The effect of stochastic interest rates on a firm’s capital structure under a generalized model

Abstract

The lattice approach derived by Broadie and Kaya (J Financ Quant Anal 42(2):279–312, 2007) has traditionally been used to determine the capital structure of a firm in economies with constant interest rates; however, this study argues that the capital structure of a firm should be determined by considering the state of its debt simultaneously with the randomness of interest rates. This study extends the Hilliard et al. (J Financ Res 19(4):585–602, 1996) bivariate binomial model to determine the capital structure of firms, taking into account stochastic interest rates and their correlation with the asset value of the firm. Our simulation results suggest that taking stochastic interest rates into consideration reduces the equity value of a firm while increasing its debt value. The stronger the correlation between variations in the asset value of the firm and the short rate, the stronger the impact of this correlation on the capital structure of the firm.

Appendices

Appendix 1: The transformed processes

Our once-transformed processes, Eqs. (3) and (5), are subsequently combined as follows:

$$X_{1} = \sigma_{r} \phi + R$$

(21)

$$X_{ 2} = \sigma_{r} \phi - R$$

(22)

The stochastic processes of the two variables in Eqs. (3) and (5) are therefore:

$$dX_{1} = \sigma_{r} d\phi + dR$$

(23)

$$dX_{2} = \sigma_{r} d\phi - dR$$

(24)

Here, the two variables are mutually orthogonal, a relationship which achieves the final purpose of:

$${\text{cov}}\left( {dX_{1} ,dX_{2} } \right) = cov\left( {\sigma_{r}^{2} - \sigma_{r}^{2} + \sigma_{r}^{2} \rho_{vr} - \sigma_{r}^{2} \rho_{vr} } \right) = 0$$

where ρ _vr is the instantaneous correlation between dZ _v and dZ _r . Hence Eqs. (23) and (24) provide the uncorrelated and constant volatility processes as:

$$\begin{aligned} dX_{1} & = \sigma_{r} \left( {mdt + dZ_{v} } \right) + \left( {a^{*} dt + \sigma_{r} dZ_{r} } \right) \\ & = \left( {\sigma_{r} m + a^{*} } \right)dt + \sigma_{r} \left( {dZ_{v} + dZ_{r} } \right) \\ & = \left( {\sigma_{r} m + a^{*} } \right)dt + \sigma_{r} \sqrt {2\left( {1 + \rho_{vr} } \right)} dZ_{1} \\ & \equiv m_{1} dt + \sigma_{1} dZ_{1} \\ \end{aligned}$$

(25)

$$\begin{aligned} dX_{2} & = \sigma_{r} \left( {mdt + dZ_{v} } \right) - \left( {a^{*} dt + \sigma_{r} dZ_{r} } \right) \\ & = \left( {\sigma_{r} m - a^{*} } \right)dt + \sigma_{r} \left( {dZ_{v} - dZ_{r} } \right) \\ & = \left( {\sigma_{r} m - a^{*} } \right)dt + \sigma_{r} \sqrt {2\left( {1 - \rho_{vr} } \right)} dZ_{2} \\ & \equiv m_{2} dt + \sigma_{2} dZ_{2} \\ \end{aligned}$$

(26)

Equations (25) and (26) also satisfy the following property:

$${\text{d}}Z_{1} \bot {\text{d}}Z_{2}$$

Therefore, as a result of the above equations, we are able to establish a lattice using the two new independent variables, (X ₁, X ₂), which are derived from the two original dependent variables, (V, r).

Appendix 2: Verifying the probability of the lattice

Fundamental probability theory creates a problem for the method of changing the variables in Hilliard et al. (1996); that is, the probability could be >1 (or <0) in those cases where the drift and volatility terms have extreme values. The definition of probability in the Cox et al. (1979) model, with Brownian motion, is:

$$p_{*} = \frac{1}{2} + \frac{{m_{*} \sqrt {\varDelta t} }}{{2\sigma_{*} }}$$

(27)

Obviously, when the drift term is greater than the volatility term, the upward probability in the Cox et al. (1979) model violates fundamental probability theory; therefore, prior to establishing the lattice, we need to make some adjustments for the magnitude of the upward probability under the following criteria:

$$p\left( {2k + 1} \right)\sigma \sqrt {\varDelta t} + \left( {1 - p} \right)\left( {2k - 1} \right)\sigma \sqrt {\varDelta t} = \mu \varDelta t$$

(28)

where, $k = 0,\, \pm 1,\, \pm 2,\, \pm 3 \ldots$.

Since the variable m, is the same as the drift term of a particular process, we fix the upward and downward difference of $2\sigma \sqrt {\varDelta t}$ in order to satisfy the expectation of a stochastic process. We must then check parameter, k, and select appropriate values. Therefore, the upward probability should be modified as follows:

$$p_{*} = \frac{1}{2} - k_{*} + \frac{{m_{*} \sqrt {\varDelta t} }}{{2\sigma_{*} }}$$

(29)

Appendix 3: The changing of the dynamic processes in our model

Following the previous section, we first need to change the dynamic processes for V and r into new processes with the property of constant volatility. This can be achieved by:

$$dX_{1} = m_{1} dt + \sigma_{1} dZ_{1}$$

(30)

$$dX_{2} = m_{2} dt + \sigma_{2} dZ_{2}$$

(31)

where

$$m_{1} = \frac{{\sigma_{r} }}{{\sigma_{v} }}\left( {r_{t} - q - \frac{1}{2}\sigma_{v}^{2} } \right) + a^{*}$$

(32)

$$m_{1} = \frac{{\sigma_{r} }}{{\sigma_{v} }}\left( {r_{t} - q - \frac{1}{2}\sigma_{v}^{2} } \right) - a^{*}$$

(33)

and

$$\sigma_{1} = \sigma_{r} \sqrt {2\left( {1 + \rho_{vr} } \right)}$$

(34)

$$\sigma_{2} = \sigma_{r} \sqrt {2\left( {1 - \rho_{vr} } \right)}$$

(35)

where a ^* = r ^−β (κ (θ − r ⁾ _t  – λ σ _r ) − (σ ² _r  β/2)r ^β–1.

We denote the upward probability in X ₁ (X ₂) as P ₁ (P ₂), with the subsequent probabilities for the next four states in our lattice approach being as follows:

$$\begin{aligned} & p_{1} p_{2} = Pr\left( {X_{1} + \sigma_{1} \sqrt {\varDelta t} ,X_{2} + \sigma_{2} \sqrt {\varDelta t} } \right) \\ & \left( {1 - p_{1} } \right)p_{2} = Pr\left( {X_{1} - \sigma_{1} \sqrt {\varDelta t} ,X_{2} + \sigma_{2} \sqrt {\varDelta t} } \right) \\ & p_{1} \left( {1 - p_{2} } \right) = Pr\left( {X_{1} + \sigma_{1} \sqrt {\varDelta t} ,X_{2} - \sigma_{2} \sqrt {\varDelta t} } \right) \\ & \left( {1 - p_{1} } \right)\left( {1 - p_{2} } \right) = Pr\left( {X_{1} - \sigma_{1} \sqrt {\varDelta t} ,X_{2} - \sigma_{2} \sqrt {\varDelta t} } \right) \\ \end{aligned}$$

(36)

Appendix 4: Some errors in Broadie and Kaya (2007)

Following on from Broadie and Kaya (2007), the payouts to a firm’s shareholders and bond holders can be expressed as:

$$\begin{aligned} If\, V_{T} & \ge \frac{C}{r} , then \\ E & = V_{T} - \frac{C}{r} \\ D & = \frac{C}{r} \\ F & = V_{T} \\ If\, V_{T} & < \left( {1 - \tau } \right)C\varDelta t + \frac{C}{r} , then \\ E & = 0 \\ D & = \left( {1 - \alpha } \right)V_{T} \\ F & = \left( {1 - \alpha } \right)V_{T} \\ \end{aligned}$$

(37)

However, their equations provide no detailed explanation of the process of inference even where there is no convergence. According to the explanation provided by the authors, the criteria in the terminal nodes have less effects than those in the non-terminal nodes. Our study, which applies the algorithm in Eq. (37), provides a new result on the criteria for approximating the infinite case, as shown in Eq. (19). The basic meaning of our new criteria is that an infinite maturity debt could be terminated by immediately repaying the residual value at the specified date, with the assumption being that the specified date expires along with the time periods. Thus, the criteria in our computation are (1) the principal is the residual value of the perpetual bond, $\left( \frac{c}{r} \right)$; and (2) the cash flow, (δ _T ) also has to be calculated.