Dynamic Risk Shifting, Costly Risk Adjustment and Asset Pricing

Transcription

1 Dynamic Risk Shifting, Costly Risk Adjustment and Asset Pricing Zhiyao (Nicholas) Chen Jan, 2011 Department of Finance, Foster School of Business, University of Washington, Seattle, Washington I am grateful for many constructive comments from Avi Kamara (chair), Jarrad Harford, Erica Xuenan Li, Ed Rice, Stephan Siegel, Lance Young and seminar participants at the University of Connecticut and University of Washington. I thank Ilya Strebulaev for sharing his Matlab codes with me. 1

2 Dynamic Risk Shifting, Costly Risk Adjustment and Asset Pricing Abstract In a dynamic contingent claim model that features endogenous default, debt restructuring, risk shifting and adjustment policies, I examine the risk shifting problem and its asset pricing implications for the cost of equity and debt. Because of limited liability, equity holders can shift downside risk to debt holders by delaying filing for bankruptcy or taking on risky projects in a hope of avoiding bankruptcy in recessions. In a repeated game, equity holders consequently bear all the costs for their opportunistic behavior if they manage to avoid bankruptcy and come back to debt markets to restructure their debt. They need to hedge asset risk back down to the original low level in order to enjoy low costs of debt and high tax benefits in expansions. Using the pricing kernel of Campbell and Viceira (QJE, 1999), I show that the countercyclical volatility endogenously generated from the firms risk shifting actions positively covaries with the market risk premium, therefore driving countercyclical default events and credit spreads. Moreover, my model provides a unified framework to reconcile two well-known empirical asset pricing results. Debt holders of the same firm who rationally anticipate the potential risk shifting problem demand high risk premiums in the form of credit spreads (Campbell and Taksler, JF, 2003). In contrast, because equity holders can delay filing for bankruptcy and further increase idiosyncratic volatility to reduce their downside risk exposure, their expected stock returns decrease with volatility (Ang et al., JF, 2006). I empirically demonstrate that the option of delaying bankruptcy significantly reduces stock returns. Keywords: Dynamic contingent claim model, risk shifting, hedging, limited liability, credit spreads and expected stock returns. JEL Classification: G12, G13, G32, G33 1

3 1 Introduction How do agency conflicts between equity and debt holders affect the cost of capital? I develop a dynamic contingent claim model to address the prominent risk shifting problem and its implications for the cost of equity and debt. This unified framework that endogenizes corporate risk management, external financing and default decisions allows me to explain two inherently related empirical observations. While Campbell and Taksler (2003) document the positive relation between idiosyncratic equity volatility and credit spreads, Ang, Hodrick, Xing, and Zhang (2006, 2009) find a negative association between idiosyncratic volatility and stock returns. 1 This contrast shows the idiosyncratic volatility has different implications for the cost of equity and debt. Idiosyncratic risk is priced in my model because the risk-sharing between equity and debt holders is asymmetric due to limited liability. Since equity holders receive nothing and are not obligated to pay anything to debt holders from their own pockets at bankruptcy, they have an incentive to shift risk to debt holders (Jensen and Meckling, 1976). There are two typical kinds of opportunistic risk shifting behaviors. The first type is static. By simply delaying filing for bankruptcy, equity holders transfer the wealth from debt holders to themselves. The longer equity holders delay, the less value of assets debt holders will receive at bankruptcy if the firm continues deteriorating. A rise in volatility increases opportunities for equity holders of a distressed firm to wait for the asset value to rebound. This is at the expense of debt holders since it induces a high likelihood of default as well. The second type is dynamic. Equity holders of a distressed firm are more likely to take on high risk projects in order to generate sufficient temporary cash flow to avoid bankruptcy (Eisdorfer, 2008). In either the static or dynamic case, debt holders who rationally anticipate equity holders opportunistic behavior demand a risk premium, namely, a high credit spread for 1 Ang, Hodrick, Xing, and Zhang (2006, 2009) document that firms with high equity volatility outperform firms with low volatility by 1.06% per month in both domestic and international stock markets. Jiang, Xu, and Yao (2009) confirm the negative relation and further argue that idiosyncratic volatility is related to the future negative earning shocks. Moreover, Huang (2009) finds similar results that firms with high cash flow volatility earn abnormally lower stock returns than their counterparts with low volatility by 1.35% per month. Other authors find the opposite evidence (Goyal and Santa-Clara (2003), Xu and Malkiel (2003) and Fu (2009)). 2

4 bearing the downside risk. Equity holders with limited liability pay the costs for potential risk shifting and receive low expected stock returns in a repeated game. To formulate my intuition, I extend the model of Leland (1994, 1998) to study the asset pricing implication of the risk shifting problem induced by limited liability. To accommodate the static type of risk shifting, an American option framework is adapted. The seminal model of Leland (1994) extends Merton (1974) to address the impact of limited liability on optimal capital structure. In Merton (1974), equity is like a European call option on the firm s assets, while debt is like a risk-free zero-coupon bond with an embedded written put. When the firm issues debt, it buys the European put from debt holders. If the asset value falls below the principal of debt at the expiration date, equity holders are not obligated to compensate debt holders for the loss. This does not reflect the intentionally opportunistic risk shifting of equity holders. In the American option framework (Leland, 1994), equity holders can endogenously choose the optimal timing of bankruptcy and therefore effectively accommodate equity holders opportunistic actions. To address the dynamic risk shifting problem, I incorporate the friction of costly risk adjustment to the dynamic capital structure model of Leland (1998). In his model, the firm endogenously restructures its capital structure upward to take advantage of tax benefits when its financial condition improves, while increasing its asset risk when its condition deteriorates. There are two drawbacks to his model. First, the amount of risk shifted from equity holders of a distressed firm is exogenously specified. This leads to a natural question of why the desperate firm with limited liability does not increase its risk as much as possible and, what is the upper limit of this risk? Second, the firm in his model adjusts its risk upward or downward costlessly. The costless setup results in only one threshold, at which the firm switches its risk up and down. However, after the firm comes out of distress, equity holders would be reluctant to immediately adjust the risk back down since equity value increases with volatility. In reality, the costs of using derivatives further deter the firm from hedging its risk. To address these two drawbacks of his model, I introduce the costs of risk adjustment and allow the firm to adjust its risk immediately prior to debt 3

5 restructuring. The rationale for risk adjustment is that, in order to reduce its costs of debt and increase its debt capacity, the firm has the incentive to decrease its risk through hedging immediately before it restructures its debt (Graham and Rogers, 2002). 2 The risk adjustment is costly. After it managed to avoid bankruptcy by taking on a risky investment, the firm has to pay a proportional cost to adjust its risk back through hedging before it restructures its debt. The costs of adjustment include commissions, transaction costs, prices paid to purchase derivatives for hedging, and operational costs. 3 Given the initial benefits of risk-shifting and subsequent costs of risk adjustment, the distressed firm determines the optimal magnitude and timing of risk shifting. If it successfully avoids bankruptcy by increasing its risk, the firm then decreases its risk by making a tradeoff between the upfront hedging cost and flow coupon payments when it comes back to the debt market. In comparative statics analysis, I show that the lump-sum hedging costs deter the firm from debt restructuring in both timing and size. Costly risk adjustment is crucial for us to understand how risk shifting affects the equity s systematic risk and its expected returns. The option of taking on a risky investment to shift downside risk to debt holders provides flexibility to equity holders of a distressed firm. However, in a repeated game where the firm needs to restructure its debt, the subsequent costs of risk adjustment reduce this flexibility to increase its risk. In the comparative statics analysis, I show that, while equity holders of a deteriorating firm can reduce their risk exposure by risk shifting, costly risk adjustment deters such opportunistic behavior and increases equity s systematic risk, particularly in economic downturns when such an option of shifting risk is more valuable to equity holders. Given their importance, the costs of risk adjustment are carefully calibrated to match the right-tailed distributions of the incremental debt tax benefits (Graham and Rogers, 2002). I take the calibrated model to study the implications of risk shifting and costly risk ad- 2 Other hedging incentives include the convexity of tax schedules (Smith and Stulz, 1985), informational asymmetry (DeMarzo and Duffie, 1995), managerial risk aversion (Smith and Stulz, 1985), increasing debt capacity (Leland, 1998). 3 The operating costs involve with maintaining a risk management operation, including the salaries, database and computer system, and so on. 4

6 justment for the time variation of credit spreads. The opportunistic risk shifting actions in my model induce the countercyclical asset volatility. Chen, Collin-Dufresne, and Goldstein (2009), Chen (2009) and Bhamra, Lars-Alexander, and Strebulaev (2009) examine the equity premium and credit spread puzzles within a consumption based asset pricing framework. While they model both consumption and production sides, the time-varying volatility in their models is exogenously specified, which does not reflect the endogenous risk management. Chen, Collin-Dufresne, and Goldstein (2009) show that their exogenously specified countercyclical default boundary or idiosyncratic volatility can generate the countercyclical default rates. Because default rates covary with the market premium positively, they are able to generate historical levels of credit spreads. Unlike their model, my model endogenously generates countercyclical evolution of volatility. 4 In my calibration exercise I show that around 80% of post-risk-shifting firms go bankrupt because the increase in idiosyncratic volatility is more likely to generate large cash flow shortfalls than to provide opportunities to avoid bankruptcy in recessions. This result suggests that the countercyclical risk shifting actions and resulting asset volatility are another potential forces that drive the countercyclical default events and credit spreads. My paper also contributes to the emerging literature that uses dynamic models to examinetheimpactsofagencyconflictsonthecrosssectionofstockreturns. 5 Albuquerue and Wang (2008) and Li (2009) examine the impacts of corporate governance on stock returns, while I study the conflicts between equity and debt holders and their impacts on stock return. I provide an economic explanation for the puzzling negative association of idiosyncratic volatility and expected stock return. The first theoretical paper that shows a negative association between asset volatility and equity s systematic risk can be dated back to Galai and Masulis (1976). Johnson (2004) presents similar results in his study on forecast dispersion and stock returns, which relies on the uncertainty of the model parameters. While both use a 4 The recovery rate and default boundary within each refinancing cycle in my model are constant, while the default boundary in Bhamra, Lars-Alexander, and Strebulaev (2009) is procyclical and the default loss rate in Chen (2009) is countercyclical. 5 A nonexclusive list of papers that study the cross section of stock returns in a dynamic model includes Berk, Green, and Naik (1999); Carlson, Fisher, and Giammarino (2004); Gomes, Kogan, and Zhang (2003); and Zhang (2005) 5

7 static European-option model with constant risk and exogenous debt financing, my dynamic model features the endogenous default, debt restructuring, risk shifting and management policies. The risk shifting problem becomes more severe in my fully fledged dynamic model because a distressed firm is more likely to take on risky projects. Consequently, equity holders who bear low systematic risk pay high cost of debt and receive low stock returns in a repeated game. In short, my dynamic model demonstrates that idiosyncratic volatility increases credit spreads (Campbell and Taksler, 2003) and decreases the systematic risk of equity and expected stock returns (Ang, Hodrick, Xing, and Zhang, 2006). Empirically, I explicitly calculate the option value of delaying bankruptcy and demonstrate that the delaying option decreases equity holders downside risk exposure, resulting in low stock returns. 6 The remainder of the paper proceeds as follows. Section 2 gives a simple static model to illustrate the the risk shifting problem and it implications for the cost of equity and debt. Section 3 presents the dynamic model and performs comparative statics analysis to demonstrate how costly risk adjustment affects optimal corporate polices. Section 4 calibrates the dynamic model to study the cyclical variation of asset volatility and corporate events. Section 5 calculates the option of delaying bankruptcy and provides empirical support for the model predictions. Section 6 concludes the paper. 2 A Simple Static Model I start with a simple static model to illustrate that the limited liability feature is crucial to understanding the negative relationship between idiosyncratic volatility and expected stock returns. This model is static in the sense that it does not allow debt restructuring and risk switching. After the firm is established, the only decision to be determined is the optimal timing of filing for bankruptcy, which leads to immediate asset liquidation. I do not consider the renegotiation before bankruptcy. 6 I obtain a closed form solution for the option of delaying bankruptcy in a simple static model so that I am able to calculate the value of the delaying option and its beta implication. However, there is no closed form solution for the options of risk shifting and debt restructuring in a dynamic model. 6

8 2.1 Market Portfolio and Risk Premium Following other dynamic models that study the cross-section of returns(e.g., Berk, Green, and Naik (1999) and Carlson, Fisher, and Giammarino (2004)), I exogenously specify the market portfolio and time varying risk premium based on Campbell, Chacko, Rodriguez, and Viceira (2004), which is a continuous time version of Campbell and Viceira (1999). 7 The process of the market portfolio M t is governed by dm t = (r +λ t )M t dt+σ M M t dw M, (1) where r is the constant after-tax risk-free rate, σ M is the constant volatility of the market portfolio, and W M is standard Brownian motion. The time varying market risk premium, λ t is dλ t = κ( λ λ t )dt+ρσ λ dw M + 1 ρ 2 σ λ dw λ, (2) where κ is the mean reverting speed of the market risk premium, λ is the long run mean, and ρ is the correlation coefficient between the market return dmt M t and the risk premium λ t. The estimate of ρ by Campbell, Chacko, Rodriguez, and Viceira (2004) is = , indicating the market risk premium is countercyclical. 2.2 Asset Value At time 0 the firm installs the assets and produces a perpetual payout flow δ t that is governed by the process dδ t δ t = ˆµ t dt+σdw t, (3) where ˆµ t is the time varying physical growth rate of the payout flow, σ is the constant volatility, and W t is a standard Brownian motion under the physical measure. I assume 7 Although it is of interest to study the consumption side as in Chen (2009), Chen, Collin-Dufresne, and Goldstein (2009) and Bhamra, Lars-Alexander, and Strebulaev (2009), I leave it to future extension since I do not study the equity premium puzzle in this exercise. 7

9 that the time varying drift ˆµ t = µ+λ t β V, where µ is the constant risk neutral counterpart of asset drift ˆµ t and β V represents the systematic risk exposure of the assets. The value of assets-in-place under the risk-neutral Q measure is 8 V t = δ t r µ. (4) Since the denominator r µ is constant, the dynamics of asset value can be expressed as dv t = µv t dt+σv t dw Q t, (5) where W Q t is a standard Brownian motion under the risk neutral measure. Alternatively, the dynamic under the physical measure is represented by dv t = (r +β V λ t δ)v t dt+β V σ M V t dw M +νv t dw i, (6) where δ = δ t /V t is the constant payout rate, ν is the idiosyncratic volatility, and W M and W i are standard orthogonal Brownian motions, implying σ = (β V σ M ) 2 +ν 2. A simple tax structure is assumed, including individual and corporate taxes. Corporate profits are taxed at τ c, dividends at τ d, and individual tax at τ i, with full tax loss offset provisions. Hence, the effective tax rate is τ eff = 1 (1 τ c )(1 τ d ). The intertemporaral cash flow to debt holders is the coupon payment (1 τ i )c. At bankruptcy debt holders take over the assets after transaction costs and receive future cash flows. Hence, the default payment at bankruptcy threshold of V B is the residual value of the firm net of taxes and default costs, (1 τ eff )(1 α)v B, where α denotes the fraction of liquidation costs. The cash flow received by equity holders is the entire payout δ net of coupon payments c to debt holders and tax payments, (1 τ eff )(δ c). Equity holders receive nothing at bankruptcy V B. 8 The risk neutral valuation is independent of the time varying risk premium (Huang and Huang, 2003). 8

10 2.3 Financing and Default Policies Since the simple model is solved by backward induction, I first show how to determine optimal default policies for given amounts of debt and coupon payments. I then present the optimal financing policy that maximizes the value of equity before debt issuance. The optimal bankruptcy threshold V B is important in determining who bears downside risk. When debt is protected under positive net economic-worth covenants, V B is determined by debt holders (Leland, 1994). When debt is unprotected or the enforcement of the bankruptcy by debt holders is not effective, equity holders with limited liability choose the optimal bankruptcy threshold V B to maximize the equity value, E(V). Davydenko (2008) documents that the majority of negative net-worth firms do not default for at least a year, and that equity holders of distressed firms renegotiate with debt holders and violate bond covenants. Therefore, I allow equity holders to choose the optimal bankruptcy threshold V B by imposing the boundary conditions as follows: E(V B ) = 0 (7) E(V) = 0, V V=VB (8) where equation (7) is the value matching condition, which states that equity holders receive nothing at bankruptcy, 9 and equation (8) is the smooth pasting condition, which enables equity holders to choose their optimal bankruptcy threshold at the expense of debt holders. The subscript t is suppressed for notational ease. The optimal amount of debt to be issued at time 0 is determined by the coupon that maximizes the equity value and the cash flow received by debt issuance net of flotation cost, which is a fraction φ of debt value. The optimal coupon is determined ex ante by 9 It is easy to introduce a Nash bargaining game at default as in Fan and Sundaresan (2000) and Garlappi and Yan (2008). However, the qualitative results remain unchanged. 9

11 maximizing the firm value before the initial debt issuance c = argmax c E(V 0 )+(1 φ)d(v 0 ). (9) The valuation functions of equity E(V) and debt D(V) are provided in Appendix A.1. The credit spread (cs) is defined as cs t = c D(V t ) r 1 τ i. (10) 2.4 Expected Stock Returns Proposition 1 Before bankruptcy V t > V B, the expected stock return in excess of the aftertax risk-free rate r is rt e = λ t β t. (11) where V E β t = β V E V (12) = β V (1+ c /r (1 ω 1 ) [c /r V B ] ( V ) ω 1), }{{} E E V }{{ B } (13) Financial Leverage Option of Delaying bankruptcy The optimal bankruptcy threshold V B for the endogenous bankruptcy is given by V B = (c /r)ω 1 (ω 1 1). (14) Equity value E and ω 1 < 0 are defined in Appendix A.1. Proof: See Appendix. Equation (11) expresses the stock return and equation (13) represents its time varying beta. Even though the volatility σ is constant, the equity beta changes over time because of the stochastic value of assets V. Essentially, the equity beta consists of three components: The first is the revenue beta normalized to one, which is from the underlying assets that 10

12 generate the perpetual payout flow; the second is the financial leverage; the last is the option of delaying the filing of bankruptcy. Evidently, the equity beta increases with the financial leverage. Because ω 1 < 0 and (V/V B ) ω 1 > 0, how the option of delaying bankruptcy affects the equity beta depends on the sign of V B c /r. Corollary 1 The bankruptcy threshold V B chosen by equity holders is lower than the equivalent face value of debt, and decreases with the asset idiosyncratic volatility V B c r ; (15) V B ν < 0. (16) Theperpetuityvalueofthecouponpayment, c/r, canberegardedasthefacevalueofthe equivalent risk-free debt. 10 Intuitively, since equity holders receive nothing at bankruptcy, they want to delay filing for bankruptcy when the asset value falls below the face value of debt. Increased idiosyncratic volatility in cash flows improves the probability of the asset value bouncing back. Equity holders always want to take advantage of waiting under uncertainty, since they receive nothing even if they go bankrupt earlier, and pay nothing from their own pocket (due to limited liability) even if the net-worth is negative. Corollary 2 Because of limited liability, the systematic risk of equity, β t, decreases with idiosyncratic volatility ν of assets given constant market volatility, β t ν < 0. (17) As shown in Corollary 1, if the unprotected debt or the enforcement of bankruptcy is not effective, equity holders with limited liability optimally select V B < c/r and V B / ν < 0. Because ω 1 < 0, the last component of equation (13) is always negative, which indicates that the option of delaying filing for bankruptcy decreases the systematic risk of equity. 10 When V, the value of debt is close to c/r (Leland, 1994). 11

13 I use a numerical example to illustrate this corollary. The values of the parameters are from the extant literature in Table 1. Figure 1 plots the full equity betas β Full t (as in equation (13)) against idiosyncratic volatility ν. To emphasize the impacts of the option of delaying bankruptcy on equity betas, I also plot the financial-leverage-only betas β Lev t exclude the delaying option in equation (13). that β Full t It is evident that the financial-leverage-only betas β Lev t are almost double the full betas across all the idiosyncratic volatility level in Panel A, given the standard parameters in the literature. More important, the relation between the financial-leverage-only beta and volatility is U-shaped. The β Lev t decreases with idiosyncratic volatility first in the low volatility area and then increases in the high volatility area. The endogenously determined coupon payment, c is the key to understanding this pattern of the financial-leverage-only beta. For a low risk firm, the increase in idiosyncratic volatility has less positive effect on the cost of debt, namely, the coupon payments, than on the equity value. Hence, the financial leverage and beta decreases in idiosyncratic volatility for a low risk firm. On the contrary, for a high risk firm that has a low debt capacity, a small increase in asset volatility increases the coupon payment much more than it does the equity value, causing the positive relation between asset volatility and financial-leverage-only betas β Lev t. The option of delaying bankruptcy plays an important role in reducing the systematic risk of equity even for a high risk firm. The full beta that includes the option, β Full t, decreases with idiosyncratic volatility monotonically. The negative association between them is consistent with the empirical results documented by Ang, Hodrick, Xing, and Zhang (2006). This option is not free. Panel B shows that debt holders who rationally anticipate the risk shifting problem charge a risk premium, resulting a high credit spread. The credit spread increases with idiosyncratic asset volatility monotonically, consistent with the empirical findings by Campbell and Taksler (2003). It is worth noting that the increasing slope of the credit spread is much faster than the decreasing slope of the beta β Full t. The simple static model illustrates that equity holders with limited liability transfer wealth from debt holders to themselves by simply delaying filing for bankruptcy and conse- 12

14 quently bear all the cost and receive low stock returns. Furthermore, they can increase the asset risk to further shift downside risk to debt holders. To address the dynamic risk shifting problem, I extend the dynamic capital structure model of Leland (1998) to incorporate costly asset risk adjustment. 3 Dynamic Model In the dynamic model, I allow the firm to restructure debt upward when its condition improves and to increase asset risk when its condition deteriorates leading to bankruptcy as in Leland (1998). After avoiding bankruptcy by risk shifting, the firm adjusts its asset risk back to the previous level by costly hedging. The risk adjustment lowers their costs of debt financing and increases debt capacity to take advantage of tax benefits immediately prior to restructuring debt. In this repeated game, the firm that anticipates the subsequent costs of risk adjustment determines the optimal magnitude and timing of risk shifting and adjustment. To illustrate the dynamic model, Figure 2 plots three possible paths the firm could take in one refinancing cycle. At time 0 a firm with a certain technology is born. The initial risk profile of the technology is chosen by nature, that is, σ Σ [σ L,σ H ], where the volatility σ L isthetechnologicallyfeasibleminimumandσ H denotesthehighvolatility. Thefirmwith a high risk technology uses derivatives to hedge the risk level σ H to σ L in order to lower initial financing costs such as credit spreads. After hedging, the initial capital structure is chosen. In observing its dynamic asset value, the firm rebalances its capital structure. Path 1 shows that, when its asset value exceeds the upper threshold V U, the firm calls back its outstanding debt at par and issues a greater amount of debt to take advantage of tax benefits. In contrast, when the firm becomes distressed, the firm takes riskier projects. Its risk increases to σ H at the risk shifting threshold V S, allowing the firm a chance to avoid bankruptcy. As shown in path 2, the increase in asset risk may quickly release the firm from financial distress and eventually leads to a subsequent debt restructuring at the same 13

15 restructure threshold V U, as if it had never increased its asset risk. However, to enjoy the samelowrestructuringthresholdandcostoffinancingasifthefirmcamefrompath1, ithas to hedge its riskiness back to its previous level with a cost in addition to flotation costs. 11 Path 3 shows that the new risky projects may cause a more severe cash flow shortfall. The deteriorating firm finally goes bankrupt at the threshold of V B, which is probably earlier than it would have gone bankruptcy without increasing asset risk. According to the dynamic paths of the underlying assets, I assume that the order of the optimal thresholds within each refinancing cycle is V B < V S < V 0 < V U. (18) The dynamic model has two regimes of asset risk (low risk σ L and high risk σ H ) within each refinancing cycle: σ L, if s = L, σ t = σ H = σ L +ǫ, if s = H, (19) where s indicates the state of the high (H) or low (L) risk regimes and ǫ denotes the amount of risk increments. It is worth noting that the firm stays at the high risk regime in my model longer than in Leland (1998) s model. Since the risk shifting and adjustment are costless in his model, there is only one risk switching threshold, V s, at which the firm can change its risk profile back and forth. Since I introduce the asymmetric risk adjustment, the firm increases its risk costlessly but lowers its risk with a cost. Consequently, it has no incentive to reduce its risk even if its condition improves and reaches the switching threshold V s. The firm waits to adjust its risk until it needs to restructure its debt at the threshold V u to take advantage of tax benefits. Asset volatility consists of two components: systematic and idiosyncratic volatility. In 11 I explicitly assume that the restructuring points are the same for the firm that come along Paths 1 and 2 to reduce the number of the optimal policies to be solved numerically. There are 6 policies in total to be solved for each firm in the model. It is computationally intensive to perform the cross-sectional analysis for a large set of firms, such as 1000 firms, used in the Monte Carlo exercise. 14

16 this partial equilibrium model, I assume that the distressed firms can change their idiosyncratic volatility only to avoid bankruptcy and do not alter their exposure to market risk β V. The market volatility σ M is constant. Hence, the corresponding idiosyncratic volatility is ν L = σl 2 ν t = (β Vσ M ) 2, if s = L, ν H = σh 2 (β Vσ M ) 2, if s = H. (20) When the firm is solvent, the inter-temporal payout flow of the firm is the sum of cash flows to equity and debt holders (δ c)(1 τ eff )+c(1 τ i ) = δ(1 τ eff )+c(τ eff τ i ). (21) Following other theoretical models (Strebulaev, 2007), I allow the negative dividend, (δ c)(1 τ eff ) < 0, but assume that it is costless for a firm to issue new equity to cover its debt service when its revenue is less than the required coupon, δ < c, in the my model. Tax structure is the same as in the static model. Debt is issued at par, which is the initial value at the time of issuance, and is retired at the time of debt restructuring. The firm s initial debt structure remains the same until the firm goes bankrupt or restructures with newly issued debt. At the restructuring point of V U, the asset risk is restored to the previous level and the time is set back to zero. Hence, the dynamic problem is transformed into a fixed-point static problem, which allows me to take advantage of the scaling property (Goldstein, Ju, and Leland, 2001). For instance, the initial asset value of period 1 equals the upper boundary value of period 0. Lemma 2 Scaling Property After resetting the asset volatility back to the previous level at the beginning of each refinancing cycle, the optimal coupon, default, restructuring and risk shifting thresholds, and the values of debt and equity are all homogeneous of degree one in asset value V. It is worth noting the optimal increments of risk ǫ remain the same in each refinancing 15

17 cycle. 3.1 Valuation of Claims In comparison to claims in the simple static model with constant volatility, the claims in the dynamic model are valued in different risk regimes. I start with the dynamic model of Leland (1998) that assumes costless risk adjustment, and then introduce the costly risk adjustment, observed in the recent financial crisis as a difficulty of undoing credit risk Firm Valuation without Costly Risk Adjustment The valuation function of the firm has different coefficients for high (H) and low (L) risk regimesand depends on whether the firm adjusts its risk before restructuring. ˆF(V) denotes for the firm value after initial debt issuance without considering the risk adjustment cost, while F(V) = ˆF(V) Adj(V) stands for its counterpart that includes risk adjustment, where Adj(V) is the adjustment cost function. The valuation function of the firm after the initial debt issuance is FL(V) ˆ = V(1 τ eff )+ c r (τ eff τ i )+f 1L V ω 1L +f 2L V ω 2L, if (s = L) (1 A = 0), ˆF(V) = FLS(V) ˆ = V(1 τ eff )+ c r (τ eff τ i )+f 1LS V ω 1L +f 2LS V ω 2L, if (s = L) (1 A = 1), FH(V) ˆ = V(1 τ eff )+ c r (τ eff τ i )+f 1H V ω 1H +f 2H V ω 2H, if (s = H), (22) where ω 1L,ω 2L,ω 1H and ω 2H are two pairs of roots of the characteristics function (A4) (given in Appendix A.2) in the low and high risk regimes respectively. is the intersection operator. 1 A is the indicator of whether the firm that increased its risk level to σ H needs to adjust its risk back down to the original level σ L at the beginning of each refinancing cycle. The boundary conditions are, respectively, FL(V ˆ U ) = u( FL(V ˆ 0 ) φp), (23) ˆ FH(V B ) = (1 α)(1 τ eff )V B, (24) 16

18 where u = V U /V 0 is the scaling scalar. The firm restructures its debt only in the low risk regime, where the bankruptcy costs are low. The first boundary condition utilizes the scaling property. It states that the firm value ˆ FL(VU ) at the new refinancing threshold equals the product of a scaling scalar and the firm value prior to the initial debt issuance, ˆ FL(V 0 ) φp, in which φp is the cost of initial debt issuance. The second condition states that the firm value at bankruptcy V B is the residual value of assets net of liquidation costs and taxes after equity holders of the deteriorating firm increase risk. As in Leland (1998), the firm costlessly switches its asset risk from low to high at the threshold V S. 12 I assume that the change in capital assets or business line to increase its risk profile is irreversible. However, the firm can employ the derivatives to hedge the risk of its cash flows, if necessary. I do not consider the costly capital adjustment in this setup in order to streamline the presentation of costly risk adjustment. After increasing to high risk, equity holders have no incentive to switch back to low risk since equity value increases with asset volatility due to its call-option feature. Unlike the firm in Leland (1998) that switches back to low risk whenever the asset value reaches V S, the firm in my model returns to its previous low risk level right before it needs to restructure its debt at the time of V U. This modification is particularly important after I introduce costly risk adjustment in the next section, which further deters the firm from reducing its risk profile. Hence, the firm stays at the high risk regime longer in my model than in Leland (1998). Thecoefficientsetf = (f 1L,f 2L,f 1LS,f 2LS,f 1H,f 2H )isdeterminedbytheaboveboundary conditions, combined with the value-matching and smooth-pasting conditions in Appendix A Costly Risk Adjustment The subsequent adjustment of risk is costly if the firm manged to avoid bankruptcy and come back to debt market to restructure its debt. The hedging costs of risk adjustment 12 A firm can costlessly sell the current assets and use the proceeds to purchase new assets with greater risk. Alternatively, the firm can shift its partial productive assets into different higher risk operations without additional costs. 17

19 include commissions, transaction costs, prices paid to purchase derivatives for hedging, and operational costs. The operating costs involved with maintaining a risk management operation include the salaries, database and computer system, and so on. After the risk adjustment, the firm enjoys the same restructuring cost and threshold as if it had never increased its asset risk before. My adjustment cost function is consistent with empirical evidence. The total cost of risk adjustment is assumed to be an increasing function of the amount of risk increment, ǫ, and the asset size, V. Hence, the total adjust cost is θǫv, where θ > 0 is the cost coefficient. The proportional cost function of risk adjustment allows me to take advantage of the scaling property in a same fashion as for the debt restructuring cost. More importantly, my specification is close to the empirical design of Graham and Rogers (2002), which enables me to calibrate the costs of risk adjustment. There is no inter-temporal hedging cost. Given the exogenous adjustment cost θ, the firm determines the optimal magnitude and timing of risk shifting, by balancing the tradeoff between the benefits of lower credit spreads and the costs of risk adjustment. The function Adj(V) of risk adjustment costs, excluding the initial risk adjustment, is defined in both low and high volatility regimes. If the firm is born with a high risk technology or stays on the path of low volatility afterwards, there are no lump-sum adjustment costs at the time of restructuring. Otherwise, it pays the hedging costs to lower its risk immediately prior to restructuring. AdjL(V) = a 1L V ω 1L +a 2L V ω 2L, if (s = L) (1 A = 0), Adj(V) = AdjLS(V) = a 1LS V ω 1L +a 2LS V ω 2L, if (s = L) (1 A = 1), (25) AdjH(V) = a 1H V ω 1H +a 2H V ω 2H, if (s = H). The boundary conditions for the adjustment cost function are similar to the ones for 18

20 firm value, which invoke the scaling property. AdjLS(V U ) = u(adjls(v 0 )+θǫv 0 ), (26) AdjH(V B ) = 0. (27) Switching from the low to high risk regime at V S does not incur a cost for the firm, while there is a proportional cost θǫv U for the firm to move from high risk regime back to low risk regime immediately before the debt restructuring threshold V U. Combined with the value matching and smoothness conditions in Appendix A.2, the boundary conditions determine the coefficient vector a = (a 1L,a 2L,a 1LS,a 2LS,a 1H,a 2H ) Debt Valuation Similar to the function of the firm value, the debt value is expressed as DL(V) = c r (1 τ i)+d 1L V ω 1L +d 2L V ω 2L, if (s = L) (1 A = 0), D(V) = DLS(V) = c r (1 τ i)+d 1LS V ω 1L +d 2LS V ω 2L, if (s = L) (1 A = 1), (28) DH(V) = c r (1 τ i)+d 1H V ω 1H +d 2H V ω 2H, if (s = H). The boundary condition at bankruptcy V B is the same as in the static case, in which debt holders receive the residual value of assets after liquidation costs and taxes. At the restructuring point V U, debt is retired at par P. DH(V B ) = (1 τ eff )(1 α)v B, (29) DL(V U ) = P. (30) The coefficient set d = (d 1L,d 2L,d 1LS,d 2LS,d 1H,d 2H ) is determined by the above condition, combined with the value matching and smooth pasting conditions in Appendix A.2. 19

21 3.1.4 Equity Valuation and Optimal Policies Equity value immediately before initial debt issuance equals firm value minus debt value, initial flotation costs, and initial risk adjustment as follows EL(V) = FL(V) DL(V) φp, if (s = L) (1 A = 0), E(V) = ELS(V) = FLS(V) DLS(V) φp θǫv 0, if (s = L) (1 A = 1), (31) EH(V) = FH(V) DH(V) φp, if (s = H). ThefunctionEL(V)isforthefirmthatdoesnotshiftriskandcomestotherestructuring directly; ELS(V) for the firm that adjusts its risk to the technological minimum σ L ; EH(V) for the firm that increases its risk to σ H. Following Goldstein, Ju, and Leland (2001), the firm precommits its debt policy c and P, switchingpolicyv S, debtrestructuringpolicyv U andtheamountofriskǫtobeincreased at risk shifting or decreased at debt restructuring. Since the policy ǫ the firm implements to maximize the firm value at restructuring is ex ante, the same amount of risk ǫ shifted by equity holders and its associated risk shifting threshold V S both have to be ex ante. 13 Intuitively, when facing the subsequent costs of risk adjustment, the firm becomes precautionary and willing to precommit to the risk-shifting policies. Hence, there is no violation of bond covenants and no costs of rebuilding its reputation when the firm comes back to debt markets. The only costs for equity holders is the hedging costs to mitigate asset risk. The firm chooses the optimal policies (c,p,v U,V S,V B,ǫ) to maximize ex ante the firm value before initial debt issuance arg max c,p,v U,V S,V B,ǫ FL(V 0 ) φp, if 1 A = 0, F(V 0 ) = FLS(V 0 ) φp θǫv 0, if 1 A = 1. (32) 13 The firm can commit the risk shifting policy ex ante or ex post when there is no risk adjustment costs. Leland (1998) shows the agency costs, the difference in firm values due to different ex ante and ex post risk shifting policies, are trivial. I also experiment the ex post risk shifting using the condition of Leland (1998) and the super contact condition of Hennessy and Tserlukevich (2008). 20

22 Since the riskiness of the technology is chosen by nature when the firm is born at V 0, the high risk firm adjusts the risk to the feasible minimum σ L with the cost θǫv 0, before issuing initial debt with flotation costs φp. 14 I consider the endogenous default policy to address the risk shifting problem as in the static model. The bankruptcy threshold V B is determined endogenously by equity holders to maximize the equity value at the bankruptcy threshold EH(V) V = 0. (33) V=VB The model is solved numerically. To avoid a local maximum, I use twelve sets of initial values for solving the six policies (c,p,v U,V S,V B,ǫ) in the nonlinear optimization. 3.2 Impacts of Costly Risk Adjustment on Optimal Policies I use the comparative statics analysis to investigate how costly risk adjustment affects optimal policies. The parameter values for the comparative statics analysis are listed in Table 1 and their justifications are provided in Appendix A.3. Being aware of the subsequent cost of risk adjustment in the repeated game, a distressed firm would be more cautious in the optimal magnitude and timing of risk shifting, both of which are determined simultaneously in my model. Panel A of Figure 3 shows that, given an initial risk level σ L, the optimal amount of risk increment, ǫ, declines with the cost coefficient, θ, of risk adjustment. The decline is the steepest for the most risky firm with σ L = The increment of asset risk is positively associated with its initial risk level σ L when the adjustment cost is low. However, this pattern is reversed when the adjustment cost is high, as shown at the left end of the panel. This reverse suggests that costly risk adjustment deters the firm from increasing risk. Panel B illustrates the effect of costly risk adjustment on the timing of the risk shifting. 14 It is possible that the initial risk σ H of the firm s technology is lower than the high risk level, to which the firm will increase, when it is in distress. But the qualitative results remain same if the reader modify it by taking a lower initial high risk level. 21

23 Given a certain cost of risk adjustment, the risk shifting threshold V S decreases in initial asset volatility σ L. This indicates that the firm is more cautious in risk shifting when aware of the subsequent, costly risk adjustment. Controlling for initial risk, the risk shifting threshold is also slightly decreasing with the adjustment coefficient θ, even for the most risky firm that substantially decreases the size of its risk shifting. Put together, the riskier the firm, the more sensitive it is to the cost of risk adjustment that occurs in the subsequent refinancing, both in the timing and magnitude as shown in Panels A and B. For a distressed firm whose condition continues deteriorating after risk shifting, it can further delay filing for bankruptcy because of limited liability, as discussed in the static case. The observation in Panel C confirms the argument that, given an adjustment cost θ, equity holders of a riskier firm tend to endogenously select lower bankruptcy thresholds. The costs of risk adjustment have a small, indirect effect on bankruptcy thresholds through their impact on the post-shifting risk level σ H = σ L +ǫ. Since the post-shifting risk level σ H declines with adjustment cost as shown in Panel A, the corresponding post-shifting bankruptcy threshold increases slightly with the adjustment cost. In contrast, potential costly risk adjustment has adverse impacts on the firm s future restructuring policy. Panel D illustrates that the upfront cost of risk adjustment causes the firm to delay the debt restructuring threshold V U. This adverse effect becomes more severe with the initial risk profile σ L. After avoiding bankruptcy by risk shifting, the firm would attempt to reset asset risk back to the previous low level in order to enjoy lower credit spreads and high debt capacity, as if it had never increased the asset risk. Given the cost of risk adjustment, the firm chooses the optimal coupon c and amount of debt P to be issued. The firm makes a tradeoff between the lump-sum risk adjustment cost and the flow coupon payments at the time of debt restructuring. Panel E shows the effects of upfront risk adjustment costs θ on the flow costs of debt financing, e.g, credit spreads. Because in this endogenous framework firms that face higher adjustment costs have less likelihood to increase their risk (as shown 22

24 in Panels A and B), they pay less risk premium ex ante via credit spreads. 15 Consequently, as shown in Panel F, the upfront lump-sum costs of risk adjustment prevent the firm from issuing more debt, causing low optimal leverage particularly for the high-risk firm. This is also consistent with our intuition that the upfront adjustment costs deter the firm from lowering its risk to increase its debt capacity and tax benefits. In short, costly risk adjustment deters firms from debt restructuring in both timing and magnitude. This corresponds to the long recovery process observed in the recent financial crisis. The firm pays lower costs of debt financing when they are less likely to shift risk to debt holders. Next, I examine the impact of costly risk adjustment on the systematic risk and costs of equity. 3.3 Systematic Risk of Equity This section is to examine the impacts of dynamic risk shifting and costly risk adjustment on the cost of equity via comparative statics. I first contrast equity betas before and after risk shifting to understand how equity holders lower their risk exposure in my model. I then compare betas from my costly adjustment model with those implied by the costless adjustment model of Leland (1998) to examine the impacts of costly risk adjustment on equity risk. Using equation (12), the equity beta for the dynamic case is derived in the same fashion as for the static case, but has no closed form solution Equity Betas before and after Risk Shifting After increasing its risk to avoid bankruptcy, the firm stays on the high risk course until it reaches the next refinancing threshold. I examine how the equity betas decrease after risk shifting by comparing them within a range of asset value from 50 to 150, during which the firm can be either in the low or high risk regime as shown in Figure 2. Figure 4 depicts the time varying equity betas β t against the underlying asset values V t. I consider a risk adjustment cost coefficient θ = 0.01 and exogenously specify two initial risk levels, such as 15 An alternative explanation is that, by paying the higher upfront, lump-sum cost to adjust asset risk back down, the firm faces lower credit spreads. 23

25 σ L = 0.15,0.45. The increased asset volatility is determined by σ H = σ L + ǫ, where the increment of ǫ is optimally determined. I plot the systemic risk β t of equity before and after the risk switching to illustrate the changes of systematic risk of equity. The solid line is for the systematic risk β t (σ L ) of equity before risk shifting and the dotted line for the risk β t (σ H ) after that. A simple glance over Panels A and C shows that a firm with less assets is more risky than its counterpart with more assets. The comparison between them suggests that the firm with a greater risk σ L = 0.45 (Panel C) has substantially lower market risk than its counterpart with low risk σ L = 0.15 (Panel A), consistent with my conclusion in the simple static case. Equity holders of a deteriorating firm further shift downside risk to debt holders. Panel A shows that the systematic risk β t shifts downward after the asset risk increases from σ L = 0.15 to σ H = To gain more insight, Panel B plots the decrease in systematic risk, β Decr = β t (σ L ) β t (σ H ). The decrease in equity risk β Decr declines significantly from 0.33 to with the underlying asset value, which is driven by the economic and firm conditions. This indicates that, for a low risk firm, equity risk can be substantially reduced by shifting the downside asset risk to debt holders, particularly in economic downturns or when the firm is in distress. This option of shifting risk become less valuable when the economy or the firm are in good states. Panels C and D show the same pattern for a firm with a higher initial risk σ L = It increases its underlying asset risk by ǫ = to reduce its equity risk. However, the magnitude of the decrease in equity risk in Panel D is much smaller than for a low risk firm in Panel B. This result is consistent with our intuition that, given a subsequent adjustment cost, the high risk firm has less room to increase its asset risk and therefore reduces its equity risk, relative to the low risk firm. Taken together, equity holders of a deteriorating firm have the option of shifting downside risk to debt holders. Given a certain adjustment cost, such an option provides more benefits to a lower risk firm in economic downturns. I answer the question of how costly 24