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Marketable and non-hedgeable risk in a duopoly framework with hedging

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Abstract

Today’s volatile markets make the issue of hedging risk more important than ever. The purpose of this paper is to contribute to the growing body of literature on corporate hedging and to investigate the duopoly case under multiple uncertainty. We model a framework with marketable as well as non-hedgeable risk and derive optimal financial risk management decisions under (μ, σ)-preferences. We study two settings: First, we consider the case of additive background risk. It is shown that production and the Nash-equilibrium are not affected by the background risk and the separation theorem is valid in this case. Second, multiplicative technological risk is analyzed. We show that the multiplicative version of the non-hedgeable risk violates the separation property and the market equilibrium depends on the stochastic dependence of the marketable and the non-hedgeable risk. In both settings, primarily the stochastic dependence of the two risks affects the hedging decision.

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Notes

  1. See for example Franke et al. (2004).

  2. Previous studies differentiate between additive and multiplicative background risk. In this paper we consider the case of additive background risk and multiplicative technological risk. Thus, the technological risk in this paper can also be seen as multiplicative background risk. For several studies dealing with non-hedgeable risk see for example Viaene and Zilcha (1998), Franke et al. (2004, 2011), Eichner and Wagener (2003) or Eichner and Wagener (2009).

  3. See for example Kajüter (2012), Glaum and Klöcker (2009) or ISDA (2009).

  4. Holthausen (1979) introduces hedging opportunities for the competitive firm and derives the famous separation and full hedge theorems. Meyer and Robison (1988) consider the competitive firm under (μ, σ)-preferences. As the location and scale-condition holds, this setup is equivalent to the expected utilities approach taken by Holthausen. Hey (1987) analyses the competitive firm within a multi-period setting and introduces the possibility to keep inventory. The author shows that separation is also valid within the multi-period setting. Several other studies include Broll et al. (1995) who consider indirect hedging opportunities. Wong and Xu (2006) examine the impact of liquidity risk on the behavior of the competitive firm and include option contracts for hedging purposes. Franke et al. (2011) discuss the impact of additional background risk and Viaene and Zilcha (1998) discuss the competitive firm under multiple uncertainty.

  5. Throughout the paper, a tilde denotes a random variable. When the tilde is missing, the variable signifies the realization of the stochastic parameter.For further approaches to model uncertain demand see for example Leland (1972).

  6. The index B references the situation with background risk.

  7. For more information and justification concerning the (μ, σ)-approach see for example Battermann et al. (2002), pp. 525 or Robison and Barry (1987), pp. 6 and 71.

  8. The additional assumption states that marginal costs of the competitors do not differ too much. Concerning the existence of a solution, see Vives (1999), p. 16. The assumptions of Theorem 2.1 are fulfilled. Concerning the uniqueness compare Vives (1999), p. 47. Again, the assumptions are met.

  9. The index C references the situation under background risk with a simultaneous hedging opportunity.

  10. See for example Hughes and Kao (1997).

  11. The index SH references the situation under background risk with a strategic hedging device.

  12. The index T references the situation under technological uncertainty.

  13. Concerning the existence of a solution, see Vives (1999), p. 16. The assumptions of Theorem 2.1 are fulfilled. Concerning the uniqueness compare Vives (1999), p. 47. Again, the assumptions are met.

  14. See for example Broll et al. (2011) or Pelster (2013).

  15. The index KR labels the situation under demand uncertainty and technological risk.

  16. See Vives (1999) again, p. 16 resp. p. 47. All assumptions are met. Existence is given as \(Z^{i} > -\frac {2b}{\alpha ^{i}}\). To ensure uniqueness \(Z^{i} > -\frac {b}{\alpha ^{i}}\) has to be fulfilled. However, Z i is positive as \(Z^{i} \cdot (q^i)^{2} = \text{var} \left (\tilde \pi ^{i}_{KR}\right )\).

  17. Obviously, it is possible that \(\frac {M \sigma _{\varepsilon }}{2 c^{i} \sigma _{\xi ^{i}}} >1\) and the interval is empty.

  18. For example, consider the following case: p(Q) = 10 − 2Q, \(\text{var} (\tilde \xi ^i) =\text{var} (\tilde \xi ^j)= 2.25\), \(\text{var} (\tilde \varepsilon ) = 0.09\), \(\rho _{\varepsilon , \xi ^{i}} =\rho _{\varepsilon , \xi ^{j}}= 0.7\) and c i = 4, c j = 1. 5 with α i = 0. 1 resp. \(\alpha ^j=\frac {1}{3}\).

  19. The index for this setting is ST.

  20. See Viaene and Zilcha (1998), p. 599.

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Acknowledgments

The author would like to acknowledge the helpful suggestions of the participants of the 20th International Business Research Conference in Dubai on earlier versions of this paper. I thank two anonymous referees for helpful comments and I also thank the Editor of this Journal.

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  1. University TU Dortmund, 44221, Dortmund, Germany

    Matthias Pelster

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Pelster, M. Marketable and non-hedgeable risk in a duopoly framework with hedging. J Econ Finan 39, 697–716 (2015). https://doi.org/10.1007/s12197-013-9273-z

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