Equity value, bankruptcy, and optimal dividend policy with finite maturity - variational inequality approach with discontinuous coefficient
- Xiaoru Han^{1, 2}Email author and
- Fahuai Yi^{2}
https://doi.org/10.1186/s13660-015-0588-5
© Han and Yi; licensee Springer. 2015
Received: 7 August 2014
Accepted: 3 February 2015
Published: 24 February 2015
Abstract
This article examines the value of equity, optimal bankruptcy boundary, and optimal dividend policy in a continuous-time framework with finite time maturity. The model of equity value is formulated as a parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal bankruptcy boundary. We present an analytical approach to analyze the behaviors of the free boundary. The regularity of the value function and the optimal dividend policy are studied as well. The main feature and difficulty are the discontinuity of the coefficient in the variational inequality.
Keywords
equity value optimal bankruptcy boundary finite horizon optimal dividend policyMSC
35R35 47J20 93E201 Introduction
The value of debt and equity and default covenants have long been interested in corporate finance literature. Merton [1] pioneers the study of corporate debt and equity value under the assumption that bankruptcy cannot be triggered before maturity, he obtained the value of debt and equity. Black and Cox [2], Leland [3], Anderson and Sundaresan [4], Bruche [5] and Zhou [6] have extended the original Merton [1] model to incorporate a more realistic assumption: the possibility of early default. By assuming a time-independent debt structure, Leland [3] derived closed-form solutions for the value of debt and for optimal capital structure.
Unfortunately in these models dividend policy is not explicitly considered. The firms either simply do not have a policy with respect to dividends (Leland [3]) or they pay out all residual cash flows as dividends (Anderson and Sundaresan [4], Leland and Toft [7]). Fan and Sundaresan [8] and Gryglewicz [9] study the optimal dividend policy. In [8], dividends are treated as a control variable in the firm’s cash flow generating process. They obtained the optimal dividend policy. Gryglewicz [9] presented a model of a firm that optimally chooses capital structure, cash holding, dividends, and default. However, most of these papers were restricted to dealing with a perpetual debt.
It is challenging to take the finite horizon case into consideration since the debt values are no longer time homogeneous. The fundamental valuation equation will explicitly depend on time.
In this paper, we aim to develop a continuous-time model for the finite horizon corporate equity, provide a theoretical analysis of the behaviors of the optimal bankruptcy boundary, and obtain the optimal dividend policy. One paper related to the present work is Han et al. [10] where they obtained the optimal reorganization boundary by using a variational inequality approach, but they did not consider the dividend policy; they paid out all residual cash flows as dividends. However, in our formulation, dividend is optional when cash payout exceeds the promised coupon rate. That is to say, when \(\beta V\geq c\), stockholders have a decision to make: they can pay all the residual cash flows as dividends to themselves, or they can reinvest a fraction to the firm. They choose their dividend policy by acting to maximize their equity value. Whenever \(\beta V< c\), the firm is under a liquidity constraint and no dividend can be paid (see (2.2) and (2.4)), which causes the function \(f(v)\) (in (2.6)) to be not continuous at the point \(V=c/\beta\) and the inequality \(\partial _{x} u\geq0\) does not hold. But the condition \(\partial _{x} u\geq0\) is critical to prove the smoothness and monotonicity of the free boundary. At this point, it brings difficulty for analyzing the behaviors of the free boundary.
The rest of the paper is organized as follows. In Section 2, we set up the model. In Section 3, we study the behavior of the solution and prove that when \(\beta V\geq c\), as the payout ratio increases from \(c(1-\gamma)/V\) to β, the firm’s equity value is increasing, in other words, it would be optimal for the equity holders to pay all the cash flows available as dividends. Section 4 is devoted to an analysis of the behaviors of the free boundary in case of \(c(1-\gamma)-rP<0\), we prove that the free boundary is decreasing and infinitely differentiable. In Section 5, we deduce the boundedness of the free boundary by the comparison principle and show that it is not always monotonic in some case. In Section 6 we provide some numerical results and some financial interpretations. We conclude in Section 7. The main contribution of this paper lies in the following. 1. We give a rigorous derivation of variational inequality (2.6) by stochastic analysis (Section 2). 2. We prove that the value of equity is increasing with the aggregate payout ratio δ and obtaining the optimal dividend policy (Section 3). 3. In the absence of \(\partial _{x} u \geq0\), we show the monotonicity and infinite differentiability of the bankruptcy boundary for the case of \(c(1-\gamma)-rP\leq0\) (Section 4). 4. We analyze the loss of monotonicity of the reorganization boundary in some cases for the case of \(c(1-\gamma)-rP> 0\) (Section 5). 5. We give some numerical results and financial interpretations (Section 6). The main feature and difficulty are the discontinuity of the coefficient \(f(v)\) in variational inequality (2.6), which brings about a lot of trouble.
2 Formulation of the model
In this section, we develop a model of equity value with finite time maturity at time T. The model is set in a continuous-time framework. The following assumptions underlie the model:
(1) There is a firm which has equity and a single issue of debt which promises a flow rate of coupon c per unit time. The principal amount of the debt is P.
(2) To focus attention on default risk, we assume that the default-free term structure is flat and the instantaneous risk-free rate is r per unit time.
(3) When the firm pays its contractual coupon c, it is entitled to a tax benefit of γc (\(0\leq\gamma\leq1\)). During the default period, the tax benefits are lost.
(4) Asset sales for dividend payments are prohibited.
(5) The firm is not otherwise constrained by covenants, bankruptcy will occur only when the firm cannot meet the required (instantaneous) coupon payment: that is, when the equity value falls to zero. In fact, in continuous time, the coupon \((cdt)\) paid over the infinitesimal interval, dt, is itself infinitesimal. Therefore the value of equity simply needs to be positive to avoid bankruptcy over the next instant.
Most models in the literature tend to assume that the residual cash flow are simply paid out as dividends. In our formulation, as Fan and Sundaresan in [8], dividends, or equivalently the total payout ratio, denoted by δ, are treated as a control variable in the firm’s cash flow generating process. Stockholders will choose their dividend policies by acting to maximize their equity value.
When the cash payout βV exceeds the promised coupon rate c, stockholders have a decision to make: they can pay all the residual cash flows as dividends to themselves, or they can reinvest a fraction into the firm. The motivation for such an action is simple: by foregoing current dividends, the stockholders can avoid costly liquidations that may arise in the future. This feature is modeled in the following manner.
Definition
Theorem 2.1
Proof
3 The behavior of solution of problem (2.10)
Lemma 3.1
Proof
Lemma 3.2
Proof
Theorem 3.1
Proof
Define \(u=u^{R}\) if \(x\in[-R,R]\) for each \(R>0\), it is clear that u is reasonable defined in \(\overline{\Omega}_{T} \) and \(u\in C(\overline {\Omega_{T}})\cap W_{p}^{2,1}(\Omega_{T}^{R}\backslash B_{\delta}(P_{0}))\) is the solution of problem (2.10). Equations (3.6) and (3.7) are consequences of (3.2) and (3.5).
In fact, the inequality in (3.9)_{1} is deduced from (2.3). Using the strong maximum principle, we deduce that \(u>0\) in \(Q_{T}\). □
Theorem 3.2
The solution of problem (2.10) is increasing with the aggregate payout ratio δ, it is to say that when \(\beta e^{x}\geq c\), as the payout ratio δ increases from \(c(1-\gamma)e^{-x}\) to β, the firm’s equity value is increasing. Thus we conclude that when \(\beta V\geq c\), it is optimal to pay all the residual cash flows as dividends. When \(\beta V< c\), it is optimal to pay no dividends.
Proof
Remark 3.1
In the following, we devote the analysis to the behavior of the free boundary (i.e. optimal bankruptcy boundary).
From (3.8), we can see that the region \(\mathbf{BR}\subset \{x\leq\ln\frac{c}{\beta}\}\).
4 Free boundary in the case of \(c(1-\gamma)-rP\leq0\)
In this section, we aim to characterize the regularities of the free boundary between the region CR and the region BR. It is worthwhile pointing out that, without consideration of the dividend policy, Han et al. in [10] deduced that \(\partial _{x} u\geq0\), which plays an important role in the analysis of the regularities of the free boundary. But it is not true in this paper. Fortunately, we have the following lemma, which enables us to define the free boundary another way.
Lemma 4.1
Proof
Lemma 4.2
If \(c(1-\gamma)-rP\leq0\), the free boundary \(F(x)\) is decreasing, moreover, \(F(x)\) is strictly decreasing and continuous in the region \(\{x:0< F(x)<T\}\).
Proof
We divide the proof into four steps.
Step 1: We will deduce that if \(F(x)>0\), then \(x\leq\min\{\ln \frac{c(1-\gamma)}{\beta},\ln P\}\).
From the initial condition \(u(x,0)=(e^{x}-P)^{+}\) and the estimation (4.2), we can conclude that \(u>0\) when \(x>\ln P\). On the other hand, when \((x,\tau)\in\mathbf{BR}\), \(u=0\), then from estimation (3.8) we have \(x\leq\ln\frac{c}{\beta}\) and \(g(x)=\beta\), combining with (2.10)_{2}, we deduce that \(x\leq \ln\frac{c(1-\gamma)}{\beta}\). From the free boundary definition of (4.4) we have the conclusion.
Step 2: We will prove that \(F(x)\) is decreasing in \(\{x: F(x)>0 \}\).
Step 3: We infer \(F(x)\) is strictly decreasing in \(\{x: 0< F(x)<T \}\).
Step 4: We conclude \(F(x)\) is continuous in \(\{x: 0< F(x)<T \}\).
Theorem 4.1
Proof
Since the free boundary \(\tau=F(x)\) is strictly decreasing, its inverse function \(h(\tau)\) is continuous and strictly decreasing.
Now we prove \(h(0)=\lim_{\tau\rightarrow0^{+} }h(\tau)=\min \{\ln P,\ln\frac{c(1-\gamma)}{\beta} \}\). From estimation (3.8), we know \(h(\tau)\leq\ln\frac{c}{\beta}\), combining its monotonicity we conclude the limitation exists. In the same way as in Lemma 4.2, we can prove the conclusion.
Because \(h(\tau)\) is decreasing in \([0,T]\), we can deduce that \(x_{\infty}=\lim_{\tau\rightarrow+\infty} h(\tau)\leq h(\tau)\leq h(0)\) for any \(\tau\in(0,T]\).
At last, we prove \(h(\tau)\in C^{\infty}(0,T]\). Since \(\partial _{\tau}u\geq0\), using the method developed by Friedman [16], it is not difficult to prove \(h(\tau)\in C^{0,1}(0,T]\). At this point we can use the result of the Stefan problem [17] to find that \(h(\tau )\in C^{\infty}(0,T]\). □
5 Free boundary in the case of \(c(1-\gamma)-rP>0\)
In this section, we aim to investigate the behaviors of the free boundary in the case of \(c(1-\gamma)-rP>0\). However, when \(c(1-\gamma )-rP>0\), the inequality \(\partial _{\tau}u\geq0\) is no longer satisfied. A numerical example is presented in Section 6. In [10], Han et al. obtained a similar property to \(\partial _{\tau}u\geq0\) by making the transformation, which is no longer true since the function \(g(x)\) is a piecewise function. In addition, \(g(x)\) is discontinuous on the point \(x=\ln\frac{c}{\beta}\), which causes it to be difficult to prove that \(\partial _{x} u\geq0\). So we cannot prove the free boundary is differentiable. Fortunately, by establishing linkages between problem (2.10) and (3.12), we deduce that the free boundary is bounded and no longer monotonic in some cases.
Han et al. [10] completely characterized the behaviors of \(\widetilde{h}(\tau)\), which is summarized as follows.
Lemma 5.1
Theorem 5.1
Proof
When \((x,\tau)\in\mathbf{BR}\), \(u(x,\tau)=0\), combining the estimation (3.8), we deduce that \(x\leq\ln\frac{c}{\beta}\), then \(g(x)=\beta\). From (2.10)_{2}, we have \(x\leq\ln\frac {c(1-\gamma)}{\beta}\). Hence, we obtain the right hand side of inequality (5.4).
On the other hand, from [10], we know that when \(x\leq \widetilde{h}(\tau )\) (where \(\widetilde{h}(\tau)\) is the optimal reorganization boundary of problem (3.12)), \(u^{\beta}(x,\tau)=0\). From (3.11), we have \(u(x,\tau)\leq u^{\beta}(x,\tau)\), noticing that \(u(x,\tau)\geq 0\), then we can infer that \(u(x,\tau)=0\) for all \(x\leq \widetilde{h}(\tau)\). Thus we conclude that \(\widetilde{h}(\tau)\leq h(\tau)\). By Lemma 5.1, \(\ln( -\frac{P\alpha_{1}}{1-\alpha_{1}})\leq \widetilde{h}(\tau)\), so we obtain inequality (5.4).
Next, we prove that the free boundary is not monotonic in some cases.
Theorem 5.2
Proof
Since α is the negative root of (4.6), we can infer that \(\ln[\frac{-\alpha}{1-\alpha}\cdot\frac{c(1-\gamma)}{r}]<\ln \frac{c(1-\gamma)}{\beta}\), thus if \(\ln P< x_{\infty}=\ln[\frac {-\alpha}{1-\alpha}\cdot\frac{c(1-\gamma)}{r}]\), we deduce that \(h(0)= \min\{\ln P, \ln\frac{c(1-\gamma)}{\beta}\}=\ln P\).
First, we prove that there exists a positive \(\tau_{0} \) such that \(h(\tau)< h(0)\) for any \(\tau\leq\tau_{0}\).
6 Numerical results
Remark 6.1
The numerical results in Figure 5 and Figure 6 reveal that the bankrupt boundary \(h(\tau)\) is decreasing with respect to the volatility σ. The financial meaning is this: when the volatility increases, the firm need not bankrupt at once, because of a possibility of increasing for V due to the big volatility.
Remark 6.2
The numerical results in Figure 7 and Figure 8 show that the bankrupt boundary is decreasing with respect to the risk-free rate r. The financial explanation is very simple. The increase of the risk-free rate r results in the increase of V. Hence \(E(v,t)\) increases, then bankruptcy is behind of schedule.
Remark 6.3
Figure 9 and Figure 10 show that, as the payout ratio δ increases from \(\beta(1-\gamma)\) to β, the bankrupt boundary is decreasing. The financial explanation is very simple. If the residual cash flows are invested back as retained earning, they become accessible by the debt holders upon bankruptcy. It would be optimal for the equity holders to pay all the cash flows available as divided.
7 Conclusion
In this paper, we study the value of equity, optimal bankruptcy and dividend policy in a continuous-time framework with finite time maturity. Most of the previous works either take only an infinite time horizon into consideration or pay out all residual cash flows as dividend.
Mathematically the model of equity value is formulated as a parabolic variational inequality with discontinuous coefficient, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal bankruptcy boundary. We aim to investigate the behaviors of the free boundary and optimal dividend policy.
As we know, the results in this paper are the first integral one for optimal dividends due to the use of the PDE technique. First we rigorously established variational inequality model (2.6) by stochastic analysis. we prove that the solution is increasing with the aggregate payout ratio and obtain the optimal dividend policy. The results are perfect in the case of \(c(1-\gamma)-rP\leq0\). In Section 5 we deduced that in the case of \(c(1-\gamma)-rP>0\) the bankruptcy boundary is bounded and we show its loss of monotonicity in some cases. We presented some numerical results and financial interpretations in Section 6.
At time t, if the assert value of the firm v is in the continuation region, then the firm should not go bankrupt, and if v is in the bankruptcy region, then the firm should go bankrupt at once.
Declarations
Acknowledgements
The project is supported by National Natural Science Foundation of China (Nos. 11271143, 11371155 and 11326123), University Special Research Fund for Ph.D. Program of China (20124407110001 and 20114407120008) and Foundation for Distinguished Young Talents in High Education of Guangdong (No. 2014KQNCX181).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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