Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

We consider the optimal dividend problem under a habit-formation constraint that prevents the dividend rate from falling below a certain proportion of its historical maximum, a so-called drawdown constraint. Our problem is an extension of Duesenberry's optimal-consumption problem under a ratcheting constraint, studied by Dybvig [Rev. Econ. Stud., 62 (1995), pp. 287--313], in which consumption is restrained to be nondecreasing. Our problem also differs from Dybvig's in that the time of ruin could be finite in our setting, whereas ruin is impossible in Dybvig's work. We formulate our problem as a stochastic control problem with the objective of maximizing the expected discounted utility of the dividend stream until bankruptcy, in which risk preferences are embodied by power utility. We write the corresponding Hamilton--Jacobi--Bellman variational inequality as a nonlinear, free-boundary problem and solve it semiexplicitly via the Legendre transform. The optimal (excess) dividend rate $c^*_t$---as a function of the company's current surplus $X_t$ and its historical running maximum of the (excess) dividend rate $z_t$---is as follows: There are constants $0 < {w_\alpha} < w_1 < w^*$ such that (1) for $0 < {\rm X}_t \le {w_\alpha} z_t$, it is optimal to pay dividends at the lowest rate $\alpha z_t$, (2) for ${w_\alpha} z_t < {X}_t < w_1 z_t$, it is optimal to distribute dividends at an intermediate rate $c^*_t \in (\alpha z_t, z_t)$, (3) for $w_1 z_t < {X}_t < w^* z_t$, it is optimal to distribute dividends at the historical peak rate $z_t$, (4) for ${\rm X}_t > w^* z_t$, it is optimal to increase the dividend rate above $z_t$, and (5) it is optimal to increase $z_t$ via singular control as needed to keep ${X}_t \le w^* z_t$. Because, the maximum (excess) dividend rate will eventually be proportional to the running maximum of the surplus, “mountains will have to move” before we increase the dividend rate beyond its historical maximum.