Optimal Dividend and Capital Structure with Debt Covenants

Abstract

We consider an optimal dividend and capital structure problem for a firm, which holds a certain amount of debt to which is associated a financial ratio covenant between the firm and its creditors. We study optimal policies under a bankruptcy framework, using a mixed reduced and structural approach in modeling default and liquidation times. Once in default, the firm is given a grace period during which it may inject more capital to correct the situation. The firm is liquidated if, by the end of the grace period, assets do not exceed the debt. Under this setup, we maximize the discounted amount of dividends distributed minus the capital injected up to the time of liquidation. It gives rise to a two-dimensional singular control problem leading to a non-standard system of variational inequalities. Beyond the usual viscosity characterization, we completely solve this problem and obtain a description of the continuation, dividend and capital injection regions enabling us to fully characterize the optimal policies. We conclude the paper with numerical results and illustrations.

Appendix: Proof of Theorem 3.1

Proof

Equations 19 and (20) follow from Proposition 2.2. The proof of the supersolution property is classical, so we omit it and focus on the subsolution property.

Let $(\varsigma ,x,y) \in \mathring{\mathcal S}^1\times \mathring{{\mathcal {S}}}^0$. Let $\psi _0 \in C^{2}(\mathring{{\mathcal {S}}}^0)$ and $\psi _1 \in C^{1,2}(\mathring{{\mathcal {S}}}^1)$ such that $\psi _1(\varsigma ,x) = v_1(\varsigma ,x)$, $\psi _0(y)=v_0(y)$ and $\psi _i \ge v_i,$ $i=0,1.$ We have to prove that $(\psi _0,\psi _1)$ satisfies (15)–(18) with $=$ replaced by $\ge .$ Suppose that it is not true:

Case (1) Assume that inequality associated with (16) is not satisfied.

There exists $\eta >0$ such that

$$\begin{aligned} \rho \psi _1(\varsigma ,x) - {\mathcal {L}}\psi _1(\varsigma ,x) -\frac{\partial \psi _1}{\partial \varsigma }(\varsigma ,x)> \eta , \text{ and } (1-\kappa _1) \frac{\partial \psi _1}{ \partial x} (\varsigma ,x)< & {} 1-\eta . \end{aligned}$$

(47)

Since $\psi _1 \in C^{1,2}(\mathring{{\mathcal {S}}}^1)$, we can assume that there exists $\epsilon >0$ ($\epsilon <\min (x,D-x,\varsigma ,\delta -\varsigma )$) such that the above inequality is also satisfied for all (s, z) such that $|x-z|^2 + |s-\varsigma | \le \epsilon .$

Let $(Z,K) \in {\mathcal {A}}$. Then, if we set $\theta := \inf \{t \ge 0 : |X^x_t - x |^2 + \varXi _t - \varsigma > \epsilon \}$, we have $|X_\theta -x |^2 - \epsilon + \theta \ge 0.$ Furthermore, $\theta < T\wedge \theta ^{D,+}$ and $|X_t-x|^2 + \varXi _t-\varsigma \le \epsilon$ for all $t\le \theta .$ Notice that on $[0,\theta ]$, $d\varXi _t= dt$ then, by Itô’s formula, we get

$$\begin{aligned} \psi _1(\varsigma ,x)= & {} \mathbb {E}_{\varsigma ,x,1} \Big [{\mathrm{e}}^{-\rho \theta } \psi _1(\varXi _\theta , X_\theta ) + \int _0^\theta {\mathrm{e}}^{-\rho t}(\rho \psi _1(\varXi _t,X_t) -{\mathcal {L}} \psi _1(\varXi _t,X_t)\\&-\, \,\frac{\partial \psi _1}{\partial \varsigma }(\varXi _t,X_t) ){\mathrm{d}}t \\&-\, \int _0^\theta {\mathrm{e}}^{-\rho t} (1-\kappa _1) \frac{\partial \psi _1}{\partial x}(\varXi _t,X_t) {\mathrm{d}}K^c_t \\&-\, \sum _{0 \le t\le \theta } {\mathrm{e}}^{-\rho t} \left( \psi _1(\varXi _t,X_{t-} + (1-\kappa _1) \varDelta K_t) - \psi _1(\varXi _t,X_{t-}) \right) \Big ]. \end{aligned}$$

It follows from (47) that

$$\begin{aligned} (1-\kappa _1) \frac{\partial \psi _1}{\partial x}(\varXi _t,X_t)< & {} 1-\eta \quad \text { and }\quad \psi _1(\varXi _t,X_{t-} + (1-\kappa _1) \varDelta K_t) - \psi _1(\varXi _t,X_{t-}) \\< & {} (1 - \eta ) \varDelta K_t. \end{aligned}$$

Consequently, we get

$$\begin{aligned} \psi _1(\varsigma ,x)\ge & {} \mathbb {E}_{\varsigma ,x,1} \Big [ - \int _0^\theta {\mathrm{e}}^{-\rho t} {\mathrm{d}}K_t + {\mathrm{e}}^{-\rho \theta } \psi _1(\varXi _\theta , X_\theta ) \Big ] \\&+\, \mathbb {E}_{\varsigma ,x,1} \Big [ \int _0^\theta {\mathrm{e}}^{-\rho t}(\rho \psi _1(\varXi _t,X_t) -{\mathcal {L}} \psi _1(\varXi _t,X_t)\\&-\, \frac{\partial \psi _1}{\partial \varsigma }(\varXi _t,X_t) ){\mathrm{d}}t + \eta \int _0^\theta {\mathrm{e}}^{-\rho t} {\mathrm{d}}K_t \Big ]. \end{aligned}$$

Combining the previous inequality with the fact that $\psi _1 \ge v_1$ and (47), we get

$$\begin{aligned} v_1(\varsigma ,x)\ge & {} \mathbb {E}_{\varsigma ,x,1} \left[ -\int _0^\theta {\mathrm{e}}^{-\rho t} {\mathrm{d}}K_t + {\mathrm{e}}^{-\rho \theta } v_1(\varXi _\theta , X_\theta ) + \eta \int _0^\theta {\mathrm{e}}^{-\rho t}\left( {\mathrm{d}} t + {\mathrm{d}} K_t \right) \right] . \nonumber \\ \end{aligned}$$

(48)

Let $\varphi (s,z) = C ((z-x)^2-\epsilon + (s-\varsigma )),$ for some positive constant $C < \min ( \frac{1}{2 \sqrt{\epsilon }}, \frac{1}{\rho \epsilon + 2 {\bar{\mu }} \sqrt{\epsilon } + {\bar{\sigma }}^2 + 1})$ in which ${\bar{\mu }} = \sup _{x\in [0,D]} \mu (x), {\bar{\sigma }} = \sup _{x\in [0,D]} \sigma (x).$

By Itô’s formula,

$$\begin{aligned} C \epsilon\le & {} \mathbb {E}_{\varsigma ,x,1} \left[ {\mathrm{e}}^{-\rho \theta } \varphi (\varXi _\theta , X_{\theta }) - \varphi (\varsigma , x) \right] \\\le & {} \mathbb {E}_{\varsigma ,x,1} \left[ \int _0^{\theta } {\mathrm{e}}^{-\rho t} (-\rho \varphi (\varXi _t,X_t) + 2 \mu (X_t) C (X_{t^-}-x) + C \sigma ^2(X_t) + C ) {\mathrm{d}}t \right] \\&+ \,\mathbb {E}_{\varsigma ,x,1} \left[ - \int _0^{\theta } {\mathrm{e}}^{-\rho t} 2 C (X_{t^-}-x)(1-\kappa _1) {\mathrm{d}}K_t \right] \\\le & {} \mathbb {E}_{\varsigma ,x,1} \left[ \int _0^{\theta } {\mathrm{e}}^{-\rho t} {\mathrm{d}} t + \int _0^{\theta } {\mathrm{e}}^{-\rho t} {\mathrm{d}}K_t \right] . \end{aligned}$$

Taking a supremum over $(Z,K) \in {\mathcal {A}}$ in (48) and applying the dynamic programming principle (11), we find the following contradiction: $v_1(t,x) \ge v_1(t,x) + \eta C \epsilon .$

Case (2) Assume that inequality coming from (17) is not verified for some $(\varsigma ,0) \in [0,\delta ]\times \{0\}$: There exists $\eta$ and $t >0$ such that for all $\epsilon >0$, $0\le x \le \epsilon$, and $\varsigma \le s \le \varsigma +t$

$$\begin{aligned} v_1(s,x) > \eta ,\quad \frac{\partial \psi _1}{\partial x} (s ,x)< \frac{1}{1-\kappa _1} - \eta \text{ and } \psi _1(s,0) < \psi _1(\varsigma ,0) + C (s-\varsigma ) ,\nonumber \\ \end{aligned}$$

(49)

due to the continuity of $v_1$ and $\frac{\partial \psi _1}{\partial x}.$ Here, $C = \sup _{\varsigma \le s \le \varsigma +t} \frac{\partial \psi _1(s,0)}{\partial s}.$ Assume further that $\epsilon < \min (D, v_1(\varsigma ,0)/\eta )$ and $t < \delta - \varsigma$.

We consider $\theta :=\inf \{s\ge 0: X_s\ge \varepsilon \}$ and deduce from (49) and the dynamic programming principle (11) that

$$\begin{aligned} v_1(\varsigma ,0)= & {} \sup _{(Z,K) \in {\mathcal {A}}, Z\equiv 0} \mathbb {E}_{\varsigma ,0,1} \Big [- J_1 +{\mathrm{e}}^{- \rho \theta } v_1(\varXi _\theta ,\epsilon ) \mathbb {1}_{\{\theta< T \wedge t\}} + {\mathrm{e}}^{- \rho t} v_1(\varXi _t,X_t) 1_{\{t \le \theta \wedge T\}} \Big ] \\\le & {} \sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} \Big [- J_1 + {\mathrm{e}}^{- \rho \theta } \psi _1(\varXi _\theta ,\epsilon ) \mathbb {1}_{\{\theta < T\wedge t \}} + A_0 1_{\{t \le \theta \wedge T\}} \Big ] \end{aligned}$$

where $A_0 = \sup _{0\le x \le \epsilon } v_1(\varXi _t,x)$, a finite quantity, and $J_1$ stands for $\int _0^{\theta \wedge T \wedge t} {\mathrm{e}}^{- \rho s} {\mathrm{d}} K_s$. We can therefore bound the value $v_1(\varsigma ,0)$ above by

$$\begin{aligned}&\sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} \Big [- J_1 + {\mathrm{e}}^{- \rho \theta }( \psi _1(\varXi _\theta ,0) + \left( \frac{1}{1-\kappa _1} - \eta \right) \epsilon ) \mathbb {1}_{\{\theta< T \wedge t\}} + A_0 1_{\{t \le \theta \wedge T\}} \Big ] \\&\quad \le \psi _1(\varsigma ,0) - \eta \epsilon + \sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} \Big [- J_1 + {\mathrm{e}}^{- \rho \theta }( C \theta + \frac{1}{1-\kappa _1} \epsilon ) \mathbb {1}_{\{\theta< T \wedge t\}} + A_0 1_{\{t \le \theta \wedge T\}} \Big ]\\&\quad \le v_1(\varsigma ,0) - \eta \epsilon + \sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} \Big [- J_1 + C \theta + {\mathrm{e}}^{- \rho \theta } \frac{1}{1-\kappa _1} X_{\theta } \mathbb {1}_{\{\theta < T \wedge t\}} + A_0 1_{\{t \le \theta \wedge T\}} \Big ] \\&\quad \le v_1(\varsigma ,0) - \eta \epsilon + \sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} \Big [C \theta + \int _0^{\theta \wedge T \wedge t} {\mathrm{e}}^{- \rho s} \left( \frac{\mu (X_s) - \rho X_s}{1-\kappa _1}\right) {\mathrm{d}} s + A_0 1_{\{t \le \theta \wedge T\}} \Big ]\\&\quad \le v_1(\varsigma ,0) - \eta \epsilon + (C+C') \sup _{K \in {\mathcal {A}}^0} \mathbb {E}_{\varsigma ,0,1} [\theta ]+ A_0 \mathbb {P}(t \le \theta ) \\&\quad \le v_1(\varsigma ,0) - \eta \epsilon + \frac{\epsilon ^2 (C+C'+ A_0/t)}{\underline{\sigma }^2}, \end{aligned}$$

where $C' = \sup _{x< D} \left( \frac{\mu (x) - \rho x}{1-\kappa _1}\right) < \infty$ and Burkholder–Davis–Gundy inequality has been applied. Finally, if we simplify by $v_1(\varsigma ,0)$ and divide by $\varepsilon$, we get $\eta \le \frac{\epsilon (C+C'+ A_0/t)}{\underline{\sigma }^2}$. We find a contradiction by sending $\epsilon$ to 0.

The inequality coming from (15) is proved with a line of reasoning similar to Case (1) and (18) by a proof similar to Case (2). $\square$