Financial Distress and Corporate Risk Management: Theory & Evidence

Transcription

1 Financial Distress and Corporate Risk Management: Theory & Evidence Amiyatosh Purnanandam (Job Market Paper) First Draft : August, 2003 This Draft : January, Sage Hall, Johnson Graduate School of Management, Cornell University, Ithaca, NY akp22@cornell.edu. I would like to thank Warren Bailey, Sudheer Chava, Thomas Chemmanur, John Graham, Robert Goldstein, Yaniv Grinstein, Jerry Haas, Pankaj Jain, Robert Jarrow, Kose John, Haitao Li, Roni Michaely, Maureen O Hara, Mitch Petersen, Bhaskaran Swaminathan, David Weinbaum and seminar participants at Cornell University and Lehman Brothers Finance Fellowship Competition for valuable comments and suggestions. All remaining errors are mine. 1

2 Abstract This paper develops a theory of corporate risk-management in the presence of deadweight losses caused by financial distress and tests its implications using a comprehensive dataset of over 3000 non-financial firms. Unlike extant theories that explain only the ex-ante riskmanagement behavior of a firm, I show that the shareholders optimally engage in ex-post risk-management activities even without a pre-commitment to do so. I generate new crosssectional predictions by relating firm characteristics such as leverage and deadweight losses from financial distress to its risk-management incentives. The model predicts a positive relationship between leverage and hedging for moderately leveraged firms. This relationship reverses, however, for highly leveraged firms. Similarly the model produces a non-monotonic relationship between leverage and hedging for high market-to-book value firms. The empirical findings are consistent with these predictions. The empirical study presents the first large-sample evidence on the extent of hedging by non-financial firms and provides many new findings. I find that large and small firms hedge for different reasons. While both groups hedge in response to the financial distress costs and exhibit economies of scale in hedging, large firms also hedge in response to underinvestment costs and tax-convexity, as predicted by the existing theories. 2

3 1 Introduction This paper develops and tests a theory of corporate risk-management in the presence of deadweight losses caused by financial distress. The existing literature shows that hedging can lead to firm value maximization by limiting deadweight losses of bankruptcy (see Smith and Stulz (1985)). 1 These models justify only ex-ante 2 risk-management behavior on the part of the firm. Ex-post, shareholders of a levered firm may not find it optimal to engage in hedging activities due to their risk-shifting incentives (Jensen and Meckling (1976)). I extend the current literature by explaining the ex-post risk-management motivation of the firm. The model generates new cross-sectional predictions by relating firm characteristics such as leverage, deadweight losses and project maturity to risk-management incentives. I test these predictions with hedging data of COMPUSTAT-CRSP firms, meeting some reasonable sample selection criteria, for fiscal years The empirical study presents the first large sample evidence on the determinants of firms hedging activities 3 and provides new findings. The key assumption underlying my theory is the distinction between financial distress and insolvency. I assume that apart from the solvent and the insolvent states, a firm faces an intermediate state called financial distress. Financial Distress isdefined as a low cash-flow state of the firm in which it incurs deadweight losses without being insolvent. The notion that financial distress is a different state from insolvency has some precedence in the literature. Titman (1984) uses a similar assumption to study the effect of capital structure on a firm s liquidation decisions. The idea of a financially distressed firm in my model is consistent with the notion of an illiquid but solvent firm in Diamond (1991) or a firm with low cash-flow in Froot, Scharfstein and Stein (1993). 1 Other motivations for corporate hedging include convexity of taxes, managerial risk-aversion (Stulz (1984), Smith and Stulz (1985)), underinvestment costs (Froot, Scharfstein and Stein (1993)) and information asymmetry between the managers and outsiders of the firm (DeMarzo and Duffie (1991,1995)). 2 Throughout the paper, I use the terms ex ante and ex post with respect to the time of borrowing. 3 Due to data limitations, the earlier studies have used small samples to investigate the determinants of firms hedging activities (see Nance, Smith and Smithson (1993), Tufano (1996), Geczy, Minton and Schrand (1997), Haushalter (2000) and Graham and Rogers (2002)). Mian (1996) provides large-sample evidence on the yes-no decision of hedging. In contrast, I use data on the extent of hedging as well and provide many new findings. 3

4 There are three important sources of deadweight losses from financial distress. First, a financially distressed firm may lose customers, valuable suppliers and key employees. 4 Opler and Titman (1994) provide empirical evidence that financially distressed firms lose significant market share to their healthy counterparts in industry downturns. Secondly, a financially distressed firm is more likely to violate its debt covenants 5 or miss coupon/principal payments without being insolvent. 6 These violations impose deadweight losses in the form of financial penalties, accelerated debt-repayment, operational inflexibility and managerial time and resources spent on negotiations with the lenders. For example, when Delta airlines violated a debt-to-equity ratio covenant in 2002, it was required by its lenders to maintain a minimum of $1 billion in cash and cash equivalents at the end of every month from October 2002 until June Finally, a financially distressed firm may have to forego positive NPV projects due to costly external financing, as in Froot et al. (1993). I develop a dynamic model of a firm that issues equity capital and zero coupon bonds to invest in a risky asset. The firm makes an initial investment with the consent of its bondholders. At a later date, shareholders can modify the firm s investment risk by replacing the existing asset with a new one or by entering into derivatives transactions. The asset value evolves according to a stochastic process. The firm is in financial distress if the asset value falls below some lower threshold during its life. In this state, the firm is unable to realize its full upside potential due to lost real options, loss of customers and suppliers or losses imposed by its bondholders. Insolvency occurs on the maturity date if terminal asset value is below the face value of debt and, consequently, debtholders gain control of the firm. Shareholders 4 For example, in the mid-1990s Apple Computers had financial difficulties leading to speculations about its long-term survival (see Business Week, January, 29 and February 5, 1996). Software developers were reluctant to develop new application software for the Mac-users, which was partly responsible for a decline of 27% in the unit sales of Mac computers from 1996 to 1997 (see Apple s K filings with the SEC). Similarly, when Chrysler faced financial difficulties in the early 1980s, Lee Iacocca (former CEO of the company) observed that "its share of new car sales dropped nearly two percentage points because potential buyers feared the company would go bankrupt" (quoted from Titman (1984)). 5 Lenders often impose debt-covenants such as maintenance of minimum net-worth or maximum debt-to-equity ratio by the borrowing firms. See Smith and Warner (1979), Kalay (1982) and Dichev and Skinner (2001). 6 Moody s Investor Service Report (1998) shows that during about 50% of the longterm publicly traded bond defaults (including missed or delayed payment of coupon and principal) didn t result in bankruptcy filings. 7 See Delta s K filings with the SEC. 4

5 final payoffs depend on the terminal asset value as well as on the path taken by the firm s asset 8 over its life. The optimal level of ex-post investment risk, from the shareholders perspective, is determined by a trade-off between deadweight losses from financial distress and value associated with the limited liability of the firm s equity. 9 Unlike the risk-shifting models such as Jensen and Meckling (1976), equity-value is not always an increasing function of firm risk in my model. While a high risk project increases the value of equity s limited liability, it also imposes a cost on the shareholders by increasing the expected cost of financial distress. Due to these losses, the shareholders find it optimal to implement a risk-management strategy expost even in the absence of an explicit pre-commitment to do so. 10 If losses in the products market (such as lost customers and suppliers) or losses due to suboptimal investment strategy are large, shareholders engage in risk-management strategy even without debt-covenants. Otherwise, in a rational expectation framework, the bondholders can set these covenants exante such that shareholders implement a desired risk-management strategy ex-post. The optimal investment risk in my model depends on firm leverage, the financial distress boundary, the time horizon of the project and deadweight losses incurred in financial distress. Asintheextantmodels(SmithandStulz(1985)),Ishow thatafirm with high leverage (financial distress) has a higher incentive to engage in hedging activities. However, by explicitly modelling the shareholders risk-shifting incentives, my model shows that risk-management incentives disappear for firms with extremely high leverage. Similarly, the relationship between leverage and hedging is predicted to be non-monotonic for a firm with high deadweight losses from financial distress (such as a high market-to-book value firm). The model shows 8 This approach is similar to valuation of equity as a path-dependent (down-and-out call) option. The equity value in my model differs from the corresponding barrier option by the amount of deadweight losses incurred in financial distress. Brockman and Turtle (2003) provide some empirical evidence in support of equity s valuation as a path-dependent option. 9 In the context of swap markets, Mozumdar (2000) demonstrates the trade-off between riskshifting and hedging incentives in the presence of information asymmetry about the firm type. His model relates hedging incentives to firm type. 10 Other papers analyzing the ex-post risk-management decisions of the shareholders include Leland (1998) and Morellec and Smith (2003). Leland (1998) provides a justification for the firm s ex-post hedging behavior in the presence of tax-benefits of debt. In Morellec and Smith (2003), the manager-shareholder conflict reduces the ex-post asset-substitution incentives of the shareholders. My model, on the other hand, is based on the cost of financial distress and provides new empirical predictions. 5

6 that hedging incentives increase with project maturity and financial distress barrier. Riskmanagement motivation in my model arises from deadweight losses incurred by the firm in states where it hits the financial distress barrier but remains solvent on the maturity date. If there are no deadweight losses, risk-management incentives disappear. On the other hand, when deadweight losses are very high, the distinction between financial distress and insolvency diminishes along with any ex-post risk-management motivations. Intermediate levels of deadweight losses create risk-management incentives within the firm. Therefore, my model predicts a U-shaped relationship between deadweight losses and hedging. When bondholders have full control over the determination of the financial distress barrier, they endogenously set the barrier such that it increases with the amount of leverage. The distress barrier set by the bondholders is non-monotonic in deadweight losses. With endogenously determined distress barriers, the U-shaped relationship between deadweight losses and risk-management disappears. However, the relationship between leverage and risk-management remains non-monotonic since the optimal ex-post response of the shareholders depends on the actual realization of the asset-value at a later date. The predictions of my model have important implications for the empirical research. To test the existing theories, empirical studies regress some measure of financial distress (such as leverage or interest coverage ratio) on firms risk-management activities. If firms with extreme distress are less likely to hedge, these models may be mis-specified. The bias can be particularly severe in small sample studies. It is not surprising that the existing empirical studies find mixed evidence in support of the distress-cost based theories of hedging. 11 I contribute to the empirical risk-management literature by analyzing interest rate and foreign currency risk-management activities of a comprehensive sample of non-financial firms. The earlier studies have either used small samples 12 or focused only on the binary (i.e., yesno) hedging decisions. 13 Since the risk-management theories provide predictions about the extent of hedging, a test based on yes-no decision of hedging is not suitable. I test the 11 For example, while Haushalter (2000) and Graham and Rogers (2002) find a positive relationship between the two variables, Nance et al. (1993), Mian (1996) and Tufano (1996) fail to find such evidence. 12 Such as 372 firms with 154 hedgers used by Geczy et al. (1997) and about 400 firms with 158 hedgers used by Graham and Rogers (2002). 13 Such as the study by Mian (1996) that uses a sample of about 3000 firms with 771 hedgers. 6

7 predictions of my model as well as the predictions of other current theories with data on the extent of hedging of a large cross-section of COMPUSTAT-CRSP firms for the fiscal year My study is free from sample selection bias and provides the first large-sample evidence on why firms hedge. There are 3,239 non-financial firms in the sample, of which 751 firms use interest rate or foreign currency derivatives for risk-management purposes. Consistent with my theory, I find strong evidence that firms with higher leverage hedge more. The hedging incentives disappear for firms with very high leverage. In line with my theory, I find a similar non-monotonic relationship between leverage and hedging for high market-to-book value firms. The empirical study provides several new findings. I find that large (above median size) and small (below median) firms hedge for different reasons. While firm size and leverage remain important determinants of the hedging behavior of both groups, large firms also provide evidence in support of underinvestment cost theories of hedging (Froot et al. (1993)). Consistent with these theories, firms with higher research and developmental expenditure and lower levels of liquid assets hedge more. In the large firm sample, I also find some evidence in support of the tax based incentives (Smith and Stulz (1985)) of hedging. The rest of the paper is organized as follows. In Section 2, I provide the model description. Section 3 analyzes the optimal risk-management policy of the firm when the financial distress barrier is exogenous. This model corresponds to the case in which losses in the products market or losses due to suboptimal investment strategy are sufficient to generate risk-management incentives within the firm even in the absence of debt-covenants. Section 4 presents a model in which rational bondholders set the financial distress barrier endogenously such that the shareholders implement a desired risk-management policy ex-post. The empirical tests are provided in Section 5. Section 6 discusses the main findings and concludes the paper. 2 Model I consider a continuous trading economy with a time horizon [t 0,T ]. On this time horizon, there is a filtered probability space (Ω, (z t ), z,p) satisfying the usual conditions. I assume 7

8 an arbitrage-free market. This guarantees the existence of an equivalent martingale measure Q. For the sake of simplicity, 14 I set the risk-free interest rate to zero. The value of any self-financing trading strategy can be computed by taking the expectation of future cashflows under the equivalent martingale measure. In what follows, I denote the indicator function of an event X by 1 {X}. There are three important dates in the model. At t = t 0,thefirm makes its capital structure decision and invests in a risky asset A i (i stands for the initial investment). These decisions are taken with the consent of the debt-holders of the firm. The risky asset (A i ) is acquired at the market-determined rate and financed through a mix of zero-coupon debt and equity capital. The capital structure decision is exogenous in the model. Let L be the face value of the zero-coupon debt, payable at time T, and E t be the time t value of the firm s equity. The asset value A i t evolves according to a stochastic process adapted to the filtration z t. At some later time t = t 1 (t 1 (t 0,T)), the shareholders (or managers acting on their behalf) make a risk-management decision. At this time they have an opportunity to change the asset s risk without the bondholder s approval. To capture the risk-shifting incentives (Jensen and Meckling (1976)), I assume that the bondholders are unable to re-contract with the shareholders at t = t 1. The shareholders can change the asset s investment risk in many ways including, but not limited to, transactions in derivative instruments. After the risk-management decisions have been made, I denote the risky asset by A, which evolves according to the following geometric Brownian motion adapted to the filtration z t : da t = µa t dt + σa t dw t I assume that the change in the investment risk of the asset (from A i to A) has no cash-flow impact on the firm at t = t 1. This provides an initial boundary condition in the model, namely A t1 = A i t 1. The final payoffs are realized at t = T. The shareholders receive the liquidating dividends and the bondholders receive the face value of debt (L) ifthefirm remains solvent on the maturity date t = T. 15 Otherwise they receive the residual value of 14 This is without any loss of generality due to the Numeraire Invariance Theorem (see Duffie(1996)). 15 Other maturity structures are possible. To illustrate the main results of the paper in its simplest form, I prefer to work with zero coupon debts. 8

9 the firm. The model can be represented with the following time-line: t = t 0 t = t 1 t = T Capital Structure Risk-Management Initial Investment Decisions Payoffs This modelling framework allows me to address the issue of ex-ante vs. ex-post riskmanagement behavior of the firm in the presence of the risk-shifting incentives of the shareholders. I now discuss the main assumption of the paper, namely the distinction between the financial distress and the insolvency. 2.1 Financial Distress and Insolvency If any time during (t 0,T) the asset value falls below a boundary K, 16 the firm is in the state of financial distress. Insolvency, on the other hand, occurs on the terminal date T if the terminal asset value is less than the debt obligations. Therefore, in the state of financial distress, control of the firm does not shift to the bondholders immediately. A firm in financial distress incurs deadweight losses. Opler and Titman (1994) show that financially distressed (highly leveraged) firms lose significant market share to their healthy competitors during industry downturns. In a sample of 31 high leveraged transactions (HLTs), Andrade and Kaplan (1997) isolate the effectofeconomicdistressfromfinancial distress and estimate the cost of financial distress as 10-20% of firm value. Therearethreeimportantsourcesofdeadweightlossesduetofinancial distress - each one of them consistent with the interpretation of deadweight losses in my model. First, a financially distressed firm may lose valuable customers, suppliers and key employees (see Titman (1984), Shapiro and Titman (1986)). The evidence presented by Opler and Titman (1994) belongs to this class. The drop in sales faced by Apple Computers and Chrysler during periods of financial difficulties provide further anecdotal evidence in support of such 16 I refer to K as distress barrier in the rest of this paper. 9

10 deadweight losses. Prior to K-Mart s Chapter 11 filings, suppliers were reluctant to extend trade credit to the firm, fearing that they would not be able to recover their dues. In Appendix 1, I provide a sample of such anecdotal evidences from the popular press and 10-K filings of the firms. The lenders of a firm often impose restrictive covenants such as maintenance of minimum networth or maximum debt-to-equity ratio by the borrowing firm. 17 A firm in financial distress is more likely to breach these covenants. Such firms are also more likely to miss coupon and principal payments to their lenders. Moody s Investor Service Report (1998) documents that only half of the long-term publicly traded bond defaults (including missed coupon or principal payments) ultimately resulted in bankruptcies over the years Thus, a defaulted firm (i.e., a financially distressed firm in my model) doesn t necessarily become insolvent. Certain features of the bankruptcy codes also support the distinction between financial distress and insolvency. For example, a firm may file for Chapter 11 protection even when it is solvent. The idea that default and insolvency are different states has been implicitly or explicitly used in Robicheck and Myers (1966), Anderson and Sundaresan (1996), Mella-Barral and Perraudin (1997), Mella-Barral (1999) and Jarrow and Purnanandam (2003) among others. When a firm breaches its debt-covenants or misses coupon/principal payments, it is likely to face deadweight losses in the form of financial penalties (such as increased interest rates or higher collateral), accelerated repayment of the debt, higher monitoring by firm outsiders and managerial time and resources spent on negotiations with the lenders. For example, in 2002 Delta Airlines breached one of its debt-to-equity covenants. Consequently it was required by its lenders to maintain cash and cash equivalents of $ 1 billion at the end of every month from October 2002 until June 2003 (see 10-K filings of Delta). Finally, a financially distressed firm may have to forego positive NPV projects due to costly external financing (Myers and Majluf (1984) and Froot et al. (1993)). In the presence of asymmetric information between the insiders and the outsiders of the firm or deadweight costs incurred by the firm in raising external funds (such as commissions paid to investment bankers), external funds become more costly than the internally generated funds. This 17 See Smith and Warner (1979) and Dichev and Skinner (2001). 10

11 imposes a cost on the financially distressed firms by forcing them to adopt a suboptimal investment strategy. The empirical literature provides evidence in support of this form of deadweight losses (see Whited (1992) and Lamont (1997)). A financially distressed but solvent firm in my model can also be identified as an illiquid but solvent firm as in Diamond (1991). 2.2 Exogenous vs. Endogenous Distress Boundary I solve the model recursively. In the first step, I solve for the optimal investment risk (from the shareholders perspective) of the firm at t = t 1, assuming an exogenous distress boundary. This model provides a complete description of the ex-post risk-management behavior of the firm when losses in the products market or losses on account of suboptimal investment strategy are sufficiently large. In such cases, rational bondholders do not need to impose costs in terms of debt-covenants. However, when these losses are not sufficient, the rational bondholders anticipate the shareholders action at t = t 1 and endogenously set the distress boundary K at t = t 0. The distress boundary is set by the bondholders such that, in a rational expectation sense, it implements a desired risk-management strategy ex-post. I solve this model in Section 4. 3 Model with Exogenous Distress Boundary I solve for the optimal investment risk of the firm at t = t 1. The firm is in financial distress at t 1 if the asset value (A i t) hits the distress boundary K at some point in [t 0,t 1 ]. First, I consider a firm that is not in the state of financial distress at t 1. The other case is considered later in the section. 3.1 Definition of Financial Distress Ontheterminaldate(T ), the firm is insolvent if the asset value at time T is less than thefacevalueofdebt(l). Ifthefirm s asset value never breaches the distress boundary K during t [t 1,T], the terminal asset value is A T. However, if the distress boundary is hit, the firm incurs deadweight losses and the terminal asset value falls to f(a T ),where 11

12 f(a T ) <A T. The function f represents the deadweight losses caused by financial distress. A wide range of functional forms can be introduced depending on factors such as the nature of the business, industry structure and market conditions. 3.2 Valuation of Equity The shareholders receive liquidating dividends at T. Due to equity s limited liability, the final payoff to the shareholders (ξ T ) is zero if the terminal asset value is below L. Let us inf define: A t 1 t T t m T for the minimum value of the asset during [t 1,T]. In the event of no distress (i.e. m T >K) and solvency on the terminal date (i.e. A T >L), the shareholders get a liquidating dividend of (A T L). If financial distress is experienced (i.e. m T K), but on the terminal date the firm remains solvent (i.e. f(a T ) >L), the shareholders receive liquidating dividends of f(a T ) L. In the event of insolvency, they receive nothing. The shareholders payoff under different states is given by the following: State at t = T Corresponding Asset Values Payoff to Shareholders Healthy A T >L,m T >K A T L Financial Distress f(a T ) >L,m T K f(a T ) L Insolvency A T L, m T >K 0 Insolvency f(a T ) L, m T K 0 Proposition 1 The equity valuation at t=t 1 is given by the following: ξ t1 = E Q [(A T L) (A T f(a T ))1 {f(at )>L,m T K} +(L A T ){1 {AT L} +1 {f 1 (L)>A T >L,m T K} }] (1) Proof. See Appendix 2.A. The equity value, as shown in Proposition 1, has three components. The first term (E Q [A T L]) represents the net asset value of the firm. This is the equity value without the distress costs and the limited liability feature. The second term (E Q [(A T f(a T ))1 {f(at )>L,m T K}]) represents the deadweight losses caused by financial distress. The shareholders of a financially distressed but solvent firm bear this cost and therefore the equity value reduces by this amount. The risk avoidance incentive results from this cost. The third term 12

13 Healthy Asset Value A t1 K Financially Distressed f -1 (L) L Insolvent t 1 τ Time (t) T Figure 1: This figure plots three paths for the evolution of the asset value (A t )ofthefirm. These paths correspond to three states of the firm in my model. In the top-most path, the asset value never hits the financial distress barrier (K). This corresponds to the Healthy state. The middle path represents the state where the distress barrier is hit (at time τ), butthe firm remians solvent at time T. This is the state of Financial Distress. In this state the terminal asset value, net of deadweight losses (i.e., f(a T )), remains above the face value of debt (i.e, L). Thus this is the state where f(a T ) >Lor alternatively A T >f 1 (L), as depicted in the figure. Finally, the bottom-most path corresponds to the state of Insolvency. (E Q [(L A T ){1 {AT L} +1 {f 1 (L)>A T >L,m T K}}]) represents the savings enjoyed by the shareholders of a levered firm due to the limited liability feature of equity. This term captures the risk-shifting incentives of the shareholders. By increasing the asset risk, the shareholders can make themselves better off by increasing the call option value (the third term). At the same time, however, the expected loss in the event of financial distress also increases with an increase in the asset risk. The optimal level of investment risk is determined by the trade-off between the two. In Smith and Stulz (1985), deadweight losses are incurred after the insolvency. By engaging in low-risk projects, a firm can lower the expected deadweight cost of bankruptcy, which benefits the bondholders. The shareholders can get better terms on their borrowings by committing to low-risk projects. However, ex-post, the risk-avoidance incentives disappear in their model. My paper extends their model by explicitly modeling a mechanism that provides an ex-post risk-management motivation. The trade-off between the risk-avoidance and risk-seeking incentives provides many interesting cross-sectional predictions in the model. 13

14 3.2.1 Deadweight Losses Proposition 1 provides a general valuation formula in my model. To proceed further I need to be explicit about the form of deadweight losses that is borne by the shareholders of a financially distressed firm. I assume that in the event of distress (i.e. m T K), thefirm s operations are adversely affected such that it is unable realize its full upside potential. In particular, I assume that in the event of distress, the terminal asset value (A T ) becomes bounded above by an arbitrary constant U<. Therefore, the deadweight losses come in the form of lost upside potential. 18 U can be made arbitrarily large, which implies that even a small loss in the asset-value is sufficient to derive the main results of the paper. This representation of deadweight losses is consistent with the view that a financially distressed firm is unable to capitalize on its real options, retain all its customers or make optimal investments. The operational inflexibility faced by such firmsandthemanagerial time and resources spent on negotiations with the lenders provide further justification for a loss in the upside potential of a financially distressed firm. This assumption allows me to derive an analytical expression for the optimal investment risk of the firm. The main results of the paper can be derived for other reasonable forms of the deadweight loss function as well. For notational simplicity, let us express U = L + M for some M>0. The liquidating dividends to the shareholders, for this specification of the deadweight loss function, is given by the following: States Payoff to Shareholders A T >L,m T >K A T L A T >L,A T L + M,m T K A T L A T >L+ M,m T K M A T L 0 The deadweight losses, therefore, can be expressed as (A T M).1 {AT >L+M,m T K}. A higher value of M corresponds to lower deadweight losses in the model. In line with proposition 1, the equity value can be expressed as follows (see Appendix 2.B): 18 In the real world, deadweight losses may be incurred by the firm at any time after the distress barrier is hit. For analytical simplicity, I assume that the net effect of all these losses is captured by assuming that the terminal asset value (A T ) is reduced. The model can be analyzed for other deadweight loss functions as well. 14

15 Equity Value L Equity Value in my model Equity Value In Healthy State 0 L L+M Asset Value at T Equity Value in Financial Distress Figure 2: This figure plots the equity value as a function of the terminal asset value of the firm. The equity value in my model is depicted by the solid line. The upper dotted line represents the equity value for the Healthy state. The lower dotted line depicts the equity value in the state of Financial Distress. The equity value in my model is a weighted average (weight is decided by the relative likelihood of the two states) of the equity value in these two states. ξ t1 = E Q [(A T L)1 {AT >L,m T >K} +(A T L)1 {AT >L,A T L+M,m T K} +M1 {AT >L+M,m T K}] (2) Figure 2 plots the equity value as a function of the terminal asset value of the firm. As shown in the diagram, the equity value is not a strictly convex function of the underlying firm value as in the classical approach where equity is valued as a call option on firm value. The deadweight loss of distress introduces a concavity in the equity value, which results in risk-management incentives within the firm. 3.3 Optimal Choice of Investment Risk At t = t 1, the shareholders make a decision about the optimal investment risk of the firm. There are two possibilities for changing the investment risk: (a) the firm can directly choose an optimal level of σ at t = t 1 or(b)theassetrisk(σ) may be fixed and the firm can alter its risk profile by buying derivative contracts such as futures and options. I analyze the problem of finding optimal σ assuming that investment risks can be costlessly modified. If asset risk is fixed or costly to change, derivative instruments can be used to alter the asset risk such that the risk of the combined portfolio (asset and hedging instruments) attains the desired 15

16 optima. In such cases, keeping all else equal, higher investment risk would correspond to lower risk-management incentives. Proposition 2 The shareholders have a well-founded motivation to engage in risk-management activities ex-post. At t = t 1, the shareholders would optimally choose a level of risk σ in the interior of all possible risks. Proof. At t = t 1, the shareholders choose an optimal risk level such that it maximizes the equity value given in Expression 2. Since the firm is not in financial distress at t = t 1, IhaveK < A t1. I also assume that the distress barrier is below the face value of debt, i.e. K<L. 19 With these conditions, the shareholders optimization problem reduces to the following (see Appendix 2.B for a detailed proof): where max σ E t 1 = {A t1 Φ(h 1 ) LΦ(h 2 )} {KΦ(c 1 ) A t 1 (L + M) Φ(c 2 )} (3) K h 1 = ln( A t1 σ2 )+ T 0 L 2 σ ; h 2 = h 1 σ T 0 and T 0 = T t 1 T 0 c 1 = ln( K2 σ2 A t1 )+ T 0 (L+M) 2 σ and c 2 = c 1 σ T 0, T 0 where Φ stands for the cumulative density function of the standard normal distribution. The optimum level of investment risk is obtained by the following first-order condition (see Appendix 2.C for the proof): A t1 φ(h 1 )=Kφ(c 1 ) (4) Where φ stands for the probability density function of the standard normal distribution. Further simplification leads to the following closed-form solution: (σ 2 ) = 1 T 0 ln( K2 )ln( K 2 L ) L(L+M) A 2 t (L+M) 1 ln( L+M L ) (5) The second-order condition is satisfied at this optima as shown in Appendix 2.D. QED. Proposition 2 shows that the choice of asset risk is not irrelevant to the equity valuation. The shareholders of a levered firm finds it optimal to manage the asset s risk even after the 19 The other case (i.e. when K > L) produces similar results (analysis is available from the author). 16

17 debt has been raised by the firm. Asaresultofthetrade-off between the risk-shifting and risk-avoidance incentives, an interior solution for the optimal risk is obtained in the model. This result differs from that of the earlier models. In risk-shifting models such as Jensen and Meckling (1976), the shareholders take as much risk as possible, whereas in risk-management models such as Smith and Stulz (1985), the optimal risk is obtained at σ =0. By obtaining an interior solution for the optimal investment risk of the firm, my model provides insights into the risk-management policies of the firm, as discussed below. Proposition 3 The firm chooses a lower level of investment risk if (a) it faces a higher distress barrier (K), (b) it has a lower starting asset value (A t1 ) and (c) it has a longer project maturity (T =T t 1 ). The relationship between the deadweight losses and the optimal q investment risk is U-shaped. Let M c = L exp 2( ln( A t 1 K )ln(l K )) L. When M>M c, the optimal investment risk decreases with an increase in the deadweight losses, otherwise it increases with an increase in the deadweight losses. Proof. The proof follows from direct differentiation of the optimal solution for σ given in Expression 5 (see Appendix 2.E). QED. The investment risk decreases (i.e. the risk-management incentive increases) with the distress boundary (K). As expected, a higher boundary increases the likelihood of financial distress. Therefore, the shareholders optimally choose a lower investment risk to avoid the deadweight losses. For a similar reason, when the firm s initial asset value (A t1 ) is closer to the distress boundary, the firm chooses lower investment risk. My results show that the firm with a longer horizon of operations (T 0 = T t 1 ) finds it optimal to engage in higher risk-management activities. There is a considerable empirical evidence that large firms hedge more than small firms. The pursuit of economies of scale has been suggested as one possible explanation for this empirical fact. Another explanation, consistent with my model, is the time horizon of operations. If firms with longer time horizons grow bigger across time, the researcher would find a positive association between risk-management activities and firm size at any given point in time. Finally, I find a U-shaped relationship between the risk management incentives and the deadweight losses incurred in financial distress. Recall that the deadweight losses in my 17

18 model are parametrized by Ṁ (losses are given by (A T M).1 {AT >L+M,m T K}). In the event of financial distress, the firm loses its upside potential beyond L + M. Thus, the higher the M, the lower the lost upside potential and therefore the lower the deadweight losses. If the deadweight losses are absent (i.e., if M = ), the shareholders lose nothing in the state of financial distress. Therefore there is no risk-management incentive. On the other hand, when deadweight losses are very high (i.e., when M =0) the distinction between default and insolvency disappears 20 along with the risk-management incentives. It s the intermediate cases that generate risk-management incentives in the model. Figure 3 illustrates this relationship. Figure 3: This figure plots the optimal investment risk as a function of the deadweight losses. The model has been calibrated with the following parameter values: A t1 =2,L=1,T 0 =1and K =0.5. On the x-axis, I plot the value of M. M measures the upside potential lost by the firm in the event of financial distress. I plot M from higher-to-lower value so that the deadweight lossesincreaseasonemovesalongthex-axis. 3.4 Leverage and Risk Management To study the relationship between leverage and risk management, I apply the implicit function theorem on Equation 4 (i.e., the first-order condition (FOC) of the optimality). Since 20 In this case, the equity value becomes similar to a down-and-out barrier option. Since the value of this option is increasing in the volatility of the underlying assets, the share-holders do not have any risk-management incentives at t 1. 18

19 the second-order conditions are satisfied at the optima, we get the following relationship: sign( σ L )=sign( (FOC) L ) After some simplifications, the above condition leads to the following: sign( σ L )=(h 1 L c 1 L + M ) K σ T (1 2c 1 0 σ T ) K 0 L c 1 K L + M σ M (6) T 0 L As shown in Expression (6), there are three ways in which leverage can affect the investment risk of the firm. The first term captures the direct effect of leverage, which is always positive. All else equal, higher leverage leads to higher investment risk due to the limited liability feature of equity. The second term captures the impact of leverage on the investment risk via the distress boundary. Finally, the third term corresponds to the effect of leverage on the investment risk via its effect on the deadweight losses. It is reasonable to expect that the distress barrier and deadweight losses are increasing functions of leverage (i.e., K/ L > 0 and M/ L < 0). 21 Thelasttwotermsproduceanegativerelationship between the level of debt and the investment risk of the firm. For simplicity, I first assume that the deadweight losses are fixed at M. The risk-management incentives increase with firm leverage if the following inequality holds : K L > ( h1 L c 1 L+M ) K σ T 0 (1 2c 1 σ T 0 ) (7) Condition (7) states that the rate of increase in K (with respect to L) must satisfy a lower bound 22 in order to produce a positive relationship between leverage and the riskmanagement activities. The lower bound is a function of the amount of debt, the investment risk, the maturity date and the deadweight losses. In order to further demonstrate the relationship between leverage and investment risks, I need to be explicit about the functional form of K(L). There are two important restrictions on K(L) :(a)itmustbeanincreasingfunctionofl, i.e. K 0 (L) > 0; and (b) the firm should 21 This assumption is consistent with the empirical findings of Opler and Titman (1994). 22 The lower bound is always positive. This follows from three inequalities that always hold: (i)h 1 >c 1, (ii)l <L+ M and (iii)2c 1 <σ T. 19

20 not be in financial distress at t = t 1, i.e. K(L) <A t1. I assume the following: 23 K(L) =u A t1 (1 e vl ) (8) In the above expression, parameters u [0, 1] and v (a positive constant) control the slope and curvature of the distress boundary. By changing u and v, the sensitivity of the distress boundary with respect to leverage can be changed. These parameters can be calibrated to generate a reasonable distress boundary for various firm, industry and market characteristics. As required by the model, the distress boundary given in Equation (8) is always an increasing function of the leverage (i.e. K 0 (L) =uv A t1 e vl > 0) and it ensures that the firm is not in distress at t 1 (i.e., K<A t1 ). I provide numerical results for the relationship between leverage and investment risk in the remainder of this section. The model is calibrated for the following parameter values: M =10,u =0.5,v =1,A t1 =2and T 0 =1. Icompute the optimal investment risk (σ) ofthefirm for various levels of leverage (L) andplotit against the debt-asset ratio in Figure 4. Figure 4: This figure plots the optimal investment risk of the firm against the debt-asset ratio. calibrated for the following parameter values: u =0.5,v =1,M =10,T 0 =1and A t1 =2. The model has been 23 My results are not specific to this functional form of K(L). Various other forms of distress boundaries are possible in this model. For example, K(L) can be made a linear function of L subject to an upper cap such that the firm is not in distress at t 1. This structure produces qualitatively similar results to those of the case that I discuss in the paper. 20

21 For a moderate level of leverage, the deadweight loss component of the equity dominates its limited liability feature. This produces a positive relationship between risk-management incentives and leverage. When leverage is very high, however, Condition (7) fails to hold (i.e., risk-shifting incentives start to dominate) and consequently the relationship between leverage and risk-management becomes negative. Therefore, the model produces a nonmonotonic relationship between the two. 24 The level of leverage at which the relationship between the leverage and risk management incentives becomes negative is termed as the leverage inflection point in the rest of the paper. 25 If deadweight losses increase with the leverage, there is another source of non-monotonicity in the relationship between leverage and the risk-management activities. In such cases, with L large enough, M may fall below the critical value M c (i.e., the deadweight losses become large) as defined in Proposition 3. As shown in Proposition 3, for this level of deadweight losses (i.e., when M<M c ), the risk-management incentives disappear Leverage and Risk Management for Firms with high deadweight losses In this section I consider the relationship between leverage and the optimal investment risk of a firm that has high deadweight losses in financial distress. For such firms, the riskshifting incentive of the shareholders can dominate the risk-avoidance motivation even at a lower level of leverage. The intuition is simple. With high deadweight losses, the distinction between default and insolvency narrows. This implies that the deadweight losses to be borne by the shareholders, in the states when the firm is in financial distress but not insolvent, is 24 Fehle and Tsyplakov (2003) study a firm s decision to initiate or terminate hedging contracts. They find a non-monotonic relationship between leverage and hedging in the presence of transaction costs of hedging. Unlike their paper, this paper provides an ex-post justification of corporate riskmanagement. Their paper focuses on how to implement a hedging decision once the firm has decided to hedge. The other predictions of my model are new. 25 In the model, I assume that the distress boundary is below the initial asset value (i.e. the firm is not already in financial distress at time t 1 ). Mechanically, one can generate a functional form for K such that K/ L is very high and Condition (7) is always satisfied. But for such functional form of K, the distress boundary would quickly approach the initial asset value (A t1 ). This corresponds to a firm that is already in financial distress at time t 1. As I show later in Proposition 5, the riskmanagement incentives disappear for these firms. Therefore, the relationship between leverage and risk-management becomes non-monotonic for these specifications of distress boundaries as well. 21

22 lower for such firms. This in turn results in risk-shifting incentives at a relatively lower level of debt. In my model, with increasing deadweight losses the lower bound on K/ L (as in Condition 7) required to produce a negative relationship between leverage and hedging increases. Thus when deadweight losses are high, the relationship between the leverage and risk-management incentives may become negative even for smaller leverage ratios (i.e. the leverage inflection point is smaller). Figure 5: This figure plots the leverage inflection point against the deadweight losses. I plot M from higher-to-lower value on the x-axis so that the deadweight losses increase as one moves along the axis. The y-axis plots the level of debt beyond which the relationship between the leverage and risk management incentive becomes negative. The model has been calibrated with the following parameter values: A t1 =2and T 0 =1. Figure 5 depicts this relationship for the distress boundary used in the earlier example (i.e. when K(L) =u A t1 (1 e vl )). I plot deadweight losses against the level of debt beyond which the relationship between the risk-management incentives and leverage becomes negative. For a reasonable level of debt (i.e., the level below the leverage inflection point ), my model predicts a positive relationship between leverage and the risk-management activities of a firm with high deadweight losses (such as a firm with high level of intangibles assets or high market-to-book value). This is in line with the predictions of Froot et al. (1993). However, when leverage is very high (i.e., above the leverage inflection point ) this relationship is reversed in my model. 22

23 The effect of leverage on the risk-management policies of the firm is formally summarized below: Proposition 4 Risk-management incentives increase with leverage; this relationship reverses for extremely high levels of debt. For firms with high deadweight losses of financial distress, risk-management incentives increase with leverage; this relationship reverses for very high leverage. 3.5 Optimal Risk For a Financially Distressed Firm I now consider the case of a firm that experiences financial distress during (t 0,t 1 ).Forsuch a firm, the deadweight losses have been incurred during t (t 0,t 1 ). Therefore, at t 1, the equity valuation collapses to a standard call option. It is trivial to show the following: Proposition 5 If the firm is already in financial distress at t = t 1, the risk-management incentives disappear. Proof. There are no more deadweight losses to be borne by the shareholders of the firm. The time t 1 value of equity is equivalent to a call option with the face value of debt as thestrikeprice.theriskmanagementincentivesdisappearsincethevalueofthisoptionis increasing in the asset variance. QED. 3.6 Risk Management Using Derivatives In the analysis so far, I analyze the optimal investment risk of the firm assuming that it can costlessly change its investment risk (σ). The analysis can be easily extended to the situations where the asset volatility is fixed or costly to change. Instead of changing the asset s volatility (σ), the firm can now buy derivative instruments to change the risk of its overall payoff. Assuming no frictions in the derivatives market, one can obtain similar results for the risk-management policies of the firm. Keeping all else constant, higher investment risk would correspond to lower risk management using derivatives. In practice, firms can 23

24 use other means such as borrowing in foreign currencies and setting-up operations in foreign countries to alter their risk (see Petersen and Thiagarajan (2000) for a case study). All these alternatives would be consistent with my model. However, in the empirical tests presented in Section 5, I focus on the risk-management activities using financial derivatives since the data on other ways of managing risks is either not readily available or identifiable. This approach to testing my theory is consistent with a large body of empirical research in the corporate risk-management literature. 4 Endogenous Distress Boundary The model discussed so far assumes an exogenous distress boundary. When losses in the products market or losses on account of suboptimal investment decisions of the firm are sufficiently large, the debt-covenants may not be required. However, when these losses are not large enough, the bondholders can set the distress boundary endogenously at t = t 0 in response to the rationally anticipated actions of the shareholders at t = t 1. In this section, I analyze the rational bondholders ex-ante (t = t 0 ) response. I want to emphasize that a complete analysis of the debt-covenants is beyond the scope of this paper since these covenants are set for various other frictions (such as managerial self-interest) as well. In the presence of these debt-covenants, from the risk-management perspective, the distress boundary can still be taken as exogenous to the model. The results of this section, therefore, should be narrowly interpreted as a rational response of the bondholders in the context of therisk-managementdecisionsofthefirm. At t = t 0, the shareholders invest in a risky asset (A i ) with the consent of the bondholders. I assume that the firm is endowed with a special technology and the initial asset A i at t = t 0 represents the optimal investment of the firm based on this unique skill. I assume that the risk of this asset is denoted by σ i. The evolution of the risky asset A i overthetimeperiod [t 0,t 1 ] can be represented by a stochastic process A i t adapted to the filtration z t. At t = t 0, the rational bondholders price the firm s debt based on this level of risk and set the distress boundary such that the optimal risk chosen by the shareholders at t = t 1 stays at this level. One can easily relax this assumption to introduce other levels of asset 24