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Who hedges more when leverage is endogenous? A testable theory of corporate risk management under general distributional conditions

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Abstract

This paper develops a theory of a firm’s hedging decision with endogenous leverage. In contrast to previous models in the literature, our framework is based on less restrictive distributional assumptions and allows a closed-form analytical solution to the joint optimization problem. Using anecdotal evidence of greater benefits of risk management for firms selling “credence goods” or products that involve long-term relationships, we prove that those optimally leveraged firms, which face more convex indirect bankruptcy cost functions, will choose higher hedge ratios. Moreover, we suggest a new approach to test this relationship empirically.

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Notes

  1. See Graham and Rogers (2002), pp. 817–823, and Brown and Toft (2002), pp. 1283–1287 for an overview of the corporate hedging literature.

  2. Smith and Stulz (1985), p. 396. Ignoring the issue of bankruptcy costs, MacMinn (1987) suggests the protection of depreciation charges and tax credits as a rationale for corporate hedging. Froot et al. (1993) propose that hedging can help to avoid the use of expensive external financing by coordinating the firm’s investment and financing policy. Recently, their framework has been specified further by Spanò (2004), who derives the convexity of the external finance cost function from the firm’s bankruptcy risk.

  3. Géczy et al. (1997), p. 1328.

  4. See the unpublished working paper by Ross (1996), p. 3, the non-technical article by Stulz (1996), p. 16, and the Presidential Adress to the American Finance Association by Leland (1998), p. 1237.

  5. cf. Graham and Rogers (2002), p. 820.

  6. See Bradley et al. (1984), p. 864 for further references.

  7. Note that in corporate risk management, volatility is often equated with variance, which is fairly restrictive as to the underlying distributional assumptions. See e.g. Brown and Toft (2002), p. 1319 and Hahnenstein and Röder (2003), p. 325. See also Shaffer and DeMaskey (2005) for hedging policies based on the Mean-Gini framework.

  8. See Sandmo (1971), p. 66 for a justification of the exclusion of negative prices. It should be noted that the assumption of a continous density function is restrictive, since discrete jumps are ruled out. Our analysis could be completely reproduced under even more general distributional assumptions in a more abstract mathematical setting (Lebesgue integrals, …), but this would not add much in terms of economic insights.

  9. Of course, r could be set to zero without loss of generality. However, as this would imply s0 =  f, the presentation would lose much of its intuition. See Chen and Huang (2002) for a treatment of forward prices under stochastic interest rates.

  10. See Smith and Stulz (1985), p. 396 and Bessembinder (1991), p. 521 for an identical assumption.

  11. Cooper and Mello (1999, p. 195), whose analysis concentrates on forward contracts subject to default risk, point out that in practice, corporations with low ratings are excluded from the forward markets unless collateral is provided.

  12. cf. e.g. the Credit MetricsTM Technical Document by Gupton, Finger and Bhatia (1997), p. 87f. and the recent study on the default boundary by Davydenko (2005).

  13. See e.g. Smith and Stulz (1985), p. 396 and Bessembinder (1991), p. 523f. who make use of the same tax schedule. As has already been pointed out by Smith and Stulz (1985) p. 396 in footnote No. 12, a more realistic treatment of taxation that grants a tax shield only to the coupon would make an analysis more complex. E.g. with the firm’s default risk deminished by hedging, the required coupon rate will generally be lower and so will the tax shield. See Leland (1994) for a more sophisticated treatment of debt contracts.

  14. cf. e.g. Altman (1984), p. 1084, Ross (1985), p. 639 and MacMinn (1987), p. 1179.

  15. See Nance, Smith and Smithson (1993), Géczy, Minton and Schrand (1997), Graham and Smith (1999) and Graham and Rogers (2002). Judge (2004) provides a topical overview of the empirical literature.

  16. See also Brown and Toft (2002), p. 1290 f., who propose an exponential function, whereas Turnbull (1979), p. 934, Smith, Smithson and Wilford (1990), p. 133 as well as Cooper and Mello (1999), p. 199 argue in favour of fixed bankruptcy costs. The latter discuss alternative specifications on p. 220 f. Adam (2002), p. 248 includes both a fixed and a proportional component.

  17. See Hall (2002, p. 38) in a similar context: “... one would intuitively expect the cost of financial distress to curve upwards as the bank is forced into ever more draconian action to survive.” See also Turnbull (1979), p. 933, Dionne and Garand (2003), p. 45 and Chen and Merville (1999) on this point.

  18. Note that when taking over the firm, the creditors’ liability for financial distress costs is not limited. They will even have to contribute additional funds in t = 1, if the value of the remaining assets is insufficient to cover the claims of the trustees, courts etc. Suppose e.g. that s1 is very low and N → 0, then according to Eq. (4) V1 is also very small and hence the extent of distress may be very high. Thus, we might have V1 <  m · (D1− V1), so that the “beneficiaries of bankrupty” can only be partially compensated. While this implication might seem not very realistic at first glance, one should keep in mind that, in commercial banking, it may well happen that during a firm’s liquidation, an unforeseen deterioration in collateral value takes place, which finally leads to the realization of proceeds at lower value than the costs of the lawsuit.

  19. See Bessembinder (1991), p. 531 and Leland (1998), p. 1219 for covenants concerning the firm’s hedging policy. Practical examples include e.g. “gold loans” (see Tufano (1996), p. 1100) and “oil-linked bonds” (see Culp, Furbush and Kavanagh (1994), p. 75).

  20. Apart from a direct collision with criminal law, a violation of the covenants by management or the equityholders themselves, would lead to a loss of reputation that could be used to justify the absence of shareholder-bondholder-conflicts within a multiperiod framework.

  21. See Breeden and Litzenberger (1978), p. 627 f. or Aït-Sahalia and Lo (1998), p. 503 f. for the relationship between state-price-densities and option pricing models. In particular, recall that the state-price density function is equal to the second partial derivative of any call option pricing formula with respect to the strike price and that no specific assumptions have to be made as to the stochastic process governing the movement of the underlying. Nevertheless, the marginal density function cannot be chosen completely arbitrarily: In order to ensure the existence of an equivalent martingale measure, it is typically assumed that the random variables are square integrable and that Novikov’s condition holds.

  22. See Harrison and Kreps (1979), p. 383 and p. 390.

  23. Note that this approach to valuation allows us to price all claims considered in our model in relation to the current value of the production good s0 and to the riskless rate r, but that it does not imply that the claims themselves are actually traded continuously in frictionless markets. See Merton (1973), p. 162 and especially Merton (1995), p. 422, who points to the fact that the technique of valuation by arbitrage can be applied to bankruptcy costs and corporate taxes and who gives further references.

  24. Note that such a debt policy could actually make more sense from the shareholder point of view than D1 < Y · f, when the parameter value for m is small in relation to τ. But, as we will state later on, it is of course always dominated by D1 = Y · f.

  25. cf. Haley and Schall (1979), p. 202 ff.

  26. cf. Modigliani and Miller (1958), p. 268, Eq. (3), Hirshleifer (1966), p. 267, Eq. (4), Merton (1977), p. 247 f., MacMinn (1987), p. 1171, Proposition 1 and Benninga and Oosterhof (2004), p. 9, Theorem 1.

  27. As in Table 2, we cannot give comparative statics for these situations.

  28. cf. e.g. Majd and Myers (1987), p. 346.

  29. See Merton (1973), Theorem 8 on p. 149 and Theorem 15 on p. 168. While in the more general Theorem 8, increasing risk is defined by adding an error term, Theorem 15 uses the variance as the most common measure of dispersion. A correction of both theorems, which makes use of the underlying’s risk-neutral density function, has been provided by Jagannathan (1984).

  30. See Sect. 4 for a numerical counterexample to Eq. (35), when leverage is endogenous.

  31. Stulz (1996), p. 16. See also Dolde (1995), p. 192 and Graham and Rogers (2002), p. 837.

  32. In contrast to our framework, these authors include a personal tax that is payable at the investor level. Moreover, they do not consider a “real world” hedging contract, but simply utilize the standard deviation of the firm’s future cash flows as a measure of business risk.

  33. Stulz (1996), p. 8 even calls the variance-minimization-paradigm the “prevailing academic theory of risk management” and Froot, Scharfstein and Stein (1993), p. 1630 summarize that “much of the previous work has the extreme implication that firms should hedge fully—completely insulating their market values from hedgeable risk.”

  34. A possible economic explanation of the property ∂2L0/ ∂N2 > 0 of the cost function may lie in the reaction of the forward price to an increased supply in the forward market: When the producer sells a larger portion H of its future output Y, the decline in the forward price f can lead to temporary market friction, as the arbitrage relation (11) between the spot and forward markets is violated. This friction is likely to increase with H.

  35. See e.g. Breeden and Litzenberger (1978), p. 630, equation (5) or Aït-Sahalia and Lo (1998), p. 504, Eq. (4). Generally, a state-price density function can be derived from any option pricing model by simply taking the second partial derivative of the call option value with respect to the strike price.

  36. With the development of liquid credit default swap markets, the availability of high-frequency time-series data has greatly improved. Bloomberg quotes for credit default swap spreads are now available for more than 1,500 companies. The relationship between credit default swap spreads, bond yields and Moody’s credit rating announcements is analyzed by Hull et al. (2004). See also Manning (2004), who explores the relationship between corporate bond spreads and default probabilities.

  37. See Dwyer et al. (2004) for an exposition of the Moody’s KMV EDFTM concept.

  38. Note that we changed the sign of the regression coefficient, because the parameter m in the theoretical model refers to bankruptcy costs which reduce shareholder wealth.

  39. Of course, the regression equations must not be interpreted as causal relationships, but they rather reveal the role of common underlying factors.

  40. Aggarwal and Simkins (2004), p. 66 provide a summary of US-GAAP disclosure requirements for derivative financial instruments during the 90s.

  41. See for example McElroy and Burmeister (1988) for the Iterated Nonlinear Seemingly Unrelated Regression (ITNLSUR) method, which avoids the errors-in-the-variables problem of the two-stage approach. See also Glosten and Jagannathan (1994), pp. 142 ff. for a discussion of parametric as well as nonparametric methods that are available for estimating a function whose functional form is not known.

References

  • Adam T. R. (2002). Risk management and the credit risk premium. Journal of Banking and Finance, 26, 243–269

    Article  Google Scholar 

  • Aggarwal, R., & Simkins, B. J. (2004). Evidence of voluntary disclosures of derivatives usage by large US companies. Journal of Derivatives Accounting, 1, 61–81

    Article  Google Scholar 

  • Aït-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499–547

    Article  Google Scholar 

  • Altman, E. I. (1984). A further empirical investigation of the bankruptcy cost question. The Journal of Finance, 39, 1067–1089

    Article  Google Scholar 

  • Benninga, S. Z., & Oosterhof, C. M. (2004). Hedging with forwards and puts in complete and incomplete markets. Journal of Banking and Finance, 28, 1–17

    Article  Google Scholar 

  • Bessembinder, H. (1991). Forward contracts and firm value: Investment incentive and contracting effects. Journal of Financial and Quantitative Analysis, 26, 519–532

    Article  Google Scholar 

  • Bradley, M., Jarrell, G. A., & Kim, E. H. (1984). On the existence of an optimal capital structure: Theory and evidence. The Journal of Finance, 39, 857–878

    Article  Google Scholar 

  • Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of Business, 51, 621–651

    Article  Google Scholar 

  • Brennan, M. J., & Schwartz, E. S. (1978). Corporate income taxes, valuation, and the problem of optimal capital structure. Journal of Business, 51, 103–114

    Article  Google Scholar 

  • Brown, G. W., & Toft, K. B. (2002). How firms should hedge. The Review of Financial Studies, 15, 1283–1324

    Article  Google Scholar 

  • Castanias, R. (1983). Bankruptcy risk and optimal capital structure. The Journal of Finance, 38, 1617–1635

    Article  Google Scholar 

  • Chen, G. M., & Merville, L. J. (1999). An analysis of the underreported magnitude of the total indirect costs of financial distress. Review of Quantitative Finance and Accounting, 13, 261–272

    Article  Google Scholar 

  • Chen, N., Roll, R., & Ross, S. A. (1986). Economic forces and the stock market. The Journal of Business, 59, 383–403

    Article  Google Scholar 

  • Chen, R., & Huang, J. (2002). A note on forward price and forward measure. Review of Quantitative Finance and Accounting, 19, 261–272

    Article  Google Scholar 

  • Colquitt, L. L., & Hoyt, R. E. (1997). Determinants of corporate hedging behavior: Evidence from the life insurance industry. The Journal of Risk and Insurance, 64, 649–671

    Article  Google Scholar 

  • Cooper, I. A., & Mello, A. S. (1999). Corporate hedging: The relevance of contract specifications and banking relationships. European Finance Review, 2, 195–223

    Article  Google Scholar 

  • Culp, C. L., Furbush, D., & Kavanagh, B. T. (1994). Structured debt and corporate risk management. Journal of Applied Corporate Finance, 7, 73–84

    Article  Google Scholar 

  • Davydenko, S. A. (2005). When do firms default? A study of the default boundary. Working paper, London Business School

  • Dionne, G., & Garand, M. (2003). Risk management determinants affecting firms’ values in the gold mining industry: New empirical results. Economics Letters, 79, 43–52

    Article  Google Scholar 

  • Dolde, W. (1995). Hedging, leverage, and primitive risk. The Journal of Financial Engineering, 4, 187–216

    Google Scholar 

  • Dwyer, D. W., Kocagil, A. E., & Stein, R. M. (2004). The Moody’s KMV EDFTM RiskCalcTM v3.1 Model – Next-generation technology for predicting private firm credit risk. Working paper, Moody’s KMV

  • Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of Political Economy, 81, 607–636

    Article  Google Scholar 

  • Fok, R. C. W., Carroll, C., & Chiou, M. C. (1997). Determinants of corporate hedging and derivatives: A revisit. Journal of Economics and Business, 49, 569–585

    Article  Google Scholar 

  • Froot, K. A., Scharfstein, D. S., & Stein, J. C. (1993). Risk management: Coordinating corporate investment and financing policies. The Journal of Finance, 48, 1629–1658

    Article  Google Scholar 

  • Géczy, C., Minton, B. A., & Schrand, C. (1997). Why firms use currency derivatives. The Journal of Finance, 52, 1323–1354

    Article  Google Scholar 

  • Glosten, L. R., & Jagannathan, R. (1994). A contingent claim approach to performance evaluation. Journal of Empirical Finance, 1, 133–160

    Article  Google Scholar 

  • Graham, J. R., & Smith, C. W. (1999). Tax incentives to hedge. The Journal of Finance, 54, 2241–2262

    Article  Google Scholar 

  • Graham, J. R., & Rogers, D. R. (2002). Do firms hedge in response to tax incentives? The Journal of Finance, 57, 815–839

    Article  Google Scholar 

  • Gupton, G. M., Finger, C. C., & Bhatia, M. (1997). CreditMetricsTM – Technical Document. J. P. Morgan & Co. Incorporated, New York

    Google Scholar 

  • Hahnenstein, L. (2001). Hedging mit Termingeschäften und Shareholder Value. Wiesbaden (Germany): Gabler Edition Wissenschaft

    Google Scholar 

  • Hahnenstein, L., & Röder, K. (2003). The minimum variance hedge and the bankruptcy risk of the firm. Review of Financial Economics, 12, 315–326

    Article  Google Scholar 

  • Haley, C. W., & Schall, L. D. (1979). The theory of financial decisions (2nd ed.). New York

  • Hall, C. (2002). Economic capital: towards an integrated risk framework. Risk (October), 33–38

  • Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408

    Article  Google Scholar 

  • Haushalter, G. D. (2000). Financing policy, basis risk, and corporate hedging: evidence from oil and gas producers. The Journal of Finance, 55, 107–152

    Article  Google Scholar 

  • Hirshleifer, J. (1966). Investment decision under uncertainty: Applications of the state-preference approach. The Quarterly Journal of Economics, 80, 252–277

    Article  Google Scholar 

  • Holthausen, D. M. (1979). Hedging and the competitive firm under price uncertainty. American Economic Review, 69, 989–995

    Google Scholar 

  • Hull, J., Predescu, M., & White, A. (2004). The relationship between credit default swap spreads, bond yields, and credit rating announcements. Journal of Banking and Finance, 28, 2789–2811

    Article  Google Scholar 

  • Jagannathan, R. (1984). Call options and the risk of underlying securities. Journal of Financial Economics, 13, 425–434

    Article  Google Scholar 

  • Judge, A. (2004). The determinants of foreign currency hedging by UK non-financial firms. Working paper. London: Middlesex University

  • Kale, J. R., & Noe, T. H. (1990). Corporate hedging under personal and corporate taxation. Managerial and Decision Economics, 11, 199–205

    Article  Google Scholar 

  • Kale, J. R., Noe, T. H., & Ramirez, G. G. (1991). The effect of business risk on corporate capital structure: Theory and evidence. The Journal of Finance, 46, 1693–1715

    Article  Google Scholar 

  • Kraus, A., & Litzenberger, R. H. (1973). A state-preference model of optimal financial leverage. The Journal of Finance, 28, 911–922

    Article  Google Scholar 

  • Leland, H. E. (1994). Corporate debt value, bond covenants, and optimal capital structure. The Journal of Finance, 49, 1213–1252

    Article  Google Scholar 

  • Leland, H. E. (1998). Agency costs, risk management, and capital structure. The Journal of Finance, 53, 1213–1243

    Article  Google Scholar 

  • Majd, S., & Myers, S. C. (1987). Tax Asymmetries and corporate income tax reform. In M. Feldstein (Ed.), The effects of taxation on capital accumulation (pp. 7–54). Chicago (Illinois)

  • MacMinn, R. D. (1987). Forward markets, stock markets, and the theory of the firm. The Journal of Finance, 42, 1167–1185

    Article  Google Scholar 

  • Manning, M. J. (2004). Exploring the relationship between credit spreads and default probabilities. working paper. Bank of England

  • McElroy, M. B., & Burmeister, E. (1988). Arbitrage pricing theory as a restricted nonlinear multivariate regression model: Iterated nonlinear seemingly unrelated regression estimates. Journal of Business and Economic Statistics, 6, 29–42

    Article  Google Scholar 

  • Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4, 141–183

    Article  Google Scholar 

  • Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance, 29, 449–470

    Article  Google Scholar 

  • Merton, R. C. (1977). On the pricing of contingent claims and the Modigliani-Miller Theorem. Journal of Financial Economics, 5, 241–249

    Article  Google Scholar 

  • Merton, R. C. (1995). Continuous-time finance. Cambridge (Mass.) and Oxford (UK)

  • Mian, S. L. (1996). Evidence on corporate hedging policy. Journal of Financial and Quantitative Analysis, 31, 419–439

    Article  Google Scholar 

  • Modigliani, F., & Miller, M. H. (1958). The cost of capital, corporation finance and the theory of investment. American Economic Review, 48, 261–297

    Google Scholar 

  • Nance, D. R., Smith, C. W., & Smithson, C. W. (1993). On the determinants of corporate hedging. The Journal of Finance, 48, 267–284

    Article  Google Scholar 

  • Rawls, S. W. III., & Smithson, C. W. (1990). Strategic risk management. Journal of Applied Corporate Finance, 2, 6–18

    Article  Google Scholar 

  • Ross, M. P. (1996). Corporate hedging: What, why and how? Working paper. Berkeley: University of California

  • Ross, S. A. (1985). Debt and taxes and uncertainty. The Journal of Finance, 40, 637–658

    Article  Google Scholar 

  • Sandmo, A. (1971). On the theory of the competitive firm under price uncertainty. American Economic Review, 61, 65–73

    Google Scholar 

  • Shaffer, D. R., & DeMaskey, A. (2005). Currency hedging using the mean-gini framework. Review of Quantitative Finance and Accounting, 25, 125–137

    Article  Google Scholar 

  • Shapiro, A. C., & Titman, S. (1985). An integrated approach to corporate risk management. Midland Corporate Finance Journal, 3, 41–56

    Google Scholar 

  • Smith, C. W., & Stulz, R. M. (1985). The determinants of firms’ hedging policies. Journal of Financial and Quantitative Analysis, 20, 391–405

    Article  Google Scholar 

  • Smith, C. W., Smithson, C. W., Wilford, D. S. (1990). Financial engineering: Why hedge? In C. W. Smith & C. W. Smithson (Eds.), The handbook of financial engeneering: New financial product innovations, applications and analyses (pp. 126–137). New York

  • Spanò, M. (2004). Determinants of hedging and its effects on investment and debt. Journal of Corporate Finance, 10, 175–197

    Article  Google Scholar 

  • Stulz, R. M. (1996). Rethinking risk management. Journal of Applied Corporate Finance, 9, 8–24

    Article  Google Scholar 

  • Tufano, P. (1996). Who manages risk? An empirical examination of risk management practices in the gold mining industry. The Journal of Finance, 51, 1097–1137

    Article  Google Scholar 

  • Turnbull, S. M. (1979). Debt capacity. The Journal of Finance, 34, 931–940

    Article  Google Scholar 

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Authors and Affiliations

  1. IKB Deutsche Industriebank AG, 40002, Düsseldorf, Germany

    Lutz Hahnenstein

  2. Lehrstuhl für Betriebswirtschaftslehre, insbesondere Finanzdienstleistungen, Universität Regensburg, 93053, Regensburg, Germany

    Klaus Röder

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  1. Lutz Hahnenstein
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  2. Klaus Röder
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Correspondence to Lutz Hahnenstein.

Additional information

Earlier versions of this paper were presented at the 5th Conference of the Swiss Society for Financial Market Research (SGF), at the 9th Annual Meeting of the German Finance Association (DGF), at the 2004 Basel Meeting of the European Financial Management Association (EFMA), the 2005 WHU Campus for Finance Conference on Options and Futures, the 12th Global Finance Conference (GFC) in Dublin, the 2005 Annual Meeting of the Northern Finance Association (NFA) in Vancouver and at the 2006 Annual Meeting of the Eastern Finance Association (EFA) in Philadelphia. We especially appreciate the valuable comments of Tim R. Adam, Axel F. A. Adam-Müller, Rakesh Bharati, René Garcia, Amrit Judge, Olaf Korn, Gunter Löffler, Lars Norden, Larry D. Wall, Josef Zechner and two anonymus RQFA referees. Of course, any remaining errors are our own.

Appendix

Appendix

1.1 Comparative statics for the deadweight claims with respect to hedging:

$$ \begin{aligned} \frac{\partial \hbox{T}_0}{\partial\hbox{N}}&=\tau \cdot \frac{\partial}{\partial \hbox{N}}\left[{\int_{\frac{{\rm D}_1 -{\rm N}\cdot{\rm f}}{{\rm Y}-{\rm N}}}^\infty {\left( {\hbox{N}\cdot \hbox{f}+(\hbox{Y}-\hbox{N})\cdot \hbox{s}_1 -\hbox{D}_1}\right)\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1}}\right]\\ &=\tau \cdot \int_{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}}^\infty {(f-\hbox{s}_1 )\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1 } -\tau \cdot \underbrace {\left( {\hbox{N}\cdot \hbox{f}+(\hbox{Y}-\hbox{N})\cdot \frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}-\hbox{D}_1 } \right)}_{=0}\\&\quad \cdot \hbox{g}\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)\cdot \frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{(\hbox{Y}-\hbox{N})^2}\\ &=\tau \cdot \left[ {\int_0^\infty {(\hbox{f}-\hbox{s}_1 )\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1 } -\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {(\hbox{f}-\hbox{s}_1 )\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1 } } \right].\\ \end{aligned} $$

From the conditions stated in Eqs. (10) and (11), which result from the absence of potential arbitrage opportunities, we conclude that:

$$ \int_0^\infty {(\hbox{f}-\hbox{s}_1 )\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1 } =\hbox{f}\cdot \left. {\hbox{G}(\hbox{s}_1 )} \right|_0^\infty -\hbox{s}_0 =\frac{\hbox{f}}{1+\hbox{r}}-\hbox{s}_0 =0 $$

must hold. Therefore, using partial integration, the above expression for ∂T0/∂N is equivalent to Eq. (32):

$$ \begin{aligned} \frac{\partial \hbox{T}_0 }{\partial \hbox{N}}&=-\tau \cdot \int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {(\hbox{f}-\hbox{s}_1 )\cdot \hbox{g}(\hbox{s}_1 )\,\hbox{ds}_1 } =-\tau \cdot \left[ {\hbox{f}\cdot \left. {\hbox{G}(\hbox{s}_1 )} \right|_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} -\left( {\left. {\hbox{s}_1 \cdot \hbox{G}(\hbox{s}_1 )} \right|_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} -\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } } \right)} \right]\\ &=-\tau \cdot \left[ {\hbox{f}\cdot \hbox{G}\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)-\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}\cdot \hbox{G}\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)+\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } } \right]\\ &=\tau \cdot \left[ {\hbox{G}\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)\cdot \frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}-\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } } \right]. \\\end{aligned} $$

Eq. (33) can be derived in the same way.

As can be concluded from Eq. (7), the sign of Eq. (32), which is always identical to that of Eq. (33), takes on the following values for D1 < Y · f:

$$ \begin{aligned} &\tau \cdot \left[ {\underbrace {\hbox{G}\underbrace {\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)}_{ > 0}}_{ > 0}\cdot \underbrace {\frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}}_{ < 0}-\underbrace {\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } }_{ > 0}} \right] < 0,\,\hbox{N} < \frac{\hbox{D}_1 }{\hbox{f}},\\ &\tau \cdot \left[ {\underbrace {\hbox{G}\underbrace {\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)}_{\le 0}}_{=0}\cdot \underbrace {\frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}}_{ < 0}-\underbrace {\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } }_{=0}} \right]=0,\,\hbox{N}\ge \frac{\hbox{D}_1 }{\hbox{f}}.\\ \end{aligned} $$

For D1 =  Y · f, we receive:

$$ \tau \cdot \left[ {\underbrace {\hbox{G}\underbrace {\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)}_{ > 0}}_{ > 0}\cdot \underbrace {\frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}}_{=0}-\underbrace {\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } }_{ > 0}} \right] < 0. $$

For D1 > Y · f, no unequivocal sign emerges from a first inspection:

$$ \tau \cdot \left[ {\underbrace {\hbox{G}\underbrace {\left( {\frac{\hbox{D}_1 -\hbox{N}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}} \right)}_{ > 0}}_{ > 0}\cdot \underbrace {\frac{\hbox{D}_1 -\hbox{Y}\cdot \hbox{f}}{\hbox{Y}-\hbox{N}}}_{ > 0}-\underbrace {\int_0^{\frac{{\rm D}_1 -{\rm N}\cdot {\rm f}}{{\rm Y}-{\rm N}}} {\hbox{G}(\hbox{s}_1 )\,\hbox{ds}_1 } }_{ > 0}} \right] < 0. $$

A proof of the above claim, which is somewhat tedious and involves the use of the “Put-Call-Parity”, is given by Hahnenstein (2001), pp. 179–181.

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Hahnenstein, L., Röder, K. Who hedges more when leverage is endogenous? A testable theory of corporate risk management under general distributional conditions. Rev Quant Finan Acc 28, 353–391 (2007). https://doi.org/10.1007/s11156-007-0017-z

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