Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

32 Pages Posted: 11 Jul 2018  

Bahman Angoshtari

University of Washington

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

V.R. Young

University of Michigan at Ann Arbor - Department of Mathematics

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

Suggested Citation

Angoshtari, Bahman and Bayraktar, Erhan and Young, V.R., Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates (June 19, 2018). Available at SSRN: https://ssrn.com/abstract=3199294 or http://dx.doi.org/10.2139/ssrn.3199294

Bahman Angoshtari (Contact Author)

University of Washington ( email )

Department of Applied Mathematics
Lewis Hall #202, Box 353925
Seattle, WA WA 98105
United States

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Virginia R. Young

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
734-764-7227 (Phone)

Register to save articles to
your library

Register

Paper statistics

Downloads
11
Abstract Views
84
PlumX