Optimal Dividend and Capital Structure with Debt Covenants
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Optimal Dividend and Capital Structure with Debt Covenants Etienne Chevalier1 · Vathana Ly Vath2
· Alexandre Roch3
Received: 23 February 2020 / Accepted: 24 September 2020 / Published online: 9 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Viscosity solutions · System of variational inequalities · Singular controls · Optimal stopping · Bankruptcy · Debt covenants
Communicated by Lars Grüne.
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Vathana Ly Vath [email protected] Etienne Chevalier [email protected] Alexandre Roch [email protected]
1
Laboratoire de Mathématiques et Modélisation d’Evry, CNRS, ENSIIE, Univ Evry, Université Paris-Saclay, Evry-Courcouronnes, France
2
Laboratoire de Mathématiques et Modélisation d’Evry, CNRS, Univ Evry, Université Paris-Saclay, Evry-Courcouronnes, France
3
Faculty of Management, University of Quebec at Montreal (UQAM), Montreal, Canada
123
536
Journal of Optimization Theory and Applications (2020) 187:535–565
Mathematics Subject Classification 60G40 · 91B70 · 93E20
1 Introduction 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. To be more precise, we study dividend and issuance of equity policies under a new bankruptcy framework. The theory of capital structure is at the center of corporate finance problems. A firm’s capital structure is an inherent feature of many mathematical models of credit risk and optimal dividend and investment problems. Made popular by Merton [1], the structural approach to credit risk consists in modeling the dynamics of a firm’s assets and debts and making assumptions about the default and liquidation mechanism. The Merton model assumes a very simple debt design that consists of a single bond with a fixed maturity. The firm defaults, if its assets are lower than the face value of the debt at the tim
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