Financial distress and corporate risk management

ARTICLE IN PRESS Journal of Financial Economics 87 (2008) 706–739 www.elsevier.com/locate/jfec Financial distress and corporate risk management: Theory and evidence$ Amiyatosh Purnanandamà Ross School of Business, University of Michigan, Ann Arbor, MI 48109, USA Received 28 February 2006; received in revised form April 2007; accepted 10 April 2007 Available online 14 December 2007 Abstract This paper extends the current theoretical models of corporate risk-management in the presence of financial distress costs and tests the model’s predictions using a comprehensive data set I show that the shareholders optimally engage in expost (i.e., after the debt issuance) risk-management activities even without a pre-commitment to so The model predicts a positive (negative) relation between leverage and hedging for moderately (highly) leveraged firms Consistent with the theory, empirically I find a non-monotonic relation between leverage and hedging Further, the effect of leverage on hedging is higher for firms in highly concentrated industries r 2008 Elsevier B.V All rights reserved JEL classification: G30; G32 Keywords: Hedging; Risk-shifting; Asset substitution; Derivatives Introduction This paper develops and tests a theory of corporate risk management in the presence of financial distress costs 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 risk-management $ This paper is based on a chapter of my Ph.D dissertation at Cornell University I would like to especially thank an anonymous referee for several useful suggestions during the reviewing process I am grateful to George Allayannis, Warren Bailey, Sugato Bhattacharya, Sreedhar Bharath, Sudheer Chava, Thomas Chemmanur, Wayne Ferson, Ken French, John Graham, Robert Goldstein, Yaniv Grinstein, Jerry Haas, Pankaj Jain, Kose John, Haitao Li, Roni Michaely, M.P.Narayanan, Maureen O’Hara, Paolo Pasquariello, Mitch Petersen, Uday Rajan, William Schwert (the editor), David Weinbaum, Rohan Williamson, and seminar participants at Boston College, Cornell, Darden, Emory, London Business School, University of Michigan, Notre Dame, University of Rochester, The Lehman Brothers Finance Fellowship Competition 2003, and the Western Finance Association’s 2005 meetings for valuable comments and suggestions I am particularly grateful to Bob Jarrow and Bhaskaran Swaminathan for their advice All remaining errors are mine ÃTel.: +1 734 764 6886; fax: +1 734 936 8715 E-mail address: amiyatos@umich.edu 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 (DeMarzo and Duffie, 1991, 1995) See also Breeden and Viswanathan (1996) and Stulz (1996) 0304-405X/$ - see front matter r 2008 Elsevier B.V All rights reserved doi:10.1016/j.jfineco.2007.04.003 ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 707 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).2 I extend the current literature by explaining the ex-post risk-management motivation of the firm.3 I provide a simple model that generates new cross-sectional predictions by relating firm characteristics such as leverage, financial distress costs, and project maturity to risk-management incentives I test the key predictions of the model with hedging data of COMPUSTAT-CRSP firms meeting some reasonable sample selection criteria for fiscal years 1996–1997 The empirical study presents the first large-sample evidence on the determinants of the extent of firms’ hedging activities 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 is defined as a low cash-flow state in which the firm incurs 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 There are three important sources of financial distress costs 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 Using data from the supermarket industry, (Chevalier 1995a, b) finds evidence that debt weakens the competitive position of a firm Second, a financially distressed firm is more likely to violate its debt covenants5 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.7 Finally, a financially distressed firm may have to forgo positive NPV projects due to costly external financing, as in Froot, Scharfstein, and Stein (1993) In this paper I focus on the first of these costs, i.e., the product market-related costs of financial distress 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 The firm’s 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 loses market share to its competitors and therefore is unable to realize its full upside potential, even when the industry condition improves at a later date Insolvency occurs on the maturity date if terminal firm value is below the face value of debt, in which case debtholders gain control of the firm Shareholders’ final payoffs depend on the terminal asset value as well as on the path taken by the firm’s asset over its life.8 Throughout the paper, I use the terms ex ante and ex post with respect to the time of borrowing Other papers analyzing shareholders’ ex-post risk-management decisions 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 shareholders’ ex-post asset-substitution incentives My model, in contrast, is based on the cost of financial distress and provides new empirical predictions For example, in the mid-1990s Apple Computers had financial difficulties leading to speculation about its long-term survival (see Business Week, January 29 and February 5, 1996) Software developers were reluctant to develop new application software for Mac-users, which led in part to a decline of 27% in the unit sales of Mac computers from 1996 to 1997 (see Apple’s 1998 10-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) Lenders often impose debt covenants such as maintenance of minimum networth or maximum debt-to-equity ratio by the borrowing firms See Smith and Warner (1979), Kalay (1982), and Dichev and Skinner (2001) Moody’s Investor Service Report (1998) shows that during 1982–1997 about 50% of the long-term publicly traded bond defaults (including missed or delayed payment of coupon and principal) didn’t result in bankruptcy filings 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 2003 See Delta’s 2002 10-K filings with the SEC This approach is similar (but not the same) 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 losses incurred in financial distress Brockman and Turtle (2003) provide some empirical evidence in support of equity’s valuation as a path-dependent option ARTICLE IN PRESS 708 A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 The optimal level of ex-post investment risk, from the shareholders’ perspective, is determined by the tradeoff between the costs of financial distress and value associated with the limited liability of the firm’s equity.9 Unlike in 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 shareholders by increasing the expected cost of financial distress Due to these losses, the shareholders find it optimal to implement a risk-management strategy ex-post even in the absence of an explicit pre-commitment to so The optimal investment risk in my model depends on firm leverage, the financial distress boundary, the time horizon of the project, and the costs of financial distress As in the extant models (Smith and Stulz, 1985), I show that a firm with high leverage has a higher incentive to engage in hedging activities However, the riskmanagement incentives disappear for firms with extremely high leverage The incentive to hedge arises from the product market-related financial distress costs and these costs are more likely to be present when a firm is vulnerable to losing market share to its competitors Empirical studies by Opler and Titman (1994) and Chevalier (1995a, b) show that debt weakens the competitive position of a firm in its industry Further, the adverse consequences of leverage are more pronounced in concentrated industries Motivated by these studies my model argues that industry concentration provides a good proxy for financial distress costs Highly leveraged firms in concentrated industries are more likely to experience a deterioration in their competitive position in the event of financial distress i.e., are expected to incur higher financial distress costs Thus, the model predicts a stronger hedging incentive for highly levered firms in concentrated industries The model shows that hedging incentives increase with project maturity because the likelihood of experiencing financial distress as well as the expected loss of default increases with the life of the asset Riskmanagement motivation in my model arises from costs incurred by the firm in states in which the firm hits the financial distress barrier but remains solvent on the maturity date If there are no financial distress costs, riskmanagement incentives disappear On the other hand, if these costs are very high, the distinction between financial distress and insolvency diminishes along with any ex-post risk-management motivations Intermediate levels of losses create risk-management incentives within the firm Therefore, my model predicts a U-shaped relation between financial distress costs and hedging 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 (typically leverage) on firms’ riskmanagement activities If firms with extreme distress are less likely to hedge, these models may be misspecified The bias can be particularly severe in small-sample studies It is not surprising that existing empirical studies find mixed evidence in support of the distress cost-based theories of hedging.10 I contribute to the empirical risk-management literature by analyzing foreign currency and commodity riskmanagement activities of a comprehensive sample of nonfinancial firms Since data on firms’ hedging activities (by means of derivatives) are not readily available, empirical studies in this area are based on small samples or investigate only the yes–no decision to hedge.11 This has created two major challenges First, our current understanding is mostly based on analyses that treat firms with different hedging intensities as similar, which limits our ability to investigate firms’ hedging motivations Second, we have been able to gain only limited insight into the effect of industry-specific factors on hedging decisions I test the predictions of my model with data on the extent of hedging of more than 2,000 firms for the fiscal year 1996–1997 Due to the large sample size drawn from different industries, I provide new empirical evidence relating industry structure to hedging decisions Consistent with the theory, I find strong evidence that firms with higher leverage hedge more, although the hedging incentives disappear for firms with very high leverage Also in line with my theory, I find that financially distressed firms in highly concentrated industries hedge In the context of swap markets, Mozumdar (2001) demonstrates the trade-off between risk-shifting and hedging incentives in the presence of information asymmetry about the firm type His model relates hedging incentives to firm type 10 For example, while Haushalter (2000) and Graham and Rogers (2002) find a positive relation between the two variables, Nance, Smith and Smithson (1993), Mian (1996), and Tufano (1996) fail to find such evidence 11 For example, Geczy, Minton, and Schrand (1997) use 372 firms with 154 hedgers; Graham and Rogers (2002) use about 400 firms with 158 hedgers Studies by Mian (1996) and Bartram, Brown, and Fehle (2003) use large samples to investigate the yes–no decision of hedging Tufano (1996) and Haushalter (2000) provide detailed evidence from gold and oil & gas industries, respectively Brown (2001) provides evidence from a detailed case study Purnanandam (2007) investigates the risk-management decisions of commercial banks ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 709 more My empirical results are robust to alternative proxies of financial distress (such as leverage, industryadjusted leverage and Altman Z-score), alternative ways of measuring the hedging activities (yes–no decision to hedge and total notional amount of hedging) and various controls for nonderivative-based hedging strategies Further for a subsample of 200 manufacturing firms, I obtain data on the firms’ hedging activities for fiscal years 1997–1998 and 1998–1999 and show that the basic results remain similar for a regression model involving changes in hedging activities While firms with a moderate increase in leverage increase their hedging activities, firms with an extreme increase in leverage decrease their hedging positions As long as firms not frequently change their operational hedging strategies (such as opening plants in foreign countries to hedge their foreign currency risk), the analysis based on change regressions provides a robust control for nonderivative-based hedging strategies of the firm The change regressions also allow me to partially disentangle the effects of ex-ante and ex-post hedging incentives The rest of the paper is organized as follows In Section 2, I provide the model description Section analyzes the optimal risk-management policy of the firm The empirical tests are provided in Section 4, and Section concludes the paper Without any loss in continuity, readers mostly interested in the empirical part of the paper can skip to Section 3.1, which provides a self-contained summary of the key features of the theoretical model Model I consider a stylized model of a continuous trading economy with time horizon ½t0 ; TŠ There are three important dates in the model discussed below Though a discrete time model can also be used to capture the key feature of my model, the continuous time version allows for an easier analytical solution at the expense of additional mathematical overhead In addition, the continuous time model provides additional prediction relating the time to maturity of the firm’s project to its hedging incentives At t ¼ t0 , the firm makes its capital structure decision and invests in risky asset Ai (i stands for the initial investment), which I refer to as an ‘‘EBIT-generating machine’’ (Goldstein, Ju and Leland, 2001) These decisions may or may not be made with the consent of the firm’s debtholders The risky asset ðAi Þ is acquired at the market-determined price and financed through a mix of zero-coupon debt and equity capital Let L be the face value of the zero-coupon debt, payable at time T, and Et be the time t-value of the firm’s equity There is a tax benefit of debt, which provides the incentive to issue debt in my model For simplicity the tax benefit is assumed to be a fraction t of the face value of debt L Optimal capital structure is determined by a trade-off between the tax benefit of debt and bankruptcy costs For simplicity, I not endogenize the capital structure decisions However, the key predictions of the model remain similar for a more general model (unreported) that solves for capital structure decisions as well The cash generated by the machine and its asset value Ait are driven by a Brownian motion with the usual properties At some later time t ¼ t1 ðt1 ðt0 ; TÞÞ, the shareholders (or managers acting on their behalf) make a riskmanagement decision At this time, which can be an instant or days or months after the capital structure decisions, they have an opportunity to change the asset’s risk without the bondholder’s approval To capture the risk-shifting incentives, I assume that the bondholders are unable to recontract with the shareholders at t ¼ t1 Further, I assume that the two parties cannot contract on the risk-management choice at time t0 through the use of bond covenants This latter assumption is what gives rise to the risk-shifting incentive in my model This assumption is in the spirit of a large literature on incomplete contracting in economics and finance (see for example, Bolton and Dewatripont, 2005) The premise here is that it is too costly to specify every state of the world and write down debt covenants that will limit shareholders behavior with respect to firm risk in each of those states Even if such covenants could be written to tie down the manager’s risk-management behavior, it would be too costly to implement them especially in very high leverage states when shareholders have a large incentive to default on covenants.12 This assumption is in the spirit of Jensen and Meckling’s argument that ‘‘To completely protect the bondholders from the incentive effects, these provisions would have to be incredibly detailed and cover most 12 As long as there are nontrivial costs in writing, monitoring and enforcing these contracts, some residual risk-management decisions are always optimally left with the shareholders/managers, which is sufficient to generate the main results of my model ARTICLE IN PRESS 710 A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 operating aspects of the enterprise including limitations on the riskiness of the projects undertaken The costs involved in writing such provisions, the costs of enforcing them and the reduced profitability of the firm (induced because the covenants occasionally limit management’s ability to take optimal actions on certain issues) would likely be nontrivial In fact, since management is a continuous decision making process it will be almost impossible to completely specify such conditions without having the bondholders actually perform the management function.’’ After the risk-management decisions have been made, the firm acquires a new EBIT-generating machine This EBIT-generating machine generates cashflows dt forever that evolves according to a geometric Brownian motion The value of this EBIT-generating machine, i.e., the value of a similar unlevered firm, is denoted by At 13 One can think of dt as the state vector representing the state of the firm’s industry I assume that the change in the investment risk of the asset (from Ai to AÞ has no cashflow impact on the firm at t ¼ t1 This provides an initial boundary condition in the model, namely At1 ¼ Ait Further, for analytical simplicity I assume that the total payout (to debtholders and shareholders) by the firm is zero during ½t0 ; TÞ, with the final payoffs realized at t ¼ T The shareholders receive the terminal equity value of the firm.14 The bondholders receive the face value of debt (L) if the firm remains solvent on the maturity date t ¼ T 15; otherwise they receive the residual value of the firm The model can be represented by the following timeline: m t ¼ t0 Capital structure Initial investment m t ¼ t1 Risk-management decisions m t¼T Payoffs This modeling framework allows me to address the issue of ex-ante vs ex-post risk-management behavior of the firm in the presence of the shareholders’ risk-shifting incentives I now discuss the main assumption of the paper, namely, the distinction between financial distress and insolvency 2.1 Financial distress and insolvency If during (t0 ; TÞ the firm’s asset value At falls below a boundary KðLÞ;16 the firm is in the state of financial distress Insolvency, on the other hand, occurs on the terminal date T if the terminal firm value ðV T ) is less than the debt obligations Therefore, in the state of financial distress, control of the firm does not shift to the bondholders immediately, but the firm does incur costs that increase with leverage Opler and Titman (1994) show that financially distressed (highly leveraged) firms lose significant market share to their healthy competitors during industry downturns The drop in sales faced by Apple Computers and Chrysler during periods of financial difficulty provide anecdotal evidence in support of such deadweight losses In a sample of 31 high-leveraged transactions (HLTs), Andrade and Kaplan (1998) isolate the effect of economic distress from financial distress and estimate the cost of financial distress as 10–20% of firm value Asquith, Gertner and Scharfstein (1994) show that on average financially distressed firms sell 12% of their assets as part of their restructuring plans Chevalier (1995a, b) uses detailed information from the local supermarket industry to provide evidence in support of predatory behavior in this market She shows that following supermarket leveraged buyouts 13 The value of the levered firm of my model differs from At by the amount of the tax benefit of debt as well as the costs associated with financial distress and bankruptcy Throughout this paper I denote the value of the levered firm by V t and the value of its assets (EBITgenerating machine) by At 14 For analytical simplicity I assume that the model’s terminal date corresponds to the maturity date of the firm’s debt, at t ¼ T This assumption should not be confused with the assumption that the firm’s life is finite It simply states that at time T initial shareholders sell the firm to some other investors at the fair market value of the firm as an ongoing concern 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 16 I refer to K as the distress barrier in the rest of this paper K is assumed to be an increasing function of leverage This definition of financial distress is equivalent to assuming that when industry conditions deteriorate, firms with high leverage become financially distressed ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 711 Asset Value Healthy Financially Distressed At1 f-1(L) L K Insolvent t1 τ T Time (t) Fig This figure plots three paths for the evolution of the firm’s asset value I assume zero tax shield of debt for this presentation 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 in which the distress barrier is hit (at time tÞ, but the firm remains solvent at time T This is the state of ‘Financial Distress.’ In this state the terminal firm value, net of deadweight losses (i.e., f ðAT Þ), remains above the face value of debt (i.e., L) Thus, this is the state where f ðAT Þ4L or alternatively AT 4f À1 ðLÞ, as depicted in the figure Finally, the bottom-most path corresponds to the state of ‘Insolvency.’ (LBOs), prices fall in local markets in which rival firms have low leverage and are concentrated Further, these price drops are associated with LBO firms exiting the local market These findings suggest that rivals attempt to prey on LBO chains Phillips (1995) studies the interactions between product market and financial structure for four industries and finds evidence consistent with debt weakening the competitive positions of firms (see also Kovenock and Phillips, 1997; Arping, 2000) Using deregulation of the trucking industry as an exogenous shock, Zingales (1998) studies the interplay between financial structure and product market competition and provides evidence that leverage reduces the probability of a firm’s survival after an increase in competition The overall message from these papers is that financial distress may impose a real cost on firms by weakening their competitive position in the product market Motivated by the empirical findings of above papers and anecdotal evidence, I assume that a firm in financial distress loses a fraction of its market share to its healthy competitors.17 In my model, this is achieved by assuming that the financially distressed firm’s EBIT-generating machine produces less cashflow resulting in a lower value for the distressed firm If the firm does not experience financial distress during t ½t1 ; TŠ, the terminal firm value is V T However, if the distress boundary is hit, the terminal value falls to f ðV T Þ, where f ðV T ÞoV T (see Fig 1) The function f represents the losses caused by financial distress 2.2 Valuation of equity The shareholders receive liquidating dividends at T Due to equity’s limited liability, the final payoff to the shareholders ðxT Þ is zero if the terminal firm value is below L Let us define inf t1 ptpT At  mT for the minimum value of the asset during ½t1 ; TŠ In the event of no distress (i.e., mT 4K) and solvency on the terminal date (i.e., V T 4L), the shareholders get a liquidating dividend of ðV T À LÞ If financial distress is experienced (i.e., mT pK), but on the terminal date the firm remains solvent (i.e., f ðV T Þ4L), the shareholders 17 In a more general industry equilibrium setting, firms can make strategic decisions about their leverage, investment risk, and hedging (see e.g., Adam, Dasgupta, and Titman, 2004; Nain, 2006) My model abstracts from such considerations and focuses on the firm’s decision, taking industry structure as given ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 712 receive liquidating dividends of f ðV T Þ À L In the event of insolvency, shareholders receive nothing and firm value drops by the fraction g ½0; 1Š The shareholders’ payoff under different states is summarized as State at t ¼ T Corresponding firm values Payoff to shareholders Healthy Financial distress Insolvency Insolvency V T 4L; mT 4K f ðV T Þ4L; mT pK V T pL; mT 4K f ðV T ÞpL; mT pK VT À L f ðV T Þ À L 0 Proposition Under mild technical conditions, the equity valuation at t ¼ t1 is given by: xt1 ¼ eÀrT E Q ½ðV T À LÞ À ðV T À f ðV T ÞÞ1ff ðV T Þ4L;mT pKg þ ðL À V T Þf1fV T pLg þ 1ff À1 ðLÞ4V T 4L;mT pKg gŠ (1) Proof See Appendix A.1 & The equity value, as shown in Proposition 1, has three components The first term ðE Q ½V T À LŠÞ represents the equity value without the distress costs and the limited liability feature The second term ðE Q ½ðV T À f ðV T ÞÞ1ff ðV T Þ4L;mT pKg ŠÞ represents the cost of financial distress Because the shareholders of a financially distressed but solvent firm bear this cost, the equity value decreases by this amount The risk avoidance incentive results from this cost The third term ðE Q ½ðL À V T Þf1fV T pLg þ 1ff À1 ðLÞ4V 4L;m pKg gŠÞ represents the T T savings enjoyed by the shareholders of a levered firm due to the limited liability feature of equity This term captures shareholders’ risk-shifting incentives By increasing the asset’s 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 asset risk The optimal level of investment risk is determined by the trade-off between the two 2.2.1 Financial distress costs Proposition provides a general valuation formula in my model To proceed further I need to be explicit about the form of financial distress cost that is borne by the shareholders of a financially distressed firm In addition, I make some simplifying assumptions for analytical tractability I assume that in the event of distress (i.e., mT pK), the firm’s cashflows drop to ldt ; l ð0; 1Š and never reach beyond some arbitrary upper bound Uo1 at time T, i.e., dT pU Therefore, the losses take the form of lost upside potential This representation of financial distress cost is motivated by existing empirical findings and anecdotal evidence, and captures the intuition that distressed firms lose cashflows due to lost sales to competitors If industry conditions improve in the future, the distressed firms continue to feel the negative effect of distress due to lost customers This representation of distress is also consistent with the view that when financially distressed firms restructure themselves by selling assets (Asquith, Gertner and Scharfstein, 1994), their EBIT-generating machine produces lower contemporaneous cashflows and in addition it limits their ability to capitalize on very good industry conditions in the future To concentrate on the effect of financial distress costs (as opposed to tax-motivated incentives of hedging as in Leland, 1998), in the rest of the paper I set t ¼ 0.18 Under this assumption and the assumption l ¼ 1, the distressed firm’s asset value can be represented as19: f ðAT Þ ¼ AT if fdT pUg; and M if fdT 4Ug for some constant M (2) 18 In unreported analysis, I solve the model with tax benefits and obtain the firm’s optimal capital structure However, to keep the focus of this paper on risk-management decisions, I not present these results in the paper With tax benefits, the firm’s payoffs increase by tL without qualitatively changing the results of the analysis 19 If lo1, then financial distress costs are even higher and the results become stronger This assumption is made only for analytical simplicity ARTICLE IN PRESS Equity Value A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 713 Equity Value In Healthy State Equity Value in my model L Equity Value in Financial Distress L L+M Asset Value at T Fig This figure plots the equity value as a function of the terminal asset value of the firm For illustrative purposes I set the tax rate to zero and g ¼ for this diagram 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 Let us denote the asset value ðAT Þ corresponding to dT ¼ U by L þ M The shareholders’ liquidating dividends are given as States Payoff to shareholders Firm value AT 4L; mT 4K AT 4L; AT pL þ M; mT pK AT 4L þ M; mT pK AT pL AT À L AT À L M AT AT LþM gAT The financial distress costs can be expressed as ðAT À MÞ:1fAT 4LþM;mT pKg 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: xt1 ¼ eÀrT E Q ½ðAT À LÞ1fAT 4L;mT 4Kg þ ðAT À LÞ1fAT 4L;AT pLþM;mT pKg þ M1fAT 4LþM;mT pKg Š ð3Þ Fig plots the equity value as a function of the terminal asset value of the firm As the diagram shows, 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 for the firm Optimal choice of investment risk Without loss of generality, I set the risk-free interest rate to zero in the rest of the analysis At t ¼ t1 , 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 s at t ¼ t1 , or (b) the asset’s risk, s, 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 s assuming that investment risks can be costlessly modified Proposition The shareholders have a well-founded incentive to engage in risk-management activities ex-post At t ¼ t1 , the shareholders optimally choose a level of risk sà in the interior of all possible risks ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 714 Proof As shown in Appendix A.2 and A.3, the optimum level of investment risk is obtained by the following first-order condition: At1 fðh1 Þ ¼ Kfðc1 Þ (4) pffiffiffiffiffi0 pffiffiffiffiffi0 2 =LÞ where ffiffiffiffiffi0 þ ðs =2ÞT Þ=s T , h2 ¼ h1 À s T , T ¼ T À t1 , c1 ¼ lnðK =At1 ðL þ MÞÞ þ ðs =2ÞT = pffiffiffiffiffi0 h1 ¼ ðlnðAt1p s T , c2 ¼ c1 À s T ; and f stands for the probability density function of the standard normal distribution Further simplification leads to the following closed-form solution: !   K2 K 2L ln ln LðL þ MÞ A2t ðL þ MÞ Ã  1 ðs Þ ¼ : & (5) LþM T ln L As a result of the trade-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 riskmanagement models such as Smith and Stulz (1985), the optimal level of risk is obtained at s ¼ 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.20 Proposition The firm chooses a lower level of investment risk if (a) it faces a higher distress barrier (K), and (b) it has a longer project maturity ðT ¼ T À t1 Þ The relation between the deadweight losses and the optimal investment pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð lnðAt =KÞ lnðL=KÞÞ risk is U-shaped Let M c ¼ L exp À L When M4M 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 s given in expression (see Appendix A.5) & 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 financial distress costs The results show that the firm with a longer operational horizon ðT ¼ T À t1 Þ finds it optimal to engage in increased risk-management activities With longer time-horizon, the probability of hitting the lower barrier increases Further, consequent to entering the state of distress expected losses increase with time to maturity because there is a higher probability of improvements in industry conditions and the distressed firm will not be able to capitalize on these opportunities There is 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 regularity My model suggests another explanation: the time horizon of operations If firms with longer time horizons grow larger over 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 relation between the risk management incentives and the cost of financial distress Recall that the deadweight losses in my model are parameterized by M (losses are given by ðAT À MÞ:1fAT 4LþM;mT pKg ) In the event of financial distress, the firm loses its upside potential beyond L þ 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., M ¼ 1), the shareholders lose nothing in the state of financial distress and hence there is no risk-management incentive On the other hand, if deadweight losses are very high (i.e., M ¼ 0) the distinction between default and insolvency disappears along with the risk-management incentives.21 It’s the intermediate cases that generate risk-management incentives in the model Fig illustrates this relation 20 With nonzero tax rates (in unreported analysis), the optimal s is even lower The additional incentives for risk reduction, in the presence of the tax-benefit of debt, comes from the potential loss in the tax shield of debt for a bankrupt firm This additional effect generates ex-post hedging as in Leland (1998) See also Fehle and Tsyplakov (2005) 21 In this case, 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 shareholders not have any risk-management incentives at t1 ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 715 2.94 2.92 2.9 Investment Risk 2.88 2.86 2.84 2.82 2.8 25 00 75 50 25 00 75 50 25 00 75 9 .5 10 11 12 12 13 14 15 0 2.78 Deadweight Loss Parameter (M) Fig This figure plots the optimal investment risk as a function of deadweight losses The model has been calibrated with the following parameter values: At1 ¼ 2; L ¼ 1; T ¼ and K ¼ 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 losses increase as one moves along the x-axis Investment Risk vs Leverage 14 12 Investment Risk 10 95 91 87 83 0 79 75 71 67 63 59 55 51 47 0 43 39 35 31 27 23 19 15 0 11 Leverage Fig This figure plots the optimal investment risk of the firm against the debt-asset ratio For this graph I assume the following structure on the distress boundary and deadweight losses: K ¼ À expÀ0:1Ãlev and M ¼ À exp2Ãlev Amount of debt raised at time zero (L) if fixed at lev equals L scaled by At1 T is set to one Leverage and risk management: To study the relation between leverage and risk management, I differentiate the optimal s with respect to firm leverage at time ðlev ¼ L=AÞ The details are provided in Appendix A.5 After some simplification it can be shown that the optimal sigma decreases (i.e., risk-management incentives increase) with an increase in leverage for a wide range of specifications of the distress boundary and ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 725 Table Summary statistics This table presents the descriptive statistics for the key explanatory variables used in the analysis Panel A presents the median characteristics of users and nonusers of foreign currency (FX) derivatives based on 1,781 observations that are identified as firms with exposure to foreign currency risk Panel B is based on commodity (CM) derivatives (1,238 observations with exposure to commodity price risk), and Panel C is based on the usage of any of these two derivatives (2,256 observations) In every panel, I provide the median characteristics of hedgers and nonhedgers as well as the entire sample The last row in each sample gives the p-Value for the test that median characteristics for hedger and nonhedger groups are equal based on a Wilcoxon-Mann-Whitney test Sales represent the total sales of the firm as reported under item 12 of COMPUSTAT tapes mv stands for market value obtained by multiplying COMPUSTAT item 25 by item 199 lev measures the ratio of total liabilities (sum of COMPUSTAT items and 34) to total assets (item 6) Quick ratio is constructed as the ratio of cash and short-term investments (item 1) to current liabilities (item 5) fsale represents the ratio of foreign sales to total sales of the firm The foreign sales data are obtained from the COMPUSTAT geographical segments file inst measures the percentage institutional ownership in the firm rnd stands for percentage research and development expenses (item 46) scaled by the sales of the firm (item 12) mtb stands for the market-to-book ratio of the firm’s assets (COMPUSTAT (item minus 60 plus ð25 à 199Þ) scaled by item 6) sales mv lev quick fsale inst rnd mtb Panel A: FX derivatives Nonhedgers 228.5150 Hedgers 1147.0000 All 334.4900 p-Value 0.01 256.6120 1313.4053 392.5571 0.01 0.1744 0.2082 0.1877 0.05 0.2632 0.2159 0.2460 0.04 0.0685 0.3480 0.1416 0.01 44.1235 58.0480 47.7017 0.01 0.0000 2.1435 0.7773 0.01 1.6588 1.7048 1.6723 0.02 Panel B: Commodity derivatives Nonhedgers 212.0220 Hedgers 768.4550 All 244.8135 p-Value 0.01 214.9539 798.7656 263.7081 0.01 0.2194 0.2829 0.2331 0.01 0.2378 0.1412 0.2091 0.01 0.0000 0.0000 0.0000 0.33 38.3827 53.0766 41.1219 0.01 0.0000 0.0000 0.0000 0.01 1.5692 1.5024 1.5501 0.09 Panel C: Any derivatives Nonhedgers 201.7550 Hedgers 917.1540 All 285.0805 p-Value 0.01 207.9030 977.1712 305.7399 0.01 0.1983 0.2267 0.2071 0.04 0.2505 0.1995 0.2271 0.02 0.0000 0.2828 0.0241 0.01 39.3921 56.2881 44.4876 0.01 0.0000 1.0701 0.0000 0.01 1.6002 1.6659 1.6136 0.06 for nondebt tax shields, is negative and significant Further, consistent with my model firms with higher profitability have lower leverage as indicated by a negative and significant coefficient on net income to sales ðni=salesÞ These results are consistent with the motivations behind the use of these variables in the leverage regression model Other results are in line with the earlier empirical literature Once I obtain the predicted values of leverage from the first-stage model, I use it in the second-stage model to explain a firm’s foreign currency and commodity hedging decision To save space, I not present the results from the first stage estimation in the rest of the paper 4.4.2 Foreign currency hedging I start with the firm’s foreign currency hedging decision and subsequently analyze the commodity hedging decisions Yes/No decision: I present the results of a second-stage Logit regression in Table The dependent variable equals one if a firm uses foreign currency derivatives and zero otherwise The model is estimated with only those firms that have a pre-defined exposure to foreign currency risks In the first model, leverage is positive and significant at 1% whereas leverage2 is negative and significant at the 1% level For easier interpretation, I present the marginal effect (on the probability of hedging) of the explanatory variable evaluated at the mean rather than the raw estimated coefficient from the logit model In the next model, I include the interaction of leverage and a dummy indicating whether the firm belongs to a highly concentrated industry or not I find a positive coefficient on the interaction of leverage and industry concentration (concd) As expected, the ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 726 Table Foreign currency hedging—yes/no decision This table presents logistic regression results for foreign currency hedging by means of derivatives In the first stage I estimate an OLS regression model for leverage The estimation results from this regression are presented in the first two columns In addition to the coefficients reported in this table, this regression also includes industry dummies based on two-digit SIC code (coefficients suppressed) In the second stage, a logistic model is estimated with firm’s foreign currency derivative usage as the dependent variable (one for hedgers and zero for nonhedgers) levà denotes the predicted value of leverage from the first-stage regression The marginal effect of explanatory variables (evaluated at the mean) on the probability of hedging along with associated t-Values are presented in the table Columns 3-8 present results from the second-stage estimation of hedging model size represents the log of total sales of the firm quick is the ratio of cash and short-term investments to current liabilities rnd stands for research and development expenses scaled by the sales of the firm concd is a dummy variable based on the four-firm concentration ratio of the firm’s industry (based on three-digit SIC code) concd equals one if the firm belongs to an industry with a concentration ratio above the median, zero otherwise fsale represents foreign sales as a percentage of total sales inst measures the percentage institutional ownership in the firm mtr stands for the historical average of firm’s marginal tax rates ppe/ta stands for plant, property, and equipment scaled by total assets Modified Z is the Altman Z-score without the leverage effect ni/sales stands for the ratio of net income to total sales taxconvexity measures the dollar tax benefit from a 5% volatility reduction in the firm’s income scaled by the sales of the firm mtb stands for the market-to-book ratio of the firm segno stands for the number of geographical segments in which the firm operates The number of observations and R2 (for OLS regression) are provided at the end of the table Leverage size levà levÃ2 lev à concd quick rnd concd fsale inst mtr ppe=ta modifiedz ni=sales da=ta taxconvexity mtb segno R2 N FX Derivatives Estimate t-Value Estimate t-Value Estimate t-Value Estimate t-Value 0.0116 (3.61) 0.1348 1.1181 À2.3759 (12.70) (3.04) (À3.34) (9.80) (2.52) (À2.95) (À5.73) (À6.52) (1.01) (À0.94) (À2.91) (2.79) (2.56) (À12.90) (À2.95) (À2.64) 0.0203 0.0103 À0.0479 0.3457 0.0007 (1.10) (5.20) (À1.75) (9.34) (1.20) (12.65) (1.94) (À3.23) (2.42) (0.82) (5.19) (À2.87) (9.39) (1.16) 0.1170 0.9519 À2.1029 À0.0295 À0.0060 0.0112 À0.0123 À0.0006 0.2634 0.1105 À0.0783 À0.1729 À0.6776 0.1346 0.7657 À2.3041 0.5295 0.0155 0.0104 À0.1706 0.3497 0.0007 0.0149 0.0101 À0.0499 0.1918 0.0006 (0.78) (4.95) (À1.83) (3.95) (1.00) À0.7850 À0.0074 0.0670 (À1.54) (À0.62) (4.60) 0.402 1,421 1,421 1,421 1,418 introduction of this interaction term lowers the significance of leverage variable, but it still remains significant at almost the 5% level These findings are consistent with the key predictions of the model Other results indicate that firms with higher foreign currency sales are more likely to use hedging products, indicating that highly exposed firms have higher incentives to hedge There is a strong relation between growth opportunities as measured by R&D expenses and hedging This finding is consistent with the theoretical predictions of Froot, Scharfstein, and Stein (1993) and earlier empirical findings of Geczy, Minton, and Schrand (1997) I find a positive relation between institutional shareholdings and hedging Assuming an inverse relation between institutional shareholdings and the extent of information asymmetry between the insiders and outsiders of the firm, this result is inconsistent with information asymmetry-based models of hedging However, more analysis is needed to draw stronger inferences for this theory since the measurement of information asymmetry remains a difficult task for empirical researchers In the final model I include three more control variables: tax convexity, market-to-book ðmtbÞ, and the number of geographical segments (segno) in which the firm operates All key results remain similar I don’t find evidence in support of tax-based ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 727 Table Foreign currency hedging—extent of hedging This table presents the Tobit regression results for foreign currency hedging by means of derivatives In the first stage (unreported), I estimate an OLS regression model for leverage In the second stage, a Tobit model is estimated with firm’s foreign currency derivative usage as the dependent variable This variable takes the value of the notional amount of foreign currency derivatives scaled by total sales of the firm (zero for nonhedgers) I provide the second-stage estimation results for three different model specifications in the table below levà denotes the predicted value of leverage from the first-stage regression The marginal effect of explanatory variables (evaluated at the mean) on the expected value of uncensored observations along with associated t-Values are presented in the table size represents the log of total sales of the firm quick is the ratio of cash and short-term investments to current liabilities rnd stands for research and development expenses scaled by the sales of the firm concd is a dummy variable based on the four-firm concentration ratio of the firm’s industry (based on three-digit SIC code) concd equals one if the firm belongs to an industry with concentration ratio above the median, zero otherwise fsale represents foreign sales as a percentage of total sales inst measures the percentage institutional ownership in the firm taxconvexity measures the dollar tax benefit from a 5% volatility reduction in the firm’s income scaled by the sales of the firm mtb stands for the market-to-book ratio of the firm segno stands for the number of geographical segments in which the firm operates The number of observations is provided at the end of the table size levà levÃ2 levà à concd quick rnd concd fsale inst taxconvexity mtb segno N Estimate t-Value Estimate t-Value Estimate t-Value 0.0117 0.1435 À0.2741 (10.51) (3.21) (À3.19) (8.47) (3.19) (À3.17) (1.46) (4.21) (À1.05) (8.27) (1.37) (10.45) (2.27) (À3.14) (2.21) (1.33) (4.17) (À2.37) (8.33) (1.36) 0.0104 0.1459 À0.2721 0.0032 0.0010 À0.0033 0.0370 0.0001 0.0116 0.1074 À0.2699 0.0572 0.0029 0.0010 À0.0162 0.0372 0.0001 0.0036 0.0009 À0.0041 0.0176 0.0001 0.0226 0.0005 0.0086 (1.66) (3.55) (À1.31) (3.06) (1.49) (0.46) (0.38) (4.99) 1,421 1,421 1,418 motivations for hedging Finally, firms operating in more diverse foreign markets hedge more as evident by a positive and highly significant coefficient on this variable This result points toward the possibility that derivative instruments act as complements to a firm’s natural hedging strategies Extent of hedging: In Table 4, I present results from the Tobit estimation with the notional amount of foreign currency derivatives scaled by total sales of the firm as the dependent variable Since the estimated coefficients in a Tobit model don’t represent the marginal effect of explanatory variables on the observed dependent variable, for easier economic interpretation I report the slope coefficients at the mean level I present results from three different model specifications and find that firms with high leverage hedge more and the relation between hedging and leverage reverses at very high levels of leverage Highly levered firms in concentrated industries have higher hedging incentives as well These results are in line with both the model’s predictions and the results obtained from the logit model described earlier, as well as with the earlier theoretical models based on bankruptcy costs (Smith and Stulz, 1985) Economically, these results suggest that if leverage increases from 10% to 20%, the firm increases its foreign currency derivative holdings by approximately 6.4%, which is about 60% of the average level of foreign currency derivatives held by the sample firms (see Table 1; these are only rough estimates with linear extrapolation around the mean.) Earlier studies provide mixed evidence in support of hedging theories based on financial distress costs Mian (1996) studies the binary (i.e., yes–no) hedging decision for a large sample of firms and finds no support for the distress cost theories.33 My results suggest that linear models seeking to test theories of risk-management, 33 There are three possible explanations for the different results between my study and Mian (1996) First, his sample comes from 1992— a period before the strict FASB regulation on derivatives disclosure Second, his model doesn’t test for nonlinearity and therefore it doesn’t have any quadratic terms Finally my modeling technique is different as I use a two-stage estimation technique that considers leverage and hedging as endogenous variables ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 728 especially if conducted on small sample sizes, may fail to detect a positive relation between financial distress and derivative activities for moderately leveraged firms To understand the nonmonotonic relation between hedging and leverage, I conduct a semi-parametric test I break the sample of firms into two groups based on whether their predicted leverage is below or above the 70th percentile of the empirical distribution of leverage in my sample For this estimation I run a Tobit regression with the same set of variables as in Model of Table after dropping leverage2 The spline regression results show that for the first group, i.e., for the group with moderate leverage the marginal effect of leverage on hedging is positive with a slope coefficient of 0.0452, which is significant at the 1% level However, the marginal effect of leverage on hedging becomes negative for firms in the other group, i.e., for firms with leverage in top 30% of the sample For this group, the marginal effect of leverage on hedging is estimated to be À0:1504 with a significance level of 2% The semi-parametric test confirms the non-monotonic relation obtained in parametric regressions 4.4.3 Commodity hedging Table provides logistic regression results for the commodity hedging decision This regression is estimated on a sample of firms with exposure to commodity price risk only As in the foreign currency derivative regression, I find a positive and significant coefficient on leverage, and a negative and significant coefficient on leverage2 Both these relations are significant at the 1% level When I include the interaction of leverage with high concentration industry, I find the coefficient on the interaction term to be positive and significant at the 6% level These results show that the predictions of my theory are supported by both foreign currency and commodity hedging data As in the case of foreign currency hedging, larger firms are more likely to use commodity derivatives as well However, in this regression the coefficient on the quick ratio becomes positive and significant, while it is positive but insignificant in the foreign currency hedging models Commodity hedgers keep more liquid assets as well, which can be taken as evidence that hedgers complement their hedging policies with liquid assets The most noticeable difference between the two models is the coefficient on the R&D variable This variable has a positive and highly significant coefficient in the logit and Tobit model of foreign currency hedging Table Commodity hedging—yes/no decision This table presents logistic regression results for commodity hedging by means of derivatives In the first stage (unreported) I estimate an OLS regression model for leverage In the second stage, a logistic model is estimated with firm’s commodity derivative usage as the dependent variable (one for hedgers and zero for non-hedgers) levà denotes the predicted value of leverage from the first stage regression The marginal effect of explanatory variables (evaluated at the mean) on the probability of hedging along with associated t-Values are presented in the table size represents the log of total sales of the firm quick is the ratio of cash and short-term investments to current liabilities rnd stands for research and development expenses scaled by the sales of the firm concd is a dummy variable based on the fourfirm concentration ratio of the firm’s industry (based on three-digit SIC code) concd equals one if the firm belongs to an industry with concentration ratio above the median, zero otherwise inst measures the percentage institutional ownership in the firm taxconvexity measures the dollar tax benefit from a 5% volatility reduction in the firm’s income scaled by the sales of the firm mtb stands for the market-to-book ratio of the firm The number of observations is provided at the end of the table size levà levÃ2 levà concd quick rnd concd inst taxconvexity mtb N Estimate t-Value Estimate t-Value Estimate t-Value 0.0360 0.9815 À1.7470 (5.20) (3.23) (À3.46) (4.79) (3.13) (À3.40) (2.13) (À6.76) (1.23) (2.16) (5.25) (2.47) (À3.50) (1.94) (2.20) (À7.19) (À1.27) (2.20) 0.0354 0.9680 À1.7337 0.0229 À0.0216 0.0206 0.0008 0.0355 0.7556 À1.7267 0.3107 0.0229 À0.0220 À0.0737 0.0008 0.0224 À0.0215 0.0212 0.0008 À0.0764 À0.0013 (2.04) (À6.56) (1.26) (2.02) (À0.33) (À0.15) 948 948 947 ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 729 Table Alternative models This table presents the results for derivatives usage for various alternative model specifications The first two models use alternative definitions of financial distress, whereas the last two models use leverage as the measure of financial distress but use bootstrapped standard errors in estimating the t-Values FD stands for the measure of financial distress for the given model and FD2 is its squared term The first model uses the industry-adjusted leverage ratio based on two-digit SIC codes as a measure of FD For this model FD2 equals leveragesquared if the firm’s leverage is above industry average, zero otherwise In the second model, I use Altman’s Z-score as a measure of the firm’s financial distress and estimate a logistic model using the firm’s usage of foreign currency or commodity derivatives as the dependent variable (one for users, zero for nonusers) For consistency with other models, I set FD equal to the inverse of the Z-score such that a higher value of FD corresponds to firms closer to financial distress The third model replicates the base-case logistic regression with bootstrapped standard errors The dependent variable is one for the users of commodity or foreign currency exposure, zero otherwise In the fourth model I estimate a Tobit model for the extent of foreign currency derivatives using bootstrapped standard errors For these last two models FD stands for predicted leverage from the first-stage regression, and FD2 is simply the squared predicted leverage All regression results provide the slope estimates evaluated at the mean of the explanatory variable along with corresponding t-statistics size represents the log of total sales of the firm quick is the ratio of cash and short-term investments to current liabilities rnd stands for research and development expenses scaled by the sales of the firm concd is a dummy variable based on the four-firm concentration ratio of the firm’s industry (based on three-digit SIC code) concd equals one if the firm belongs to an industry with a concentration ratio above the median, zero otherwise fsale represents foreign sales as a percentage of total sales inst measures the percentage institutional ownership in the firm Number of observations used for the estimation is provided in the last row Estimate t-Value N 0.1089 0.2536 À0.7713 0.0111 0.0039 À0.0255 0.2833 0.0010 (13.20) (3.07) (À2.70) (0.89) (2.35) (À1.16) (9.07) (2.05) 2,089 Estimate Z-score Ind Leverage size FD FD2 quick rnd concd fsale inst t-Value Estimate 0.0433 0.0812 À0.0265 0.0039 0.0019 À0.0095 0.1168 0.0004 2,049 t-Value Estimate LOGIT (1.98) (3.83) (À11.11) (0.79) (1.70) (À0.95) (1.97) (1.48) 0.1167 1.3070 À2.4002 0.0415 0.0057 À0.0250 0.3056 0.0012 1,769 t-Value TOBIT (12.05) (4.01) (À4.18) (2.63) (2.81) (À1.09) (8.47) (2.07) 0.0111 0.1162 À0.2276 0.0029 0.0009 À0.0034 0.0345 0.0001 (5.88) (2.57) (À2.71) (1.41) (3.58) (À1.47) (4.56) (1.49) 1,421 decisions However, here it is negative and insignificant While high growth firms manage their foreign currency hedging more aggressively, they are less likely to manage their commodity risk exposure Although exploring the differences in hedging incentives across different types of risks is beyond the scope of this paper, this finding is suggestive of firm’s facing conflicting incentives in managing various forms of risk These conflicting incentives may be driven by factors such as differences in the correlation between the risk being hedged and the firm’s investment opportunity set For example, the argument behind a positive relation between growth opportunities and hedging relies on the assumption that high growth firms may need funds to undertake projects in bad cashflow states If firms not have good investment opportunities in states with poor realizations of cashflows, then this incentive disappears At the extreme, if the investment opportunity set is highly positively correlated with the realizations of cashflows, then such firms may have a disincentive to hedge A better understanding of these issues is left for future research that explicitly incorporates these correlations in the analysis 4.5 Alternative model specifications Industry-adjusted leverage ratios: The results presented so far in the paper are based on a two-stage specification that requires an assumption about the structural model determining a firm’s leverage choice For robustness, I test a model that does not require such a specification Specifically, I conduct the analysis with firms’ industry-adjusted leverage ratios as industry adjustment provides a simpler and perhaps more robust way to classify firms into moderately and highly leveraged The industry-adjusted leverage ratio of a firm is defined as the difference between the firm’s leverage and the industry median based on the two-digit SIC code ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 730 I reestimate all my results with these ratios For this model, leverage variable can be either positive or negative depending on whether the firm is above or below industry-median Thus, I cannot use leverage2 as an explanatory variable to test for the nonlinearity predicted by the model Instead, I use a variable that equals leverage2 þ for firms with higher than industry median leverage and zero otherwise To conserve space, I pool foreign currency and commodity hedging decisions to estimate these robustness results I estimate a logit model on a sample of firms that are exposed to either of these two types of risks and present the results in Model of Table 34 My key results remain the same with this definition of leverage Altman Z-score: I use the Altman Z-score as an alternative proxy of financial distress Lower Z-score values correspond to financially weaker firms I, therefore, transform them by taking their inverse to be consistent in the presentation of results Results are presented in Model of Table I find a nonmonotonic relation based on this measure as well Bootstrapped standard errors: Since I use a two-stage estimation methodology in the logit and Tobit regressions, there is a potential for overstated t-statistics due to the sampling error of first-stage estimation (see Maddala, 1983) To account for this possibility I reestimate my models with bootstrapped standard errors In every replication I create a pseudo-random sample by drawing observations from the base sample with replacement Thus, in every replication some of the observations appear more than once and some not appear at all With 100 such replications, I generate an empirical distribution of estimated coefficients in the logit and Tobit models The standard deviations of these estimates are then used to obtain bootstrapped p-Values for my base estimation This methodology does not rely on any structural form for the estimation of the variance-covariance matrix and has the advantage of benchmarking base estimates against their empirical distributions In Models and of Table 6, I present the Logit and Tobit model estimates with bootstrapped errors As shown, all my key results are robust to bootstrapped standard error estimation Alternative IV regression specification: Wooldridge (2002) suggests an alternative instrumental variable regression model for models involving functions of an endogenous variable (such as leverage2 in the secondstage estimation) Potentially, this technique provides econometrically better estimates than the model that uses predicted values of leverage and its function in the second stage In this method, rather than using the squared value of predicted leverage in the second-stage regression, both leverage and leverage2 are treated as endogenous variables and instrumented with their own instruments To achieve identification, I add the squared terms of all exogenous variables entering the leverage model as instruments for leverage2 These instruments are the squared values of: mtr, da=ta, ppe=ta, ni=sales and modified_z In addition, the squared predicted value of leverage from the first stage estimation is used as an additional instrument for leverage2 With both leverage and leverage2 as endogenous variables and these instruments in hand, I estimate an instrumental variable model in a two-stage regression framework I estimate the binary decision to hedge foreign currency or commodity derivatives with an IV Probit model and the extent of foreign currency hedging with an IV Tobit model The results are presented in Table Note that the parameter estimates in this model are not directly comparable to the earlier tables since, due to computational simplicity, I report the coefficients from the regressions directly rather than the slope coefficients presented earlier I find that my key results remain robust to this alternative IV estimation technique All other results remain similar to the earlier basecase specification 4.6 Dynamic analysis Change regression: I focus on a three-year panel of 200 manufacturing firms to exploit the variations in the individual firm’s leverage and hedging intensities in a dynamic setting This empirical strategy closely resembles my theoretical model and presents several econometric advantages The change regression controls for unobserved firm-specific factors In particular, unless a firm’s nonderivative-based hedging strategies have changed substantially over this time period, change regressions control for operational and natural hedging in a more precise manner Second, the endogeneity argument is less severe for change regressions since by construction it removes firm-specific unobservable effects that could be correlated with both hedging and 34 Individual regressions estimated separately on foreign currency and commodity samples are qualitatively similar ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 731 Table Alternative IV model This table presents the second-stage results for derivatives usage for an alternative instrumental variable regression model following Wooldridge (2002) Rather than using the square of predicted values of leverage in the second-stage regression, this specification uses both leverage and its squared term as endogenous variables Leverage is instrumented with da/ta (depreciation and amortization scaled by total assets), mtr (marginal tax rates), ppe (property, plant and equipment scaled by total assets), modified Z-score and ni/sales (net income to total sales) The squared terms of each of these variables are used as additional instruments for leverage2 In addition, the squared value of the predicted leverage is also used as an instrument for leverage2 as suggested by Wooldridge (2002) Models and provide the secondstage results from the instrumental variable Probit models with foreign currency derivatives and commodity derivatives, respectively In Model 3, an instrumental variable Tobit model is estimated The dependent variable in Tobit regressions is the notional amount of foreign currency derivatives (scaled by total sales) for the users of derivatives and zero for the rest of the firms Size represents the log of total sales of the firm rnd stands for research and development expenses scaled by the sales of the firm quick is the ratio of cash and short-term investments to current liabilities concd is a dummy variable based on the four-firm concentration ratio of the firm’s industry (based on three-digit SIC code) concd equals one if the firm belongs to an industry with concentration ratio above the median, zero otherwise fsale represents foreign sales as a percentage of total sales inst measures the percentage institutional ownership in the firm For all three regressions, the parameter estimate and p-Values are provided Number of observations used for the estimation is provided in the last row t-Value Estimate FX Yes/No size lev lev2 rnd quick concd fsale inst N 0.3628 6.6451 À11.0198 0.0348 0.0937 À0.1774 1.0809 0.0020 1421 Estimate t-Value Commodity Yes/No (9.10) (2.93) (À3.12) (4.72) (1.36) (À1.94) (8.25) (0.93) 0.1976 9.5649 À11.4499 À0.0890 0.3032 0.2141 (4.20) (4.22) (À3.65) (À3.40) (3.53) (1.57) 0.0045 (1.59) 948 t-Value Estimate FX-Extent 0.0412 1.0180 À1.5277 0.0047 0.0186 À0.0187 0.1584 0.0004 (6.59) (2.77) (À2.71) (3.91) (1.63) (À1.27) (7.32) (1.24) 1421 leverage at any given point in time Third, this model allows me to relate changes in this year’s hedging intensities to both current and past year’s leverage changes, allowing me to establish some evidence on ex-ante and ex-post hedging incentives I obtain data on derivative holdings for a random subsample of 200 manufacturing firms (one-digit SIC code 2) for the fiscal years ending in 1997–1998 and 1998–1999, i.e., for two more years after the initial sample period I focus on manufacturing firms to ensure that the sample is homogenous The choice of a smaller subsample is purely dictated by the requirements of manually collecting data After dropping one firm-year observation due to the non-availability of its 10-K on EDGAR and restricting attention to firms with nonmissing observations, I obtain 394 first-differenced firm-year observations The accounting characteristics such as size, leverage and market-to-book ratio of this subsample are qualitatively similar to those of the overall sample (unreported) and thus the sample is a reasonable representation of COMPUSTAT-CRSP firms For this sample, I obtain data on their foreign currency derivatives and investigate a subset of firms that have either increased or decreased the intensity of their foreign currency hedging Since there are very few initiators or terminators of commodity derivatives in my sample, the change regression is not feasible for commodity hedges In this sample, there are 42 firm-year observations with an increase and 39 firm-year observations with a decrease in the extent of hedging (notional amount of foreign currency derivatives as a percentage of sales) The remaining firm-year observations have no changes mostly due to zero hedging positions across the period In my analysis, I focus on only those observations that have either increased or decreased their hedging positions to avoid inferences based on a majority of firms that remain nonhedgers during the sample period Focussing on the sample of active hedging decisions (increase or decrease) allows for a sharper identification of hedging incentives in response to changes in firm-level variables A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 732 Table Change regression This table presents logistic and OLS regression results based on yearly changes in foreign currency derivatives holding of a sample of manufacturing firms (SIC code 2) for 1996–1997 to 1998–1999 period Changes in derivative holding is regressed on changes in leverage and various other firm characteristics The dependent variable is one if firm increases its hedging intensity as measured by the ratio of foreign currency derivatives to total sales, zero if decreases it Dlev is the change in book leverage over the same year Dlev2 equals the square of Leverage change if leverage has increased over the year and zero otherwise Dlaglev is the previous year’s change in book leverage All other variables used in the regression are based on changes for the corresponding year size represents the log of total sales of the firm quick is the ratio of cash and short-term investments to current liabilities rnd stands for research and development expenses scaled by the sales of the firm fsale represents foreign sales as a percentage of total sales Models and are logistic estimations (slope coefficients evaluated at mean are reported in the table), whereas Models and are OLS (a linear probability model) estimates Number of observations is provided at the end of the table All standard errors are clustered at firm level to account for correlation in error terms of same firms across multiple years Estimate t-Value Estimate t-Value Estimate t-Value Logit Dsize Dlev Dlev2 Dlaglev Dquick Drnd Dfsale N 0.5618 1.0473 À12.0573 (0.75) (1.84) (À2.31) 0.1807 0.5362 À0.0314 (0.75) (0.16) (À0.14) 81 Estimate t-Value 0.2502 0.9642 À8.9984 0.8826 0.0444 0.3883 0.0339 (0.48) (1.92) (À2.69) (2.34) (0.91) (0.11) (0.16) OLS 0.4991 1.3793 À15.5853 1.2980 0.1849 0.1901 0.1794 81 (0.76) (2.14) (À2.72) (2.40) (0.78) (0.06) (0.64) 0.4124 0.8890 À9.5736 (0.68) (1.81) (À3.01) 0.0554 0.7597 À0.0532 (1.03) (0.20) (À0.26) 81 81 I conduct logit as well as OLS regressions (linear probability model) to estimate the impact of changes in leverage on changes in derivatives holding The following model is estimated: X Dhedgej;t ¼ a0 þ a1 Dlevj;t þ a2 Dlev2j;t þ aControl j;t þ j;t (9) The dependent variable takes a value of one for an increase in hedging and zero for a decrease; Dlevj;t measures the change in leverage of firm j in year t; ½DðlevÞj;t Š2 is the squared change in leverage for firms with an increase in leverage, zero otherwise, and all other control variables in the model are also first differenced I include all control variables in this model that enter the earlier regression except for variables that are unlikely to change much on a yearly basis, namely the industry concentration ratio and institutional shareholdings Including these variables in the model does not change any results The results are provided in Table In the first (logit) and third (OLS) model, I find that firms with a moderate increase in leverage are more likely to increase their hedging positions as evident by a positive and significant coefficient on Dlev In contrast, firms with a very high increase in leverage are more likely to decrease their hedging intensities Though the statistical significance of the coefficient on Dlev is weaker as compared to the cross-sectional case, it remains significant at the 7% level The coefficient on Dlev2 on the other hand remains statistically strong (at 2%) as in the cross-sectional case In fact changes in leverage remain the most significant determinant of changes in hedging intensities as compared to other covariates that enter this model Ex-ante vs ex-post incentives: Due to data limitations, my base model is estimated with cross-sectional data At any given point in time, an empiricist observes a firm’s hedging position, which is a mixture of both ex-ante and ex-post actions (i.e., hedging decisions taken before/together with debt issuance and those taken after debt issuance) Ex-ante theory predicts a positive association between leverage and hedging, whereas ex-post theory predicts a nonmonotonic relation In cross-sectional data, therefore, ex-ante decisions bias my study against finding a nonmonotonic relation This happens because, if all decisions are taken ex-ante, then the hedging motivations should be strongly positively associated with leverage even at very high levels of leverage, making the task of finding a negative relation between the two variables harder Though change regressions lead to an improvement over the cross-sectional model, it is still possible that changes in leverage and hedging occur at the same time in the spirit of ex-ante hedging theory To analyze this issue further, I regress Dhedgej;t on Dlevj;t ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 733 along with Dlevj;tÀ1 (i.e., the lagged value of leverage change) and other control variables used in the earlier regression Thus, I regress innovations in hedging intensities on contemporaneous as well as lagged changes in leverage While the contemporaneous change in leverage contains a mix of ex-ante and ex-post decisions, the coefficient on the lag change can be reasonably attributed to the hedging decisions consequent to debt issuance Given that we not observe reporting of hedging at a high frequency, establishing a link between past leverage change and current change in derivatives based on annual data is a challenging task Still, the results from columns and in Table show that the past year’s leverage change is a significant predictor of the current year’s hedging activities The coefficient on the lagged value of leverage change (i.e., Dlaglev) is positive and significant at the 2% level This is encouraging since, due to the sequence of decision making (i.e., last year’s leverage decision and this year’s hedging), this specification is very likely to detect causation between the hedging and leverage variable Conclusion This paper develops a theory of corporate risk management in the presence of financial distress costs By distinguishing ‘‘financial distress’’ from ‘‘insolvency’’, I provide a justification for the ex-post riskmanagement behavior of the firm Due to financial distress costs, the shareholders engage in ex-post riskmanagement activities even without a pre-commitment to so The theory is based on a trade-off between shareholders’ risk-shifting incentives due to equity’s limited liability and their risk-avoidance incentives due to financial distress costs I obtain a closed-form solution for the optimal level of investment risk based on this trade-off The model generates several testable predictions It predicts a nonmonotonic relation between leverage and hedging and a U-shaped relation between financial distress costs and hedging Financially distressed firms in highly concentrated industries are predicted to have higher hedging incentives I test the key predictions of my model with one of the most comprehensive samples used in the literature I model a firm’s leverage and hedging in an endogenous framework using a sample of more than 2,000 nonfinancial firms I find evidence in support of a positive relation between leverage and foreign currency and commodity hedging Consistent with the theory, this relation becomes negative for firms with very high leverage Financially distressed firms in highly concentrated industries hedge more Finally, I show that the key results remain similar for a dynamic analysis based on a change regression for a smaller subset of firms Appendix A In the theoretical model of the paper I consider a continuous trading economy with a time horizon ½t0 ; TŠ and filtered probability space ðO; ð t Þ; ; PÞ satisfying the usual regularity conditions I assume a complete and arbitrage-free market This guarantees the existence of an equivalent martingale measure Q I assume a deterministic short interest rate process given by r In what follows, I denote the indicator function of an event X by 1fX g I assume that the unlevered asset value of the firm can be expressed as a Q-Brownian Motion (under a martingale measure) as follows: dAt ¼ rAt dt þ sAt dW t A.1 Proof of Proposition Proof Shareholders’ payoff (CF) on the terminal date ðTÞ is given by CF T ¼ ðV T À LÞ1fV T 4L;mT 4Kg þ ðf ðV T Þ À LÞ1ff ðV T Þ4L;mT pKg ¼ ðV T À LÞð1fV T 4Lg À 1fV T 4L;mT pKg Þ þ ðf ðV T Þ À LÞ1ff ðV T Þ4L;mT pKg ¼ ðV T À LÞ1fV T 4Lg À ðV T À LÞ1fV T 4L;mT pKg þ ðf ðV T Þ À LÞ1ff ðV T Þ4L;mT pKg ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 734 ¼ ðV T À LÞ1fV T 4Lg À ðV T À LÞ½1ff ðV T Þ4L;mT pKg þ 1ff À1 ðLÞ4V T 4L;mT pKg Š þ ðf ðV T Þ À LÞ1ff ðV T Þ4L;mT pKg ¼ ðV T À LÞ1fV T 4Lg À ðV T À f ðV T Þ1ff ðV T Þ4L;mT pKg þ ðL À V T Þ1ff À1 ðLÞ4V T 4L;mT pKg ¼ ðV T À LÞ À ðV T À f ðV T Þ1ff ðV T Þ4L;mT pKg þ ðL À V T Þf1fV T pLg þ 1ff À1 ðLÞ4V T 4L;mT pKg g Under mild technical restrictions, the equity value at t ¼ t1 ðxt1 Þ is simply the expectation of this payoff under the martingale measure Taking the expectation of the terminal payoff gives the desired result & A.2 Equity valuation As shown in expression in the paper, the equity valuation is given by the following: Et1 ¼ E Q ½ðAT À LÞf1fAT 4L;mT 4Kg þ 1fAT 4L;AT pLþM;mT pKg g þ M1fAT 4LþM;mT pKg Š ¼ E Q ½ðAT À LÞf1fAT 4L;mT 4Kg þ 1fAT pLþM;mT pKg À 1fAT pL;mT pKg g þ M1fAT 4LþM;mT pKg Š ¼ E Q ½ðAT À LÞf1fAT 4Lg À 1fAT 4LþMg þ 1fAT 4LþM;mT 4Kg g þ M1fAT 4LþM;mT pKg Š ðA:1Þ The first two components of the equity value, namely, E Q ½ðAT À LÞ1fAT 4Lg Š and E Q ½ðAT À LÞ1fAT 4LþMg Š, can be computed using the standard Black-Scholes formula for the valuation of European call options Let F and f stand for the normal cumulative density function (cdf) and probability density function (pdf), respectively For notational simplicity I set At1 ¼ A0 and T ¼ T À t1 Then the first two terms result in the following expression: E Q ½ðAT À LÞ1fAT 4Lg Š ¼ A0 Fðh1 Þ À LFðh2 Þ and E Q ½ðAT À LÞ1fAT 4LþMg Š ¼ A0 Fðd Þ À LFðd Þ, where   A0 s2 ln þ T0 L pffiffiffiffiffi h1 ¼ s T0 and  A0 s2 ln þ T0 LþM pffiffiffiffiffi d1 ¼ s T0 pffiffiffiffiffi h2 ¼ h1 À s T ,  and pffiffiffiffiffi d ¼ d À s T The last two terms, i.e., E Q ½ðAT À LÞ1fAT 4LþM;mT 4Kg Š and E Q ½M1fAT 4LþM;mT pKg Š, require the knowledge of the joint distribution of the running minima and the terminal value of the geometric Brownian motion Such distributions have been widely used for the pricing of path-dependent options I use the following lemma (see Harrison, 1985 or Musiela and Rutkowski, 1998) to obtain the expression for the valuation of these two path-dependent expressions: Lemma Let KoL and KoA0 , then the joint density of the terminal asset value ðAT Þ and the running minima of the geometric Brownian motion ðmT Þ, under the martingale measure, is provided by the following formula:  2   A0 s2 K s2   ln ln À T0 À T0 B B L C A0 L C C C À A0 FB pffiffiffiffiffi0 QðAT 4L; mT XKÞ ¼ FB (A.2) pffiffiffiffi0 @ A @ A K s T s T ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 735 Using (A.2), I compute the expectation of the last two terms of expression (A.1) as follows:   A0 L Fðc2 Þ , E Q ½ðAT À LÞ1fAT 4LþM;mT 4Kg ފ ¼ fA0 Fðd Þ À LFðd Þg À KFðc1 Þ À K E Q ½M1fAT 4LþM;mT pKg ފ ¼ A0 M Fðc2 Þ, K where d and d are as given before and   K2 s2 ln þ T0 A0 ðL þ MÞ pffiffiffiffiffi c1 ¼ and s T0 pffiffiffiffiffi c2 ¼ c1 À s T Collecting the above results and simplifying the expressions further, I get the following expression for the valuation of the firm’s equity at t ¼ t1 :   A0 ðL þ MÞ Fðc2 Þ (A.3) Et1 ¼ fA0 Fðh1 Þ À LFðh2 Þg À KFðc1 Þ À K A.3 Proof of Proposition At t ¼ t1 ; the shareholders choose an optimal risk level such that it maximizes the equity value given in expression (A.3) I am assuming that the firm is not in financial distress at t ¼ t1 , i.e., KoA0 I also assume that the distress barrier is below the face value of debt, i.e., KoL At the optimum: qEt1 ¼ qs Differentiating expression (A.3) gives the following:     qEt1 qh1 qh2 qc1 A0 L qc2 A0 M qc2 fðc2 Þ fðc2 Þ ¼ A0 fðh1 Þ À Lfðh2 Þ À À Kfðc1 Þ þ K K qs qs qs qs qs qs (A.4) Note that: A0 fðh1 Þ, L K2 fðc1 Þ fðc2 Þ ¼ A0 ðL þ MÞ fðh2 Þ ¼ ðA:5Þ Eqs (A.4) and (A.5) lead to       qEt1 qh1 qh1 pffiffiffiffiffi0 qc1 KL qc1 pffiffiffiffiffi0 fðc1 Þ ¼ A0 fðh1 Þ À A0 fðh1 Þ À T À Kfðc1 Þ À À T qs qs qs qs ðL þ MÞ qs   KM qc1 pffiffiffiffiffi0 fðc1 Þ þ À T ðL þ MÞ qs Simplification leads to the following:      pffiffiffiffiffi qEt1 qc1 KL qc1 pffiffiffiffiffi0 KM qc1 pffiffiffiffiffi0 ¼ A0 fðh1 Þ T À fðc1 Þ K À À T À T À ðL þ MÞ qs qs qs ðL þ MÞ qs Thus, I have the following first-order condition for the optimal investment risk of the firm, pffiffiffiffiffi pffiffiffiffiffi A0 fðh1 Þ T À fðc1 ÞK T ¼ (A.6) ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 736 Thus, A0 fðh1 Þ À Kfðc1 Þ ¼ (A.7) A.4 Second-order condition Differentiating expression (A.6) gives the second-order optimality condition: pffiffiffiffiffi qh1 pffiffiffiffiffi qc1 q2 Et1 þ c1 fðc1 ÞK T ¼ Àh1 A0 fðh1 Þ T qs qs qs Using the first-order condition and simplifying, the above expression, at the optimum, reduces to (A.8) pffiffiffiffiffi qh1 pffiffiffiffiffi qc1 q E t1 þ c1 fðc1 ÞK T ¼ À h1 fðc1 ÞK T qs qs qs  pffiffiffiffiffi c  pffiffiffiffiffi pffiffiffiffiffi h1  0 ¼ fðc1 ÞK T Àh1 T À þ c1 T À s s pffiffiffiffiffi pffiffiffiffiffi fðc1 ÞK T ðh1 À c1 Þðh1 þ c1 À s T Þ ¼ s Using the following equalities,     A ðL þ MÞ K2 ln ln pffiffiffiffiffi LðL þ MÞ K L pffiffiffiffiffi0 pffiffiffiffiffi ðh1 À c1 Þ ¼ 40 and ðh1 þ c1 À s T Þ ¼ o0, s T s T0 it follows that q2 Et1 o0 qs2 Thus, the second-order condition for the maximization problem is satisfied (A.9) A.5 Comparative statistics Comparative statistics are obtained by a direct differentiation of the optimal solution for s given in expression (a) Sensitivity with respect to default boundary (K): (  !)  qðs2 Þà K2 K 2L   ln ¼ þ ln o0 LþM qK LðL þ MÞ A2t ðL þ MÞ T K ln L The inequality follows from the facts that KoL; KoAt1 and M40 (b) Sensitivity with respect to time-to-maturity ðT ¼ T À t1 Þ: qðs2 Þà ðs2 Þà ¼ À o0 qT T0 (c) Sensitivity with respect to the deadweight loss parameter M: Direct differentiation of the optimal investment risk leads to the following: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnðAt =KÞ lnðL=KÞ qðs2 Þà À L 40 if M4Lexp qM ARTICLE IN PRESS A Purnanandam / Journal of Financial Economics 87 (2008) 706–739 737 (d) Sensitivity with respect to leverage: Let X ¼ lnðK =LðL þ MÞÞ, Y ¼ lnðK L=A2t ðL þ MÞÞ and Z ¼ ln ððL þ MÞ=LÞ Denote leverage by lev ¼ L=At1 After some algebra it can be shown that       qs2 T qK qM 1 Y qK qM ¼ À ÀA þ þ À þA À ql K ql ql L þ M L LþM Z K ql ql L þ M L LþM   X qM AM XY À  À Z ql L þ M LðL þ MÞ Z In this model, leverage affects investment risk via its affect on distress boundary ðKÞ and deadweight loss à à parameter ðMÞ Consider K ¼ À expÀ0:1 lev and M ¼ À exp2 lev This specification corresponds to a concave distress boundary A highly levered firm faces higher distress boundary, meaning such firms are more likely to get into distress and become the target of predatory behavior of their rivals even for relatively small additional downturn in their financial health The specification of M corresponds to a model in which higher leverage imposes higher costs on the firm again due to reasons such as lost customers to rivals The model has been solved for different values of leverage using the analytically derived formula for s For this specification, I set debt to one and vary the asset value to obtain different levels of leverage in the model T has been set to The results are plotted in Fig and show the nonmonotonic relation between leverage and investment risk I also use several other parametric specifications on K and M and obtain similar results A.6 Tax-convexity measure I use the methodology suggested by Graham and Smith (1999) to measure the tax-convexity incentive of hedging Using the simulation methods and considering the 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