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International Conference on Dynamics, Games and Science

DGS 2014: Modeling, Dynamics, Optimization and Bioeconomics II pp 183–222Cite as

Corporate Asset Pricing Models and Debt Contracts

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 195))

Abstract

Our contribution aims to provide an introduction to the theory of corporate asset pricing models and explain the potential of their usage in the design of credit contracts. We describe the evolution of structural models starting from the basic Mertonian framework through the introduction of a default barrier, and ending with stochastic interest rate environment. Further, with the use of game theory analysis, the parameters of an optimal capital structure and safety covenants are examined. Furthermore an EBIT-based structural model is introduced that considers stochastic default barrier. Such set-up is able to catch the different optimal capital structures in various business cycle periods, as well as bankruptcy decisions dependent on the state of the economy. The effects of an exogenous change in the risk-free interest rate on the asset value, probability of default, and optimal debt ratio are also explained.

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Notes

  1. 1.

    The probability measure that reflects the true probabilities is called the physical measure.

  2. 2.

    The assumptions are written exactly in a way as Merton wrote them, except for the symbols used.

  3. 3.

    This process is called Geometric Brownian Motion.

  4. 4.

    For the details about the construction of this portfolio, and for the complete derivation of Eq. (7) see [42] pp. 451–452.

  5. 5.

    For pricing of more complex capital structures and the issue of contractural design see the original work of [8].

  6. 6.

    The solution of (13) can be found in [8] p. 356.

  7. 7.

    Here we use the subscript D in order to distinguish this pay-out from c, which was the payout ratio to equity holders.

  8. 8.

    For the mathematical derivation see [7] p. 81 and the preceding calculations.

  9. 9.

    See Sect. 4.

  10. 10.

    In fact this inequality is not explicitly wrote down by [36], however it is implicitly assumed.

  11. 11.

    Note that this set-up can easily catch Absolute Priority Rule (APR) violations.

  12. 12.

    See [7] pp. 105–106.

  13. 13.

    See [47], or [10].

  14. 14.

    More on this see, for example [2].

  15. 15.

    See, for example, [25, 26, 39, 43, 54].

  16. 16.

    See [18, 55].

  17. 17.

    \(F_0\) denotes the fair value of the loan at time 0, as it will be described in more details later.

  18. 18.

    Otherwise the present value of the interest payments would converge to infinity.

  19. 19.

    The optimal default levels from the debtor’s and the creditor’s points of view are derived in [57], pp. 48–49.

  20. 20.

    Note that early bankruptcy means no tax deductibility in the future, and therefore it decreases the current value of the tax shield.

  21. 21.

    Here “plain vanilla” refers to the absence of additional clauses defining loan covenants.

  22. 22.

    A comprehensive analysis of covenants and their effect on debt pricing can be found in the work of Reisel [49].

  23. 23.

    For the mathematical derivation of this statement see [57], pp. 58–59.

  24. 24.

    See [57], pp. 67–68.

  25. 25.

    As Nejadmalayeri and Singh [46] showed, the US tax code’s loss carry provisions affect the equity holders’ bankruptcy decision.

  26. 26.

    At this point we do not concentrate on the problem how the DB is chosen; that issue will be covered in Sect. 4.7.

  27. 27.

    Recall that a default barrier of 0.3 means triggering default when the instantaneous earnings are at 30% of the coupon rate.

  28. 28.

    This holds only at the moment when the contract is signed. Later on both the debt and equity holders profit from an increase in the firm value.

  29. 29.

    That is the one with parameters set to their base levels.

  30. 30.

    More about risk shifting in the next section, where—in contrast with the present situation—it is possible.

  31. 31.

    See Table 1.

  32. 32.

    All these values are rounded: as we want to illustrate the decision process, the accurate numbers are not important. In real the DB is one number (between the mentioned 0.4 and 0.5) not an interval, and the FV that corresponds to the maximal firm value given this DB is determined unambiguously as well.

  33. 33.

    This could be lower expected EBIT growth, or some risk of being exposed, for example.

  34. 34.

    Source: author’s calculations using Financial Derivatives Toolbox.

  35. 35.

    Higher interest payments imply higher DB in absolute terms. The DB of the x axis on Fig. 5 is a ratio of the instantaneous interest payments.

  36. 36.

    For \(\kappa =0\) this is intuitive: the defaultable corporate bond can be represented as a risk-free bond with the same parameters minus the expected losses caused by default. Since the price of a riskless bond that pays continuous interest is always equal to its face value, it is not dependent on the current interest rate.

  37. 37.

    These parameters were the same as in the basic setting, with the exception of lower recovery rate (5 yearly EBITs), and higher correlation between the EBIT and interest rate processes (\(\rho =0.5\)). These modifications were made in order to make the results more sensible on the selection of the DB. Furthermore the number of iterations was doubled to increase the significance of small deviations between the two settings.

  38. 38.

    As it was discussed in Sect. 4.7, observable primitive variable implies low default triggering level. Consequently there is insignificant difference in the values produced by the two DB types.

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Acknowledgements

The research leading to these results was supported by the European Union’s Horizon 2020 Research and Innovation Staff Exchange programme under the Marie Sklodowska-Curie grant agreement No 681228. The authors further acknowledge financial support from the Czech Science Foundation (grants number 16–00027S and 15–00036S). Karel Janda acknowledges research support provided during his long-term visits at Australian National University, Toulouse School of Economics, New Economic School, McGill University and University of California, Berkeley. The views expressed here are those of the authors and not necessarily those of our institutions. All remaining errors are solely our responsibility.

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Authors and Affiliations

  1. Institute of Economic Studies, Faculty of Social Sciences, Charles University in Prague, Prague, Czech Republic

    Martin Dózsa & Karel Janda

  2. Department of Banking and Insurance, Faculty of Finance and Accounting, University of Economics, Prague, Czech Republic

    Karel Janda

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  1. Martin Dózsa
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Correspondence to Karel Janda .

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  1. Department of Mathematics and LIAAD-INESC TEC, University of Porto, Porto, Portugal

    Alberto A. Pinto

  2. Department of Agricultural and Resource Economics, University of California, Berkeley, California, USA

    David Zilberman

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Dózsa, M., Janda, K. (2017). Corporate Asset Pricing Models and Debt Contracts. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics II. DGS 2014. Springer Proceedings in Mathematics & Statistics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-55236-1_11

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