Optimal Dividend and Capital Structure with Debt Covenants

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.