Abstract
This paper presents an option-theoretic framework for considering capital structure issues. The framework allows for illustration of the effect of leverage using numerical examples that are objective within the model. That is, once the framework is established and reasonable parameter estimates are made, there remains no subjectivity in the effects the model illustrates. As a result, the outcome of this paper amounts to a benchmark case against which one can consider the effects on value and cost of capital should real world conditions deviate from the assumptions we make. We do not conduct empirical tests. While our setup can reproduce earlier theoretical results, it also finds exceptions to the finding that the value curve is concave, as is often referenced in other studies. In addition, we demonstrate that choosing the level of leverage to maximize value is not always equivalent to choosing the level of leverage to minimize cost of capital. Finally, we compare outcomes for weighted average cost of capital based on state-claim expected returns to outcomes for weighted average cost of capital when it is calculated as taught in business schools. We find that in some settings there is a significant difference between the outcomes. Thus, our results suggest that traditional determinations of weighted average cost of capital in high leverage situations may produce misleading results, and that this caution ought to be part of the pedagogical agenda.
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Notes
Other option pricing platforms could be used, such as Merton (1976), Heston (1993) or Christoffersen (2015), but the basic narrative would not differ materially and the algebraic proofs would be more difficult, hence our decision to use Black and Scholes (1973). For completeness, we also cite Merton (1973).
It is acknowledged that there are other settings in which the deviation of expected return on debt from nominal return on debt can be studied.
Stiglitz (1974) considers the interaction of corporations and the household sector, but his analysis is conducted in the absence of both corporate and personal taxation, and in the absence of the possibility of bankruptcy (see Stiglitz, 1974, p. 857). As a result, it contributes only tangentially to the current discussion.
Note that we conducted our entire analysis in a Cox et al. (1979) setting in which the state space is discrete and the distributional assumptions over asset prices and returns are a very close approximation to the distributional assumptions in a Black/Scholes/Merton setting and our results go through identically in that setup.
We prefer the term ‘reorganization costs’ because we see them as being triggered by financial distress, which may or may not involve legal bankruptcy (i.e. a firm in proceedings under a bankruptcy statute). We do not think ‘bankruptcy’ is a particularly good synonym for financial distress because it leaves open to interpretation whether the user means legal bankruptcy or financial distress without legal bankruptcy.
Hirshleifer (1966, P. 25) refers to bankruptcy penalties but does not reflect them explicitly in the presented analysis.
Ross et al. (2019) comments on this style of analysis on page 645, referring to it as “the extended pie model”.
To be specific, we imagine the formation of a firm at time t, the ongoing efforts of the firm over the period T – t following which the firm has a terminal cash outcome or payoff available for distribution to claimants of VT.
Although simple and somewhat obvious the algebra is presented in Appendix 1.
This simplification also makes it easier to model a switch between triggering reorganization fees if: (a) \({V}_{T}\) is less than the bond’s face value; versus, (b) if \({V}_{T}\) is less than the bond’s face value plus interest owing at T.
Capital losses are more fully specified below. It is acknowledged that this approach assumes that there is sufficient income available from other sources to make the shelter useful.
The fact that it is necessary to make this assumption is not terribly debilitating in our approach because we are not seeking to determine what \(\mu\) ought to be. Rather, we are seeking to determine how a given return \(\mu\) is carved up among contingent claims depending on the level of leverage.
These are the usual Black/Scholes/Merton expressions. They are presented collectively as Eq. (13) in Black and Scholes (1973, p. 644). The put expression is also the usual Black/Scholes/Merton expression and follows from put-call parity. Further, we assume there is no continuously paid dividend during the life of the equity claim.
The call pricing expression and d1 and d2 are the usual Black/Scholes/Merton expressions. They are presented collectively as Eq. (13) in Black and Scholes (1973, p. 644). The put expression is also the usual Black/Scholes/Merton expression and follows from put-call parity. Further, we assume there is no continuously paid dividend during the life of the equity claim.
The amount available for dividends to shareholders is \(\left( {1 - \tau } \right)Call\left( {A_{t} e^{{q\left( {T - t} \right)}} + B_{t} } \right)\) In an integrated tax system, tax paid in the corporation as added back to dividends to create the tax base for the shareholders. Tax payable by applying the tax rate on dividends to this tax base is reduced by the amount of tax paid in the corporation, leaving the shareholders to pay the balance. When the tax rate on profits in the corporation is set equal to the tax rate on dividends to shareholders, the tax position of the shareholder after both corporate income tax and personal tax on dividends is as shown in Eq. (9).
In a Black/Scholes/Merton setting, the value of a put is specified as \({\text{Put}}_{t}={e}^{-r\left(T-t\right)}KN\left(-d2\right)-{V}_{t}N(-d1)\) where r is the continuously compounded riskless interest rate, t is the current time and T is the maturity date of the option, K is the strike price, \({d}_{1} \mathrm{and} {d}_{2}\) are as specified in Black and Scholes (1973), page 644, and \(N\left(\bullet \right)\) is the cumulative normal density function. The second term of the option pricing expression is desired discounted expectation such that \(Reorg={\alpha e}^{r(T-t)}E\left({V}_{T}|{V}_{T}<D\right)=\alpha {V}_{t}N(-d1)\)
As stated earlier, it is acknowledged that there are other settings in which the deviation of expected return on debt from nominal return on debt can be studied.
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McIntyre, M.L. Capital structure in an option-theoretic setting. SN Bus Econ 2, 109 (2022). https://doi.org/10.1007/s43546-022-00263-w
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DOI: https://doi.org/10.1007/s43546-022-00263-w