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I. Introduction

The modeling of default is instrumental in determining the values of corporate securities and the firm’s financing decisions since it affects firm value and its sharing among claimholders. Because of the limited liability principle, default is the shareholders’ decision and can be endogenized by identifying the default policy that maximizes equity value. This point, first made by Black and Cox (1976), is developed in more recent works by Mello and Parsons (1992), Leland (1994), Goldstein, Ju, and Leland (2001), and Morellec (2001). In most of these models, default leads to an immediate liquidation of the firm’s assets. In practice, however, a firm in financial distress can either liquidate its assets under Chapter 7 of the U.S. Bankruptcy Code or try to renegotiate outstanding debt in terms that are more affordable.

The possibility of out‐of‐court renegotiations has been recognized by a stream of models, including those of Anderson and Sundaresan (1996), Mella‐Barral and Perraudin (1997), and Fan and Sundaresan (2000). These papers incorporate, under the assumption of costless renegotiation, debt reliefs granted to firms in financial distress. Since liquidation is costly and bondholders bear liquidation costs, there is room for strategic default, that is, situations in which the firm pays less than the contractual debt service. Strategic default occurs each time the state variable—typically the value of the firm’s assets—falls below some endogenous default threshold. In a rational expectations model, bondholders anticipate the opportunistic behavior of shareholders and reflect the associated wealth extraction in the pricing of corporate debt.

While contingent claims models of out‐of‐court renegotiations provide a useful framework to analyze private workouts, their applicability to court‐supervised debt renegotiation is more limited. Indeed, renegotiations under Chapter 11 of the U.S. Bankruptcy Code differ from private workouts in a number of dimensions that affect the renegotiation process and, hence, the values of corporate securities. To our knowledge the only two contingent claims models that analyze the impact of Chapter 11 on debt values are those by Franks and Torous (1989) and Longstaff (1990). These both model Chapter 11 as the right to extend (once) the maturity date of the debt (a zero‐coupon bond). The longer this extension privilege, the more valuable it is to shareholders and hence the larger the credit spread on corporate debt. Unfortunately, these articles do not analyze optimal leverage nor the decision to go bankrupt.

This article analyzes the impact of Chapter 11 on asset prices, capital structure choices, and the decision to default. For so doing, the framework we develop captures the following features of the U.S. bankruptcy procedure (see White 1996):

Section II presents the valuation framework. In our setting, default can lead either to liquidation of the firm’s assets or to renegotiation of the debt contract. In bankruptcy, the firm incurs costs of financial distress, and the cash flows it generates are shared among claimholders. Moreover, for firms that choose to renegotiate their claims under the court's protection, the default date is the starting point of a period during which the parties involved in the process (the claimholders and the court) observe the evolution of the value of the firm’s assets. The firm emerges from financial distress if the value of its assets shows signs of recovery during the observation period. Otherwise, liquidation is pronounced at the end of the period.

In Section III we analyze the main implications of the model for asset prices and capital structure decisions. We show that the possibility to file for Chapter 11 and to renegotiate debt contracts has an ambiguous effect on firms’ leverage choices. By contrast, the impact of debt renegotiation on shareholders' default incentives and credit spreads is unambiguous. Specifically, debt renegotiation encourages early default and increases credit spreads on corporate debt. Moreover, both the time and costs constraints associated with the renegotiation process affect the magnitude of these effects. The analysis in this article also reveals that the quantitative impact of claimholders' bargaining power on corporate financing decisions is important. By contrast, credit spreads on corporate debt show little sensitivity to this very parameter.

II. Chapter 11 and Asset Prices

This section presents the valuation framework. In order to emphasize the effects of Chapter 11 on asset prices and capital structure decisions, we take as benchmarks the models developed by Leland (1994) and Fan and Sundaresan (2000). Consequently, the model relies on the following assumptions.

Assumption 1. Assets are continuously traded in arbitrage‐free and complete markets.

Assumption 2. The default‐free term structure is flat with an instantaneous riskless rate r, at which investors may lend and borrow freely.

Assumption 3. Management acts in the best interests of shareholders.

Assumption 4. The firm has issued perpetual debt with contractual coupon payment c.1

Assumption 5. Corporate taxes are paid at a rate τ, and full offsets of corporate losses are allowed.2

Assumption 6. Upon liquidation, the firm incurs costs that represent a proportion α of the value of remaining assets. The proceeds from liquidation are distributed according to absolute priority.

Assumption 7. The value of the firm’s assets is independent of capital structure choices and is governed, under the risk‐neutral measure Q, by the process In equation (1), δ is the firm’s payout rate, σ is the constant volatility of asset returns, and is a standard Brownian motion defined on the probability space .

Although the firm may operate its assets forever, it can also choose to default on its debt obligations. Within our framework, default leads either to liquidation of the firm’s assets (under Chapter 7 of the U.S. Bankruptcy Code) or to renegotiation of the debt contract. While firms can renegotiate their debt privately, we focus hereafter on Chapter 11 filings, that is, court‐supervised debt renegotiation. (We show below that our model encompasses those of Leland [1994] and Fan and Sundaresan [2000], which apply, respectively, to Chapter 7 filings and private workouts.)

Chapter 11 differs from Chapter 7 and private workouts in a number of dimensions that affect shareholders’ incentives to default and, hence, the values of corporate securities. Key among these differences is the treatment of claimholders in default. When default leads to a Chapter 7 filing, the firm’s assets are liquidated, and no renegotiation is possible. By contrast, private workouts allow claimholders to renegotiate their claims after default without any time constraint. Chapter 11 filings typically lie in between these two alternatives. Upon default, the court grants the firm a period of observation during which the firm can renegotiate its claims. At the end of this period, the court decides whether the firm continues as a going concern or not.

Empirical studies show that most firms emerge from Chapter 11. Only a few firms (5%, according to Gilson, John, and Lang [1990] and Weiss [1990]), and between 15% and 25%, according to Morse and Shaw [1988]) are eventually liquidated under Chapter 7 after filing Chapter 11. Why do some firms recover while some others do not? It is generally acknowledged (see Wruck 1990 or White 1996) that there exist two types of defaulting firms. First, firms that are economically sound promptly recover under Chapter 11. Default was only due to a temporary financial distress. Second, firms that are economically unsound keep on losing value under Chapter 11. Their financial indicators remain in the red. In that same spirit, we assume hereafter that the firm is liquidated only if the value of its assets reaches a level —indicative of financial distress—and remains below that same level for a time interval larger than the period of observation. This specification for default and liquidation implies that firms emerging from Chapter 11 do not bear real economic costs (see Maksimovic and Phillips [1998] for empirical evidence). It also captures the fact that a large proportion of firms that emerge from Chapter 11 reenter bankruptcy in the following years (see, e.g., Hotchkiss 1995 or Gilson 1997).

In order to determine the value of corporate securities in this setting, we define the following random variables: where is the last time before t that the value of the firm’s assets reaches the default threshold and is the liquidation time, that is, the first time the value of the firm’s assets has spent d units of time consecutively below the default threshold. Figure 1 illustrates our modeling of the liquidation date. In our setup, shareholders hold a Parisian down‐and‐out call option on the firm’s assets. That is, shareholders have a residual claim on the cash flows generated by the firm’s assets unless the value of these assets reaches the default threshold and remains below that threshold for the exclusivity period.3

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Fig. 1.— The liquidation criterion

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We assume in what follows that claimholders anticipate the same length d of the observation period, reflecting the balance of power between the court, the manager, and claimholders. The following examples are of special interest since they pertain to previous contributions of the literature.

Example 1. When , default leads to an immediate liquidation and the liquidation date is described by the first passage time of the state variable to V_B: . Thus when , we get as a special case the standard modeling of default and liquidation (see Leland 1994).

Example 2. When , default never leads to liquidation, and we get as a special case the standard modeling of default and renegotiation (see Anderson and Sundaresan 1996 or Fan and Sundaresan 2000).

In order to examine the implications of debt renegotiation for the values of corporate securities and the firm’s default decision, we have to determine the impact of a successful renegotiation on the cash flows accruing to claimholders. This impact depends on the value of the firm’s assets triggering default, the costs of renegotiation, and the sharing rule for the cash flows from the firm’s assets during strategic default.

As discussed above, we presume that the firm renegotiates its debt obligations whenever the value of its assets falls below a constant threshold . In particular, we follow Fan and Sundaresan (2000, p. 1062) by considering that, when the firm is in default, “borrowers stop making the contractual coupon and start servicing the debt strategically until the firm’s asset value goes back above the threshold again.” In our framework, we first evaluate corporate securities for a given renegotiation boundary and then determine the selected threshold endogenously by maximizing equity value. The sharing rule for the cash flows from assets in place during default typically results from a bargaining game between firm claimants. Again, we follow Fan and Sundaresan (2000) by considering a Nash bargaining game. We denote the bargaining power of shareholders by η and the bargaining power of bondholders by . In contrast to previous models, we also incorporate in our analysis proportional costs borne by the firm during the renegotiation process, which we denote by φ. This assumption captures the fact that, even though the costs implied by private workouts generally are low (0.32% of the value total assets in the study by Gilson, John, and Lang [1990]), Chapter 11 filings are associated with large costs of financial distress (see Andrade and Kaplan 1998) that may affect shareholders’ default decision.

Using the above set of assumptions, we can determine the allocation of the renegotiation surplus between firm claimants as follows. Assumption 6 implies that, if default leads to liquidation, equity is worthless upon default, while the value of bondholders’ claims is . By contrast, if default leads to renegotiation, firm value is equal to and is shared between firm claimants. Denoting the sharing rule for firm value upon default by θ, the incremental value accruing to shareholders is and the incremental value to bondholders is .4 Therefore, the sharing rule for the renegotiation surplus upon default satisfies the solution to which is

The solution to the bargaining game between bondholders and shareholders can be interpreted as follows. The surplus associated with debt renegotiation is equal to . Therefore, equation (5) shows that the outcome of the renegotiation process allocates this surplus between financial stakeholders according to their bargaining power. Specifically, shareholders get and bondholders get Equation (7) shows that the value of bondholders’ claim upon default is equal to the liquidation value of the firm ( ) plus a fraction of the surplus associated with renegotiation. This, in turn, implies that if shareholders have no bargaining power ( ), bondholders’ claim is equal to firm value upon default. An equally important result is that the surplus shareholders extract from renegotiation increases with those very parameters that increase the difference between the ongoing value of the firm and its liquidation value. In particular, we show below that shareholders’ surplus increases with the exclusivity period d, liquidation costs α, and the corporate tax rate τ and decreases with costs of financial distress φ and the volatility of returns on the firm’s assets σ.

Equations (6) and (7) demonstrate that the value of shareholders’ claim upon default depends on firm value. Before valuing corporate securities, it will thus be necessary to determine firm value. Within our model, firm value is equal to the value of the firm’s unlevered assets plus the present value of the tax advantage of debt less the present value of costs of financial distress and liquidation costs. Assumption 1 ensures that there exists a unique probability measure Q under which the price of any asset, discounted at the riskless rate, is a martingale (see Harrison and Pliska 1981). Therefore, we can write firm value as where is the indicator function of the event ω.

This equation can be interpreted as follows. The first term on the right‐hand side accounts for the cash flows generated by the firm until liquidation. This term reflects the fact that (i) costs of financial distress are incurred by the firm whenever it files Chapter 11 (third term in the square brackets) and (ii) tax benefits are akin to a security that pays a constant coupon as long as the firm has not declared Chapter 11 (second term).5 By contrast, liquidation costs are borne only when the firm files Chapter 7 (second term on the right‐hand side).

Solving this equation yields the following result.

Proposition 1. When claimholders take into account the possibility of Chapter 11 filings, firm value satisfies for a given default threshold : with , , and . In equation (8), we have and with the standard normal cumulative distribution function.

Proof of proposition 1. See appendix B.

The next step is to solve for the equity and debt values. For doing so, it is necessary to determine the impact of Chapter 11 filings on the cash flows accruing to claimholders. The above discussion shows that, within the present model, Chapter 11 filings affect the payoffs accruing to bondholders in two ways. First, Chapter 11 filings change the average liquidation time and firm value upon liquidation. In particular, the payoff accruing to bondholders upon liquidation (i.e., to be received at time ) is . Second, bondholders lose part of the interest payments each time the firm files for Chapter 11. Specifically, if we denote by the strategic debt service when the firm is in Chapter 11, coupon payments are defined at any time u, , by whereas dividend payments are defined by

The payoffs to shareholders in and outside bankruptcy allow for a determination of the values of corporate securities. In particular, when claimholders anticipate the possibility of Chapter 11 filings, the values of corporate debt and equity are given by and

In order to solve these equations and determine the values of corporate securities, we proceed as follows. The value of equity can be written as the value of the perpetual stream to the current flow of income ( ) plus the option to default on the firm’s debt obligation. This implies that, using the strong Markov property of Brownian motion, we can write equity value as where a is a constant and the first two terms on the right‐hand side represent the unlimited liability value of equity. Consider now the value of the option to default. The sharing rule for the surplus upon default implies that the value of equity at the default time satisfies where firm value is given by equation (8). Combining equations (9) and (10) gives the value of equity for a given default threshold . The value of corporate debt is then given by

Solving these equations leads to the following proposition.

Proposition 2. When claimholders take into account the possibility of Chapter 11 filings, the values of corporate debt and equity satisfy for a given default threshold and where the renegotiation surplus at the time of default satisfies and the factors , , and are defined in proposition 1.

The closed‐form expressions reported in proposition 2 for the values of corporate debt and equity can be interpreted as follows. The values of corporate debt and equity are equal to a perpetual entitlement to the current flow of income (respectively given by c and ) plus the change in values occurring in default. Because this change in values occurs if and when the firm defaults on its debt obligations, it has to be weighted by the Arrow Debreu price of this event ( ).

It is noteworthy that these expressions extend those obtained by Leland (1994) and Fan and Sundaresan (2000) to incorporate the possibility of a positive but finite maximum period of renegotiation. Indeed, when or , equations (11) and (12) reduce to equations (7) and (13) in Leland (1994). These equations hold for situations in which (i) default leads to an immediate liquidation of the firm’s assets or (ii) shareholders have no bargaining power and hence do not extract any surplus from renegotiation. When and renegotiation is costless ( ), equations (11) and (12) are similar to those provided by Fan and Sundaresan (2000) where default never leads to liquidation.

Within our framework, we can also derive the probability of liquidation. This probability is given in the following proposition.

Proposition 3. The probability of liquidation obeys where stands for the mean total return on the firm’s assets under the physical probability measure and equals the product of the physical probability of default and the physical probability of liquidation, given default.

Proof of proposition 3. See appendix B.

The expression reported in proposition 3 for the liquidation probability again generalizes those of Leland (1994) and Fan and Sundaresan (2000) to account for a strictly positive but finite maximum period of renegotiation. Indeed, when , we have , and this probability reduces to the probability that the process (1) ever reaches in the future. When , the second term on the right‐hand side of equation (13) tends to zero, and so does the probability of liquidation.

III. Applications and Analysis

We now turn to the implications of the model for the decision to default, credit spreads, and optimal leverage. While most of the results are derived analytically, part of the analysis relies on numerical examples. Input parameter values for these numerical simulations are set as follows:6 the initial value of the firm’s assets, ; the riskless interest rate, ; the net tax advantage of debt, ; liquidation costs, ; costs of financial distress, ; the constant payout rate, ; the exclusivity period, ; and the mean total rate of return on V, . For each simulation, the coupon is set so as to maximize equity value at the time of debt issuance. In addition, we consider that the bargaining power is balanced equally between firm claimants, that is, .

A. The Timing of Default

When the decision to default is endogenous, the timing of default depends on the default threshold selected by shareholders. This default threshold can be determined by characterizing the default policy that maximizes equity value. Using equation (12), we get the following result.

Proposition 4. When claimholders take into account the possibility of Chapter 11 filings, the default threshold selected by shareholders is given by

Proof of proposition 4. See appendix B.

The expression provided in proposition 4 for the default threshold extends those obtained by Leland (1994) and Fan and Sundaresan (2000) to incorporate the possibility of Chapter 11 filings. Indeed, when default leads to an immediate liquidation, we have and equation (14) reduces to equation (14) in Leland: This equation reveals that, if default leads to liquidation, it is optimal for shareholders to default when equity is worthless and the default threshold does not (directly) depend on bankruptcy costs.

When default never leads to liquidation and renegotiation is costless, we have and . Equation (14) then reduces to the expression determined by Fan and Sundaresan (2000) for private workouts: where . More generally, proposition 4 reveals that the default threshold should depend on the maximum length of the renegotiation process, d. This result is not surprising since this parameter reflects the extent to which claimants can bargain over firm value.

The closed‐form expression of the default threshold reported in proposition 4 allows us to compare the default threshold associated with Chapter 11 filings with those associated with private workouts or Chapter 7 filings. Specifically, using the inequalities and we immediately get the following result.

Proposition 5. The default threshold selected by shareholders satisfies for any and : In addition, a sufficient condition for the threshold to be lower than the liquidity threshold is

Proposition 5 characterizes the default policy that maximizes equity value for a given—exogenous—debt level c. Specifically, this proposition reveals that the default threshold associated with private workouts ( ) is greater than the default threshold associated with Chapter 11 filings ( ), which in turn is greater than the default threshold associated with Chapter 7 filings ( ). If default leads to liquidation, the decision to default is irreversible and it is optimal for shareholders to default when equity is worthless. When it is possible to renegotiate the debt contract, the decision to default no longer is irreversible and shareholders have incentives to default earlier in order to extract concessions from bondholders. Because the liquidation probability decreases with the length of the observation period, the shareholders' incentives to default early increase with d, inducing a negative relation between the anticipated length of the observation period and the default threshold .7

An additional implication of proposition 5 is that, for most input parameter values, the borrower will default on its debt obligations in the region where the firm is cash constrained ( ). This implies that equity will be issued to finance the contractual coupon during this period when V belongs to the interval . To explore the determinants of the region of asset values where the firm is cash constrained, we can examine the impact of the input parameter values on its relative size defined by This expression reveals that the relative size of the interval decreases with those very parameters that increase shareholders’ incentives to default early. In other words, the relative size of this interval (i) decreases with liquidation costs α, the length of the renegotiation process d, and shareholders’ bargaining power η, and (ii) increases with the corporate tax rate and costs of financial distress φ. Equation (15) also shows that the size of this region of asset values where the firm is cash constrained depends on the volatility of returns on assets in place, the firm’s dividend yield, and the risk‐free rate. In particular, because more volatility implies a greater value for the option of waiting to default, additional volatility increases the relative size of the region where the firm issues equity to finance the contractual coupon.

Additional implications concerning the behavior of the default threshold at optimal leverage are reported in table 1. As shown in this table, the default threshold, and thus equity value at the time of default, increases with the corporate tax rate τ and decreases with shareholders’ bargaining power η, liquidation costs α, costs of financial distress φ, firm risk σ, and the total payout rate to claimholders δ. It is noteworthy that some of these effects go in the opposite direction to that predicted by equation (14) for a given coupon payment. In other words, these effects are essentially driven by the impact of these very parameters on optimal leverage, as we now illustrate.

Table 1 Comparative Statics of the Model

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B. Optimal Leverage

Before turning to the analysis of leverage decisions, it is important to make a clear distinction between the value of equity ex ante—at the time of debt issuance—and ex post—after debt has been issued. The value of equity ex post is given by the present value of the cash flows accruing to shareholders after the debt has been sold (see eq. [12]). The value of equity ex ante is given by the sum of the value of equity ex post and the market value of debt at the time it is issued. As a result, although the default threshold typically is selected ex post to maximize equity value, optimal leverage is determined ex ante to maximize firm value.

Within the present model, investment policy is fixed and capital structure decisions result from a trade‐off between the tax advantage of debt and bankruptcy costs. Optimal leverage is defined by where is the value‐maximizing coupon payment. The following proposition relates this coupon payment to the various parameters of the model.

Proposition 6. When claimholders take into account the possibility of Chapter 11 filings, the value‐maximizing coupon payment satisfies where

Equation (16) reveals that the bargaining power of shareholders η, liquidation costs α, the corporate tax rate τ, and costs of financial distress φ affect the value‐maximizing coupon payment. In particular, leverage should increase with corporate tax rate τ (the benefit of debt) and decrease with liquidation costs α and costs of financial distress φ (the costs of debt). In addition, the larger the bargaining power of shareholders, the greater the cost of debt and hence the lower optimal leverage. These effects are illustrated in table 1 and in figure 2c, which report the firm’s leverage choices in different economic environments.

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Fig. 2.— Shareholders’ bargaining power and corporate debt. Parts a, b, and c plot, respectively, the equilibrium default threshold, credit spreads (in basis points), and optimal leverage as a function of the exclusivity period for various shareholders’ bargaining powers. Parameters are set as in the base case. Shareholders’ bargaining power is (long‐dashed line), (short‐dashed line), and (thin line).

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While it is not possible to determine analytically the impact of debt renegotiation on the debt level selected by shareholders, it is clear from equation (16) that optimal leverage depends on the exclusivity period. In other words, one of the predictions of the present model is that the treatment of claimholders in default has an impact on the capital structure selected ex ante by firms. Figure 2 plots optimal leverage, credit spreads, and the selected default threshold as a function of the exclusivity period for different levels of shareholders’ bargaining power. In this figure, we consider three different scenarios: a take‐it‐or‐leave‐it offer by bondholders ( ), a take‐it‐or‐leave‐it offer by shareholders ( ), and a game where the bargaining power is balanced equally between firm claimants ( ).

In our base case, optimal leverage is roughly 57% when liquidation immediately follows default. Allowing for renegotiation affects optimal leverage, either increasing it or decreasing it, as figure 2c illustrates. Allowing for renegotiation after default reduces the present value of liquidation costs. On the other hand, every observation period is associated with costs of financial distress and tax shield reductions that make debt less attractive. Figure 2c shows that the impact of these two effects on optimal leverage strongly depends on shareholders’ bargaining power. If debtors have most of the bargaining power, the first effect dominates and leverage decreases as the constraints put on the renegotiation process decrease. By contrast, if creditors have most of the bargaining power, the second effect dominates and leverage increases with the length of the exclusivity period. Overall, the analysis suggests that the bargaining power of claimholders in default has a large impact on ex ante leverage choices. It also shows that bankruptcy procedures may help explain the wide variety of debt levels observed in practice.

IV. Conclusion

This article presents a contingent claims model that captures some of the fundamental features of Chapter 11 and investigates their impact on corporate financing decisions. In our setting, Chapter 11 filings allow shareholders to benefit from the automatic stay of assets during a fixed period of time granted by the court. Defaulting on the debt contract implies costs of financial distress to be paid during the observation period. In return, it avoids immediate liquidation and redefines the payoffs that, respectively, accrue to both shareholders and bondholders. In this context, the decision to default depends on the degree of protection offered to shareholders by Chapter 11 pondered by the inevitable costs of financial distress.

The model shows that the possibility to file for Chapter 11 and to renegotiate debt contracts has an unambiguous effect on both shareholders’ default incentives and credit spreads. Specifically, debt renegotiation encourages early default and increases credit spreads on corporate debt. By contrast, the impact of debt renegotiation on firms’ leverage choices is ambiguous. In particular, the model shows that, when debtors (creditors) have most of the bargaining power, we can expect leverage to decrease (increase) with the constraints put on the renegotiation process. The analysis in this article also reveals that the quantitative impact of claimholders' bargaining power on corporate financing decisions is important. By contrast, credit spreads on corporate debt show little sensitivity to this very parameter. Finally, although simulations have been calibrated with respect to U.S. estimates for Chapter 11 modeling, our framework is flexible enough to capture other bankruptcy procedures that involve observation periods with the automatic stay of assets.