# Dynamic risk management: investment, capital structure, and hedging in the presence of financial frictions.

ABSTRACTThis article develops a dynamic risk management model to determine a firm's optimal risk management strategy. This strategy has two elements. First, for low-leverage values, the firm fully hedges its operating cash flow exposure, due to the convexity of its cost of capital. When leverage exceeds a very high threshold, the firm gambles for resurrection and stops hedging. Second, the firm manages its capital structure through dividend distributions and investment. When leverage is low, the firm replaces depreciated assets, fully invests in opportunities if they arise, and distribute dividends, all of these together to achieve its optimal capital structure. As leverage increases, the firm stops paying dividends, while fully investing. After a certain leverage, the firm also reduces investment until it stops investing completely. The model predictions are consistent with empirical observations.

INTRODUCTION

Risk management is a critical issue for all firms. Over the last 15 years, many financial and nonfinancial firms have adopted an integrated approach to measure and manage all their risks, called enterprise risk management (ERM). The definition of risk management is now broader; it includes not only derivatives usage, but also the choice of capital structure, the constitution of cash reserves and lines of credit, the structuring of the insurance portfolio, and sometimes operational policies (see, e.g., Leautier, Rochet, and Villeneuve, 2007; Pettit, 2007; Hyot and Liebenberg, 2011; Paape and Spekte, 2012 and the references they contain).

A rich academic literature (reviewed later in this section) has accompanied this corporate interest in risk management. Numerous articles have identified fundamental financial frictions that justify risk management (e.g., tax shield from debt, bankruptcy and business disruption costs, costly external financing, asymmetry of information between managers/insiders and investors/outsiders), and have derived the optimal risk management strategy, given these frictions.

This article attempts to better relate corporate practice and finance theory of risk management. It proposes a reduced form model that represents what managers of large publicly traded firms actually do. As reported by Graham and Harvey's (2001) survey, chief financial officers (CFOs) are agnostic as to the origin of financial frictions. They simply observe that the weighted average cost of capital (WACC) is a U-shaped function of the firm's leverage ratio (Graham and Harvey, 2001; Cohen, 2004; Pettit, 2007, pp. 110-111, 141-159). Taking this observation as given, they make capital budgeting, dividend distribution, and hedging decisions to maximize the expected net present value (NPV) of the free cash flows, facing uncertainty about both future cash flows and future investment opportunities, and the possibility of bankruptcy. This representation of managerial decision-making cannot be derived from micro-foundations. Yet, it provides valuable insights for it captures most of the real features of corporate decision-making; hence, the analysis' predictions can be compared against actual firms' behavior, as captured by previous empirical studies.

This article first determines analytically the optimal risk management strategy, that is, the mix of hedging, dividend distribution, refinancing, and investment policies. Second, it illustrates the optimal strategy for a "representative" industrial firm, using estimates of the main model parameters. Finally, it shows that the model's predictions are consistent with empirical observations.

The main result of this article is the optimal risk management strategy, which is surprisingly simple. First, dividend distribution and investment jointly follow four regimes (Proposition 1). For low leverage, the firm enjoys full financial flexibility: it fully finances its investment needs and distributes dividends to reach its optimal leverage ratio. For intermediate leverage, the firm faces financial tightness: it still fully finances its investment needs but no longer distributes dividends, as leverage increases from one period to the next. For higher leverage, the firm faces a financial constraint: it is no longer able to fully finance its investment needs. The portion it finances is determined to reach a target leverage, after which the marginal value of investing becomes negative. Finally, for high leverage, the firm faces financial hardship: it is no longer able to finance any of its investment needs, not even depreciation.

Second, full hedging is optimal unless leverage gets higher than some threshold, in which case gambling for resurrection becomes optimal (Proposition 2). These results differ from previous work (e.g., Bolton, Chen, and Wang, 2011; Rochet and Villeneuve, 2011), who find that when the firm's cash reserve (or cash-to-capital ratio in Bolton et al., 2011) is high enough, the firm becomes risk neutral and, since hedging is costly, stops hedging. In our model, the tax shield drives the concavity of the value function; hence, the optimality of full hedging. By choosing leverage as the state variable, we are able to capture the tax shield from debt, a real effect absent from Rochet and Villeneuve (2011) and Bolton et al. (2011).

An essential finding of the analysis is that if the firm's expected profitability is lower than a threshold (function of the magnitude of its investment opportunity), the firm does not exhaust the benefits of the tax shield; rather, it keeps an equity cushion (Proposition 3): the optimal leverage target is lower than the static optimum that minimizes the cost of capital.

The analysis enables us to estimate the value of hedging: a firm that fully hedges its risk can increase its optimal leverage, which results in lower cost of capital, hence, higher value.

The implications of the model are consistent with previously reported empirical findings. First, Proposition 1 predicts that, ceteris paribus, dividend distribution decreases when firms are less profitable on average or face higher investment opportunities, which is confirmed by Fama and French (2002). Second, the importance of corporate taxes in the decision to hedge (Proposition 2) is confirmed empirically by Graham and Rogers (2002). Third, Graham (2000) reports that even profitable firms with low expected cost of financial distress hold an equity cushion, consistent with our model (Proposition 3). Finally, the numerical example developed in the article suggests the value of hedging is around 5 percent on average, consistent with Allayanis and Weston (2001).

As mentioned earlier, this article builds on a rich theoretical and empirical literature. Most of the results hinge on the fact that the firm's value is a concave function of leverage with a unique maximum. In this model, the concavity of this function is caused by the convexity of the expected return required by investors, itself motivated by empirical considerations. Three families of theoretical models also produce value functions of similar shape.

The first family of models develops the trade-off theory of capital structure (Leland, 1994, 1996; Leland and Toft, 1996): optimal capital structure trades-off the tax advantage of debt against the direct and indirect bankruptcy costs. Since interest are tax deductible, debt creates a tax shield. While there is no consensus on the appropriate discount rate for the tax shield (Welch, 2008, pp. 504-507), all agree that the value of the tax shield; hence, the value of the firm, increases with the leverage. On the other hand, as leverage increases, so do the probability of bankruptcy and business disruption costs. In addition, firms may engage in risk-shifting behavior, choosing riskier strategies that benefit shareholders at the expense of bondholders. Incorporating all these effects, Leland (1996) derives the value of risky debt, the value of equity, and the optimal capital structure and risk management strategy.

The second family of models relies on costly external financing. The interaction between costly external financing, underinvestment, and risk management was first modeled in a two-period environment by Froot, Sharfstein, and Stein (1993) and Froot and Stein (1998). The former considers a firm facing random cash flows, random investment opportunities, and convex cost of external financing. At the optimum, the firm fully hedges if cash flows and investment opportunity are uncorrelated, and reduces its hedging as the correlation between both sources of uncertainty increases. The latter introduces capital structure as a risk management device. A marginal increase in equity raises the firm's capacity to pursue risky investments. On the other hand, it generates deadweight costs, arising, for example, from the tax deductibility of interest payments. The optimal equity level balances these two effects.

More recent articles have examined this issue in multiperiod models, for example, Mello and Parsons (2000) and Rochet and Villeneuve (2011). The latter develops an infinite-horizon, continuous-time model, where a constant-size firm faces exogenous cash flow shocks and stringent financial frictions: the firm is liquidated as soon as its cash reserve becomes negative. At each instant, the firm selects its dividend payment and decides its hedging ratio or insurance coverage for discrete risks. Rochet and Villeneuve restate the risk management problem as an inventory management problem, in which the cash reserve is the state variable, and dividend payment and risk transfer decisions are the control variables. They then show that the firm pays dividends if and only if the cash reserve exceeds a threshold, and it fully hedges if the cash reserve is below the threshold. In addition, they show that the firm insures small risks but not large ones.

Bolton et al. (2011) extend Rochet and Villeneuve (2011), most notably by including investment and growth, and less stringent financial frictions; that is, refinancing is possible, albeit costly. They first characterize the optimal dividend distribution, investment, and refinancing policies. Rochet and Villeneuve (2011) and Bolton et al. (2011) find that the firm optimally distributes dividends if and only if its cash reserve (as a percentage of its size) exceeds a given threshold. They also prove that the optimal investment policy is to set the marginal cost of adjusting physical capital equal to the ratio of the marginal Tobin's q over the marginal cost of financing, a departure from the Modigliani and Miller rule, which is to equalize the marginal cost of physical capital to the marginal Tobin's q. Finally, they determine the optimal hedging policy that balances the marginal benefits and costs of hedging.

A third family of articles has provided a micro-foundation for financial frictions and concave value functions. In a striking reversal of Miller and Modigliani propositions (1958, 1963), Holmstrom and Tirole (2000) and Tirole (2006) prove that information asymmetry between managers/insiders and investors/outsiders limits the ability of firms to raise external fund; thus, not all value-creating firms or projects are financed: profitable--but cash-constrained--firms may not be able to refinance themselves after a negative shock to their cash flow, hence, may go bankrupt; firms with insufficient internal funds may have to forego profitable investment opportunities, an issue known as the underinvestment problem.

A series of articles have then expanded Holmstrom and Tirole (2000) model: they have derived dynamically optimal financial contracts resolving the underlying repeated moral-hazard problem between managers/insiders and investors/outsiders, and explored their implementation through standard securities (see Biais, Mariotti, and Rochet, 2011, for a synthetic model and literature review). Rochet and Villeneuve (2011) and Biais et al. (2007) introduce cash reserve as a state variable, since the limited liability constraints of the manager/insider and financier/ outsider impose that negative operating cash flow cannot be financed by the agents, and prove that the market- to-book ratio of equity is a concave function of cash reserves, with a unique maximum. (1)

In our model, we allow intermediate refinancing: when the firm's free cash flow is negative, it can borrow to finance the shortfall. Hence, the cash reserve in Biais et al. (2007) is replaced in our model by the borrowing capacity, held outside of the firm, not inside. With this reinterpretation, the shape of the market-to-book ratio of equity with respect to cash reserves derived by Biais et al. translates into the shape of market-to-book ratio with respect to leverage.

While the second and third families of articles provide significant conceptual contributions, they do not capture essential real-world features. First, corporate taxes and leverage are absent from the analysis, even though they play an important role in corporate decision making, as illustrated, for example, by Graham and Rogers (2002) econometric analysis of firms' determinants of hedging. Second, as observed in Graham and Harvey's (2001) survey, financial executives and managers at large, publicly traded firms use the NPV of the free cash flows, discounted at the WACC, to make capital budgeting decisions, while managers in these articles maximize the NPV of dividends. Finally, at least a fraction of investment opportunities appears to be stochastic, as was modeled by Froot et al. (1993). Firm's growth is shaped by the availability of investment opportunities as well as by real frictions in adding capital.

We are not aware of any model derived from micro-foundations that incorporates these essential features. Furthermore, micro-founded models rely on information asymmetry, whose parameters are by nature difficult to estimate; hence, these models' predictions are difficult to test empirically (Graham and Harvey, 2001; Graham and Rogers, 2002).

Thus, this article takes a different perspective: we observe, as practitioners do, that the WACC is a convex function of leverage, and consequently derive the optimal risk management strategy consistent with that observation. This model, an extension of Leautier et al. (2007), is strongly grounded in the trade-off theory of capital structure, yet it is consistent with costly external financing and micro-founded models.

This article is structured as follows. The second section presents the model. The third section presents the estimation of the parameters used to illustrate the analysis. The fourth section presents the optimal risk management strategy. The fifth section examines the robustness of the optimal risk management strategy to various changes. The sixth section concludes and discusses future research directions. Throughout the article, proofs are presented in the Appendix, while results' intuition is presented in the main text.

THE MODEL

Timing, Decisions, and Free Cash Flow

For t [greater than or equal to] 0, period (t + 1) runs from dates t to (t + 1). At date t, the firm's invested capital is [I.sub.t], which depreciates at constant rate [delta] during each period. Depreciation during period (t + 1) is therefore [delta] [I.sub.t].

At date t, an investment opportunity arises. The magnitude of the opportunity [i.sub.t] is expressed as a fraction of the invested capital [I.sub.t]. Therefore, the nominal opportunity is [i.sub.t] [I.sub.t]. Senior managers do not know in advance when or whether a new investment opportunity will arise or materialize, as this occurrence depends on a variety of factors, for example, regulatory approval, operational limitations. Therefore, investment opportunities {[i.sub.t] : t [member of] N) are random variables, assumed to be independently and identically distributed.

At date t, the firm makes four decisions. First, it selects its dividend payout [d.sub.t][I.sub.t] for the next period; that is, the firm commits at the beginning of a period to a minimum dividend. As will be discussed later, dividend rates are assumed to be nonnegative, that is,

[d.sub.t] [greater than or equal to] 0. (1)

Second, the firm chooses its investment [g.sub.t][I.sub.t] for the next period. The firm has the possibility to replace partially or completely the capital [delta][I.sub.t] depreciated throughout the period and to invest partially or completely in the opportunity [i.sub.t][I.sub.t], thus,

0 [less than or equal to] [g.sub.t] [less than or equal to] [i.sub.t] + [delta]. (2)

Taking depreciation and investment into account, invested capital at date (t + 1) is

[I.sub.t+1] = (1 + [g.sub.t] - [delta])[I.sub.t].

Third, the firm sets [[eta].sub.t], its hedging ratio to the underlying source of risk, as described in the "Hedging Technology" section. Finally, the firm may distribute extraordinary dividends [[??].sub.t][I.sub.t-1] under conditions described in the "Leverage Dynamics" section.

Throughout period (t + 1), the invested capital It generates stochastic return on invested capital (ROIC) (2) [x.sub.t+i], expressed in percent. The net operating profit after adjusted taxes (NOPAT) realized during period t is [[pi].sub.t+i] = [x.sub.t+1][I.sub.t]. The free cash flow (FCF) is the NOPAT minus the net increase in invested capital [I.sub.t+1] - [I.sub.t] = ([g.sub.t] - [delta]) [I.sub.t]:

[FCF.sub.t+1] = ([x.sub.t+1] - [g.sub.t] + [delta])[I.sub.t].

Hedging Technology

The random variable [Z.sub.t+1] represents the primitive source of uncertainty that affects the ROIC during period (t + 1). For example, for an oil company, [Z.sub.t+1] is the wholesale crude oil price (multiplied by its per period production, less operating costs, taxes, and depreciation, divided by invested capital). Returns on invested capital {[z.sub.t] : t [member of] N) are assumed to be independently and identically distributed (i.i.d.), and independent from the investment opportunities. Since old and new assets are exposed to same underlying risk [Z.sub.t+1], the firm does not diversify its risk through its investments.

We also assume that the firm can costlessly hedge its entire exposure to risk [Z.sub.t+i] at a forward price equal to the expected spot price. This assumption may appear unreasonable. We acknowledge this limitation and consider this simplifying assumption as a first step toward a more general model. We argue in the "Robustness of the Results" section that relaxing this assumption does not significantly modify the structure of the risk management strategy. The ROIC [x.sub.t+1] is then:

[x.sub.t+1] = [[eta].sub.t]E[z] + (1 - [[eta].sub.t])[z.sub.t+1]

E[[x.sub.t+i]] = E[z] for all values of r]t? hedging does not affect expected profitability. However, as expected, hedging modifies the profitability's volatility:

Var([x.sub.t+i]) = [(1 - [[eta].sub.t]).sup.2] Var(z).

We assume that the board of directors, concerned that derivatives are used to speculate, prevent the firm from using derivatives to either (i) increase its exposure, or (ii) become short its underlying exposure. For example, an oil company that buys oil forward, that is, sets [[eta].sub.t] < 0, increases its exposure, as (1 - [[eta].sub.t]) > 1. On the other hand, an oil company that sells more oil forward than its production, that is, sets [[eta].sub.t] > 1, becomes short the underlying risk as (1 - [[eta].sub.t]) < 0. Hence, we assume:

0 [less than or equal to] [[eta].sub.t] [less than or equal to] 1. (3)

In the "Robustness of the Results" section, we consider departure of this hedging constraint to include possible short sales ([[eta].sub.t] < 0).

Leverage Dynamics

The firm's capital structure at date t is a combination of debt [D.sub.t] and equity [E.sub.t]. The firm's leverage ratio is [[lambda].sub.t] = [D.sub.t]/[I.sub.t].

The firm is assumed to not issue shares, nor to hold cash reserves. The first assumption is consistent with empirical evidence: Rajan and Zingales (2003) report that the fraction of gross fixed-capital formation raised via equity (including initial and seasoned equity offerings) in 1999 was only 12 percent in the United States, 9 percent in the United Kingdom and France, 8 percent in Japan, and 6 percent in Germany. The second assumption is obviously not consistent with the data, as firms do hold cash reserves. Again, these simplifying assumptions constitute a first step toward a more general model. The impact of relaxing them is discussed briefly in the "Robustness of the Results" section.

Denoting the after-tax cost of debt by r([[[lambda].sub.t]), the financing flow for period (t + 1) is the after-tax interest payments r([[[lambda].sub.t])[D.sub.t], minus changes in financial structure [D.sub.t+i1 - [D.sub.t], plus anticipated and extraordinary dividends paid during the period ([d.sub.t] + [[??].sub.t+i]) [I.sub.t]:

[FF.sub.t+i] = r([[lambda].sub.t])[D.sub.t] - ([D.sub.t+1] - [D.sub.t]) + ([d.sub.t] + [[??].sub.t+i])[I.sub.t].

Free cash flow equals financing flow for the period; hence,

([x.sub.t+1] - [g.sub.t] + [delta])[I.sub.t] - r([[lambda].sub.t]) [D.sub.l] - ([D.sub.t+1] - [D.sub.t]) + ([d.sub.t] + [[??].sub.t+1)[I.sub.t].

Dividing by It and observing that [I.sub.t+1] = (1 + [g.sub.t] - [delta])[I.sub.t] yields:

[x.sub.t+1] - [g.sub.t] + [delta] = [mu]( [[lambda].sub.t]) - [[lambda].sub.t]

where [mu]([[lambda].sub.t]) = [[lambda].sub.t](l + r([[lambda].sub.t])) is the debt and its interest per unit of invested capital. Isolating [[lambda].sub.t] in the previous equation leads to the following definition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[y.sub.t-1] = [mu]([[lambda].sub.t]) + [g.sub.t] - [delta] + [d.sub.t] - E[z]/1 + [g.sub.t] - [delta] (4)

and [[epsilon].sub.t+1] = [z.sub.t+1] - E[z]/[sigma] is a random variable of mean zero and standard deviation unity.

Extraordinary dividends [[??].sub.t+1] are distributed if and only if there is still an excess cash flow after interest and all outstanding debt has been paid; that is, extraordinary dividends are

[[??].sub.t+1] = max([sigma](1 - [[eta].sub.t])[[epsilon].sub.t+1] (1 + [g.sub.t] - [delta]) [y.sub.t+1,0]),

thus [[LAMBDA].sub.t+1] = max([y.sub.t+1] - [sigma] 1 - [[eta].sub.t]/1 + [g.sub.t] - [delta] [[epsilon].sub.t+1], 0),1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The firm is considered bankrupt whenever [[LAMBDA].sub.t+1] [greater than or equal to] 1, then [[lambda].sub.t+1] is set to 1, and

[[lambda].sub.t+1] = min(max([y.sub.t+1] - [sigma] 1 - [[eta].sub.t]/1 + [g.sub.t] - [delta] [[epsilon].sub.t+1], 0),1). (5)

Equations (4) and (5) shows that [[lambda].sub.t+1] increases with [[lambda].sub.t] and [d.sub.t], and decreases with [epsilon].sub.l+1]. The effect of a higher investment level [g.sub.t] is mixed: it increases (decreases) the leverage when (1 - [[lambda].sub.t] + [X.sub.t+1]) is larger (smaller) than the dividends plus interest payments per unit of invested capital ([d.sub.t] + [[??].sub.t+1] [[lambda].sub.t]r ([[lambda].sub.t])).

Managers' Objective Function

This article assumes that managers of large publicly traded firms maximize [V.sub.t], the total value of the firm at date t, that is, the net present value of the free cash flows.

This assumption differs from Rochet and Villeneuve (2011) and Bolton et al. (2011), who assume managers maximize the value to shareholders, that is, the NPV of dividends, and may appear contrary to managers' fiduciary duties. However, it is consistent with managerial practices: Graham and Harvey (2001) report that 75 percent of the CFOs they surveyed always or almost always use the NPV as the primary capital budgeting method, and that this effect is stronger for larger firms. This objective function is also consistent with ex ante optimization of the value of the firm, as discussed by Welch (2008, pp. 452-454): if a management team announces ex ante it will maximize the value of shareholders at the expense of other stakeholders, another team that commits to maximizing the value for all stakeholders can raise more capital to purchase the assets, hence, replace the former.

Firms use a two-step approach for valuation: cash flows are explicitly computed for the first T periods (usually 5 years), and a continuing value is estimated for cash flows arising from date (T + 1) onward. For t [less than or equal to] T, [V.sub.t]), the value of the firm at date t, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [V.sub.t+1] is the continuing value, D [([[lambda].sub.t]) = (1 + w([[lambda].sub.t])).sup.-1] is the discount factor for the period ]t, t + 1], assumed to be a function of the leverage ratio [[lambda].sub.t], and the conditional expectation [E.sub.t] is taken with respect to the information (3) available at date t. Financial frictions are incorporated in the value of the firm through the cost of capital (4) w([[lambda].sub.t]). Expression (6) is equivalent to the recursive definition:

[V.sub.t] = D([[lambda].sub.t])[E.sub.t][FCF.sub.t+1] + [V.sub.t+1]]. (7)

Discount Rate. The discount rate w([[lambda].sub.t]) is the expected rate of return required by investors. Absent financial frictions, w([lambda]) is constant: the required rate of return (or equivalently, the value of the firm) is independent of the leverage ratio. This is the Miller and Modigliani (1958) irrelevance proposition.

Financial frictions alter this result. First, the tax deductibility of interest payments generates a tax shield, that increases as leverage increases. For example, if leverage is constant and the debt is risk free, this friction can be represented by a linearly decreasing w([lambda]) (see, e.g., Copeland, Weston, and Shastri, 2005; Welch, 2008).

If w([lambda]) was decreasing for all values of [lambda], firms would be all debt financed, which is of course inconsistent with reality. Information asymmetries between insiders/managers and outsiders/investors imply that, as leverage increase, the latter require a higher expected return than under the Modigliani and Miller hypotheses. Tirole (2006, chap. 3), derives this result using a simple and general moral- hazard model where investors/ outsiders must leave rents to a manager/insider to elicit effort. External financing available to the manager cannot exceed a multiple d of her initial net worth A; that is, the cost of expected return required by investors is constant until the leverage reaches d/d+1 (taxes are ignored), then becomes infinite afterward.

Welch (2008) provides a clear discussion of these frictions and illustrates their impact on the discount rate w([lambda]). He shows that including the tax deductibility of interest payments leads to a decreasing w([lambda]), while including information asymmetries leads to convex w([lambda]). (5)

Theory suggests that leverage should be computed using market values. This creates a circularity: market value depends on the cost of capital, which itself depends on market value. For simplicity, only book ratios are used here.

Copeland et al. (2005) confirm empirically the convexity of w([lambda]). They estimate w([lambda]) for a set of companies using the classical WACC formula:

w([lambda]) = (1 - k)ke([lambda]) + [lambda] (1 - [tau])[k.sup.d]([lambda]),

where [k.sup.e]([lambda]) is the expected return on equity, and [k.sup.d]([lambda]) is the expected yield on debt. First, they start from the yield curve for bonds, that is, the relationship between the yield to maturity of 10-year bonds and their ratings. They notice a strong yield increase between BBB3, the last investment grade rating, and BB1, the first speculative grade rating. Second, they estimate the statistical relationship between debt rating (dependent variable) and leverage (independent variable). Combining these two steps provides them with an empirical estimate of [k.sup.d]([lambda]). Third, they compute the cost of equity [k.sup.e]([lambda]) using the capital asset pricing model. Combining these expressions, they compute w([lambda]) for each company in their sample. They graphically show that w([lambda]) is convex and reaches a minimum around the leverage corresponding to a BBB rating, which contradicts empirically the Modigliani and Miller hypotheses (see Figure 15.16 in Copeland et al., 2005).

Firms use a method similar to Copeland et al. (2005) to estimate their cost of capital, and recognize that the discount rate is a U-shaped function of leverage ratio (Pettit, 2007, pp. 110-111, 141-159; Cohen, 2004).

Following the above discussion, we assume that:

Assumption 1: (Shape cf iv and r)

(1) w(x) is a continuous, differentiable and convex function on [0,1] reaching a minimum at [k.sup.*].

(2) The discount factor D(x) = [(1 + w ([lambda])).sup.-1] is a concave function of [lambda] on [0,1],

(3) [lim.sub.[lambda] [right arrow]1 w ([lambda]) = [infinity].

(4) The interest rate r(-)isa continuous, increasing, convex and twice differentiable function of [lambda] with [lim.sub.[lambda] [right arrow]1 r([lambda]) = [infinity].

(5) (D[mu])" - D" [mu] is a positive function (6) of [lambda] on [0,1].

Points 1 and 2 imply that the discount factor D (*) is a continuous function, increasing to the left of [k.sup.*] and decreasing afterward. Point 3 indicates that when leverage reaches 1, the firm is no longer able to access financial mar[k.sup.e]ts and has to declare bankruptcy. To ensure continuity of the value function, the value to existing investors (shareholders and creditors) is then assumed to be equal to zero. Alternatively, we could have assumed the market for corporate control provides an exogenous liquidation value, for example, book value. This extension is discussed in the "Robustness of the Results" section. The same comment also applies to the assumption that [lim.sub.[lambda] [right arrow]1 r ([lambda]) = + [infinity]. Point 4 implies that the debt and its interest per unit of invested capital, [mu] (x), is a nonnegative, increasing and convex function of [lambda] on [0,1]. Point 5 is a sufficient technical condition for the concavity of the relative firm value on some interval [0, b] [subset or not equal to] [0,1], with b < 1.

Continuing Value. To estimate the continuing value, the firm assumes that from date (T + 1) onward, all values remain constant:

* the long-term growth rate of the invested capital is g, (7)

* the ROIC is E[z],

* the leverage ratio is [[lambda].sub.T+1]; hence, the cost of capital is w([[lambda].sub.T+1]).

Assumption 2: (Continuing value parameters)

g - [delta] < w ([[lambda].sup.*]) < E [z].

Since for all s [greater than or equal to] T + 1, [x.sub.S+i] = E[z], [g.sub.s] = g, w([[lambda].sub.s]) = w([[lambda].sub.T+1]), and [I.sub.s] = [(1 + g [delta]).sup.s-(T+1)] [I.sub.T+1], the continuing value is equivalent to the value of a growing perpetuity of the expected free cash flow at date (T + 1), discounted by the cost of capital of the firm at date (T + 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Since w([[lambda].sub.T+1]) [greater than or equal to] w([[lambda].sub.*]), Assumption 2 guarantees that the continuing value is finite and positive.

Assumptions 1 and 2 are sufficient to derive the risk management strategy presented in the "Optimal Risk Management Strategy" and "Robustness of the Results" sections. These assumptions are met for the parameters estimated in the "Data and Estimation" section.

Dynamic Programming Formulation

From Equation (6), and using the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the average Tobin's q [v.sub.t] = [V.sub.t]/[I.sub.t] - is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

with the convention that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because [I.sub.t] does not depend on future control variables, maximizing firm's value [V.sub.t] from date t onward is equivalent to maximizing the average Tobin's q [v.sub.t] = [V.sub.t]/[I.sub.t]. This problem can be cast in terms of a dynamic program. The state of the system at date t is described by the leverage [[lambda].sub.t] and the available investment opportunities [i.sub.t]. The decision variables or controls are the hedging ratio [[eta].sub.t], the dividend payout ratio [d.sub.t], the investment level [g.sub.t], and if necessary the exceptional dividend [[??].sub.t].

Conditional on the state of the firm at date t, the relative value of the firm at date t is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Using the continuing value at time T + 1 and taking t = T as the last decision point, the solution (8) to Equation (10) verifies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11b)

where

[[THETA].sub.t] ([g.sub.t], [[eta].sub.t], [d.sub.t]; [[lambda].sub.t]) = D ([[lambda].sub.t]) (E[z] - [g.sub.t] + [delta] + (1 + [g.sub.t] - [delta]) [E.sub.t] [J.sup.*.sub.t+1]([[lambda].sub.t+1] ([[lambda].sub.t], [g.sub.t], [d.sub.t], [[eta].sub.t]), [i.sub.t] + 1)]) (12)

As illustrated in Equations (11a), (11b), and (12), leverage interacts with refinancing in two ways. First, it impacts the expected return investors require to hold their investment in the firm, hence, the discount rate D ([[lambda].sub.t]). If the firm's leverage is low, increasing the leverage increases the tax shield, hence, reduces the expected return required by investor, for the same free cash flows. If leverage is high, the tax shield effect is overshadowed by a concern that managers/insiders may protect their interests before those of investors/outsiders; hence, the latter requires a higher expected return.

Second, leverage impacts the incremental cost of financing: as leverage increases, incremental refinancing becomes more expensive. Since it does not change the free cash flows, there is no direct impact on the value of the firm. It does, however, impact the value indirectly through next period's leverage: as leverage [[lambda].sub.t] increases, so does the incremental cost of refinancing r ([[lambda].sub.t]), hence, ceteris paribus next period leverage [[lambda].sub.t+1]. The firm may then need more external refinancing. This is the "death spiral," observed in practice.

Data and Estimation

We derive in the next section the optimal risk management strategy for any functions w(x) and r(x) and parameters E[z], g, and [delta] that satisfy Assumptions 1 and 2. To illustrate the analysis on a "representative" industrial firm, we estimate these functions and parameters using a balanced panel of 854 industrial firms (first two digits of The North America Industry Classification System (NAICS) between 20 and 39) with annual data covering the period 1990-2009. In total there are 17,080 firm- year observations. Firm data are taken from Compustat.

Source of Uncertainty

Since the underlying source of uncertainty is not directly observable, we employ the return on invested capital (ROIC) as a proxy for this variable. The ROIC for firm n at the end of the fiscal year t is given by

[ROIC.sub.n,t] = [OIADP.sub.n,t](1 - [tau])/[ICAPT.sub.n,t] (13)

where OIADP (operating income after depreciation) and ICAPT (invested capital total) are items from Compustat, and r is the tax rate, assumed to be constant across firms and years. Following Hennessy and Whited (2007), we select [tau] = 40 percent, corresponding to the average combined federal and state tax for firms in the top tax bracket.

For each firm n, we estimate the average ROIC over the period and its standard deviation across time. Then, to limit the impact of outliers, we use the median values of the resulting distribution for the analysis, as seen on Table 1. ROIC is then assumed to be normally distributed, with expectation E[z] = 8.4 percent and standard deviation [sigma] = 5.8 percent.

The standard deviation of the observed ROIC is probably slightly lower than the standard deviation of the underlying source of uncertainty, as the former includes firms' risk management strategies. We show in the "Robustness of the Results" section that doubling [sigma] does not change the structure of the optimal risk management strategy.

Depreciation

The depreciation rate 8 is taken as the median value of [[delta].sub.t], the cross-firm average annual depreciation rate for year f. Following Eberly, Rebelo, and Vincent (2009), [[delta].sub.t] is computed using the double-declining balance method, reflecting accelerated depreciation in the early years of an asset, that is, [[delta].sub.t] = 2/[L.sub.t] where [L.sub.t] is the average useful life of capital goods for year t (in years), defined as

[L.sub.t] = 1/N [N.summation over (i=1)] [PPE.sub.n,t-1] + [DEPR.sub.n,t-1] + [CE.sub.n,t]/[DEPR.sub.n,t]

where N is the number of firms in the sample, [PPE.sub.n,t-1] is the previous year's book value of property, plant, and equipment (PPEGT item in Compustat), [DEPR.sub.n,t-1] the depreciation expense (DP item in Compustat), and [CE.sub.n,t] is the capital expenditure incurred in the present year (CAPX item in Compustat). As reported in Table 1, the median depreciation rate is estimated at [delta] = 12 percent.

Investment Opportunity and Long-Term Growth

Firms' invested capital increase in every period depends on the arrival (or not) of an investment opportunity. Therefore, we assume that at each period, invested capital grows at an average constant rate g (adjusted for depreciation [delta]) and, with probability p, increases by a quantity m beyond its normal growth rate. This yields the following

panel model for the invested capital [I.sub.n,t+1] of the firm n at time t + 1:

[I.sub.n,t+1] = [I.sub.n,t] (1 + g - [delta] + [m.sub.Bn,t+1] + [H.sub.n,t+1]), (14)

where [{[B.sub.n,t]}.sub.n,t] are i.i.d. Bernoulli variables of parameter p, and [{[B.sub.n,t]}.sub.n,t] are i.i.d. centered normal variables of standard deviation [[sigma].sub.H]. The parameters are estimated using maximum likelihood estimation (MLE). The MLE estimates, with standard errors in parentheses, of the panel model are given in Table 2.

Equation (14) allows us to identify (g - [delta]) only. We use our previous estimate [delta] = 12 percent to estimate g = 14.1 percent. Assuming the firms in our sample always invest to the maximum, comparing equations [I.sub.t+1] = (1 + [g.sub.t] - [delta]) [I.sub.t] and (14) yields i = m + g - [delta] = 14.7 percent.

Cost of Capital and Cost of Debt

We compute the leverage of the firm as the ratio between the book value of liabilities and the total value of the firm as measured by the book value of its assets and liabilities, that is:

[[lambda].sub.n,t] = [D.sub.n,t]/[A.sub.n,t], (15)

where [D.sub.n,t] is the sum of debt in current liabilities (DLC in Compustat) and long-term debt (DLTT in Compustat), and [A.sub.n,t] the total book value of assets (AT in Compustat).

The cost of capital w([lambda]) is computed in a two-step process. In the first step, the cost of capital (WACC) is computed for each firm and for each period using the procedure described in Copeland et al. (2005); that is, the cost of capital for firm n and year t is estimated using the formula

[WACC.sub.nt] = (1 - [[lambda].sub.nt])[k.sup.e.sub.nt] + [[lambda].sub.nt] (1 - [tau]) [k.sup.d.sub.nt], (16)

where [k.sup.e.sub.nt] and [k.sup.d.sub.nt] are the cost of equity and the cost of debt for firm n and year t, and [tau] is the tax rate (40 percent).

The cost of equity [k.sup.e.sub.nt] is computed following the CAPM. Stock monthly-return data come from the Center for Research in Security Prices (CRSP). The Treasury bill rate, taken as risk-free rate, and the expected market return come from the Fama- French data base available through the Wharton Research Data Services (WRDS). The firms' beta at any point in time is estimated by regressing the firm's stock return on that of the market's return based on the previous 5 years of monthly prices.

The expected cost of debt [k.sup.d.sub.nt] is obtained as the sum of the 10-year zero-coupon yield plus the expected yield spread according to the firm's rating. Expected corporate yield spreads based on the company's rating come from Chen, Lesmond, and Wei (2007), while the 10-year risk-free zero-coupon yield is computed following the methodology described in Gurkaynak, Sack, and Wright (2007). The historical rating for a firm is obtained from the long-term rating of S&P available in WRDS. Since not all firms have a rating in the S&P rating database, the resulting data set is decreased to 330 firms.

The cost of capital is thus computed firm by firm on a year basis. Given that yield spreads in Chen et al. (2007) are computed for the period 1995-2003, the estimate for the firm's cost of capital, [WACC.sub.n], is computed as an average of the WACC's year estimates over this period.

The second step is to estimate the function w([lambda]) using [WACC.sub.n]. The functional form proposed for w([lambda]) is

n([lambda]) = ([[beta].sub.0] + [[beta].sub.1] [lambda] + [[beta].sub.2] [[lambda].sup.2])/1 - [lambda], (17)

where [[beta].sub.0], [[beta].sub.1], and [[beta].sub.2] are coefficients to estimate. As required, this functional form for the cost of capital is convex in [lambda] provided that [[beta].sub.0] + [[beta].sub.1] + [[beta].sub.2] > 0 and has an asymptote toward [infinity] at [lambda] = 1 if [[beta].sub.2] > 0. The time-aggregated means of Equations (16) and (15) across firms is then used to estimate [[beta].sub.0], [[beta].sub.1], and [[beta].sub.2] in Equation (17) with the quasimaximum likelihood estimator (QMLE). The resulting estimates are presented in Table 3. With these parameters, the cost of capital is minimized for a leverage value of [[lambda].sup.*] =40.5 percent and has a convex shape as shown in Figure 1.

The promised yield on debt [k.sup.d] is assumed to follow

[k.sup.d] ([lambda]) = [[alpha].sub.0] + [[alpha].sub.1] [lambda]/1 - [lambda],

where [[alpha].sub.0] and [[alpha].sub.1] are coefficients to estimate and r is the tax rate. This function is increasing and monotonic in [lambda] provided that [[alpha].sub.0] + [[alpha].sub.1] > 0, which reflects the fact that debt yield increases with leverage. [[alpha].sub.0] and [[alpha].sub.1] are estimated using the procedure previously described and presented in Table 4. The after tax promised yield used in the analysis is then:

r([lambda]) = (1 - [tau]) [k.sup.d] ([lambda]) = (1 - [tau]) ([alpha].sub.0] + [alpha].suub.1] [lambda])/1 - [lambda]. (18)

OPTIMAL RISK MANAGEMENT STRATEGY

In this section, the optimal hedging, investment and dividends policies are presented. We first start with a numerical example that provides intuition for the general case.

The value function [J.sup.*.sub.t] ([lambda].sub.t], [i.sub.t]) is represented in Figure 2 for both [i.sub.t] = 0 and [i.sub.t] = i. For consistency, all figures are presented for a date t selected "far enough" from the last period T that the value function is numerically stationary, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This criterion translates into (T - t) [congruent to] 20.

As seen in Figure 2, [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) admits a unique maximum: for small values of [[lambda].sub.t], increasing leverage increases the tax shield, hence, the relative value of the firm, whereas for large values of [[lambda].sub.t], increasing leverage decreases the relative value as financial frictions increase the expected return required by investors. We also observe that [J.sup.*.sub.t] ([[lambda].sub.t], i) [greater than or equal to] [J.sup.*.sub.t+1] ([[lambda].sub.t], 0): when [[lambda].sub.t] is small, the relative value of the firm is larger whenever an investment opportunity is present, while when [[lambda].sub.t] is large, the presence of an investment opportunity has no impact on the value of the firm. Figure 2 also illustrates that there exists a critical value of [[lambda].sub.t], denoted [b.sub.t], such that [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) is concave for [[lambda].sub.t] [less than or equal to] [b.sub.t] and convex for [[lambda].sub.t] [less than or equal to] [b.sub.t]. In our numerical implementation, the leverage for which the relative value reaches a maximum is 38 percent when investment opportunity is present, and 40 percent when the investment opportunity is not present. The change in convexity occurs at leverage ratio of 80 percent. The shape of the value function is responsible for the optimal risk management strategy discussed below.

Optimal Investment and Dividend Payments in the Full Hedging Case

Consider first the case [[eta].sub.t] = 1: next-period leverage (5) is completely determined by current-leverage [[lambda].sub.t] and the two other control variables [g.sub.t] and [d.sub.t]. The optimal controls are characterized by:

Proposition 1: (Investment and dividends regimes in the full hedging case) Investment and dividend payout decisions follow four regimes. Indeed, there exist thresholds

[[lambda].sup.(1).sub.t] ([i.sub.t]) [less than or equal to] [[lambda].sup.(2).sub.t] ([i.sub.t]) [less than or equal to] [[lambda].sup.(3).sub.t] [less than or equal to] [[lambda].sup.(4).sub.t],

such that:

(1) For [[lambda].sub.t] [member of] [0, [[lambda].sup.(1).sub.t] ([i.sub.t])], the firm exhibits full financial flexibility: it fully finances its investment needs and distributes dividends to reach its optimal leverage ratio.

(2) For [[lambda].sub.t] [member of] [0, [[lambda].sup.(1).sub.t] ([i.sub.t]), [[lambda].sup.(2).sub.t] ([i.sub.t])], the firm faces financial tightness: it still fully finances its investment needs but no longer distributes dividends.

(3) For [[lambda].sub.t] [member of] ([[lambda].sup.(2).sub.t], [[lambda].sup.(3).sub.t]], the firm faces a financial constraint: it is no longer able to fully finance its investment needs. The portion it finances is determined to reach its target leverage [[??].sub.t+1].

(4) Finally, for [[lambda].sub.t] [member of] ([[lambda].sup.(3).sub.t], 1], the firm faces financial hardship: it is no longer able to finance any of its investment needs, not even depreciation. Its debt ratio increases.

The precise definition of all thresholds is provided in the "Full Hedging Case-- Part 1" section. Thresholds' value corresponding to our estimates of the model's parameters are presented in Table 5. A rigorous proof of the proposition is given in the "Full Hedging Case--Part 1," "The First-Order Conditions," "Value Function at Time t + 1," and "Investment and Dividends Policies in the Full Hedging Case Given Properties of Value Function at Date (t + 1)" sections. Economic intuition is provided below.

Suppose [J.sup.*.sub.t+1] ([[lambda].sub.t+1], [i.sub.t+1]) has the same "shape" as [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]). Since the expectation is taken with respect to [i.sub.t] only, the firm expected relative value [[phi].sub.t+1] (y) = [E.sub.t] [[J.sup.*.sub.t+1] (y, [i.sub.t+1])] also has the same same shape as [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]); that is, [[phi].sub.t+1] (y) is concave on [0, [b.sub.t+1]] and then convex on the remaining part of the unit interval, and admits a unique maximum [[[bar.lambda]].sub.t+1] on [0, [b.sub.t+1]].

Thus, when possible, managers choose the investment policy [g.sub.t] and the dividend rate [d.sub.t] to reach the optimal next-period leverage [[lambda].sub.t+1]. For small values of the current leverage [[lambda].sub.t], this can be accomplished by investing as much as possible and by paying

dividends

[d.sup.*.sub.t] = E [z] + [[[bar.lambda]].sub.t+1] - (1 - [[[bar.lambda]].sub.t+1]) [i.sub.t] - [mu] ([[lambda].sub.t]). (19)

This is the first part of Proposition 1, illustrated as Region I in Figure 3.

The dividend rate [d.sup.*.sub.t] is decreasing as [[lambda].sub.t] increases. If [[lambda].sub.t] exceeds a threshold (denoted [[lambda].sup.(1).sub.t] (x)), the firm can no longer attain [[lambda].sub.t+1] as next period's leverage. Then, as long as the marginal value from increased size exceeds the marginal cost of increased leverage, the firm's managers still invest to the maximum possible, and [[lambda].sub.t+1] increases (Part 2 of Proposition 1, Region II in Figure 3). If [[lambda].sub.t] exceeds another threshold (denoted [[lambda].sup.(2).sub.t] (x)), the marginal value from increased size equals the marginal cost of increased leverage; hence, managers reduce investment to keep leverage constant (Part 3 of Proposition 1, Region III in Figure 3). If [[lambda].sub.t] exceeds another threshold (denoted [[lambda].sup.(3).sub.t]), the firm is so indebted that it is no longer profitable to invest since the marginal cost of increased leverage exceeds the marginal benefit from size (Part 4 of Proposition 1, Region IV in Figure 3).

Equation (19) shows that, ceteris paribus, when dividends are distributed, they increase with expected profits, and decrease with the magnitude of the potential investment opportunity and leverage. Dividend policy is a complex topic, which includes multiple theoretical considerations, ranging from the tax impact of payments to signaling (see, e.g., Welch, 2008). One key finding of all CFO surveys concerning dividend distribution is managers' extreme reluctance to cut dividends (Lintner, 1956; Brav et al., 2005). This model does not claim to fully explain dividend distribution. However, predictions from the model are consistent with the data.

Fama and French (2002) find that, ceteris paribus, when dividends are distributed, they increase with expected profits, and decrease with the magnitude of the potential investment opportunity. Our unreported regression of dividends on leverage and the explanatory variables used in Fama and French (2001) confirm that dividend decrease with leverage. The coefficient of leverage is negative and statistically significant. However, the estimate is much smaller than predicted by the model, due to the stickiness of dividend distributions.

Optimal Hedging

Proposition 2: (Hedging policy) For any t [less than or equal to] T,

(1) The firm fully hedges for [[lambda].sub.t] < [[lambda].sup.(4).sub.t].

(2) The firm gambles for resurrection when [[lambda].sub.t] < [[lambda].sup.(4).sub.t].

A rigorous proof is presented in the "Hedging Policy Given Properties of Value Function at Date (t + 1)" section. Intuition is as follows: if [[eta].sub.t] = 1, [[phi].sub.t+1] (y) is concave in y for y < [[lambda].sup.(4).sub.t+1]. Thus, for [[lambda].sub.t] such that [y.sub.t+1] < [[lambda].sup.(4).sub.t+1], the firm seeks to minimize the volatility of [[lambda].sub.t+1]; hence, [[eta].sub.t] = 1 is indeed optimal. Conversely, for [[lambda].sub.t] such that [y.sub.t+1] > [[lambda].sup.(4).sub.t+1], [[phi].sub.t+1] (y) is convex; hence, [[eta].sub.t] = 1 cannot be optimal. The firm seeks to maximize volatility of [[lambda].sub.t+1]. Since [[eta].sub.t] [epsilon] [0, 1] by assumption, [[eta].sub.t] = 0 is optimal.

Corollary 1: Investment and dividends policies described at Proposition 1 are optimal for leverage [[lambda].sub.t] smaller than [[lambda].sup.(4).sub.t+1].

The hedging decision at date t is driven by the concavity of [[lambda].sub.t+1] (y) = [E.sub.t] [[J.sup.*.sub.t+1] (y, [i.sub.t+1])].

For [[lambda].sub.t] < [[lambda].(4).sub.t], the firm fully hedges. This result differs from Rochet and Villeneuve (2011) and Bolton et al. (2011), who find that, when the firm's cash reserve (or cash-to-capital ratio in Bolton et al., 2011) is high enough, the firm becomes risk neutral and, since hedging is costly, stops hedging. In our model, the tax shield drives the concavity of the value function, hence, the optimality of full hedging. By choosing leverage as the state variable we are able to capture the tax shield from debt, absent from Rochet and Villeneuve (2011) and Bolton et al. (2011).

For [[lambda].sub.t] > [[lambda].sup.(4).sub.t], the firm gambles for resurrection. Since we impose the constraint [[eta].sub.t] [less than or equal to] 0, it selects [[eta].*.sub.t] = 0. The intuition is that if [[lambda].sub.t] > b, then [y.sup.*.sub.t+1] ([[lambda].sub.t]) > [[lambda].sub.t]: hedging leads to certain bankruptcy. Gambling for resurrection is then optimal. A similar effect can be found in Rochet and Villeneuve (2011), where the firm chooses not to hedge a discrete and large risk.

Equity Cushion

Proposition 3: (Equity cushion) For any t [less than or equal to] T,

(1) If the expected ROIC is lower than interest payments (per unit) and the equity fraction of the investment (per unit),

E[z] < [[lambda].sup.*]r ([[lambda].sup.*]) + (1 - [[lambda].sup.*]) i, (20)

then for any t, [[[bar.lambda]].sub.t] < [[lambda].sup.*]; that is, the leverage that maximizes the expected firm relative value [[phi].sub.t] is smaller than the leverage that minimizes the cost of capital w. The difference [[lambda].sup.*] - [[[bar.lambda]].sub.t] > 0 is the equity cushion.

(2) If E [z] [greater than or equal to] [[lambda].sup.*] r ([[lambda].sup.*]) + (1 - [[lambda].sup.*]) i, then [[[bar.lambda]].sub.t] = [[lambda].sup.*]; that is, the firm minimizes the static cost of capital.

Proof. See the "Equity Cushion" section.

We show in the "Equity Cushion" section that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Equation (21) illustrates why minimizing the cost of capital cannot be optimal. If the firm minimizes its cost of capital, that is, sets [[lambda].sub.t] = [[lambda].sup.*], the first term in Equation (21) is equal to zero. However, if an investment opportunity arises at date t, the firm can no longer optimize its capital structure at (t + 1), and selects [y.sup.*.sub.t+1] ([[lambda].sub.t]) > [[[bar.lambda]].sub.t+1], which reduces the continuing value of the firm:

[[phi]'.sub.t] ([[lambda].sup.*]) = pD ([[lambda].sup.*]) (1 + i) [[phi]'.sub.t+1] ([y.sup.*+1] ([[lambda].sup.*])) [partial derivative][y.sup.*.sub.t + 1]/[partial derivative] [[lambda].sub.t] ([[lambda].sup.*], i) < 0.

The firm therefore maintains an equity cushion to protect against that occurrence.

The estimated values of the parameters are such that Inequality (20) holds. Hence, [[[bar.lambda]].sub.t] < [[lambda].sup.*] for all t [less than or equal to] T: in order to maximize value, the firm does not minimize the cost of capital; that is, does not exhaust the benefits of the tax shield. The firm pays the static cost [LAMBDA] = w ([[[bar.lambda]].sub.t]) - w ([[lambda].sup.*]) > 0 to protect financial flexibility. [DELTA] can be interpreted as an insurance cost, or the cost of financial flexibility. This result is consistent with Rochet and Villeneuve (2011) and Bolton et al. (2009), who find that firms optimally hold cash reserves: they accept to pay the opportunity cost of holding these reserves in order to protect their financial flexibility.

This result complements the common wisdom found in the practitioners' literature. First, it confirms that the optimal capital structure is not simply the one that minimizes the cost of capital, as was sometimes incorrectly argued (e.g., Cohen, 2004). Second, it refines qualitative insights. For example, Pettit (2007) argues (pp. 154-155) that the optimal capital structure depends on the growth prospect of the firm: a high growth firm, for example, a tech firm, needs lower leverage than a mature, low-growth firm.

This result is consistent with Graham's (2000) observation that firms do not exhaust the tax advantages of debt. One striking finding in Graham's study is that even profitable, large (hence, diversified), liquid firms with low ex ante distress cost use debt conservatively, as long as they have growth options. This model explains that observation: Condition (20) does not depend on the size of the firm, or on the volatility of returns, since the firm is fully hedged. So, for equally profitable firms, the main driver of debt conservativeness is the magnitude of the anticipated investment opportunity, consistent with Graham (2000).

Condition (20) is reversed if E [z] is much higher and/or i much smaller: highly profitable firms with small investment opportunities do not maintain an equity cushion, rather minimize static cost of capital.

Hedging Premium

Suppose now that for an exogenous reason, the firms does not hedge, that is, set [[eta].sub.t] = 0. We have verified empirically that the shape of value function and the optimal investment and dividend payments strategies are qualitatively unchanged for [[eta].sub.t] = [[eta].sup.*] and [[eta].sub.t] = 0. The intuition is as follows: the logic applied to [[lambda].sub.t+1] = [y.sub.t+1], next period's leverage when firms hedge, applies to [y.sub.t+1], the expected next-period leverage when the firm does not hedge, as long as truncation is not an issue.

The leverage thresholds, however, are slightly lower, as illustrated in Table 5. Comparison of [[lambda].sup.(1).sub.t] in Table 5 shows that the presence of hedging allows the firm to pay dividends for higher leverage. Similarly, hedging implies that full investment remains optimal for higher leverage (see [[lambda].sup.(2).sub.t]) and that the firm continues to invest at a higher leverage (see [[lambda].sup.(3).sub.t]). Furthermore, since the optimal leverage ratio is higher, Equation (19) indicates that dividends are also higher when the firm is allowed to hedge.

The firm's optimal leverage ratio [[[bar.lambda]].sub.t] is higher, or equivalently, the precautionary savings is lower by about 4 percent, when hedging is allowed. This translates into higher relative firm value: the "hedging premium," defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

is presented in Figure 4. The average hedging premium 4.5 percent if an investment opportunity is present, 4.2 percent otherwise; the maximum premium is 19 percent, for [[lambda].sub.t] slightly below 80 percent, and the premium vanishes if [[lambda].sub.t] > 80 percent. When the firm reaches [[lambda].sub.t] slightly below 80 percent, the fully hedging firm's leverage can remain safely below 80 percent; hence, the firm avoids bankruptcy. On the other hand, a firm that does not hedge has roughly 50 percent of crossing the threshold, which may then lead to bankruptcy. Hedging is most valuable to firms that are financially fragile, but not yet desperate enough to gamble for resurrection.

This analysis suggests that firms hedge to take advantage of the tax shield from debt, not of the convexity of the tax schedule, as is suggested, for example, by Smith and Stulz (1985). This confirms the widely held view among practitioners that "hedging is tax advantaged equity" (Pettit, 2007, pp. 206-207). This observation is confirmed empirically by Graham and Rogers (2002): firms hedge to increase their leverage; hence, the tax shield; hence, their value. On their sample, hedging firms increase leverage by 3 percent, and value by 1.1 percent. The magnitude of the "hedging premium" is also consistent with Allayanis and Weston (2001), who estimate that hedging firms' average Tobin's q is 5.7 percent higher than nonhedging firms, and much lower than Carter, Rogers, and Simkins (2006) who estimate the hedging premium for a fully hedged airline would be close to 19 percent. Our estimate is much lower than 19 percent, since we assume that profitability is uncorrelated with investment opportunities. The value we obtain is larger than Bolton et al. (2011), who find only a 1 percent increase in value arising from hedging. Two reasons explain this difference. First, taxes are included in our analysis. Hedging enables the firm to increase leverage, hence, capture tax savings, an effect not included in Bolton et al. (2011). Second, we exclude hedging costs.

ROBUSTNESS OF THE RESULTS

Changes in the Model's Parameters

Higher Volatility. As previously mentioned, our numerical illustrations use the volatility of the actual ROICs, which are postrisk management activities by the firms, as an estimate of the volatility of [Z.sub.t+1]. Therefore, we probably underestimate the true underlying volatility. To correct for this bias, we double the volatility of ROIC.

As presented in Figure 5, the results are almost unchanged. For [[lambda].sub.t] < [[lambda].sup.(4).sub.t], the firm optimally hedges fully; hence, the underlying volatility does not matter. For [[lambda].sub.t] > [[lambda].sup.(4).sub.t], the firm's expected relative value [J.sup.*.sub.t] is higher when the volatility is higher. This is a consequence of the gambling for resurrection: the probability of achieving leverage lower than b (defined in the "Full Hedging Case--Part 1" section); hence, the value of the unhedged firm increases with the underlying volatility.

More Convex WACC. The numerical results we obtain, in particular the hedging premium, depend on our estimate of the WACC. It is extremely difficult to obtain different WACC curves by splitting our sample into subcategories, as we need a large variety of leverage values to obtain meaningful estimates of the WACC. Therefore, to test the robustness of these empirical results, we have constructed a different WACC curve. Specifically, we assume that the firm offers higher expected return, but also carries higher risk. This is represented by a higher [[beta].sub.0], the expected returns on assets, and a lower [[lambda].sup.*], the leverage that minimizes the expected return. We choose the following parameters:

[[beta].sub.0] = 0.13, [[beta].sub.1] = -0.42, [[beta].sub.2] = 0.54,

which produces [[lambda].sup.*] = 36 percent, and a more convex WACC. The resulting critical thresholds are reported on the first row of Table 6. As expected, a more convex WACC produces lower thresholds, and a higher average hedging premium, 7.1 percent if an investment opportunity is present, 6.7 percent otherwise. This analysis confirms that hedging is more valuable if a firm's WACC is more convex.

Changes in the Model's Specification

Equity Issuance and Asset Sales. We expect including these possibilities would not significantly alter the optimal strategy, rather would create an upper bound on feasible leverage.

Consider first equity issuance. As is well documented (e.g., Rajan and Zingales, 2003), equity issuance is associated with a significant reduction in share value, most likely due to a signaling effect. The firm would then issue equity if and only if this loss is compensated by a relative value increase due to lower leverage. Given the shape of the value function, this would likely mean that the firm issues equity if and only if leverage exceeds a given threshold.

The same argument applies to asset sales. If the external asset price is higher than the internal value, the firm sells assets. This is simply arbitrage. The more relevant case is when the external asset price is lower than the internal value. The firm would sell assets and use the proceeds to retire debt if and only if the value loss is compensated by a relative value increase due to lower leverage. Given the shape of the value function, this would again translate into a maximum leverage decision rule.

Cash Reserves. Introducing cash reserves will likely prove more complex, as this would create an additional state variable. The economic intuition is that cash is another asset that produces lower returns than the operating assets, and the firm can costlessly invest in and divest. We thus expect the firm to build cash reserves when leverage is low, and to deplete these reserves when leverage is high to protect the investment program while maintaining an acceptable leverage.

Further work will confirm or modify this intuition.

Hedging Technology. Our analysis assumes that the firm can (1) hedge all its exposure, and (2) do so costlessly. The first assumption is of course not met in practice: no firm can hedge all of its risk. Even if an oil company fully hedges its oil price risk, its operational risk remains. Consider, for example, the case of an airline firm whose main source of uncertainty comes from jet fuel prices. A proportion of the firm's ROIC comes from idiosyncratic risk that cannot be hedged, such as operational performance. If that is the case, only a proportion of the firm's total risk, the one related to jet fuel prices, can be hedged with financial contracts. To accommodate this case in our model, we set the upper limit for the hedging ratio to a value lower than one. To give a numerical example, we suppose that all values of the model remain the same, but we set the maximum value that the hedging ratio can reach to 0.3. The second row of Table 6 presents critical values for this case. When the hedging ratio is constrained to be in the interval [0, 0.3], critical values are lower than those of the base case, yet higher than the no hedging case. Reducing the share of total risk the firm is able to hedge does not change the structure of the risk management strategy, but deteriorates investment and dividend policies.

The assumption of costless hedging is not as unrealistic as it seems. Transaction fees, the variable costs of hedging, are small and not essential to the decision of large industrial firms. In general, these do not buy and sell derivatives repeatedly; hence, they pay transaction fees only once. The collateral firms are required to post when hedging, which is included, for example, in Bolton et al. (2011), constitutes an important cost not included here. Finally, the fixed costs of hedging are significant: hiring of the traders, systems, control, accounting, and so on. However, these do not depend on the transaction volume, so they have a limited impact on the hedging ratio, once the decision to set up a hedging group has been made.

The analyses of the no hedging and partial hedging cases yield some preliminary insights into the optimal solution in a more general hedging environment, where (1) the firm is able to hedge only a portion of its underlying risk, and (2) hedging is costly. We expect the firm's risk management strategy will remain qualitatively unchanged, even though the critical values will change. First, the no hedging case shows that the structure of the dividend/investment strategy is unchanged. Second, the hedging strategy will be altered. The optimal hedging ratio balances the marginal cost of hedging against its marginal benefit. The latter arises from the concavity of the objective function; hence, hedging will be more valuable as the function becomes more concave. If we assume that the main variable cost is the cost of collateral that need to be posted, the cost should be very low for firms with low leverage (a highly rated firm is required to post less collateral), and increase as firms become more levered. Including costs should marginally reduce the hedging ratio for low-leverage firms, and progressively reduce the optimal hedging ratio as firms are more levered. It will lead to gambling for resurrection earlier, as in Rochet and Villeneuve (2011).

Opportunity to Increase Exposure. So far, we have assumed that the firm's risk management policy constraints the hedging ratio between zero and one (see Equation (3)). What would happen if a firm were allowed to buy forward part of its production, that is, to set [[eta].sub.t] < 0? To study this case, we constrain the hedging ratio to be in the interval [-1, 1] while maintaining the other model parameters equal to those of the base case.

The last row of Table 6 shows critical values when [[eta].sub.t] < 0 is allowed. These critical values are unchanged compared to the base case. This result hinges on the fact that the region in which the firm hedges is practically the same as the one of the base case. The difference comes once the firm decides not to hedge, as the optimal solution in the new case is the lower bound established for the hedging ratio: the firm sells forward all its production by setting the hedging ratio to -1. In this region, and similar to what was observed in Figure 5, the relative firm value increases respect to the value found in the base case.

CONCLUDING REMARKS

This article develops a dynamic model to determine a firm's optimal risk management strategy that, consistent with firms' reported practices, faces a required rate of return convex as a function of leverage. The risk management strategy has two elements. First, the firm manages its capital structure through dividend distributions and investment. When leverage is very low, the firm fully replaces depreciated assets, fully invests in opportunities if they arise, and distributes dividends to reach its optimal capital structure. As leverage increases, the firm stops paying dividends, while fully investing. After a certain leverage, the firm also reduces investment, until it stop investing completely. Second, until leverage is very high, the firm fully hedges its operating cash flow exposure, due to the convexity in its cost of capital. When leverage exceeds a very high threshold, around 80 percent with the parameters we have estimated, the firm gambles for resurrection. The model predictions are consistent with empirical observations.

This analysis can be enriched by several additions. First, a more complex capital structure including equity issuance could be examined. Currently, financing needs are met by debt issuance, and excess cash is used to pay down debt, until all debt is repaid. Similarly, sale of assets as a means to reduce leverage could be introduced.

Second, we would like to examine richer hedging environment. Multiple risk factors, not all hedgeable, could be included. Except for commodity producers, very few firms face a single risk factor. Determining the overall hedging strategy and, in particular, the trade-offs between hedging the different risks would be very valuable. Similarly, nonlinear hedging strategies could be examined. Senior managers and boards are often reluctant to hedge their risk using forward contracts, as it deprives them of the potential upside, should the output price go up (or the input price go down). Investing in options might prove more acceptable to them, if the value was clearly identified.

Third, we would like to introduce cash reserves as another state variable. In this model, firms are assumed to always be able to finance themselves on external markets, albeit at an increasing cost. Holding cash reserves would enable firms to lower the debt increase required to finance their investment.

These extensions would render the analysis closer to the reality and choices faced by firms, hence, would provide clear and practical guidance as firms strive to define their risk management strategy.

Finally, we would like to continue to test empirically the main the implications from our analysis, that is, the relationship between dividend and leverage, the equity cushion, and the gambling for resurrection. Throughout the "Optimal Risk Management Strategy" section, we have already mentioned empirical evidence coherent with these findings. One potentially promising avenue for further work would be to exploit the cross-sectional variation across industries, in particular, the difference in growth rates and profitability, to better understand these effects.

APPENDIX: PROOFS OF PROPOSITIONS 1-3

The full hedging case is discussed in the first part of the appendix. The first- order conditions are derived in the second part of the appendix. The induction hypotheses on the shape of [[phi].sub.t+1] are formally defined in the third part of the appendix. The optimal controls, given the induction hypotheses, are derived in the fourth and fifth parts of the appendix. The induction hypotheses for [[phi].sub.t] are proved in the sixth and seventh parts of the appendix. The equity cushion is derived in the eighth part of the appendix.

Full Hedging Case--Part 1

We first examine the full hedging case, that is, [[eta].sub.t] = 1. To maximize the value function [THETA] ([[lambda].sub.t], [g.sub.t], [d.sub.t], [[eta].sub.t]) given by Expression (12), we compute the first partial derivatives of [THETA] and hence find critical values that are candidates for the optimal solution. General expressions are presented at Equations (A2)-(A5) of "The First-Order Conditions" section. The full hedging case provides simpler expressions for the partial derivatives:

[partial derivative][[THETA].sub.t]/[partial derivative][d.sub.t] ([[lambda].sub.t], [g.sub.t], [d.sub.t], [[eta].sub.t] = 1) = D ([[lambda].sub.t]) [[phi]'.sub.t+1] ([y.sub.t+1]), (A1a)

[partial derivative][[THETA].sub.t]/[partial derivative][g.sub.t] ([[lambda].sub.t], [g.sub.t], [d.sub.t], [[eta].sub.t] = 1) = D ([[lambda].sub.t]) [[phi].sub.t+1] ([y.sub.t+1]), (A1b)

where [y.sub.t+1] ([[lambda].sub.t], [g.sub.t], [d.sub.t]) is the expected leverage defined in Equation (4), which is equal to [[lambda].sub.t+1] since [[eta].sub.t] = 1,

[[phi].sub.t+1] (y) = [E.sub.t] [[J.sup.*.sub.t+1] (y, [i.sub.t+1])]

is the firm's relative expected value, viewed as a function of next period's expected leverage, and

[[phi].sub.t+1] (y) = [[phi].sub.t+1] (y) - 1 + (1 - y) [[phi]'.sub.t+1] (y).

The risk management strategy is derived using a backward induction argument on the shape of [[phi].sub.t+1]. Suppose that [[phi].sub.t+1] is concave on [0, [b.sub.t+1]] (Induction Hypothesis 1) and reaches a unique maximum denoted [[[bar.lambda]].sub.t+1] [member of] (0, [b.sub.t+1]) such that [[phi].sub.t+1] ([[[bar.[lambda]].sub.t+1]) > 1 (Induction Hypothesis 2). Under these assumptions, we derive the optimal controls ([g.sup.*sub.t], [d.sup.*sub.t], [[eta].sup.*sub.t]). Finally, we prove that, given these controls, [[phi].sub.t] satisfies Induction Hypotheses 1 and 2 on [0, [b.sub.t]]. The intuition for the optimal controls is presented below, while the verification of Induction Hypotheses 1 and 2 is presented in the "Investment and Dividends Policies in the Full Hedging Case Given Properties of Value Function at Date (t + 1)" and "Hedging Policy Given Properties of Value Function at Date (t + 1)" sections.

Suppose [[phi].sub.t+1] satisfies Induction Hypotheses 1 and 2 on [0, [b.sub.t+1]]. Corollary 2 of the "Value Function at Time t + 1" section shows that there exists a unique [[??]sub.t+1] [member of]

([[bar.[lambda]].sub.t+1], [b.sub.t+1]) such that [[phi].sub.t+1] is positive if and only if y < [[??].sub.t+1]. These critical values [[bar.[lambda]].sub.t+1] < [[??].sub.t+1] then lead to different thresholds that determine hedging, investment, and dividends policies.

Definition 1: Define [[lambda].sup.(1).sub.t] [less than or equal to] [[lambda].sup.(2).sub.t] [less than or equal to] [[lambda].sup.(3).sub.t] [less than or equal to] [[lambda].sup.(4).sub.t] [less than or equal to] bt [less than or equal to] b < 1 as the unique solution of the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where b is the unique solution on (0,1) of the fixed point problem

[mu](b) = E[z] + b(1 - [delta]) + [delta].

From Assumption 1, the function [mu](x) is increasing and convex with [mu](0) = 0 and [lim.sub.[lambda][right arrow][mu]] ([lambda]) [right arrow] [infinity]. Consequently, [[lambda].sup.(1).sub.t](0) and [[lambda].sup.(2).sub.t](0) always exist. If the magnitude of the investment opportunity i is very large, the right-hand side of the corresponding equality may be negative, and therefore, [[lambda].sup.(1).sub.t](i) or [[lambda].sup.(2).sub.t] (i) are ill defined. In such cases, they are set to zero. Assumption 1 guarantees existence and unicity in (0,1) of [[lambda].sup.(3).sub.t] ([[??].sub.+1]), [[lambda].sup.(4).sub.t] ([b.sub.t+1]), and b.

The inequalities in Definition 1 are justified as follows. The first inequality, [[lambda].sup.(1).sub.t] ([i.sub.t]) [less than or equal to] [[lambda].sup.(2).sub.t] ([i.sub.t]), comes from the fact that [[bar.[lambda]].sub.t+1]. For the second one, [[lambda].sup.(2).sub.t] ([i.sub.t]) [less than or equal to] [[lambda].sup.(3).sub.t], is given due to [delta] > 0. For the third inequality, [[lambda].sup.(3).sub.t] [less than or equal to] [[lambda].sup.(4).sub.t], it is justified because [[??].sub.t+1] < [b.sub.t+1]. Lemma 3 in the "Verification of the Induction Hypothesis 2 at Time t" section shows that [[lambda].sup.(4).sub.t] [less than or equal to] [b.sub.t] [less than or equal to] b. Moreover, as the right-hand side decreases with the investment opportunity, [[lambda].sup.(1).sub.t] (i) [less than or equal to] [[lambda].sup.(1).sub.t](0) and [[lambda].sup.(2).sub.t] (i) [less than or equal to] [[lambda].sup.(2).sub.t](0). The "Investment and Dividends Policies in the Full Hedging Case Given Properties of Value Function at Date (t + 1)" section shows that these thresholds characterize the dividends and investment policies, as stated in Proposition 1.

The First-Order Conditions

To optimize the firm relative value, the partial derivatives of Equation (12) with respect to each of the control variables are required. Assuming the independence between [Z.sub.t+1] and [i.sub.t+1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A4)

[partial derivative][THETA]/[partial derivative][[eta].sub.t] ([[lambda].sub.t], [g.sub.t], [d.sub.t], [[eta].sub.t] = 1) = 0 if 0 < [y.sub.t+1] < 1, (A5)

where [PHI] represents the cumulative distribution function of a standard normal random variable [A.sub.t] = ([y.sub.t+1] ([[lambda].sub.t+1], [g.sub.t], [d.sub.t]) - 1) and 1 + [g.sub.t] [delta]/[[sigma].sub.Z] (1 - [[eta].sub.t]) and [B.sub.t] ([y.sub.t+1], [g.sub.t], [d.sub.t], [[eta].sub.t]) = [y.sub.t+1] ([[lambda].sub.t], [g.sub.t], [d.sub.t]) 1 + [g.sub.t] - [delta]/[[sigma].sub.Z](1 - [[eta].sub.t]).

Sketch of the Proof

Conditional on the available information at time t, [[lambda].sub.t+1] is a truncated Gaussian random variable, that is, 0 [less than or equal to] [[LAMBDA].sub.t+1] [less than or equal to] if and only if [A.sub.t] ([[lambda].sub.t], [d.sub.t], [[eta].sub.t]) [less than or equal to] [less than or equal to]([[epsilon].subl.t+1] [less than or equal to] [B.sub.t] ([[lambda].sub.t+1], [g.sub.t], [d.sub.t]), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Expressing the conditional expectation as an integral,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the bounds [A.sub.t] and [B.sub.t] as well as the conditional mean and standard deviation of [[lambda]t+1], are functions of the control variables. Partial derivatives of [E.sub.t] [[J.sup.*.sub.t+1]([[lambda].sub.t+1])] are obtained from the application of Leibniz integration rule to each of the three integrals.

In the following, we refer to the first-order conditions the solution to [partial derivative][THETA]/[partial derivative][u.sub.t] ([[lambda].sub.t], [g.sup.*.sub.t], [d.sup.*.sub.t], [[eta].sup.*.sub.t]) = 0, [u.sub.t] being one of the three control variables.

Value Function at Time t + 1

The induction hypothesis in broken in two parts:

Induction Hypothesis 1: [[phi].sub.t+1] is concave on [0, [b.sub.t+1]).

Induction Hypothesis 2: [[phi].sub.t+1] ([[bar.[lambda]].sub.t+1]) > 1 and [[phi].sub.t+1] ([bar.[lambda]].sub.t+1]) > 1 and [[phi].sub.t+1] ([[lambda].sup.(4).sub.t+1]) < 1.

The following mild technical assumption is required in the initiation of the recursion.

Assumption 3: Shape of the continuing value

(1) w([[lambda].sup.(4).sub.T] > E[z].

(2) [J.sub.T+1] (x) is concave on [[a.sub.T+1], [b.sub.T+1]] where 0 < [a.sub.T+1] < [[lambda].sup.*] < [[??].sub.T+1] [less than or equal to] [b.sub.T+1] [less than or equal to] 1.

Point 1 is used in the "Verification of the Induction Hypothesis 2 at

Time t" section. Intuitively, if the cost of capital is too low compared to the expected ROIC, the firm has no incentive to control its indebtedness and invests as much as it can. Consequently, we assume there exists a leverage above which the expected return on investment is smaller than the cost of capital, that is, w([lambda]) > E[z] for all sufficiently large [lambda]. Point 2 starts the convexity recursion about the firm's relative expected value.

Corollary 2: There is a unique [[??].sub.t+1] [member of] [[[bar.[lambda]].sub.t+1] , [b.sub.t+1]] such that [[phi].sub.t+1] > 0 on [0, [[??].sub.t+1]), [[phi].sub.t+1] ([[??].sub.t+1]) = 0 and [[phi].sub.t+1] < 0 on (xt+i, bt+1].

Proof: The function [[phi].sub.t+1] ([lambda]) - [[phi].sub.t+1]([lambda]) - 1 + (1 - [lambda])[[phi]'.sub.t+1]([lambda]) is decreasing on [0, [b.sub.t+1]]. Indeed, [[phi].sub.t+1] being concave, [[phi]".sub.t+1] < 0

[[phi]'.sub.t+1] ([lambda]) = {1 - [lambda])[[phi]".sub.t+1] ([lambda]) < 0.

Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the last inequality is justified by Induction Hypothesis 2. Therefore, [[phi].sub.t+1]([lambda]) is positive on [0, [[bar.[lambda]].sub.t+1], remains positive for a while before it reaches 0, and becomes negative.

The "Verification of Induction Hypotheses 1 and 2 at time T" section is similar to the case t < T presented in the "Verification of Induction Hypothesis 1 at Time t (Concavity of [J.sup.*.sub.t] Over [0, [b.sub.t]])" and "Verification of the Induction Hypothesis 2 at Time t" sections.

Investment and Dividends Policies in the Full Hedging Case Given Properties of Value Function at Date (t + 1)

Proof and Illustration: In the following [y.sup.*.sub.t+1] = [y.sub.t+1]([[lambda].sub.t], [g.sup.*.sub.t], [d.sup.*.sub.t]), where [g.sup.*.sub.t] ([[lambda].sub.t]) and [d.sup.*.sub.t] ([[lambda].sub.t]) are the optimal investment level and dividends rate in the full hedging case. In the neighborhood of [y.sub.t] = 0, the partial derivative (A1b) is positive (Corollary 2 of the "Value Function at Time t + 1" section). Therefore, the investment is set to its maximal value, [g.sup.*.sub.t] = [i.sub.t] + [delta], and the dividends are chosen such that the next period leverage maximizes the firm's relative value:

[[lambda].sub.t+1] = [y.sup.*.sub.t+1] = ([mu][[lambda].sub.t]) + [g.sup.*.sub.t] - [delta] + [d.sup.*.sub.t] - E[z]/1 + [g.sup.*.sub.t] - [delta] = [mu]([[lambda].sub.t]) + [i.sub.t] + [d.sup.*.sub.t] - E[z]/1 + [i.sub.t],

which implies that

[d.sup.*.sub.t] = E[z] + [[bar.[lambda]].sub.t+1] (1 + [i.sub.t]) - [i.sub.t] - [i.sub.t] - [mu]([[lambda].sub.t]) = E[z] + [[lambda].sub.t+1] - (1 - [[bar.[lambda]].sub.t+1]) [i.sub.t] - [mu]([[lambda].sub.t]). (A6)

The right-hand side in Equation (A6) is a decreasing function of [[lambda].sub.t]. This strategy lasts until the dividend rate is null, that is, for all [[lambda].sub.t] [right arrow] [[lambda].sup.(1).sub.t] ([i.sub.t]). This produces the almost linear decay of the optimal dividends policy on the bottom right panel of Figure 2 and the expected leverage plateau on the left of Figure 3.

For [[lambda].sub.t] slightly larger than [[lambda].sup.(1).sub.t] ([i.sub.t]), the firm is still financially constrained since [partial derivative][[THETA].sub.t]/[partial derivative][g.sub.t] = D([[lambda].sub.t[)[phi])([y.sub.t+1]) > 0: the optimal control is still [g.sup.*.sub.t] = [i.sub.t] + [delta] as can be seen on the right panels of Figure 2. In that case,

[y.sup.*.sub.t+1] ([[lambda].sub.t]) = [mu]([[lambda].sub.t]) + [g.sup.*.sub.t] - [delta] + [d.sup.*.sub.t] - E[z]/1 + [g.sup.*.sub.t] - [delta] = [mu]([[lambda].sub.t]) + [i.sub.t] - E[z]/1 + [i.sub.t]

increases almost linearly, as shown in Figure 3. Since next period leverage is an increasing function of [[lambda].sub.t], this situation lasts until [partial derivative][[THETA].sub.t]/[partial derivative][g.sub.t] = D([[lambda].sub.t[)[phi])([y.sub.t+1]) reaches zero, that is, until [[lambda].sub.t] [less than or equal to] [[lambda].sup.(2).sub.t]([i.sub.t]) where [[lambda].sup.*.sub.t+1]([[lambda].sup.(2).sub.t] ([i.sub.t])) = [[??].sub.t+1].

Once [[lambda].sub.t] > [[lambda].sup.(2).sub.t] ([i.sub.t]), the optimal investment g* is chosen according to the first-order condition [partial derivative][[THETA].sub.t]/[partial derivative][g.sub.t] = D([[lambda].sub.t[)[[phi].sub.t+1])([y.sup.*.sub.t+1] = 0. This happens if [y.sup.*.sub.t+1] = [[??].sub.t+1], which produces the second plateau of Figure 3. Therefore,

[[??].sub.t+1] = [mu]([[lambda].sub.t]) + [g.sup.*.sub.t] - [delta] - E[z]/1 + [g.sup.*.sub.t] - [delta],

which implies that

[g.sup.*.sub.t] = E[z] + [[??].sub.t+1] (1 - [delta]) + [delta] - [mu]([[lambda].sub.t])/1 - [[??].sub.t+1].

This lasts until the optimal investment [g.sup.*.sub.t] reaches 0. Let [[lambda].sup.(3).sub.t] be the unique solution to [g.sup.*.sub.t] ([[lambda].sub.t]) = 0. The decay of the optimal investment policy may be observed at top right panel of Figure 2.

For [[lambda].sub.t] > [[lambda].sup.(3).sub.t], the firm is so indebted that it can not invest anymore. Consequently, [J.sup.*.sub.t] [([[lambda].sub.t], i) - [J.sup.*.sub.t]([[lambda].sub.t], 0). Next-period leverage [[lambda].sup.*.sub.t+1] increases with [[lambda].sub.t].

Hedging Policy Given Properties of Value Function at Date (t + 1)

Lemma 1: Let [[phi]".sub.t+1] stands for the second derivative of [[phi].sub.t+1], that is, [[phi]".sub.t+1](y) = [E.sub.t] [[partial derivative].sup.2][J.sup.*.sub.t+1]/[partial derivative][[lambda].sup.2.sub.t+1] (y, [i.sub.t+1])].

(1) If [[phi]".sub.t+1] ([y.sub.t+1]([lambda])) > 0 then the optimal hedging decision is [[eta].sup.*.sub.t]([lambda]) = 0.

(2) If ([[phi]".sub.t+1]([y.sub.t+1]([lambda])) < 0 then the optimal hedging decision is [[eta].sup.*.sub.t]([lambda]) = 1.

Proof of Lemma 1: Equation (A5) implies that full hedging always satisfies the firstorder condition. Equation (A4) goes to zero as [[eta].sub.t] [right arrow] [infinity]. As the hedging parameter is constrainted to be positive, it is not an admissible solution. Hence, there will be some cases where the optimal solution will stand on limit of the constraint [[eta].sub.t] [greater than or equal to] 0, that is, [[eta].sup.*.sub.t]([[lambda].sub.t]) = 0. To determine whether the full hedging decision is optimal, we consider the second derivative:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since 1/1 + w([[lambda].sub.t] [[sigma].sup.2.sub.Z]/1 + [g.sub.t] - [delta] > 0 the sign of [[phi]".sub.t+1] determines the sign of [[partial derivative].sup.2][[THETA].sub.t]/[partial derivative][[eta].sup.2.sub.t]. If [[phi]".sub.t+1] > 0, then [[eta].sub.t] = 1 minimizes [THETA] and cannot be the optimal control. Therefore, ([[phi]".sub.t+1] ([y.sub.t+1]) > 0 implies that [[eta].sup.*.sub.t]([[lambda].sub.t]) = 0. If [[phi]".sub.t+1] ([y.sub.t+1]) < 0, then [[eta].sub.t]([[lambda].sub.t]) = 1 is a local maximum. As there is no other root, it is the optimal decision.

Lemma 2: The deterministic part of the next-period leverage in the full hedging case,

[y.sup.t.sub.t+1]([[lambda].sub.t]) = [mu]([[lambda].sub.t]) + [g.sup.*.sub.t]([[lambda].sub.t]) - [delta] + [d.sup.*.sub.t]([[lambda].sub.t]) - E[z]/1 + [g.sup.*.sub.t]([[lambda].sub.t]) - [delta],

is a continuous and nondecreasing function of [[lambda].sub.t].

Proof: According to the optimal management policies of Proposition 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A7)

Continuity arises from the definitions of [[lambda].sup.(1).sub.t] ([i.sub.t]), [[lambda].sup.(2).sub.t] ([i.sub.t]), and [[lambda].sup.(3).sub.t]. Because [mu] is an increasing function, [y.sup.*.sub.t+1] is a nondecreasing function of [[lambda].sub.t].

Proof of Proposition 2: We prove in the "Proofs of Propositions 1-3" section that (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Equation (A7)); hence, [[eta].sub.t] = 1 is always a candidate for the optimal strategy, and (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Lemma 1). [[eta].sub.t] = 1 is the optimal strategy if and only if [[phi].sub.t+1] ([y.sup.*.sub.t+1]) is concave. Using Induction Hypothesis 1, this yields:

[n.sup.*.sub.t] = [left and right arrow] [y.sup.*.sub.t+1] [member of] [0, [b.sub.t+1]).

From the optimal controls derived in Proposition 1, [y.sup.*.sub.t+1] ([[lambda].sub.t]) is a nondecreasing function of [[lambda].sub.t] such that

[y.sup.*.sub.t+1] ([[lambda].sub.t]) [member of] [0, [b.sub.t+1]) [right and left arrow] [[lambda].sub.t] < [[lambda].sup.(4).sub.t].

Consequently,

[[eta].sup.*.sub.t] = 1 [right and left arrow] [[lambda].sub.t] < [[lambda].sup.(4).sub.t]).

For [[lambda].sub.t] > [[lambda].sup.(4).sub.t]), [[eta].sub.t] = 1 is not optimal. Lemma 1 shows that, in that case, [[eta].sup.*.sub.t] = 0.

Verification of Induction Hypothesis 1 at Time t (Concavity of [J.sup.*.sub.t] Over [0, [b.sub.t]]) Replacing the optimal control variables in Equation (12), the firm relative value at time t becomes [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A8)

For [[lambda].sub.t] [member of] [0, [[lambda].sup.(1).sub.t] ([i.sub.t])], [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) = D([[lambda].sub.t](E[z] - [i.sub.t] + (1 + [i.sub.t])[[phi].sub.t+1]([[bar.[lambda].sub.t+1])), which is concave on [0, [[lambda].sup.(1).sub.t]([i.sub.t])] because the discount factor is.

For [[lambda].sub.t] [member of] ([[lambda].sup.(1).sub.t]([i.sub.t]), [[lambda].sup.(2).sub.t]([i.sub.t]), the concavity is verified by looking at the second derivative of [J.sup.*.sub.t], which must be negative. Equation (A8) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

D" < 0 by Assumption 1 making the first term negative. [[lambda].sub.t] [member of] ([[lambda].sup.(1).sub.t] ([i.sub.t]), [[lambda].sup.(2).sub.t] ([i.sub.t])] implies that [[bar.[lambda]].sub.t+1] [less than or equal to] [y.sup.*.sub.t+1] ([[lambda].sub.t]) [less than or equal to] [[??].sub.t+1]. Therefore, [[phi]'.sub.t+1] ([y.sup.*.sub.t+1] (Af)) < 0. Consequently the second term is negative provided that 2D' [mu]' + (D[mu])" = D" [mu] + 2D' [mu]' + D[mu]" implies that [[phi]".sub.t+1] ([y.sup.*.sub.t+1] ([[lambda].sub.t])) < 0; therefore, the third term is negative.

Consider [[lambda].sub.t] [member of] ([[lambda].sup.(2).sub.t] ([i.sub.t]), [[lambda].sup.(3).sub.t]. First, note that [[phi].sub.t+1] ([[??].sub.t+1]) > 1. Indeed,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where the last inequality holds because [[??].sub.t+1] > [bar.[lambda]].sub.t+1]) implies that [[phi].sub.t+1] is decreasing at [[??].sub.t+1]. Second, the concavity is obtained by means of the second derivative. Indeed, applying the derivative operator to equation (A8) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

D" < 0 implies that the first term is negative. Since [[phi].sub.t+1] ([[??].sub.t+1]) > 1, then [[phi].sub.t+1]) - 1/([[??].sub.t+1]]) is positive. 2 D'[mu]' + D[mu]" = (D[mu])" - D" [mu] > 0 by Assumption 1.

Finally, let [[lambda].sub.t] [member of] ([[lambda].sup.(3).subl.t], [[lambda].sup.(4).sub.t]]. As there is no investment, the value function remains the same whenever an investment opportunity is present or not, that is, [J.sup.*.sub.t] ([[lambda].sub.t], i) = [J.sup.*.sub.t] ([[lambda].sub.t], 0).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[lambda].sub.t] [member of] ([[lambda].sup.(3).sub.t], [[lambda].sup.(4).sub.t]] implies that [y.sup.*.sub.t+1] ([[lambda].sub.t]) = [mu]([[lambda].sub.t]) - [delta] - E[z]/1 - [delta] > [[??].sub.t+1] ([y.sup.*.sub.t+1] ([[lambda].sub.t])) < 0.

D" < 0 implies that the first term of [[partial derivative].sup.2][J.sup.*.sub.t]/[partial derivative][[lambda].sup.2.sub.t] is negative. The second term is negative since 2 d' [mu]' + D[mu]" = (D[mu])" - D"[mu] > 0 by Assumption 1. Finally, Induction Hypothesis 1 implies that [[phi].sub.t+1] is concave on [0, [b.sub.t+1]]. Hence, [[phi]".sub.t+1] ([y.sup.*.sub.t+1]([[lamda].sub.t])) and, consequently, the third term are negative for [y.sup.*.sub.t+1] ([[lambda].sub.t]) = [mu]([[lambda].sub.t]) - [delta] - E[z]/1 - [delta] [less than or equal to] [b.sub.t+1], that is, for [[lambda].sub.t] [less than or equal to] [[lambda].sup.(4).sub.t].

Since the first two terms are negative and the third equal to zero, [[partial derivative].sup.2] [J.sup.*.sub.t]/[partial derivative][[lambda].sup.2.sub.t] ([[lambda].sup.(4).sub.t]) 0; hence, [[partial derivative].sup.2] [J.sup.*.sub.t]/[partial derivative][[lambda].sup.2.sub.t] < 0 on ([[lambda].sup.(3).sub.t], [[lambda].sup.(4).sub.t]]. Then [[lambda].sup.(4).sub.t] [less than or equal to] [b.sub.t] defined byu [[phi]".sub.t] ([b.sub.t]) = 0.

Verification of the Induction Hypothesis 2 at Time t

[[phi].sub.t] ([[bar.[lambda]].sub.t]) [greater than or equal to] 1. We will show that on [0, [[lambda].sup.3.sub.t]], [D.sup.-1] ([[lambda].sub.t]) [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) [greater than or equal to] 1 + E[z]. Hence, since [[lambda].sup.*] [member of] [0, [[lambda].sup.(3).sub.t]], [J.sup.*.sub.t] ([[lambda].sup.*] [i.sub.t]) [greater than or equal to] 1 + E[z]/1 + w([[lambda].sup.*]) > 1. Therefore,

[[phi].sub.t] ([bar.lambda]].sub.t] [greater than or equal to] [[phi].sub.t] ([[lambda].sup.*], 0) (1 - p) [greater than or equal to] 1.

On [0, [[lambda].sup.(1).sub.t] ([i.sub.t])], Induction Hypothesis 2 implies that [[phi].sub.t+1] ([[bar.[lambda]].sub.t+1]) > 1. Therefore, starting

[D.sup.-1] ([[lambda].sub.t]) [J.sup.*.sub.t] ([[lambda].sub.t], [i.sub.t]) = E[z] - [i.sub.t] + (1 + [i.sub.t]) [[phi].sub.t+1] ([[bar.[lambda]].sub.t+1]) [greater than or equal to] E[z] + 1.

Because [[lambda].sub.t] [member of] ([[lambda].sup.(1).sub.t] ([i.sub.t]), [[lambda].sup.(2).sub.t] ([i.sub.t])] implies that [[bar.[lambda]].sub.t+1] [less than or equal to] [[lambda].sup.*.sub.t+1] ([[??].sub.t+1]), [[phi].sub.t+1] ([[lambda].sup.*.sub.t+1] ([[lambda].sub.t])) [greater than or equal to] [[phi].sub.t+1] ([[lambda].sup.*.sub.t+1]([[lambda].sup.(2).sub.t]([i.sub.t])) = [[phi].sub.t+1] ([[??].sub.t+1]) > 1.

Therefore, Equation (A8) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the interval [[lambda].sub.t] [member of] ([[lambda].sup.(2).sub.t] ([i.sub.t]), [[lambda].sup.(3).sub.t]],

E[z] + [[??].sub.t+1] - [mu] ([[lambda].sub.t])/1 - [[??].sub.t+1] [greater than or equal to] E[z] + [[??].sub.t+1] - [mu] ([[lambda].sup.(3).sub.t])/1 - [[??].sub.t+1]

Therefore, starting from Equation (A8),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[[phi].sub.t] ([[lambda].sup.(4).sub.t]) < 1.

Lemma 3: For all t [less than or equal to] T,

[b.sub.t+1] [less than or equal to] [[lambda].sup.(4).sub.t] [greater than or equal to] [b.sub.t] [less than or equal to] b

Proof of Lemma 3: As we have seen from the analysis of the concavity of [J.sup.*.sub.t] ([lambda].sub.t], [i.sub.t]) [[lambda].sup.(4).sub.t] [less than or equal to] [b.sub.t]. We claim that [[lambda].sup.(4).sub.t] [less than or equal to] b. Indeed, because b is the fixed point of [y.sup.*.sub.t+1](b) = b, [y.sup.*.sub.t+1] ([[lambda].sub.t]) [less than or equal to] [[lambda].sub.t]) [right and left arrow] b. It implies that if [[lambda].sub.t] > b, then [y.sup.*.sub.t+1] ([[lambda].sub.t]) > [[lambda].sub.t]; hence, the fully hedging firm becomes bankrupt with probability one. Therefore, it cannot be optimal for the firm to fully hedge for [[lambda].sub.t] > b, implying that [[lambda].sup.(4).sub.t] [less than or equal to] b. From this last inequality,

[b.sub.t+1] = [y.sup.*.sub.t+1]([[lambda].sup.(4).sub.t]) [less than or equal to] [[lambda].sup.(4).sub.t] [less than or equal to] [b.sub.t],

and this is verified for all t [less than or equal to] T.

Lemma 3. implies that [[lambda].sup.4.sub.t+1] [less than or equal to] [b.sub.t+1] = [y.sup.*.sub.t+1] ([[lambda].sup.(4).sub.t]). Since [[phi].sub.t+1] is decreasing on the right of [[lambda].sub.t]+l, <pt+i([y.sup.*.sub.t+1](k|4))) < <[y.sup.*.sub.t+1](A.|4)1). Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is strictly smaller than one provided that w([[lambda].sup.(4).sub.t]) > E[z]. But w being increasing to the right of [[bar.[lambda]].sub.t+1], w ([[lambda].sup.(4).sub.T]) [greater than or equal to] E[z] where the last inequality comes from Assumption 3.

Equity Cushion

The proof proceeds in four steps. First, it is shown that if [[lambda].sup.1.sub.t](i) < [[lambda].sup.*], then [[lambda].sup.1.sub.t](i) < [[bar.[lambda]].sub.t] < [[lambda].sup.*]. Second, backward induction is used to show that if [[lambda].sup.1.sub.T](i) [[lambda].sup.*], then [[lambda].sup.1.sub.t](i) < [[lambda].sup.*] for all t. Third, a necessary and sufficient condition for [[lambda].sup.1.sub.T](i) < [[lambda].sup.*] is given, ending the induction. The fourth step examines the case [[lambda].sup.1.sub.T](i) [greater than or equal to] [[lambda].sup.*].

Part I. Suppose [[lambda].sup.(1).sub.t](i) < [[lambda].sup.*] [less than or equal to] [[lambda].sup.(1).sub.t] (0) and consider [[lambda].sub.t] [member of] ([[lambda].sup.(1).sub.t](i), [[lambda].sup.*]. Because [[lambda].sub.t] < [[lambda].sup.(4).sub.t], full hedging is optimal. If [i.sub.t] = 0, then [g.sup.*] ([[lambda].sub.t], [i.sub.t]) = [delta] and the optimal dividend rate is chosen such that [y.sub.t+1] ([[lambda].sub.t], [g.sup.*.sub.t], [d.sup.*.sub.t]) = [[bar.[lambda]].sub.t+1]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[[phi]'.sub.t] ([[lambda].sup.(1).sub.t](i)) > 0 since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being smaller that [[lambda].sup.*]. Moreover, since D' ([[lambda].sup.*]) = 0,

[[phi]'.sub.t] ([[lambda].sup.*]) = D([[lambda].sup.*]) (1 + i) [[phi]'.sub.t+1] ([y.sup.*.sub.t+1]/[partial derivative] [[lambda].sub.t] ([[lambda].sup.*], i) p,

which is negative because [partial derivative][y.sup.*.sub.t+1]/[partial derivative] [[lambda].sub.t] > 0 and [y.sup.*.sub.t+1] ([[lambda].sup.*]) [greater than or equal to] [[bar.[lambda]].sub.t+1] implies that [[phi]'.sub.t+1] ([y.sup.*.sub.t+1] ([[lambda].sup.*])) [less than or equal to] 0. By continuity of [[phi]'.sub.t], since [[phi]'.sub.t] ([[lambda].sup.(1).sub.t](i)) > 0 and [[phi]'.sub.t] ([[lambda].sup.*]) [less than or equal to] 0, there is [[bar.[lambda]].sub.t] [member of] ([[lambda].sup. (1).sub.t](i), [[lambda].sup.*]] such that [[phi]'.sub.t] ([[bar.[lambda]].sub.t]) = 0.

Part II. Assume that [[lamdba.sup.(1).sub.T](i) < [[lambda].sup.*]. A backward induction argument is used to show that [[lambda].sup.(1).sub.t](i) < [[lambda].sup.*]. Indeed, from Part I, if [[lambda].sup.(1).sub.t+1](i) < [[lambda].sup.*], then [[bar.[lambda]].sub.t+1] [member of] ([[lambda].sup.(1).sub.t+1](i), [[lambda].sup.*]]. Therefore, because [mu] ([[lambda].sup.(1).sub.t]([i.sup.t])) = E[z] + [[bar.[lambda]].sub.t+1] - (1 - [[bar.[lambda]].sub.t+1]) [i.sub.t],

[mu] ([[lambda].sup.(1).sub.T](i)) - [mu] ([[lambda].sup.(1).sub.t](i)) = [[lambda].sup.*] - [[bar.[lambda]].sub.t+1]) (1 + i) > 0.

Hence, as [mu] is an increasing function, [[lambda].sup.(1).sub.t](i)) = [[lambda].sup.(1).sub.T] (i) [less than or equal to] [[lambda].sup.*].

Part III. [[lambda].sup.(1).sub.T](i) < [[lambda].sup.*] if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is, the expected ROIC is lower than interest payments (per unit) and the equity fraction of the investment (per unit):

E[z] < [[lambda].sup.*]([[lambda].sup.*]) + 1 - [[lambda].sup.*]) i.

Part IV. What happens if [[lambda].sup.(1).sub.T](i) [greater than or equal to] [[lambda].sup.*]? One can show by induction that [[bar.[lambda]].sub.t] = [[lambda].sup.*] for any t. Indeed, if [[bar.[lambda]].sub.t+1] = [[lambda].sup.*], then

[mu] ([[lambda].sup.(1).sub.T](i)) - [mu] ([[lambda].sup.(1).sub.t](i)) = [[lambda].sup.*] - [[bar.[lambda]].sub.t+1]) (1 + i) = 0,

implying that [[lambda].sup.(1).sub.t](i) = [[lambda].sup.(1).sub.T](i) [greater than or equal to] [lambda].sup.*]. But, for any [[lambda].sub.t] [less than or equal to] [[lambda].sup.*] [less than or equal to] [[lambda].sup.(1).sub.t](i),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because D' ([[lambda].sup.*]) = 0, ([[phi]'.sub.t] ([[lambda].sub.t]) = 0 and [[phi].sub.t] ([[lambda].sub.t]) is maximized at [lambda].sup.*], that is [[bar.[lambda]].sub.t] = [[lambda].sup.*].

DOI: 10.1111/jori.12025

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(1) Biais et al. (2007) prove that the book-to-market ratio of equity is a U- shaped function of cash reserves, which is equivalent.

(2) See, for example, Copeland, Roller, and Murrin (1995).

(3) The filtration {[F.sub.t]: t = 0, ..., T} is generated by the random terms, that is, [F.sub.t] = [sigma] {[i.sub.u], [Z.sub.u]: u = 1, 2, ..., t}. The conditional expectation with respect to [F.sub.t] is denoted [E.sub.t] [*].

(4) Ai et al. (2012) propose another approach to solve the same problem: they assume that firms incorporate financial frictions through capital at risk constraints, that is, that managers attempt to maintain the 5 percent (or 1 percent) worst cash flow above a certain level.

(5) See Figures 18.1 and 19.1 in Welch (2008).

(6) f" stands for the second derivative off.

(7) g is not necessarily the growth rate during the last period gr, since the latter may not be representative of the firm's long-term growth potential.

(8) Details are available from the authors upon request.

Diego Amaya is in the Finance Department, Universite du Quebec a Montreal (UQAM), Quebec, Canada. Genevieve Gauthier is in the Department of Management Sciences, FIEC Montreal, and GERAD Quebec, Canada. Thomas-Olivier Leautier is at the Toulouse School of Economics (IDEI-IAE-CRM), France. Diego Amaya can be contacted via e-mail: amaya.diego@uqam.ca. Diego Amaya would like to thank FQRNT and [IFM.sup.2] for financial support. Genevieve Gauthier would like to thank NSERC and [IFM.sup.2] for financial support. An earlier version of this article was circulated under the title, "Coordinating Capital Structure With Risk Management Policies." We thank seminar participants at the Annual Conference on Risk Management and Corporate Governance, the Annual Australasian Finance and Banking Conference, and the Midwest Financial Association meetings for their comments on earlier versions of this article. Any remaining inadequacies are ours alone.

TABLE 1 Descriptive Statistics for ROIC and Depreciation Median (%) Mean (%) Std. Dev. (%) P1 (%) P99 (%) E[z] 8.4 7.9 11.4 -35.6 27.5 [sigma] 5.8 8.9 22.5 0.9 69.8 [delta] 12.0 14.4 8.7 8.7 46.8 Note: The median, mean, standard deviation (Std. Dev.), and percentiles 1% (P1) and 99% (P99) are based on a sample of 854 industrial firms from annual Compustat files. For each firm, the return on invested capital (E[z]), its standard deviation ([sigma]), and the depreciation ([delta]) are computed for the period 1990-2009. The return on invested capital (ROIC) and depreciation are computed according to the procedure described in the "Data and Estimation" section. TABLE 2 Investment Opportunity and Long-Term Growth Estimates g-[delta] p m [[sigma].sub.H] Point estimate (%) 2.1 21.2 12.6 6.4 Standard deviation (%) (0.32) (0.14) (0.3) (3.3) Note: Estimation of Equation (14) is based on a sample of 854 industrial firms from annual Compustat files for the period 1990 to 2009. TABLE 3 Estimates for the WACC Function [[beta].sub.0] [[beta].sub.1] [[beta].sub.2] Point estimate 0.1019 -0.2111 0.1691 Standard deviation (0.004) (0.028) (0.041) Note: Estimation for Equation (17) is based on a sample of 340 firms during the period of 1995-2003. TABLE 4 Estimates for the Cost of Debt Function [[alpha].sub.0] [[alpha].sub.1] Point estimate 0.0781 -0.0422 Standard deviation (0.002) (0.007) Note: Estimation for Equation (18) is based on a sample of 340 firms during the period of 1995-2003. Table 5 Investment and Dividends Thresholds in the No Hedging Case Thresholds (%) [[bar.[lambda]] [[??].sub.t] [[lambda].sub. .sub.t] t.sup.(1)] (i) Hedging 40 56 37 No hedging 36 51 33 Thresholds (%) [[lambda].sub. [[lambda].sub. [[lambda].sub. t.sup.(1)] (0) t.sup.(2)] (i) t.sup.(2)] (0) Hedging 45 54 59 No hedging 42 50 56 Thresholds (%) [[lambda].sub. [[lambda].sub. t.sup.(3)] t.sup.(4)] Hedging 64 80 No hedging 60 80 Note: In both cases, the parameters are as in Tables 2-4. Critical values are for a selected t "far enough" from the last period T that the value function is numerically stationary, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This criterion translates into (T - t) [congruent to] 20. The cost of capital w(x) is minimized at [[lambda].sup.*] = 40 percent. "Hedging" corresponds to the constraint 0 [less than or equal to] [[eta].sub.t] [less than or equal to] 1 while "No hedging" is associated to [[eta].sub.t] = 0. TABLE 6 Investment and Dividends Thresholds in the No Hedging Case Thresholds (%) [[bar.[lambda]] [[??]] [[lambda].sub. .sub.t+1] .sub.t+1] t.sup.(1)] (i) More convex WACC 31 40 27 Partial 38 54 35 Increased exposure 40 56 37 Thresholds (%) [[lambda].sub. [[lambda].sub. [[lambda].sub. t.sup.(1)] (0) t.sup.(2)] (i) t.sup.(2)] (0) More convex WACC 36 37 45 Partial 43 51 57 Increased exposure 45 54 59 Thresholds (%) [[lambda].sub. [[lambda].sub. t.sup.(3)] t.sup.(4)] More convex WACC 51 74 Partial 62 80 Increased exposure 64 79 Note: In both cases, the parameters are as in Tables 2-4. Critical values are for a selected t "far enough" from the last period T that the value function is numerically stationary, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This criterion translates into (T - t) [congruent to] 20. The parameters of the more convex WACC are [[beta].sub.0] = 0.13, [[beta].sub.1] = -0.42, [[beta].sub.2] = 0.54. Partial hedging is obtained under the constraint 0 [less than or equal to] [[eta].sub.t] [less than or equal to] 0.30. Increased exposure corresponds to the constraint relaxation - 1 [less than or equal to] [[eta].sub.t] [less than or equal to] 1.

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Author: | Amaya, Diego; Gauthier, Genevieve; Leautier, Thomas-Olivier |
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Publication: | Journal of Risk and Insurance |

Article Type: | Abstract |

Geographic Code: | 1CANA |

Date: | Jun 1, 2015 |

Words: | 18937 |

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