# Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

SIAM J. Financial Mathematics, to appear (2019)

34 Pages Posted: 11 Jul 2018

Date Written: June 19, 2018

### Abstract

We consider the optimal dividend problem under a habit formation constraint that prevents the dividend rate to fall below a certain proportion of its historical maximum, the so-called drawdown constraint. This is an extension of the optimal Duesenberry's ratcheting consumption problem, studied by [Dybvig1995, Review of Economic Studies 62(2), 287–313], in which consumption is assumed to be nondecreasing. Our problem differs from Dybvig's also in that the time of ruin could be finite in our setting, whereas ruin was 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 semi-explicitly via the Legendre transform. The optimal (excess) dividend rate c*t - as a function of the company's current surplus Xt and its historical running maximum of the (excess) dividend rate zt - is as follows: There are constants 0 < wa < w0 < w* such that (1) for 0 < Xt <= wa zt, it is optimal to pay dividends at the lowest rate a zt, (2) for wa zt < Xt < w0 zt, it is optimal to distribute dividends at an intermediate rate c*t in (a z_t, z_t), (3) for w0 zt < Xt < w* zt, it is optimal to distribute dividends at the historical peak rate zt, (4) for Xt > w* zt, it is optimal to increase the dividend rate above zt, and (5) it is optimal to increase zt via singular control as needed to keep Xt <= w* zt. 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.

**Keywords:** Optimal Dividend, Drawdown Constraint, Ratcheting, Stochastic Control, Optimal Control, Variational Inequality, Free-Boundary Problem

**JEL Classification:** C61, G02, G11

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