Version: June 18, 001 On the Optimal Mix of Corporate Hedging Instruments: Linear versus Non-linear Derivatives Gerald D. Gay, Jouahn Nam and Marian Turac* Abstract We examine how corporations should choose their optimal mix of linear and non-linear derivatives. We present a model in which a firm facing both business (output) and market (price) risk maximizes its expected cash flows when subject to financial distress costs. The optimal hedging position is generally comprised of linear contracts, but as the levels of business and market risk increase, the use of linear contracts will decline due to the risks associated with over-hedging. At the same time, a substitution effect occurs towards the use of non-linear contracts. The degree of substitution will depend on the correlation between output levels and market prices, the firm's financial leverage, and its transaction cost structure. Our empirical tests support various predictions of the model as related to business risk, leverage and firm size. *Gerald D. Gay is Professor of Finance at Georgia State University. Jouahn Nam is Assistant Professor of Finance at Pace University. Marian Turac is Visiting Assistant Professor at the University of South Florida. Please direct all correspondence to Gerry Gay at the Department of Finance, J. Mack Robinson College of Business, Georgia State University, Atlanta, GA 30303-3083, Voice: (404) 651-1889, Fax: (404) 651-630, e-mail: ggay@gsu.edu. We acknowledge the helpful comments and conversations we have had with Mike Rebello, Steve Smith, Alan Tucker, P.V. Viswanath, Jayant Kale, Tom Webster and seminar participants at Pace University and South Florida.
1 On the Optimal Mix of Corporate Hedging Instruments: Linear versus Non-linear Derivatives 1. Introduction A rich literature has emerged that explores the various channels through which hedging can contribute to higher firm value (see Smithson (1998) for an excellent summary and also the seminal articles of Smith and Stulz (1985), Froot, Scharfstein and Stein (1993) and DeMarzo and Duffie (1995)). While these studies have increased our understanding as to why or what motivates corporations to manage risk, less attention has been directed as to how corporations should hedge, particularly, how firms should choose their optimal mix of linear and non-linear derivative instruments. We use the term "linear derivative instrument" to refer to those products such as futures, forward and swap contracts whose payoffs are linear in the underlying asset or reference price, while option contracts are illustrative of instruments having non-linear payoffs. And though financial engineers have long recognized the ability to replicate the payoffs of linear instruments using non-linear contracts, in practice, these product groups appear to be fundamentally different and offer distinct advantages and tradeoffs (see, for example, Black (1976), Bookstaber and Clarke (1983), Block and Gallagher (1986) and Smithson, Smith and Wilford (1995)). Further supporting their apparent distinctive differences, a cursory inspection of firms derivative positions that we report in Table 1 indeed reveals a large variation in the practices of U.S. corporations regarding the use of linear and non-linear
derivatives. 1 For a sample of 671 non-financial firms reporting the use of interest rate derivatives as shown in panel (a), we find 470 firms reporting the use of only linear instruments (e.g., futures, forwards and swaps), 89 firms using strictly non-linear instruments (e.g., options, floors and caps), while 11 firms report using some combination of the two instrument groups. These patterns are similarly observed in panel (b) for users of currency derivatives as well in panel (c) for users of commodity derivatives, and give rise to several interesting questions regarding a firm s choice of instruments. For instance, why do most firms use mainly linear instruments, why do a substantially smaller number of firms use only non-linear instruments, and what firm and market circumstances dictate a specific combination of the two? We shed light on these issues and provide additional insight as to the design of a firm s optimal hedging strategy. To provide a backdrop for investigating these issues, we first present a simple hedging model of a firm facing both business risk (alternatively referred to here as output risk) and market (or price) risk, and whose objective is to maximize expected cash flows when subject to potential financial distress costs. We subsequently analyze the sensitivities of the resulting optimal linear and non-linear hedging positions to the levels of business and market risk faced by the firm, the extent of the correlation between output and market prices, the degree of the firm's financial leverage, and to the level of the fixed and variable transaction costs associated with conducting a hedging program. 1 This table, explained in greater detail in a later section, is based on information reported in the 1997 Swaps Monitor database of derivatives users.
3 We find that for corporations facing little or no business risk, the optimal hedging portfolio will be comprised largely, if not entirely, of linear instruments, a strategy which essentially pushes the effective price the firm receives for its output toward the mean (forward) price. However, as both business and market risks become an increasingly important consideration for the firm, we find that although linear instruments will continue to typically dominate the firm s hedging mix, their usage will decline. This is because the greater a firm's business risk, the greater the likelihood that the firm will experience "over-hedging" costs that can arise from the use of linear instruments. This "over-hedging problem" which is typically ignored in the risk management literature results from the firm having sold too many forward contracts relative to realized output, concurrent with high realized prices. If the firm over-hedges and prices rise, the firm suffers not only a revenue shortfall due to the lower output, but also finds that its revenue windfall due to higher realized prices is more than offset by the losses on its excess forward contract position. Thus, as the risk of over-hedging increases (due to either greater business risk or greater market price risk), the firm will reduce its optimal linear position, but to maintain protection against falling prices, the firm will increase its use of non-linear contracts (i.e., purchase long put options). We find that the degree of this substitution effect between linear and non-linear instruments will be strongly influenced by the correlation between output and prices. Along these lines, Stulz (1996) discusses motivations for the corporate use of options related to limiting potential financial distress costs arising from large price movements.
4 Under negative correlation, prices will likely be high (low) when the firm's realized output is low (high). A negative correlation thus produces a natural hedging effect because the firm's revenue shortfall (increase) due to lower (higher) output is offset to some extent by the higher (lower) prices realized on the sale of its output. Thus, a negative correlation will reduce the firm's overall demand for hedging instruments. However, a negative correlation increases the likelihood that a firm will face the problem of over-hedging. In response, the firm will further reduce its linear position and substitute instead non-linear contracts. A positive correlation between output and prices will have the opposite effect. Under positive correlation, prices will be more likely to be low in those states where a firm has over-hedged (realized low output). Under this condition, the firm's overall demand for derivatives will increase because, in addition to reducing price risk, derivatives can be used to reduce a portion of the firm's business risk. A positive correlation between output and prices will also help mitigate a firm's potential overhedging problem associated with using linear contracts. Consequently, when compared to the zero or negative correlation cases, such firms will have greater demand for linear contracts and lower demand for non-linear instruments. We also demonstrate the separate effects that the variable and fixed transaction costs associated with conducting a hedging program and executing trades will have on a firm's optimal hedging position. As one would expect, the larger the variable transaction cost, the lower will be the overall demand for both linear and non-linear contracts. However, the variable transaction cost will have a disproportionately larger
5 effect in terms of reducing the non-linear position as compared to the linear position. In contrast, the level of fixed transaction costs associated with establishing and maintaining a hedging program will have less of an effect on the composition of the firm's optimal hedging position, but rather will be an important determinant of whether the firm will undergo hedging. That is, for a given level of fixed transaction costs, there will be a corresponding threshold of business and market risk combinations below which hedging will not be justified as the benefits from hedging (i.e., the reduction in the sum of the expected financial distress and over-hedging costs) are lower than the fixed transaction costs. But conditioned on hedging being justified, the resulting mix in the optimal hedging position will be largely unaffected by the level of fixed transaction costs. We subsequently conduct empirical tests on a large sample of firms having foreign sales and examine the relationship between their currency derivatives positions and the level of business risk along with other firm characteristics. In support of the predictions of our model, we provide strong evidence that firms usage of non-linear currency derivatives is positively related to business risk while linear derivatives usage is inversely related to business risk. We also find that the extent of firms' linear (nonlinear) derivatives usage is negatively (positively) related to firm size for reasons we attribute to potential transaction cost considerations. By using disaggregated positions and controlling for business risk, these contrasting results regarding firm size help to potentially explain the conflicting results revealed in prior investigations of the
6 relationship between derivative usage and firm size. Finally, we find weak evidence that a firm's use of non-linear derivatives is concave in the firm's use of debt. The paper proceeds as follows. The next section provides a brief discussion of the related literature on the choice of linear and non-linear instruments. In Section 3 we develop a model of the firm s cash flows that incorporates both business and market risk factors. This model is used for determining a firm s optimal hedging strategy in light of potential financial distress costs. Section 4 solves for and illustrates the sensitivities of the optimal hedging positions with respect to business and market risk, the correlation between the level of output and market prices, financial leverage, and transaction costs. We also provide a detailed analysis of the effect of both linear and non-linear instruments on expected distress and over-hedging costs. Section 5 presents the empirical results of both univariate and multivariate tests of various predictions of our model while Section 6 concludes the paper.. Prior Research on the Linear/Non-Linear Instrument Choice The question of why firms hedge has been a subject of broad investigation. Popular rationales, typically premised on various departures from the classical assumptions set forth in Modigliani and Miller (MM, 1958), include arguments that hedging can increase firm value by lowering the costs of financial distress, by reducing expected taxes, or by minimizing potential underinvestment problems (see, for example, Mayers and Smith (198), Smith and Stulz (1985), and Froot, Scharfstein, and Stein (1993)). Taking a different approach, DeMarzo and Duffie (1991) show that even when the
7 MM assumptions hold, corporate hedging can be optimal if the firm is concerned about revealing strategic, proprietary information to outsiders. Since shareholders do not possess this information they cannot hedge it on their own account, and they therefore find it optimal for the firm to hedge on their behalf. The issue of how firms elect to hedge, in terms of the selection of derivative type, remains less investigated. Early comparisons of linear and non-linear instruments are presented in Black (1976) and Moriarty, Phillips, and Tosini (1981). These studies note the ability of instruments with linear payoffs such as futures and forwards to inexpensively transfer risk and to reduce the variance of cash flow. Further, as noted by Black, futures markets facilitate price discovery and enhance resource allocation in the economy by allowing individuals and corporations to make better decisions regarding production, storage and processing. On the other hand instruments with non-linear payoffs such as options can reduce downside risk while allowing upside potential and can be used for yield or income enhancement. Along these lines, Bookstaber and Clarke (1983, p. 13) note that Futures contracts will not be the appropriate hedging instrument if there is a desire to maintain a profit potential from favorable price changes or if there is uncertainty about the amount of quantity that will be held [quantity risk]. Options are the instrument to overcome these shortcomings. In support of these views, Block and Gallagher (1986) provide survey evidence that futures are perceived by managers as being advantageous in terms of their cost and efficiency of hedging. They also note that options are seen by managers
8 as providing lower risk exposure and having fewer administrative problems, and are also viewed as being more effective in protecting against contingent events. Several authors attempt to model the firm s hedging decision using an expected utility framework. For example, Detemple and Adler (1988) present a model in which risk averse managers with limited access to financing sources face both price and quantity risk. They do not, however, allow for the simultaneous choice of linear and non-linear instruments. Their model predicts that firms facing borrowing constraints and those facing higher price risk will be more active users of options. Tufano (1996), applying the Detemple and Adler model to gold-mining firms, tests whether firms facing greater financial constraints are more likely to use options. To proxy for the level of a firm s financial constraints, he uses variables such as the level of extraction costs, the market value of the firm and the level of gold reserves. Contrary to predictions, he finds no supporting evidence. Lapan, Moschini, and Hanson (1991) propose a one-period model wherein utility maximizing managers face price risk (but not production uncertainty) and choose among both forwards and options when making their hedging decisions. Using the assumption of normally distributed prices (which allows for negative prices) they show that when forward and option prices are unbiased, the optimal hedging position will consist only of forward contracts as options become redundant. Lence, Sakong and Hayes (1994) extend the Lapan, et al model into a multiperiod framework. They find that such an extension will produce a general change in the optimal forward position, but more importantly, create a hedging role for options. Sakong, Hayes, and Hallam
9 (1993) also extend the model of Lapan, et al, by allowing for production uncertainty. They find that the optimal hedging position will almost always include options in addition to forward contracts. Among studies departing from the expected utility framework, Froot, Scharfstein, and Stein (1993) propose a model wherein managers facing a single source of hedgeable risk attempt to maximize firm value. They demonstrate that the optimal choice of hedging instruments is dependent on the relative sensitivities of internally generated cash flows and investment opportunities to changes in market prices. If the sensitivities are similar, a linear strategy (futures) will be optimal, otherwise, firms may prefer options. Brown and Toft (1997) also derive an optimal hedging strategy for a profitmaximizing firm facing both price and quantity risk. Allowing for financial distress costs, they derive optimal hedging positions based on strategies first using forwards only and then options only. They find that when price and quantity are negatively correlated, put options can be superior to selling forward contracts. They also introduce a third, theoretical hedging instrument referred to as a custom exotic derivative. This product, selected as a result of an optimization exercise, will have a quadratic state contingent payoff. By construction the exotic derivatives are superior to either forwards or options in terms of hedging efficiency. Brown and Toft also explore the extent to which these exotic derivatives can be replicated with forwards or options. Mello and Parsons (000) develop a model wherein value-maximizing managers attempt to mitigate financial distress costs caused by firm illiquidity. In their
10 model they incorporate for both hedgeable output and non-hedgeable input risk. Allowing only for the use of short-term futures, they derive optimal hedging positions. Adams (001) extends this model and that of Froot, Scharfstein and Stein (1993) by focusing on the cost differential between internal and external funds. He derives the optimal payoff on a hedging security that maximizes the present value of the firm's expected net cash flows in the presence of costly external financing. For firms whose cost of external financing is relatively low, the optimal payoff is convex suggesting the firm's need to purchase put options. For firms with costly external financing the payoff will be concave suggesting the writing of calls while for intermediate cost differentials, the optimal hedging strategy will contain both elements (e.g., a collar strategy). As was the case in Froot, Scharfstein and Stein, Brown and Toft (1997) and Mello and Parsons (000), this model does not specifically translate the results into the optimal combinations of linear and non-linear instruments.
11 3. A Hedging Model with Business and Market Risk In this section we develop a model for a firm s hedging problem that allows for the simultaneous inclusion of both linear and non-linear hedging instruments in the formulation of the optimal hedging strategy. In addition, we consider both business (quantity) and market (price) risks and assume that managers maximize expected cash flows in the presence of financial distress costs. 3.1 The Firm s Cash Flow Function We consider a firm s short-run or one-period ahead hedging problem. Similar to Brown and Toft (1997) we take as predetermined the firm's investment and operating structure. While in the long run these can be adjusted to reflect the financial risks the firm is experiencing, in the short run these policies are somewhat inflexible and may indeed also be a contributing factor to distress costs. Thus, to address its financial risks the firm may choose to develop a short-run hedging policy. We consider a firm whose end of period revenues are subject to both output and market price risk. We let Z and ε denote the firm's output and market price, respectively, which together will produce revenues or sales of Z ε. To illustrate, Z can be thought of as the level of output of a commodity-producing firm (e.g., barrels of oil) which is to be sold at an uncertain spot price ε. Alternatively, Z can be thought of as a firm s foreign currency denominated sales or revenues whose final domestic currency value is subject to an uncertain spot exchange rate ε. We let π 0represent the firm s unhedged stochastic cash flows given by:
1 π ( Z, ε ) = Z ε Dr C 0 (1) where Z and ε are bivariate lognormally distributed variables having a joint probability density g(z, ε) and correlation ρ, D is the amount of firm debt outstanding, r is the fixed rate of interest on debt, and C represents the fixed dollar costs of production. 3 It is assumed that Z has an expected value of µ and standard deviation of σ while ε has an expected value of ε F and standard deviation of σ ε. The parameter µ can be thought of as the permanent component of a firm s output while ε represents the market price associated with the firm s output. Also, in addition to output or business risk (σ ) the firm's revenues are subject to market price risk ( σ ). We assume that the firm has access to forward and option (puts) contracts written on the risky commodity with both contracts expiring at the end of the period. 4 We assume that the forward price, ε F, equals the future expected spot price, that is, ε ε F = E[ ε]. And for simplicity, only one strike price for put options is considered and which is set to equal the current forward price (or rate). The firm chooses its forward and option positions such that it maximizes its end of period expected cash flows. At the beginning of the period, the firm sells X oneperiod forward contracts at a forward rate ε F. At period end, the firm settles the X 3 We assume that the repayment of the debt principal is to occur at a later date, thus allowing our focus to be on the firm s one period ahead hedging problem.
13 forward contracts at the then prevailing spot rate ε, producing a net cash flow equal to ( ε F ε )X. Similarly, the firm purchases Y puts with strike price ε F for a total end of period cost of PY where P is equal to the put's expected payoff. At period end, the put will have a payoff of either zero, if ε exceeds the strike price ε F, or a gain of ( ε F ε )Y otherwise. Thus, the firm s hedged end of period cash flow can be expressed as π = π ( Z, ε ) + ( ε ε ) X + ( ε ) ly 0 F F ε PY = Z ε Dr C + ( ε ε) X + ( ε ε) ly PY () F F where l = 1 if ε ε F and l = 0 if ε F ε >. 3. The Financially Unconstrained Firm We first consider the hedging strategy of the firm which faces no financial constraints, that is, the firm is assumed to have access to unlimited risk-free borrowing. At the beginning of period, the firm chooses the optimal mix of linear and non-linear instruments, X * and Y *, respectively, that maximizes its expected end-of-period cash flow: max X, Y E[ π ] (3) 4 We consider only forward and put contracts since each forward, put, or call contract can be replicated by a combination of the two remaining contracts and the risk-free security.
14 In the absence of financial constraints, there is no advantage to hedging as the firm s expected hedged cash flows are equal to its expected unhedged cash flows, E [π ] = E π ]. Thus, E[π ] is independent of the choice of X and Y. [ 0 This result is not surprising since in the absence of market imperfections, corporate risk management (e.g., hedging) as a means for increasing firm value should be irrelevant. In the following section we reconsider the firm s risk management decision by incorporating a market imperfection related to financial distress costs.
15 3.3 The Financially Constrained Firm We next assume that when a firm s cash flows are insufficient to satisfy its credit obligations, the firm will incur varying degrees of distress costs related to the amount of shortfall. We specify a financial distress cost function that is convex in the degree of cash flow shortfall. 5 Specifically, we assume that the distress cost function encompasses three regions. First, in those states when the firm s cash flows are sufficient to satisfy the interest coverage or times-interest-earned ratio (TIE), as required by creditors, the firm will incur no financial distress costs. This condition can be expressed as TIE Operating Profit = Interest Expense π = + Dr > β Dr which can be rewritten as π > (β - 1)Dr where β is the specified interest coverage ratio. When cash flows fall below this point and down to a level where operating losses occur but remain in excess of available net working capital (π > -NWC), we assume that the firm incurs a distress cost that is proportional to the extent of the shortfall, γ[π - (β - 1)Dr], where γ is a proportionality coefficient assumed to take on a range of values 0< γ < 1. In practice these costs may entail higher future borrowing costs (see Diamond, 1984) or business disruption costs (see Titman, 1984). When 5 In the risk management literature, various market imperfections including financial distress costs, a progressive tax schedule, and costly external financing have been shown to induce concavity into the firm s value function, a necessary condition for a firm s hedging activities to be value-enhancing. See, for example, Mayers and Smith (198), Smith and Stulz (1985), Froot, Scharfstein, and Stein (1993) and Géczy, Minton and Schrand (1997).
16 losses fall below available working capital (or alternatively cash reserves as in Mello and Parsons, 000), we assume that the firm faces the serious prospect of default. In addition to the above proportionality losses, the firm is assumed to incur fixed distress costs, K, associated with losses on asset sales, legal fees and other deadweight costs related to bankruptcy and reorganization. Based on the above assumptions we assume that managers maximize the following profit function, π DC, which is equal to expected cash flows less any potential distress costs (DC): where (4) π DC = π DC DC= 0 = γ[ π -( β -1)Dr] = γ[ π -( β -1)Dr] +K if if if π ( β -1)Dr -NWC π < ( β -1)Dr π <- NWC (5) To graphically illustrate the effect of the above distress cost function on a firm's profits, we refer to Figure 1 which is based on the following simple example. For illustration we assume the following parameter values: Dr = $1, γ = 0.1, β =, K = $0.50 and NWC = $1.30. In Figure 1, the dotted (linear) line represents for comparison purposes the firm s profit function in the absence of distress costs (i.e., γ and K both equal to zero). The solid line represents the firm's profit function after consideration of
17 only the proportional distress cost component while the dashed line also incorporates the influence of the fixed distress cost component, K. To find the optimal notional amount of linear and non-linear hedging instruments, X* and Y*, the following maximization of the expected profit of the firm is solved: max X, Y [ ] E π DC Alternatively, the solution to this problem is equivalent to that obtained from the minimization of the expected distress costs as shown by the following: (6) max X, Y E [ π DC ] = max X, Y E[ π DC] = max X, Y ( E[ π ] E[ DC] ) = = max X, Y ( E[ π 0 ] E[ DC] ) = E[ π 0 ] + max X, Y ( E[ DC] = E[ π ] min E[ DC] 0 X, Y ) = We thus solve the following minimization problem: min, E[ DC] X Y (7) To minimize the expected distress cost function one must integrate over both random variables Z and ε. However, after integrating over variable Z, we obtain an expression that cannot be further analytically integrated over the variable ε. The interested reader can find this resulting expression in Appendix A denoted as equation (A.1). Therefore, to find the optimal linear (X*) and non-linear (Y*) positions, the solution to expression (A.1) is found numerically.
18 4. Sensitivity Analysis of the General Solution In this section we provide the results of our numerical solutions to the firm s hedging problem developed in the previous section. We present the numerical solutions in the form of graphical illustrations of the sensitivities of the optimal linear (X*) and nonlinear (Y*) positions to both business risk and market risk. We subsequently extend this analysis to include correlation effects, leverage and transactions costs. 4.1 Business and Market Risk Effects To illustrate the effect of business risk and market risk on the optimal mix of linear and non-linear instruments, we utilize the following base case parameter values in arriving at our numerical solutions. We assume expected revenues or sales equal to $10 based on a level of expected output (µ) of 10 units and an expected forward price of the commodity produced (ε F ) of $1 (or which is analogous to the firm having expected foreign currency revenues of 10 FC and an expected exchange rate of $1/FC). We also assume a face value of debt (D) equal to $10; an interest rate (r) equal to 10 percent; and a level of fixed production costs (C) equal to $6. Thus, the interest payment rd is equal to $1, the expected unhedged cash flows E [ π 0 ] are equal to 3, and the expected times-interest-earned (TIE) ratio is equal to 4. In addition, we assume that the firm incurs financial distress costs when TIE becomes less than β=. In these instances, we assume a proportionality cost coefficient of γ = 0.1 and a fixed distress cost of K = 0.
19 Figure provides the first set of results under the initial assumption of a zero correlation between output and prices (i.e., ρ Z,ε = 0). The optimal notional amount of both linear (X*) and non-linear (Y*) contracts is plotted along the vertical axis while the level of business risk (σ ) is plotted along the horizontal axis. For each level of business risk, we report results for values of market risk ( σ ε ) equal to 10, 30, 50 and 70 percent. We note several observations on the level and pattern of the optimal hedging positions. First, for low and moderate levels of business risk, a firm s risk exposure is optimally hedged mainly using linear or forward contracts (represented by the solid curves in Figure ) with a relatively minor position in non-linear or option contracts (represented by the dashed curves). Second, as the level of business risk increases, the optimal linear position declines while the non-linear position increases. Third, holding constant the level of business risk, we see that optimal linear position is decreasing in market risk, that is, the greater the level of market risk, the smaller the use of forward contracts. In contrast, we also observe that that the optimal non-linear position is increasing in market risk. It appears that under normal market and business operating conditions, firms should conduct their hedging primarily using linear contracts. Further, the optimal linear position will be to sell a notional amount of forward contracts that is somewhat below the value of expected output (or expected foreign sales). That is, the firm will engage in less than full hedging. And as firms face increasingly higher levels of business risk, they will further reduce their use of linear contracts, but at the same time will use more non-
0 linear contracts. Further, this substitution effect between linear and non-linear instruments becomes greater, the higher the level of market risk. 4. Discussion Using the results observed in the above numerical exercise as a basis for discussion, we next provide further insight as to how firms should approach their hedging decision. As a general matter, because distress costs are mainly present when operating revenues become low or negative, the firm's hedging position should ideally move cash flows from those states where operating profits are high (high output, high prices) to those states where operating profits are low (low output, low prices). 6 For lower levels of business risk, this can effectively be accomplished with linear contracts since, for example, short forward positions will have negative payoffs in high price states and positive payoffs in low price states. However, as we discuss below, options are generally less effective but will become increasing beneficial as business risk increases. We first consider the firm's hedging decision in the absence of business risk whereby the firm faces only price risk with respect to a known level of production output. Following this we consider the effect on hedging behavior when the firm does face uncertainty regarding the level of its production output. 4..1 Hedging in the absence of business risk 6 Mello and Parsons (000) provide an analogous argument with respect to hedging for purposes of shifting cash flows from states wherein the shadow price of liquidity is low to states where it is high.
1 (a) Linear Contracts: Consider first the firm's hedging decision regarding the use of linear contracts under zero business risk (we later consider the role of non-linear contracts for hedging price risk in the absence of business risk). In this case the hedging decision in the face of only market or price risk is relatively straightforward. The optimal linear strategy is for the firm to sell forward a quantity of its production that will ensure a minimum guaranteed level of cash flow so as to avoid the triggering of financial distress costs. As shown in expression (5) the firm avoids triggering financial distress costs when its total cash flows p are sufficiently high to satisfy coverage ratio requirements, that is π µ (β-1)dr. Substituting for p from equation () into this condition, the firm will sell a quantity of forward contracts X that ensures a sufficient level of hedged revenue (i.e., revenues from selling output plus hedging gains and losses) to cover the sum of its fixed production costs C plus β*dr. 7 That is, the firm will choice X such that Z*ε + (ε F -ε)*x will be at least or greater than C + β*dr. Using the base case assumptions of Z equal to 10, β equal to, interest expense (Dr) equal to $1, ε F equal to $1, and fixed production costs (C) equal to $6, the firm will require a minimum guaranteed level of hedged revenues of $8. Thus, to ensure such a level of revenue, even in the worst case scenario that the realized price ε of output falls to zero, the firm 7 The firm's revenue from selling output plus hedging gains and losses, which we refer to as its "hedged revenue" is represented by Z*ε + (ε F -ε)*x. Later, when we also consider non-linear contracts, this expression is represented by Z*ε + (ε F -ε)*x + (ε F - ε)l *Y - PY.
will sell a notional quantity of 8 forward contracts. This corresponds to the optimal linear position depicted in Figure when business risk approaches zero. Though popular belief often holds that a firm's hedge position should match expected output (or foreign revenues), the above illustration makes the similar observation as in Mello and Parsons (000) that the optimal hedging position of a firm will typically be to partially hedge. The hedge position needs only to ensure that the firm is able to generate a sufficient level of revenue to avoid distress costs. In the above example, though the firm's entire output of 10 units is subject to price risk, the firm can hedge as few as 8 units of output and still avoid financial distress costs. Ignoring transaction costs, the firm is actually indifferent to hedging any quantity of units between 8 and 10, but hedging additional units beyond 8 will produce zero expected net benefits. 8 Hedging beyond 10 contracts, however, exposes the firm to additional risks and to incurring costs related to overhedging. We loosely define over-hedging with linear contracts as having determined ex post that an excessive notional quantity of linear or forward contracts were sold relative to realized output. In the absence of business risk, over-hedging simply means hedging with a quantity of forwards that exceeds the known level of output, a suboptimal activity that can be easily avoided. In our example, hedging more than 10 units of output will be suboptimal as any additional forward positions will increase expected distress costs. This is because conditioned on the firm already hedging at least 8 units, gains on additional short forward positions
3 when prices fall cannot further reduce distress costs as they will by then have been completely eliminated. However, if prices rise the firm's loss on the additional forward contracts will exceed the extra revenues generated from selling output at higher prices If prices rise significantly, the firm's hedged revenues could fall below a level that triggers financial distress costs. We refer to this increase in expected financial distress costs due to hedging with an excessive number of linear contracts as the "cost of overhedging". (b) Non-linear Contracts: Generally speaking, at the margin, the use of nonlinear instruments will likely have negligible or even negative benefit. That is, conditioned on the firm being at its optimal linear position, substituting non-linear for linear contracts may not be attractive due to (1) the relative greater number of contracts needed (thus suggesting higher transaction costs), and () limits on the extent of their usage because of the potential for the payment of option premiums to trigger financial distress costs. To see this we build on our earlier example where recall that the firm's optimal linear hedging position was 8 contracts. 9 Consider first the substitution of a long put contract for the last (i.e., 8th) forward contract. The firm's hedging position will now consist of 7 short forward contracts each having a contract price of $1 and one long put contract with an exercise price of $1. Due to the option premium, the put will not 8 We consider below the effect of transaction costs on the optimal hedging positions. 9 Though 9 and 10 contracts positions were also optimal, the 9 th and 10 th contracts were superfluous in terms of contributing to firm value.
4 provide the same level of net downside protection against potential distress costs as did the forward contract. To illustrate, assume a put value of $0.0 and that output prices fall to zero. The firm's revenues on output (Z*ε) will be zero, the profits on the 7 forward contracts will be $7 and the net profit on the put will be ($1-$0.0) or $0.80, together producing a total hedged revenue of only $7.80. The firms will thus incur financial distress costs as this number is less than the $8 (C + βdr). To avoid these costs, the firm should instead purchase options in the ratio of $1/($1-$0.0) or 1.5 puts per forward contract. In terms of hedging performance, the firm will be indifferent between using either the 1.5 puts or the 1 forward contract. However, if transaction costs per contract are comparable, the forward contract will be preferred due to the lower number of contracts required. Consider next the situation of a firm that is already close to experiencing financial distress. In the prior example, assume instead that the firm has a cost structure and covenant requirements such that it requires a level of revenue to avoid financial distress (C + βdr) of $9.90. With 10 units of output and thus $10 of expected sales, the firm can be viewed as close to experiencing financial distress. The firm's optimal linear strategy is to then sell 9.9 forward contracts, which will ensure a minimum level of hedged revenue of $9.90 regardless of how low prices fall. In terms of substituting puts, the maximum quantity that the firm can purchase and still ensure a sufficient level of hedged revenue of at least $9.90 is only 0.5 puts (at $0.0 each and thus a total cost of $0.10). Purchasing more than.5 puts will produce a hedged revenue of less than $9.90 if, for example, prices were to remain unchanged and the options expire
5 worthless. Thus, to maintain the same level of downside protection the firm's purchase of these 0.5 puts will be in lieu of 0.4 forward contracts (based on a substitution ratio of 1.5 puts per forward). The firm's resulting hedged position will then consist of 0.5 puts long and 9.5 forwards short. As can be seen, the closer the firm is to experiencing financial distress, the smaller the opportunity for using puts due to the potential effect of premiums for generating financial distress costs. Still, as long as business risk is zero, the use of puts will never provide benefits over and above those provided by forwards, even when the substitution of puts for forward contracts does not generate financial distress costs. 4.. Hedging in the presence of business risk (a) Linear Contracts: Earlier we noted that in the absence of business risk, the firm can avoid the financial distress costs related to over-hedging by simply not selling a quantity of forward contracts greater than its known output. However, when business risk becomes non-zero, the uncertainty regarding actual output levels makes avoiding the costs of over-hedging a more difficult task. Consider the situation where expected output remains at 10 units, but now because of business risk, the firm's actual output can vary from 6 to 14 units. Previously, 8 forward contracts were optimal in the case of no business risk. Now, however, if the firm continues with an 8 forward contract position, then over-hedging can occur if realized output falls between 6 and 8 units. To completely avoid the risk of over-hedging, the firm would need to use 6 or less forward contracts. However, using less than 8 contracts exposes the firm to potential additional distress costs in the
6 event that prices fall. Thus, the firm will use somewhere between 6 and 8 contracts. Generally speaking, the optimal linear position (in the absence of non-linear contracts) will be such that it minimizes the sum of the expected distress costs from falling prices and the costs of overhedging from rising prices (henceforth, "total overall expected distress costs"). This is also to say that at the optimum, the marginal expected distress and over-hedging costs from adding or subtracting contracts are equal. (b) Non-linear Contracts: In contrast to the zero business risk case, there will now be a limited, but important role for non-linear contracts because of their ability to fine-tune (tailor) the optimal risk management strategy to further reduce the sum of the expected distress and over-hedging costs. Given the optimal linear position, consider first the effect of the substitution of a long put in place of the last short forward contract. 10 Compared to the foregone forward contract, the put will be less effective for reducing expected distress costs in low price states due to the option premium. However, with respect to the over-hedging costs in high price states, the put will be more advantageous as the put will expire worthless and with its net payoff limited to the loss of the option premium. Thus, the substitution of a quantity of puts for forward contracts can alter the total expected distress costs in a positive or negative manner. The substitution will be optimal if the decrease in over-hedging costs exceeds the increase in expected distress costs thus leading to a reduction in the total overall 10 Note that substituting puts for forwards in this fashion can be thought of as somewhat analogous to buying calls. That is, buying a put and closing out or buying back a short forward contract can be replicated by buying a call. However, the optimal substitution ratio in our model, as noted earlier, will typically not be one-to-one.
7 expected distress costs. However, if the substitution were to increase the overall costs, the firm should instead engage in "reverse substitution" wherein they simultaneously add additional short forward contracts and write puts (which together can be viewed as equivalent to "writing calls"). The extent of substitution in either case and hence the extent of using options is, however, limited. To see this recall that the optimal linear position in the previous example was said to be between 6 and 8 contracts. Assume that 7 linear contracts were the optimal position. Under zero correlation, if either substitution or reverse substitution is optimal, the resulting optimal linear positions will never fall below 6 contracts (a reduction of one forward) or above 8 contracts (an addition of one forward). At a level of 6 forward contracts, all costs of over-hedging are completely eliminated while at 8 contracts the expected distress costs in low price states cannot be further reduced through additional forwards. Thus, based on a 1.5 substitution rate, the potential maximum number of puts will be limited to only 1.5 contracts (long or short). As the level of business or output risk increases, there will be a greater opportunity for substitution and hence a larger role for options. To illustrate, assume that output can range from 4 to 16 units. The optimal linear position will thus be somewhere between 4 to 8 contracts. Assume that it is 6 forward contracts. Depending on whether substitution or reverse substitution is optimal, as many as
8 forwards could be eliminated or added and, therefore, as many as.5 long or short put contracts established. Finally, in contrast to the zero business risk case, there is one additional consideration. Even if prices remain relatively stable, the firm can incur financial distress if output falls significantly. Forward contracts will not provide relief in these states, but the additional income from writing puts could provide additional protection. The implication of this is that the propensity to buy put options will be reduced, thus weakening (strengthening) the substitution (reverse substitution) effect noted above. 4.3 Correlation Effects We next investigate the effect of the correlation between output and prices (ρ Z,ε ) on the optimal hedging positions. Depending on the sign of the correlation, revenues will exhibit either greater or lower volatility. A positive correlation has the effect of exacerbating fluctuations in revenues ( Z ε ) since output levels and corresponding prices will move in the same direction. The implication for hedging is two-fold. First, the overall demand for derivatives will increase as the business risk becomes more "hedgeable" since output now moves more in line with prices. Second, the overhedging problem discussed earlier will become less severe since the likelihood of observing a simultaneous drop in output and an increase in price is diminished. A negative correlation between output and prices will produce an opposite result as it will serve to dampen fluctuations in revenues, thus producing a natural hedge effect. For example, a negative correlation for an exporting firm selling goods in a foreign country implies that it will experience increasing (declining) sales volume at
9 the same time as the domestic currency is strengthening (weakening). Alternatively, a commodity firm experiencing higher (lower) output will have the potential increase (decrease) in revenues offset to some extent by a lower (higher) market selling price. To illustrate the effect of correlation, Figure 3 provides a comparison of the optimal hedging positions corresponding to values of ρ Z,ε equal to 0.5, 0 and +0.5. Panels (a) and (b) present the optimal linear and non-linear positions, respectively, for various levels of business risk and an assumed market risk level of 30 percent. As shown in panel (a), the higher the correlation, the greater the optimal linear position for a given level of business risk. This can be attributed to the over-hedging problem becoming less (more) severe, the greater (lower) the correlation. As mentioned, the over-hedging problem becomes of greatest concern in states where output falls and prices rise. If quantity and prices are positively correlated, this occurrence becomes less likely, thus leading the firm to use more linear instruments. The corresponding optimal non-linear positions are presented in panel (b). As shown, the substitution effect of non-linear for linear contracts is inversely related to the level of correlation. That is, the greater (lower) the correlation, the lower (greater) the non-linear position. For the negative correlation case, the non-linear position is initially both greater than that observed for the zero correlation case and is increasing in business risk. Though the position is still relatively small, the larger non-linear position (than in the zero correlation case) helps mitigate the over-hedging problem and provides additional protection against large declines in prices that could otherwise generate significant financial distress costs.
30 It is also interesting to note that as the correlation becomes increasingly positive, the optimal non-linear position declines in size and can even become negative. In this case the firm switches from essentially purchasing put protection to a strategy of put writing (i.e., "reverse substitution") for purposes of generating premia income to add to its revenue stream. Intuitively, to see why a firm would pursue such a strategy of writing puts, note that the increased linear position shown in panel (a) due to the positive correlation, coupled with put writing, is equivalent to writing calls. The revenue from writing options serves to offset the lower revenue in either low price or low output states, thus reducing potential financial distress costs. In high price states the potential losses on the calls are of less concern to the firm because output and hence revenue is likely to be higher due to the positive correlation. 4.4 Leverage Effects Financial leverage has been a frequently argued determinant of derivatives usage. For example, the financial distress hypothesis advanced in Smith and Stulz (1985) and elsewhere suggests that to reduce the expected costs of financial distress, firms with greater leverage and thus financial exposure are more likely to hedge. By hedging, a firm is able to shift the distribution of its cash flows towards states where distress costs are lower. Our model allows us to further investigate this effect and to analyze the distinctive cash flow shifting abilities of linear versus non-linear instruments. Using again our earlier base case assumptions and a level of market risk of 0.3, Figure 4 illustrates the influence of the level of debt or related interest expense on the optimal linear and non-linear positions, respectively. First, consider the optimal linear