e., the amount of debt issuance). Second, the investment quantity (one of the three solutions) is independent of η. On the other hand, the investment timing (threshold) and coupon payment (two other solutions) depend on η. In addition, we show that the investment timing (threshold) for a constrained levered (debt financed) firm is not between those of the two special cases (an unlevered firm and an unconstrained levered firm), while the investment quantity and coupon payments are between those of the two special cases. Consequently, it is less costly to distort the investment timing (threshold) than the other two solutions. Third, suppose that, as a benchmark, the financing constraints are not effective. Recall then that an increase in η decreases the coupon payments and the amount of debt issuance. In addition, an increase in η decreases the credit spreads. In our model, suppose that the financing constraints are effective. An increase in η increases the coupon payments and keeps the amount of debt issuance constant. This implies that an increase in η increases the credit spreads. These results for a constrained levered firm are contrary to those for an unconstrained levered firm. Fourth, we examine how economic situation the firm would like to issue the debt with a renegotiation option (debt with renegotiation, in short). To do so, suppose that equity holders have an option to issue one of two types of debt: debt with renegotiation and debt without renegotiation. On the one hand, if the firm issues debt with renegotiation, equity holders can negotiate with debt holders to reduce coupon payments during financial distress to avoid wasteful liquidation. On the other hand, if the firm issues debt without renegotiation, the firm is immediately liquidated during financial distress. While renegotiation increases the values of equity and debt ex post (after investment) by preventing wasteful liquidation, renegotiation also incurs a cost on the equity option value ex ante (before investment). Thus, whether debt with renegotiation is preferred ex ante depends 3

on the three key parameters, although debt with renegotiation is always preferred ex post. We show that higher volatility as well as weaker equity holders’ bargaining power are likely that the firm prefers debt with renegotiation ex ante. The remainder of the paper is organized as follows. Section 2 describes the setup of our model. As a benchmark, we provide the solutions to the special case of our model. Section 3 provides the solution to our model and examines its properties. Section 4 considers the model’s implications. Section 5 investigates the availability of debt renegotiation. Section 6 concludes. Technical developments are included in four appendices. Appendix A provides derivations of the value functions. Appendix B shows proofs of the results of this paper. Appendix C demonstrates the properties of the solutions that are the same as in the previous papers. Appendix D contains the solutions in the absence of debt renegotiation.

2

Model

In this section, we begin by describing the model. We then provide the value functions after investment and formulate the financing and investment decision problem. Finally, as a benchmark, we derive solutions for two special cases in our model.

2.1

Setup

Equity holders possess the option to invest in a single project at any time. If the investment option is exercised, equity holders receive an instantaneous cash inflow δXt . Here, Xt is the stochastic price and is given by the following geometric Brownian motion: dXt = µXt dt + σXt dzt ,

X0 = x > 0,

(1)

where µ > 0 and σ > 0 are positive constants and zt denotes a standard Brownian motion. For convergence, we assume that r > µ, where r > 0 is the risk-free interest rate. We assume that the current price level X0 = x is sufficiently low such that equity holders do not undertake the investment immediately. Alternatively, δ ≥ 1 represents the investment quantity and affects the one-time fixed investment cost denoted by I(δ). We assume that the investment cost I(δ) satisfies the conditions I(1) > 0, I 0 (δ) > 0, and I 00 (δ) > 0 for any δ. These conditions are intuitively reasonable. The first condition states that the investment cost is required to initiate the project. The second and third conditions mean that I(δ) is strictly increasing and convex in δ. Note that at the time of investment, δ is endogenously chosen to maximize the firm’s profits. To ensure a unique optimal solution for δ, we further assume that the elasticity 4

of the cost with respect to the investment quantity, δI 0 (δ)/I(δ), is increasing with respect to δ.2

2.2

Value functions after investment

This subsection provides the value functions after investment. We assume that the firm issues risky debt at the time of investment. For analytical convenience, we limit the condition such that the risky debt is perpetual. If the firm issues the debt, equity holders obtain (1 − τ )(δ1 Xt − c1 ), where τ > 0 represents the corporate tax, δ1 ≥ 1 is the investment quantity under debt financing, and c1 ≥ 0 is the coupon payment, where the subscript “1” represents the debt financing with renegotiation. Once Xt becomes sufficiently small after issuing the debt, the firm should be liquidated because equity holders cannot pay c1 . Following previous research such as Leland (1994), because there is a liquidation cost defined by α(1 − τ )δ1 Xt /(r − µ), where α ∈ (0, 1), debt holders collect the residual value (1 − α)(1 − τ )δ1 Xt /(r − µ). Then, because liquidation (formally bankruptcy) is costly, when Xt is small after issuing the debt, debt holders will renegotiate the terms of the debt contract with equity holders. As in Sundaresan and Wang (2007), we assume that equity and debt holders negotiate and divide the surplus generated by avoiding liquidation between them. Let us denote by η ∈ [0, 1] and 1 − η the bargaining powers of equity and debt holders, respectively. The division of the surplus between equity and debt holders depends on their relative bargaining powers. Thus, during periods of financial distress, the coupon payments to debt holders are reduced through a costless private workout. Let us denote by T1i and T1d the investment (indicated by superscript “i”) and coupon switching (indicated by superscript “d”) timings, respectively. Mathematically, the investment and coupon switching timings are defined as T1i = inf{s ≥ 0|Xs ≥ xi1 } and T1d = inf{s ≥ T1i |Xs = xd1 }, where xi1 and xd1 denote the associated investment and coupon switching thresholds, respectively. Let us denote by E1j (Xt , c1 , δ1 ), D1j (Xt , c1 , δ1 ), and V1j (Xt , c1 , δ1 ) the equity, debt, and total firm values, respectively (j ∈ {a, b}). Here, the superscripts “a” and “b” indicate the normal and bankruptcy (coupon reduction) regions, respectively. As shown in Appendix A, we derive the equity and debt values after issuing the debt as follows.3 For any time t > T1i , E1a (Xt , c1 , δ1 ) and E1b (Xt , c1 , δ1 ) are given respectively 2 3

This condition is the same as in Wong (2010). The derivations are similar to those in Sundaresan and Wang (2007). The only difference is that our

model includes the endogenous investment quantity δ1 ≥ 1.

5

by E1a (Xt , c1 , δ1 )

(2) ´³ X ´γ c1 ³ c ηγ 1 t = max Πδ1 Xt − (1 − τ ) − (1 − αη)Πδ1 xd1 − (1 − τ − τ ) , d d r r β − γ x x1 1 ³ τ c1 γ ³ Xt ´β ´ b E1 (Xt , c1 , δ1 ) = η αΠδ1 Xt − , (3) r β − γ xd1 where Π := (1 − τ )/(r − µ) > 0, β := 1/2 − µ/σ 2 + ((µ/σ 2 − 1/2)2 + 2r/σ 2 )1/2 > 1 and γ := 1/2−µ/σ 2 −((µ/σ 2 −1/2)2 +2r/σ 2 )1/2 < 0. Note that E1a (Xt , c1 , δ1 ) and E1b (Xt , c1 , δ1 ) are defined for the normal region of {Xt > xd1 } and the bankruptcy (coupon reduction) region of {Xt ≤ xd1 }, respectively. Importantly, wasteful liquidation never occurs at the equilibrium in our model, because the reduced coupon payment in the bankruptcy region, s(Xt , δ1 ), is a linear function of Xt , i.e., s(Xt , δ1 ) = (1 − αη)(1 − τ )δ1 Xt in (A.10). The optimal coupon switching threshold is chosen to satisfy ∂E1a /∂xd1 = ∂E1b /∂xd1 , i.e., xd1 (c1 , δ1 ) =

c1 ≥ 0, κ1 δ1

(4)

where κ1 := (γ − 1)(1 − αη)Πr/(γ(1 − τ (1 − η))) ≥ 0. Note that xd1 (c1 , δ1 ) is a linear function of c1 with limc1 ↓0 xd1 (c1 , δ1 ) = 0. Importantly, xd1 (c1 , δ1 ) is decreasing with δ1 . This implies that an increase in δ1 decreases the coupon switching threshold. Substituting (4) into (2) and (3) gives ³1 − τ ³ κ δ X ´γ ´ 1 1 t E1a (Xt , c1 , δ1 ) = Πδ1 Xt − + w1 c1 , r c1 ³ τ γ ³ κ1 δ1 Xt ´β ´ E1b (Xt , c1 , δ1 ) = η αΠδ1 Xt − c1 , rβ−γ c1

(5) (6)

where w1 := (1−αη)Π/κ1 −(1−τ −τ ηγ/(β −γ))/r ≤ 0. Note that limc1 ↓0 E1a (Xt , c1 , δ1 ) = Πδ1 Xt . The debt values in the normal and bankruptcy regions, D1a (Xt , c1 , δ1 ) and D1b (Xt , c1 , δ1 ), are given by D1a (Xt , c1 , δ1 )

=

³1

³ κ δ X ´γ ´ 1 1 t − n1 c1 , r c1

D1b (Xt , c1 , δ1 ) = (1 − αη)Πδ1 Xt − (1 − η)

(7) τ γ ³ κ1 δ1 Xt ´β c1 , rβ−γ c1

(8)

where n1 := τ β/(r(β − γ)) − w1 ≥ 0. Note that limc1 ↓0 D1a (Xt , c1 , δ1 ) = 0. Finally, the total firm value is defined by V1j (Xt , c1 , δ1 ) = E1j (Xt , c1 , δ1 ) + D1j (Xt , c1 , δ1 ) (j ∈ {a, b}).

2.3

Upper limits of debt financing

In this subsection, we examine the most important assumption in our model, i.e., debt issuance constraints. 6

Recall that equity holders can issue the amount of debt at the time of investment to finance the amount of investment expenditure I(δ1 ) > 0. Equity holders issue debt because they always increase their values by doing so (see, e.g., Sundaresan and Wang (2007) and Shibata and Nishihara (2012)). Then, equity holders have the possibility of issuing D1a (xi1 , c1 , δ1 ), which is greater than I(δ1 ), in order to maximize their value. That is, the implication of D1a (xi1 , c1 , δ1 ) > I(δ1 ) is that the excess is distributed to equity holders as a dividend immediately after issuing the debt. This is illegal in practice, although it is mathematically possible. We reexamine the numerical examples of Table 1 in Sundaresan and Wang (2007).4 Figure 1 depicts the debt–investment ratio with η. We see that the debt–investment ratio is larger than 1 for η < 0.6668. As a result, we should incorporate the constraints of D1a (xi1 , c1 , δ1 ) ≤ I(δ1 ). [Insert Figure 1 about here] In addition, issuing debt encourages risk to shift from equity to debt holders. To minimize their risk, debt holders are reluctant to lend more than a certain amount (see Jensen and Meckling (1976)). That is, debt holders permit equity holders to issue an amount less than the investment cost I(δ1 ). Thus, we assume that the firm’s debt issuance limit is constrained, i.e., D1a (xi1 , c1 , δ1 ) ≤ q, I(δ1 )

(9)

for some constant q ∈ [0, 1]. The left-hand side of inequality (9) represents the ratio of debt value to investment cost. Thus, inequality (9) implies that the debt issuance upper limit is restricted. The upper limit of q < 1 might arise because of the risk-shifting problem. Consequently, the parameter q imposes restrictions on the firm’s debt issuance limit. The lower the ratio q, the more severe the debt issuance constraint. In the extreme case of q = 0, the firm is not allowed to issue debt. Conversely, in the extreme case of q = 1, the firm can issue an amount of debt equivalent to the investment cost I(δ1 ).

2.4

Financing and investment decisions problem

In this subsection, we formulate the financing and investment optimization problem, i.e., the problem for a constrained levered (debt-financed) firm. Let us denote by E1∗∗ (x) the equity option value of the constrained levered firm before investment, where the superscript “∗∗” represents the optimum for the constrained 4

They assume the basic parameters: α = 35%, r = 6%, τ = 25%, µ = 0, σ = 15%, and I = 1.

7

problem. The equity option value is defined as h n ³ ´oi ∗∗ −rT1i a a E1 (x) := sup E0 e E1 (XT1i , c1 , δ1 ) − I(δ1 ) − D1 (XT1i , c1 , δ1 ) ,

(10)

T1i ≥0,c1 ≥0,δ1 ≥1

subject to (9), where we assume that debt is fairly priced. Using the standard arguments i

in Dixit and Pindyck (1994), we have E0 [e−rT1 ] = (x/xi1 )β , where xi1 > x > 0. The optimization problem is formulated as E1∗∗ (x) =

³ x ´β n o a i V (x , c , δ ) − I(δ ) , 1 1 1 1 1 i xi1 ≥0,c1 ≥0,δ1 ≥1 x1 max

subject to

D1a (xi1 , c1 , δ1 ) ≤ q, I(δ1 )

(11) (12)

where x < xi1 . Because the triple (xi1 , c1 , δ1 ) is endogenously determined, the quadruple (xi1 , c1 , δ1 , xd1 (c1 , δ1 )) is obtained. Before solving our problem, we first briefly review two special cases.

2.5

Two special cases as benchmarks

In this subsection, we provide two benchmark solutions for an unlevered firm and an unconstrained levered firm. First, we derive the solutions for an unlevered (all-equity financed) firm. This problem is the extended version of the seminal model of McDonald and Siegel (1986), in that the investment quantity is endogenously determined. Let us denote by E0∗ (x) the equity option value of the unlevered firm before investment, where the superscript “∗” represents the optimum for the unconstrained problem and the subscript “0” indicates the unlevered firm. The optimization problem for the unlevered firm is given by E0∗ (x) =

max

³ x ´β n

xi0 ≥0,δ0 ≥1

xi0

o Πδ0 xi0 − I(δ0 ) ,

(13)

where x < xi0 . We then obtain the following. Lemma 1 Assume that the firm does not issue any debt. The solutions and value are δ0∗ = δ ∗ ,

xi∗ 0 =

θ I(δ ∗ ) , Π δ∗

E0∗ (x) =

³ x ´β (θ − 1)I(δ ∗ ), xi∗ 0

(14)

∗ where x < xi∗ 0 and δ is implicitly defined by satisfying

θ :=

β δ ∗ I 0 (δ ∗ ) = . β−1 I(δ ∗ )

8

(15)

∗ Note that xi∗ 0 is obtained through δ . We use these solutions and values as a benchmark.

Second, we assume that there is no financing constraint (12). Let us denote by E1∗ (x) the equity option value for an unconstrained levered firm, where the superscript “∗∗” is changed to “∗.” The solutions and value are as follows. Lemma 2 Assume that there is no debt financing constraints. The solutions are obtained through δ1∗ = δ ∗ ,

i∗ xi∗ 1 = ψ 1 x0 ,

c∗1 =

κ1 ∗ i∗ δ x1 , h1

d ∗ ∗ xd∗ 1 := x1 (c1 , δ ),

(16)

where ³ β ´−1/γ ³ τ (1 − αη) 1 ´−1 h1 := (1 − γ) ≥ 1, ψ1 := 1 + ≤ 1. β−γ 1 − τ (1 − η) h1

(17)

The equity option value is given by E1∗ (x) = ψ1−β E0∗ (x) =

³ x ´β (θ − 1)I(δ ∗ ), xi∗ 1

(18)

where x < xi∗ 1. i∗ ∗ ∗ Note that δ1∗ = δ0∗ = δ ∗ and xi∗ 1 ≤ x0 . In addition, we find that E1 (x) ≥ E0 (x) because ∗ i∗ ψ1 ≤ 1 and β > 1 and that E1∗ (xi∗ 1 ) = E0 (x0 ). We refer to this property as the “leverage

effect.” See Myers (1977) for more detail.

3

Model solutions

This section provides the solutions to our problem, which was described in the previous section. The following proposition provides one component δ1∗∗ of the solutions (the proof is given in Appendix B). Proposition 1 Consider the optimization problem for a constrained levered firm. Then we obtain δ1∗∗ = δ ∗ . In addition, we define the critical point for the financing constraints by ³1 ´ ∗ ∗ D1a (xi∗ γ κ1 ψ1 θ 1 , c1 , δ ) q1 (η, σ) := = − n1 h1 ≥ 0, I(δ ∗ ) r h1 Π

(19)

for a fixed η and σ. Then, if q < q1 (η, σ), the firm is financially constrained. Otherwise, it is not.

9

The first statement of Proposition 1 implies that δ1∗∗ = δ ∗ is independent of η.5 The second statement has two important implications. First, q1 (η, σ) does not depend on δ1∗∗ = ∗ ∗ a i∗ ∗ δ ∗ . In other words, although I(δ ∗ ) 6= I(1) and D1a (xi∗ 1 , c1 , δ ) 6= D1 (x1 , c1 , 1), we have ∗ ∗ ∗ a i∗ ∗ ∗∗ ∗ D1a (xi∗ 1 , c1 , δ )/I(δ ) = D1 (x1 , c1 , 1)/I(1). That is, by controlling δ1 = δ > 1, the firm

cannot change the region where it is constrained. Second, by comparing the magnitudes of q and q1 (η, σ), we determine whether the debt issuance constraint is binding. Using these properties, the following proposition yields two other components of the solutions, ∗∗ xi∗∗ 1 and c1 (the proof is given in Appendix B).

Proposition 2 Consider the optimization problem for a constrained levered firm. Sup∗∗ pose that q < q(η, σ) for any η and σ. The solutions xi∗∗ 1 and c1 are determined uniquely

by solving two simultaneous equations: ∗∗ ∗ ∗∗ ∗ f11 (xi∗∗ f13 (xi∗∗ 1 , c1 , δ ) 1 , c1 , δ ) = , ∗∗ ∗ ∗∗ ∗ f12 (xi∗∗ f14 (xi∗∗ 1 , c1 , δ ) 1 , c1 , δ )

c∗∗ 1

³1 r

− n1

³ κ δ ∗ xi∗∗ ´γ ´ 1 1 = qI(δ ∗ ), c∗∗ 1

(20)

where f11 , f12 , f13 , and f14 are given by f11 := (β −

1)Πδ ∗ xi∗∗ 1

³ κ δ ∗ xi∗∗ ´γ ´ τ³ 1 1 ∗ +β 1− c∗∗ 1 − βI(δ ), ∗∗ r c1

³ κ δ ∗ xi∗∗ ´γ 1 1 γc∗∗ 1 n1 , c∗∗ 1 ³ κ δ ∗ xi∗∗ ´γ τ³ β ´ 1 1 := 1− (1 − γ) , r c∗∗ β−γ 1 ´γ 1 ³ κ1 δ ∗ xi∗∗ 1 := − (1 − γ)n1 . r c∗∗ 1

(21)

f12 :=

(22)

f13

(23)

f14

(24)

∗ We have xd∗∗ := xd1 (c∗∗ 1 1 , δ ). Suppose, on the other hand, that q ≥ q(η, σ) for any η and i∗ ∗∗ ∗ d∗∗ σ. We obtain xi∗∗ = xd∗ 1 = x1 , c1 = c1 , and x1 1 . Substituting the solutions into the

equity value before investment yields ³ x ´β n o a i∗∗ ∗∗ ∗ ∗ E1∗∗ (x) = V (x , c , δ ) − I(δ ) , 1 1 1 xi∗∗ 1

(25)

where x < xi∗∗ 1 .

4

Model implications

In this section, we discuss some properties of the solutions. The three key parameters are equity holders’ bargaining power η, financing friction q, and volatility σ. 5

In addition, we have an invariance of δ1∗∗ = δ ∗ with q, which is the same as that of Wong (2010),

where there is an issuance limit of debt without renegotiation (strategic default) during periods of financial distress.

10

To consider the properties of the solutions, we consider some numerical examples. In order to do so, the investment cost function is assumed to be I(δ) = I0 + δ 2 .

(26)

Suppose that the basic parameters are r = 9%, µ = 2%, I0 = 4, τ = 15%, α = 40%, σ = 15%, and x = 0.4. Under the basic parameters, we have δ ∗ = 4.6002 and x < xi∗∗ 1 for any q. Subsection 4.1 considers the effects of η on the regions where the firm is financially constrained. Subsection 4.2 investigates the effects of σ on the financially constrained regions. Subsection 4.3 examines the effects of η and σ on the financing and investment strategies. Subsection 4.4 demonstrates the effects of η and σ on the leverages, credit spreads, and default probabilities. Subsection 4.5 considers the agency costs of financing frictions.

4.1

Financially constrained regions

In this subsection, we consider the effects of η and q on the financially constrained regions. Note that as equity holders’ bargaining power η is increasing, the equity holders can extract more out of the surplus from renegotiation, and debt holders anticipate higher reductions of the contractual coupon payments during financial distress. [Insert Figure 2 about here] The left panel of Figure 2 depicts the regions where the firm is financially constrained with upper limit of debt issuance in the space (η, q) for a fixed σ = 15%. The line from (η, q) = (0.8235, 1) to (η, q) = (1, 0.8880) indicates the boundary q1 (η, σ) for σ = 15% which is satisfied with (19). The unconstrained debt value, D1a∗ := D1a (x∗1 , c∗1 , δ ∗ ), is monotonically decreasing with η (see the right panel), and δ ∗ is constant with η as shown in Proposition 1. Thus, in the lower-left regions of the boundary q1 (η, σ), the upper limit constraint for debt financing is binding. The lower-left regions of the boundary represent the financially constrained regions. On the other hand, the upper-right regions represent the unconstrained regions where the upper limit constraint is not binding. We see that, for a fixed q and a fixed σ, an increase in η is more likely that the firm is not constrained by the upper limit of debt financing. We summarize the result as follows. Observation 1 Whether the upper limit constraints are binding depends on the three key parameters: q (financing friction), η (bargaining power), and σ (volatility). When η is larger for a fixed q and a fixed σ, the firm is not likely to be financially constrained. 11

Observation 1 implies the following. As η (equity holders’ bargaining power) increases, the amount of debt issuance necessary to maximize total firm value decreases. Thus, an increase in η is less likely that the firm is not financially constrained. The right panel presents the ratio of debt–investment, D1aj /I(δ ∗ ) with η, where D1aj = D1a (xij1 , cj1 , δ ∗ ) (j ∈ {∗, ∗∗}). Under the basic parameters, we have I(δ ∗ ) = 25.1619. We see that D1a∗ /I(δ ∗ ) is decreasing with η. Then, when we assume q = 1 and σ = 15%, the firm is financially constrained for η ∈ [0, 0.8325] while it is not for η ∈ (0.8325, 1]. That is, D1a∗∗ /I(δ ∗ ) is constant for η ∈ [0, 0.8325], while it is monotonically decreasing with η ∈ (0.8325, 1].

4.2

Volatility effects on the constrained regions

In this subsection, we consider how an increase in volatility influences the financially constrained regions. [Insert Figure 3 about here] Figure 3 demonstrates the effects of the volatility on the constrained regions. The thin line represents the boundary q1 (η, σ) for σ = 10%, while the thick line represents the boundary q1 (η, σ) for σ = 15%. The lower-left regions of these two boundaries correspond to the constrained regions, respectively. We see that an increase in σ pushes up the boundary q1 (η, σ) to the upper-right. That is, the lower-left regions of the boundaries are enlarged by increasing σ. We summarize the result as follows. Observation 2 An increase in volatility is likely that the firm is financially constrained by the upper limit of debt financing. Observation 2 implies that the firm with higher volatility is more likely to be financially constrained. Mathematically, q1 (η, σ) is increasing in σ for a fixed η. The reason is that the debt value is increasing with coupon payments, which are increasing with σ.6 Our result corresponds to the empirical results by Cleary (2006).

4.3

Financing and investment strategies

In this subsection, we consider financing and investment strategies with η and σ for a fixed q = 1. We assume σ = 0.15 and η = 0.8325 when examining the effects of η and σ, respectively. Thus, the firm is financially constrained for η < 0.8325 for a given σ = 0.15, while it is done for σ > 0.15 for a given η = 0.8325 (see Figure 2 for details). 6

We demonstrate that the coupon payments are increasing with σ in the next subsection.

12

∗∗ ∗∗ d∗∗ d∗∗ ∗∗ Our solutions are quadruple (xi∗∗ := xd1 (c∗∗ 1 , c1 , δ1 , x1 ) where x1 1 , δ1 ). Recall that

δ1∗∗ = δ ∗ is independent of η and q, as shown in Proposition 1. We here consider the i∗ ∗ ∗ d∗ properties of three other solutions. Recall that xi∗ 0 ≥ x1 , c1 ≥ c0 = 0, x1 ≥ 0 as shown ∗ d∗ in Lemmas 1 and 2, where xi∗ 0 ≥ 0, c0 = 0, and x0 = 0 corresponds to the solutions for ∗ d∗ an unlevered firm, while xi∗ 1 ≥ 0, c1 ≥ 0, x1 ≥ 0 are the solutions for an unconstrained

levered firm. Then, we intuitively conjecture the following inequalities: i∗ i∗ xi∗∗ 1 ∈ [x1 , x0 ],

∗ c∗∗ 1 ∈ [0, c1 ],

xd∗∗ ∈ [0, xd∗ 1 1 ].

These three inequalities imply that, the constrained levered solutions (investment thresholds, coupon payments, and default thresholds) are in between the two special solutions (the unlevered ones and the unconstrained levered ones). However, these intuitive conjectures are not always correct as shown by the following discussion. The top, middle, and bottom panels of Figure 4 depict xi (investment thresholds), c (coupon payments), and xd (default thresholds), respectively. The three left panels examine the effects of η (equity holders’ bargaining power), while the three right panels consider the effects of σ (volatility). In the three left panels, the firm is financially constrained for η ∈ [0, 0.8325] for q = 1. In the three right panels, the firm is financially constrained for σ ∈ [0, 0.15] for η = 0.8325. Importantly, in these six panels, xi∗∗ ∈ 1 i∗ 7 ∗∗ ∗ d∗∗ [xi∗ ∈ [0, xd∗ 1 , x0 ] is not always obtained although c1 ∈ [0, c1 ] and x1 1 ] are obtained. i∗ i∗ ∗∗ Thus, it is less costly to distort xi∗∗ 1 from the regions of [x1 , x0 ] than to distort c1 and

xd∗∗ from the regions of [0, c∗1 ] and [0, xd∗ 1 1 ]. These numerical results suggest the following results. [Insert Figure 4 about here] Observation 3 Suppose that the firm is financially constrained by the upper limit of debt issuance. Then, the investment is not always delayed. However, the normal coupon payments are always reduced, and the coupon reduction during financial distress is always delayed. The first result in Observation 3 is contrary to our intuition. However, the first result is consistent with the the previous papers of Wong (2010), Shibata and Nishihara (2012), and Shibata and Nishihara (2015). The two other results are obtained intuitively. Interestingly, in the top-right, middle-left, and bottom-right panels, we have contrary effects depending on whether or not the firm is financially constrained. Consider, e.g., the middle-right panel. In the middle-left panel, the unconstrained coupon payment c∗1 is 7

For the basic parameters, we have xi∗ 0 = 1.6287, which does not depend on η.

13

decreasing with η as shown in Sundaresan and Wang (2007). In contrast, the constrained coupon payment c∗∗ 1 is increasing with η ∈ [0, 0.8325] (i.e., when financing constraints are effective). The two other panels follow similarly. We define the investment–capital ratio as I(δ ∗ )/δ ∗ . The left panel of Figure 5 displays the investment–capital ratio with η for a fixed q = 1 and σ = 0.15. We see the investmentcapital ratio is constant with η. This is because the investment quantity δ ∗ is constant with η. The right panel demonstrates the investment–capital ratio with σ for a fixed q = 1 and η = 1. We see the investment-capital ratio is increasing with σ. This result is consistent with Panousi and Papanikolaou (2012). [Insert Figure 5 about here] Finally, we show that the investment thresholds have a U-shaped curve with financial friction q in Appendix C. The U-shaped relationship can be justified on the grounds that the firm issues debt with renegotiation during periods of financial distress. However, this relationship is the same as those in previous papers such as Wong (2010), Shibata and Nishihara (2012), Nishihara and Shibata (2013), and Shibata and Nishihara (2015). We conclude that the U-shaped relationship is robust.

4.4

Leverages, credit spreads, and default probabilities

This subsection considers the leverages, credit spreads, and default probabilities. The leverages, credit spreads, and default probabilities are defined as lj :=

D1a (xij1 , cj1 , δ ∗ ) , V1a (xij1 , cj1 , δ ∗ )

csj :=

cj1 − r, D1a (xij1 , cj1 , δ ∗ )

pj =

³ xij ´γ 1 dj x1

,

respectively (j ∈ {∗, ∗∗}). In particular, when there are no financial constraints, we have l∗ =

³1 r

− n1 hγ1

´³ h ´´−1 τ³ β 1 Π + 1− hγ1 , κ1 r β−γ

cs∗ =

³1 r

− n1 hγ1

´−1

− r,

p∗ = hγ1 .

∗ ∗ Note that these three values are independent of xi∗ 1 , δ , and c1 .

The top, middle, and bottom panels of Figure 6 depict lj , csj , and pj , respectively. The three left panels investigate the effects of η, while the three right panels consider the effects of σ. From these six panels, we have the following three inequalities: l∗∗ ∈ [0, l∗ ],

cs∗∗ ∈ [0, cs∗ ],

p∗∗ ∈ [0, p∗ ].

These results are summarized as follows. [Insert Figure 6 about here] 14

Observation 4 When the upper limit constraints of debt issuance are binding, leverages, credit spreads, and default probabilities are decreased. In Observation 4, the first result is straightforward. We consider the second and third results. The last two measures are regarded as measures of return and risk for debt holders, respectively. These links between risk and return are the same as those traditionally suggested in the literature and also match the results of Gomes and Schmid (2010). Interestingly, we have four cases of contrary effects depending on whether or not the firm is financially constrained. First, in the middle-left panel, cs∗∗ is increasing with η ∈ [0, 0.8325] (i.e., when financial constraints are effective), while cs∗ is always decreasing with η ∈ [0, 1]. Second, in the bottom-left panel, recall that p∗ is independent of η as shown on page 257 of Sundaresan and Wang (2007). In contrast, p∗ depends on η ∈ [0, 0.8325]. Third, in the top-right panel, l∗∗ is decreasing with σ ∈ [0.15, 0.175], while l∗ is increasing with η ∈ [0.15, 0.175]. Fourth, in the bottom-right panel, p∗∗ is decreasing with σ ∈ [0.15, 0.175], while p∗ is increasing with σ.

4.5

Agency costs

This subsection examines the effects of financing constraints on the agency costs. We define the agency costs of financing constraints by ac(x) :=

E1∗ (x) − E1∗∗ (x) ≥ 0, E1∗ (x)

(27)

i∗∗ where x < min{xi∗ 1 , x1 }.

The two panels of Figure 7 show the agency costs ac(x) with η and σ, respectively. We see that ac(x) is decreasing with η, while ac(x) is increasing with σ. We summarize these results as follows. [Insert Figure 7 about here] Observation 5 Agency costs of financing constraints are decreasing with equity holders’ bargaining power, while they are increasing with volatility. These results correspond to the fact that the firm is less likely to be financially constrained for larger η and smaller σ. In particular, as for the first statement of η, while renegotiation increases ex post total firm value by preventing costly liquidation, renegotiation also induces a cost on the ex ante equity option value. In contrast, as for the second statement of σ, an increase in volatility is more likely to result in renegotiation, which decreases the equity option value by less than the amount of debt. 15

5

Availability of debt renegotiation

This section considers the availability of debt renegotiation. To do so, we compare our solutions with those of Wong (2010) and Shibata and Nishihara (2012), where the firm is liq∗∗ ∗∗ d∗∗ uidated without renegotiation during financial distress. We denote by (xi∗∗ 2 , c2 , δ2 , x2 )

and E2∗∗ (x) the solutions and value for the firm financed by debt without renegotiation (indicated by the subscript “2”). See Appendix D for the solutions and values. Subsection 5.1 considers under what kind of financial situation equity holders prefer to issue debt with renegotiation when equity holders can issue two type of debts: debt with renegotiation and debt without renegotiation. Subsection 5.2 investigates the distortion of financial constraints.

5.1

Preference for debt with renegotiation

This subsection considers whether debt with or without renegotiation is preferred by equity holders at the time of investment. We cannot determine analytically which is i∗∗ larger, E1∗∗ (x) or E2∗∗ (x) where x < min{xi∗∗ 1 , x2 }. We will examine their magnitudes

using numerical examples. The top panel of Figure 8 depicts the regions of E1∗∗ (x) > E2∗∗ (x) in (η, q) space. The two lines indicate the boundaries of E1∗∗ (x) = E2∗∗ (x) for σ = 10% and σ = 15%, respectively. The lower-left regions of the boundaries correspond to the regions of E1∗∗ (x) > E2∗∗ (x) (i.e., equity holders prefer to issue debt with renegotiation). The reason is that E1∗∗ (x) is decreasing with η while E2∗∗ (x) is constant with η. Importantly, an increase in σ enlarges the regions in which E1∗∗ (x) > E2∗∗ (x). We summarize these results as follows. [Insert Figure 8 about here] Observation 6 Whether debt with or without renegotiation is issued depends on the three key parameters: q (friction), η (bargaining power), and σ (volatility). As the equity holders’ bargaining power increases, the firm is less likely to issue debt with renegotiation. In addition, as volatility increases, the firm becomes more likely to issue debt with renegotiation. In Observation 6, this result seems to be contrary to our intuition. However, this result is reasonable. This is why stronger equity holders’ bargaining power lowers the amount of debt issuance, which leads to a decrease in the equity option value before investment. In addition, the finding that an increase in volatility is more likely to result in a preference for debt with renegotiation is intuitively reasonable. 16

5.2

Distortion of symmetric relationship

In this subsection, we consider how financing frictions distort the symmetric relationship between investment strategies and their values. As a benchmark, suppose that q ≥ qj (η, σ) for any η and σ (j ∈ {1, 2}). Then, we have i∗ i∗ xi∗ j = ψ j x0 ≤ x0 ,

Ej∗ (x) = ψj−β E0∗ (x) ≥ E0∗ (x),

(28)

where ψ1 ≤ 1 and ψ2 ≤ 1 are given by (17) and (D.7), respectively. Because β > 1, (28) suggests the following result. Corollary 1 Suppose that q ≥ qj (η, σ) for any η and σ (j ∈ {1, 2}). Then, we have i∗ xi∗ 1 ≤ x2

if and only if

E1∗ (x) ≥ E2∗ (x).

(29)

Corollary 1 implies that the equity value is larger if the corresponding investment threshold is smaller. The middle-left and middle-right panels of Figure 8 depict xi∗ j and Ej∗ (x) where x < xi∗ j with η (j ∈ {1, 2}), respectively. We see that there is a symmetric relationship between investment thresholds and their values. By contrast, we suppose that q < q(η, σ) for any η and σ. Then, such a symmetric relationship does not necessarily hold. The bottom-left and bottom-right panels of Figure 8 depict xi∗∗ and Ej∗∗ (x) where x < xi∗∗ with respect to η (j ∈ {1, 2}), respectively. We j j assume q = 0.8, which implies that the firm is financially constrained for any η ∈ [0, 1] and σ = 15%.8 In the bottom-left panel, we have xi∗∗ ≤ xi∗∗ for η ≤ 0.9330. In the 1 2 bottom-right panel, we see that E1∗∗ (x) > E2∗∗ (x) for η < 0.9754. From these two results, we obtain xi∗∗ > xi∗∗ and E1∗∗ (x) > E2∗∗ (x) for η ∈ (0.9330, 0.9754). Our finding is as 1 2 follows. Observation 7 Suppose that q < qj (η, σ) for any η and σ (j ∈ {1, 2}). Then, there does not always exist a symmetric relationship between investment thresholds and their equity values. Observation 7 implies that financial frictions distort the symmetric relationship that is always obtained in the frictionless market. 8

The other parameters are the same. Under these parameters, by Figure 2, we confirm that the firm

is constrained for any η.

17

6

Concluding comments

This paper has incorporated upper limit constraints for debt financing and endogenous investment quantity in the financing and investment decision problem in the presence of debt renegotiation during periods of financial distress. We show how three key parameters such as η (equity holders’ bargaining power), q (financial friction), and σ (cash flow volatility) influence the regions where the firm is financially constrained as well as the interactions between the financing and investment decisions. We obtain four new results. First, an increase in η is likely that the firm is not constrained while an increase in σ is likely that the firm is constrained. Thus, for weaker η and higher σ, the firm is likely to be financially constrained. Second, investment quantity is independent of η. However, the investment threshold and coupon payment are dependent on η. In addition, the constrained levered investment threshold is not in between those of the two special cases (unlevered and unconstrained levered), while the constrained investment quantity and coupon payments are in between those of the two special cases. Thus, it is less costly to distort the investment threshold than to distort the two other solutions. Third, an increase in η increases the constrained credit spread while it decreases the unconstrained one. We have contrary results according to whether or not the constraint is effective. Finally, whether debt with renegotiation or debt without renegotiation is preferred depends on the three key parameters. We show that smaller η and larger σ are likely that the firm prefers debt with renegotiation.

Acknowledgments We thank Carol Alexander (the editor) and two anonymous referees for their useful comments and suggestions that have helped to improve the paper substantially. We also thank participants of AMMF Conference (Angers), CFE Conference (London), EURO XXVI Conference (Rome), NUS-UTokyo Workshop (Singapore), and seminar participants at Akita Prefecture University, Bank of Japan, F-bukai (Tokyo), and HKUST for providing helpful comments. This research was supported by a Grant-in-Aid from the Ishii Memorial Securities Research Promotion Foundation, the JSPS KAKENHI (Grant Numbers 26242028, 26285071, and 26350424), the JSPS Postdoctoral Fellowships for Research Abroad, and the Telecommunications Advances Foundation.

18

Appendix A. Derivation of value functions In this appendix, we provide the derivations of the value functions after investment for the firm financed by debt with renegotiation. These values are the same as those in Sundaresan and Wang (2007) and Shibata and Nishihara (2015). For any time t > T1i (after investment), the equity values are defined by E1a (Xt , c1 , δ1 ) (A.1) h Z T1d i d = sup Et e−r(u−t) (1 − τ )(δ1 Xu − c1 )du + e−r(T1 −t) E1b (XT1d , c1 , δ1 ) , T1d ≥t

t

where Et denotes the expectation operator at time t and E1b (Xt , c1 , δ1 ) (A.2) h Z T1d i d = Et e−r(u−t) ((1 − τ )δ1 Xu − s(Xu , δ1 ))du + e−r(T1 −t) E1a (XT1d , c1 , δ1 ) . t

The debt values are defined by hZ a D1 (Xt , c1 , δ1 ) = Et

T1d

−r(u−t)

e

c1 du + e

t

−r(T1d −t)

i ,

D1b (XT1d , c1 , δ1 )

(A.3)

where D1b (Xt , c1 , δ1 )

hZ

T1d

= Et t

i d e−r(u−t) s(Xu , δ1 )du + e−r(T1 −t) D1a (XT1d , c1 , δ1 ) .

(A.4)

The total firm value is defined by V1j (Xt , c1 , δ1 ) = E1j (Xt , c1 , δ1 ) + D1j (Xt , c1 , δ1 ) (j ∈ {a, b}). We begin by calculating the total firm values. Through the standard variation principle, we obtain the following differential equations: rV1a (x, c1 , δ1 ) = (1 − τ )δ1 x + τ c1 + µx

∂V1a (x, c1 , δ1 ) 1 2 2 ∂ 2 V1a (x, c1 , δ1 ) + σ x , ∂x 2 ∂x2

Xt ≥ xd1 , (A.5)

rV1b (x, c1 , δ1 ) = (1 − τ )δ1 x + µx

∂V1b (x, c1 , δ1 ) ∂x

1 ∂ + σ 2 x2 2

2

V1b (x, c1 , δ1 ) , ∂x2

Xt < xd1 , (A.6)

subject to the boundary conditions: V1a (xd1 , c1 , δ1 )

=

V1b (xd1 , c1 , δ1 ),

∂V1a (x, c1 , δ1 ) ¯¯ ∂V1b (x, c1 , δ1 ) ¯¯ ¯ d= ¯ d. ∂x ∂x x=x1 x=x1

Thus, V1a (x, c1 , δ1 ) and V1b (x, c1 , δ1 ) are obtained by τ c1 ³ β ³ Xt ´γ ´ a V1 (Xt , c1 , δ1 ) = Πδ1 Xt + 1− , Xt ≥ xd1 , r β − γ xd1 τ c1 γ ³ Xt ´β b V1 (Xt , c1 , δ1 ) = Πδ1 Xt − , Xt < xd1 . r β − γ xd1 19

(A.7)

(A.8) (A.9)

We then derive the reduced coupon payment in the bankruptcy region. Let us denote by η and 1 − η the bargaining powers of equity and debt holders, respectively. As in the derivations in Fan and Sundaresan (2000) and Sundaresan and Wang (2007), the reduced coupon payment in the bankruptcy region, s(Xt , δ1 ), is given by s(Xt , δ1 ) = (1 − αη)(1 − τ )δ1 Xt ,

Xt < xd1 .

(A.10)

The reduced coupon payment s(x, δ1 ) is a linear function of x with limx↓0 s(x, δ1 ) = 0. The lower the state variable x, the lower the reduced coupon payment s(x, δ1 ). This means that equity and debt holders do not need to consider liquidation once it is in the negotiation region. Equity and debt holders negotiate and divide the surplus to prevent wasteful liquidation. The incremental value for equity holders is wV1b (Xt , c1 , δ1 ) while that for debt holders is (1 − w)V1b (Xt , c1 , δ1 ) − (1 − α)Πδ1 Xt . The division of the surplus between equity and debt holders depends on their relative bargaining powers. Using Nash bargaining, the fraction of total firm value that equity holders receive from renegotiation satisfies ³ ´η ³ ´1−η w(Xt , c1 , δ1 ) = argmax wV1b (Xt , c1 , δ1 ) (1 − w)V1b (Xt , c1 , δ1 ) − (1 − α)Πδ1 Xt w

= η−

η(1 − α)Πδ1 Xt . V1b (Xt , c1 , δ1 )

(A.11)

Thus, the sharing rules are E1b (Xt , c1 , δ1 ) = ηV1b (Xt , c1 , δ1 ) − η(1 − α)Πδ1 Xt ,

(A.12)

D1b (Xt , c1 , δ1 ) = (1 − η)V1b (Xt , c1 , δ1 ) + η(1 − α)Πδ1 Xt ,

(A.13)

where Xt < xd1 . As a result, E1b (Xt , c1 , δ1 ) and D1b (Xt , c1 , δ1 ) are obtained by (3) and (8), respectively. Moreover, E1a (Xt , c1 , δ1 ) and D1a (Xt , c1 , δ1 ) are rewritten as (2) and (7), respectively.

Appendix B. Proof of the proposition In this appendix, we provide the proofs of Propositions 1 and 2.

Proof of Proposition 1 We show that δ1∗∗ = δ ∗ . For problem (11) subject to (12), the Lagrangian is formulated as L =

³ x ´β n xi1

o n o V1a (xi1 , c1 , δ1 ) − I(δ1 ) + λ qI(δ1 ) − D1a (xi1 , c1 , δ1 ) , 20

(B.1)

where λ ≥ 0 denotes the multiplier on the constraint. The Karush–Kuhn–Tucker conditions are given by ∂L/∂xi1 = 0, ∂L/∂c1 = 0, ∂L/∂δ1 = 0, and n o ∗∗ ∗∗ λ qI(δ1∗∗ ) − D1a (xi∗∗ , c , δ ) = 0. 1 1 1

(B.2)

If the debt issuance constraints are binding, λ > 0. By substituting (∂L/∂c1 )c1 = 0 and ∗∗ ∗∗ i i qI(δ1∗∗ ) = D1a (xi∗∗ 1 , c1 , δ1 ) into (∂L/∂x1 )x1 = 0, we obtain

³ x ´β n o a i∗∗ ∗∗ ∗∗ ∗∗ (1 − β)V (x , c , δ ) + βI(δ ) − λqI(δ1∗∗ ) = 0. 1 1 1 1 1 xi∗∗ 1

(B.3)

∗∗ ∗∗ Similarly, substituting (∂L/∂c1 )c1 = 0 and qI(δ1∗∗ ) = D1a (xi∗∗ 1 , c1 , δ1 ) into (∂L/∂δ1 )δ1 = 0

produces ³ x ´β n o n o a i∗∗ ∗∗ ∗∗ ∗∗ 0 ∗∗ ∗∗ 0 ∗∗ ∗∗ V (x , c , δ ) − δ I (δ ) − λq δ I (δ ) − I(δ ) = 0. 1 1 1 1 1 1 1 1 1 xi∗∗ 1 By removing λq from (B.3) and (B.4), we obtain n on o ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 0 ∗∗ V1a (xi∗∗ , c , δ ) − I(δ ) βI(δ ) − (β − 1)δ I (δ ) = 0. 1 1 1 1 1 1 1

(B.4)

(B.5)

∗∗ ∗∗ ∗∗ ∗∗ 0 ∗∗ ∗∗ Because V1a (xi∗∗ 1 , c1 , δ1 ) > I(δ1 ), we have β/(β − 1) = δ1 I (δ1 )/I(δ1 ), implying that

δ1∗∗ = δ ∗ . We complete the proof.

Proof of Proposition 2 We have already found that δ1∗∗ = δ ∗ . Substituting δ ∗ into (B.1), we formulate the Lagrangian. We have ∂L/∂xi1 = 0, ∂L/∂c1 = 0, and (B.2) as the Karush–Kuhn–Tucker conditions. Then, removing λ from the first two conditions produces the first equation of (20). Because λ > 0 in (B.2), the final condition is the second equation of (20). We show the proof of the result that E1∗∗ is monotonically increasing with q. For any q 0 , q 00 with q 00 ≥ q 0 ≥ 0, we assume that the respective optimal values are given by ³ x ´β {V1a (xi01 , c01 , δ ∗ ) − I(δ ∗ )}, i0 x1

³ x ´β 00 ∗ ∗ {V1a (xi00 1 , c1 , δ ) − I(δ )}, i00 x1

(B.6)

β a i00 00 ∗ ∗ (x/xi01 )β {V1a (xi01 , c01 , δ ∗ ) − I(δ ∗ )} > (x/xi00 1 ) (V1 (x1 , c1 , δ ) − I(δ ).

(B.7)

and

Then, because a firm in state q 00 can increase its value by choosing (xi01 , c01 ), which is a contradiction.

21

Appendix C. Properties of the solutions In this appendix, we demonstrate that the investment thresholds have a U-shaped relationship with financial friction q.

We assume η = 1 and σ = 0.15.

The other

parameters are the same as the basic parameters. Figure 9 displays the investment thresholds with financial friction q.

Note that the firm is financially constrained if

q < q1 (η = 1, σ = 0.15) = 0.8880 while it is not otherwise. We see that the financially constrained region is independent of δ because q1 (η = 1, σ = 0.15) is independent of δ. We see that the constrained thresholds have a U-shaped relationship with q as in the previous papers such as Wong (2010), Shibata and Nishihara (2012), and Shibata and Nishihara (2015). However, we demonstrate this relationship on the grounds that the firm issues debt with renegotiation during periods of financial distress. [Insert Figure 9 about here] ∗ ∗ i∗∗ In addition, we see that xi∗∗ 1 (δ ) and δ are larger than x1 (1) and δ = 1. These re-

sults imply underinvestment and accelerated investment for the fixed investment quantity. These results have similar effects to Lyandres and Zhdanov (2014) where convertible debt produces underinvestment and accelerated investment.

Appendix D. Related paper In this appendix, we define the investment optimization problem for firms financed by debt without renegotiation developed by Wong (2010) and Shibata and Nishihara (2012). We then derive the solution and its value. The optimization problem is formulated as E2∗∗ (x) =

³ x ´β n ³ ´o a i a i E (x , c , δ ) − I(δ ) − D (x , c , δ ) , 2 2 2 2 2 2 2 2 2 i xi2 ,c2 ,δ2 x2 max

subject to

D2a (xi2 , c2 , δ2 ) ≤ q, I(δ2 )

(D.1) (D.2)

where x < xi1 , E2a (Xt , c2 , δ2 ) and D2a (Xt , c2 , δ2 ) denote the equity and debt values after investment by a firm financed by debt without renegotiation, respectively. The equity value E2a (Xt , c2 , δ2 ) is given by E2a (Xt , c2 , δ2 )

hZ

T2d

= sup Et T2d

−r(u−t)

e

i (1 − τ )(δ2 Xu − c2 )du

t

n c2 ³ c2 ´³ Xt ´γ o = max δ2 ΠXt − (1 − τ ) − δ2 Πxd2 − (1 − τ ) , (D.3) r r xd2 xd 2 22

for any time t > T2i . Then, the optimal liquidation threshold, xd2 (δ2 , c2 ) is given by xd2 (c2 , δ2 ) = argmax E2a (Xt , c2 , δ2 ) = xd 2

c2 , κ2 δ2

(D.4)

where κ2 := (γ − 1)Πr/(γ(1 − τ )) ≥ 0. Note that limc2 ↓0 xd2 (c2 , δ2 ) = 0, as shown in Black and Cox (1976). The debt value, D2a (Xt , c2 , δ2 ), is defined by D2a (Xt , c2 , δ2 )

hZ

i d e−r(u−t) c2 du + e−r(T2 −t) (1 − α)δ2 Πxd2 (δ2 , c2 ) t ³ κ δ X ´γ ´ ³ κ δ X ´γ c2 ³ 2 2 t 2 2 t = 1− + (1 − α)Πκ−1 c . 2 2 r c2 c2

= Et

T2d

(D.5)

The total firm value is defined by the sum of these two values, i.e., V2a (Xt , c2 , δ2 ) := E2a (Xt , c2 , δ2 ) + D2a (Xt , c1 , δ2 ). Note that there does not exist a “region b” under debt financing without renegotiation because the firm is liquidated once the state variable Xt reaches xd2 (δ2 , c2 ). We derive the solutions of the investment problem for firms financed by debt without renegotiation. First, suppose that the firm does not allowed issuance of any debt, implying that D2a (Xt , c2 , δ2 ) = 0. Then the problem is the same as in (13). The solutions are given in Lemma 1. Second, suppose that there are no financing constraints. The solutions are δ2∗ = δ ∗ ,

i∗ xi∗ 2 = ψ 2 x0 ,

c∗2 =

κ2 ∗ i∗ δ x2 , h2

(D.6)

where ³ ³ 1 − τ ´´−1/γ h2 = 1 − γ 1 + α ≥ 1, τ

³ ψ2 := 1 +

τ 1 ´−1 ≤ 1. 1 − τ h2

The equity option value before investment is ³ x ´β E2∗ (x) = ψ2−β E0∗ (x) = (θ − 1)I(δ ∗ ). xi∗ 2

(D.7)

(D.8)

Third, suppose that there are financing constraints. Then, δ2∗∗ = δ ∗ (the proof is the ∗∗ same as in Proposition 1). The other solutions, xi∗∗ 2 and c2 , are obtained by solving two

simultaneous equations: ∗∗ ∗ ∗∗ ∗ f21 (xi∗∗ f23 (xi∗∗ 2 , c2 , δ ) 2 , c2 , δ ) = , ∗∗ ∗ ∗∗ ∗ f22 (xi∗∗ f24 (xi∗∗ 2 , c2 , δ ) 2 , c2 , δ )

c∗∗ 2

³1

³ κ δ ∗ xi∗∗ ´γ ´ 2 2 − n2 = qI(δ ∗ ), (D.9) ∗∗ r c2

where f21 , f22 , f23 , f24 , and n2 are given as f21 := (β − f22 :=

1)Πδ ∗ xi∗∗ 2

³ τ ³ κ δ ∗ xi∗∗ ´γ τ αΠ ´ ∗∗ 2 2 + β − (β − γ) ( + ) c2 − βI(δ ∗ ),(D.10) ∗∗ r c2 r κ2

³ κ δ ∗ xi∗∗ ´γ 2 2 γc∗∗ 2 n2 , c∗∗ 2

(D.11) 23

´γ ³τ τ ³ κ2 δ ∗ xi∗∗ αΠ ´ 2 − (1 − γ) + , r c∗∗ r κ2 2 ´γ 1 ³ κ2 δ ∗ xi∗∗ 2 := − (1 − γ)n2 , r c∗∗ 2 1 Π := − (1 − α) , r κ2

f23 :=

(D.12)

f24

(D.13)

n2

(D.14)

respectively. The equity value before investment is E2∗∗ (x) =

³ x ´β n o a i∗∗ ∗∗ ∗ ∗ V (x , c , δ ) − I(δ ) . 2 2 2 xi∗∗ 2

(D.15)

References Black, F., Cox, J. C., 1976. Valuing corporate securities: some effects of bond indenture provisions. Journal of Finance, 31(2), 351–367. Broadie, M., Chernov, M., Sundaresan, S., 2007. Optimal debt and equity values in the presence of Chapter 7 and Chapter 11. Journal of Finance, 62(3), 1341–1377. Brown, D. T., 1989. Claimholder incentive conflicts in reorganization: The role of bankruptcy law. Review of Financial Studies, 2, 109–123. Cleary, S., 2006. International corporate investment and the relationships between financial constraint measures. Journal of Banking and Finance, 30, 1559-1580. Dixit, A., Pindyck, R. S., 1994. Investment under uncertainty: Priceton University Press, Princeton, NJ. Fan, H., Sundaresan, S., 2000. Debt valuation, strategic debt service and optimal dividend policy. Review of Financial Studies, 13(4), 1057–1099. Francois, P., Morellec, E., 2004. Capital structure and asset prices: Some effects of bankruptcy procedures. Journal of Business, 77, 387–411. Gilson, S. C., John, K., Lang, L., 1990. Troubled debt restructurings: An empirical study of private reorganization of firms in default. Journal of Financial Economics, 27(2), 315–353. Gomes, J. F., Schmid, L., 2010. Levered returns. Journal of Finance, 65(2), 467–494. Jensen, M. C., Meckling, W. H., 1976. Theory of the firm: Managerial behavior, agency costs and ownership structure. Journal of Financial Economics, 3(4), 305–360. 24

Leland, H. E., 1994. Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance, 49, 1213–1252. Lyandres, E., Zhdanov, A., 2014. Convertible debt and investment timing. Journal of Corporate Finance, 24(1), 21–37. McDonald, R., Siegel, D. R., 1986. The value of waiting to invest. Quarterly Journal of Economics, 101(4), 707–727. Mella-Barral, P., Perraudin, W., 1997. Strategic debt service. Journal of Finance, 52(2), 531–556. Merton, R. C., 1974. On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 29, 449–470. Nishihara, M., Shibata, T., 2013. The effects of external financing costs on investment timing and sizing decisions. Journal of Banking and Finance, 37(4), 1160-1175. Panousi, V., Papanikolaou, D., 2012. Investment, idiosyncratic risk, and ownership. Journal of Finance, 67(3), 1113–1148. Shibata, T., Nishihara, M., 2012. Investment timing under debt issuance constraint. Journal of Banking and Finance, 36(4), 981–991. Shibata, T., Nishihara, M., 2015. Investment timing, debt structure, and financing constraints. European Journal of Operational Research, 241(2), 513–526. Sundaresan, S., Wang, N., 2007. Investment under uncertainty with strategic debt service. American Economic Review Paper and Proceedings, 97(2), 256–261. Wong, K. P., 2010. On the neutrality of debt in investment intensity. Annal of Finance, 6(6), 335–356.

25

η

0

Optimal amount of debt

0.2

1.4299 1.2917

0.4

0.6

0.8

1

1.1618

1.0394

0.9236

0.8136

Table 1: Optimal amount of debt issuance in Sundaresan and Wang (2007)

1.4299 Da/I

1

0.8138 0

η

0.6668

1

Figure 1: Optimal amount of debt issuance without upper limits a

We have D > I for η < 0.6668.

26

1

1.6026 Nonconstrained regions

Constrained regions

0.8880

Da∗∗/I(δ∗)

q1(η,σ=15%)

Da∗/I(δ∗)

0.8 1

0.75 0.75

0.8325

η

0.8883 0

1

η

0.8325

1

Figure 2: Debt renegotiation and financing constraints The left panel demonstrates the constrained regions defined by the lower-left regions of the boundary satisfying with q = q1 (η, σ) for a fixed σ = 0.15 in space (η, q). The upper right regions are defined by the unconstrained regions. The right-panel depicts debt values with η

27

1

0.8880

Constrained regions 0.7937 q1(η,σ=10%) q2(η,σ=15%)

0.5 0.5

0.6605

η

0.8325

1

Figure 3: Volatility effects Two lines represent the boundaries satisfying q = q1 (η, σ) for σ = 0.1 and σ = 0.15 in space (η, q), respectively. The lower-left regions of the boundaries correspond to the constrained regions, respectively. An increase in σ enlarges the constrained regions.

28

0.7050

0.8725 0.8534

0.75

xi∗∗ 1 i∗

xi∗∗ 1 i∗

x1

x1 0.6982

0.6609

0.6479 0

η

0.8325

0.5945 0.125

1

4.2428

0.15 σ

0.175

0.15 σ

0.175

0.15 σ

0.175

4.2525

3.8716

c∗∗ 1

c∗∗ 1

∗

∗

c1

c1

2.6065

2.6045

2.3083 0

η

0.8325

1.8207 0.125

1

0.5849

0.7140

0.5487

0.65

xd∗∗ 1

xd∗∗ 1

d∗

xd∗

x1

1

0.5792

0.3010 0

η

0.8325

0.5028 0.125

1

Figure 4: Financing and investment strategies We assume q = 1 and σ = 0.15 in the three left panels where the firm is constrained for η < 0.8325. We assume q = 1 and η = 0.8325 in the three right panels where the firm is constrained for σ > 0.15. 29

6.4383

I(δ∗)/δ∗ I(δ∗)/δ∗

5.4697

0

η

0.8325

4.9423 0.125

1

0.15 σ

0.175

Figure 5: Investment–capital ratio We assume q = 1 and σ = 0.15 (left panel), and q = 1 and η = 0.8325 (right panel).

30

59.5313

95.2757

59.4419

l∗∗

l∗∗

l∗

l∗

63.0418

52.8075 0

η

0.8325

59 0.125

1

152.1589

0.15 σ

0.175

0.15 σ

0.175

0.15 σ

0.175

183.2527

135.8213 132.7667

147.1807 cs∗∗

cs∗∗ ∗

cs

19.8727 0

135.9331

η

0.8325

cs∗

95.0186 0.125

1

59.5842 54.5575

p∗∗

54.5464

8.1502 0

p∗∗ p∗

p∗

η

0.8325

49.5048 48.9726 0.125

1

Figure 6: Leverages, credit spreads, and default probabilities We assume q = 1 and σ = 0.15 (three left panels) and q = 1 and η = 0.8325 (three right panels).

31

9.7229

0.2228

ac

0 0

ac

η

0.8325

0 0.125

1

0.15 σ

0.175

Figure 7: Agency costs of financial frictions We assume q = 1 and σ = 0.15 (left panel), and q = 1 and η = 0.8325 (right panel).

32

1

0.7350

E∗∗ =E∗∗ under σ=10% 1 2 E∗∗ =E∗∗ under σ=15% 1 2 E∗∗ >E∗∗ 1 2

0 0

η

0.7050

0.6094

0.8786

1

41.8180

41.4069 E∗1

xi∗ 1 i∗ x2

∗

E2

0.6997

0.8

0.8676

η

1

40.6345 0.8

1

η

40.5760

0.6881

xi∗∗ 1

E∗∗ 1

i∗∗

E∗∗ 2

x2

40.4393

0.6827

0.6607 0.9

0.8676

0.9330

η

1

40.3827 0.9

η

0.9754

1

Figure 8: Regions of E1∗∗ (x) > E2∗∗ (x) and distortion of symmetry by frictions The top panel depicts the regions of E1∗∗ (x) > E2∗∗ (x) in space (η, q). An increase in σ enlarges the regions. In the two middle panels (where the constraint is not binding), we i∗ have a symmetric relationship, i.e., E1∗ (x) ≥ E2∗ (x) iff xi∗ 1 ≤ x2 . In the two bottom panels

(where the constraint is binding), we do not always have symmetric relationship. That i∗∗ is, we have E1∗∗ (x) > E2∗∗ (x) iff xi∗∗ 1 > x2 for η ∈ (0.9330, 0.9754). 33

0.7577

0.7050 0.6926 0.6862

xi∗ (δ∗>1) 1 i∗

x1 (δ=1)

0.6445

0.6272 0

0.8880

1

q

Figure 9: Investment thresholds with financial frictions We assume η = 1 and σ = 0.15. Then the firm is constrained if q ≤ q1 (η, σ) = 0.8880. We have a U-shaped investment threshold with q.

34