# On a Mean Reverting Dividend Strategy with Brownian Motion

11 Pages Posted: 13 Nov 2009 Last revised: 20 Nov 2009

See all articles by Benjamin Avanzi

## Benjamin Avanzi

UNSW Australia Business School, School of Risk and Actuarial Studies

## Bernard Wong

UNSW Australia Business School, School of Risk & Actuarial Studies

Date Written: November 19, 2009

### Abstract

In actuarial risk theory, the introduction of dividend pay-outs in surplus models goes back to Bruno de Finetti (1957). Dividend strategies that can be found in the literature often yield pay-out patterns that are inconsistent with actual practice. One issue is the high variability of the dividend payment rates over time. We aim at addressing that problem by specifying a dividend strategy that yields stable dividend pay-outs over time.

In this paper, we model the surplus of a company with a Brownian risk model. Dividends are paid at a constant rate $g$ of the company's modified surplus (after distribution of dividends), which operates as a buffer reservoir to yield a regular flow of shareholders' income. The dividend payment rate reverts around the drift of the original process $\mu$, whereas the modified surplus itself reverts around the level $l=\mu/g$.

We determine the distribution of the present value of dividends when the surplus process is never absorbed. After introducing an absorbing barrier $a$ (inferior to the initial surplus) and stating the Laplace transform of the time of absorption, we derive the expected present value of dividends until absorption. The latter is then also determined if dividends are not paid whenever the surplus is too close to the absorbing barrier. The calculation of the optimal value of the parameter $l$ (and equivalently $g$) is discussed. We conclude by comparing both barrier and mean reverting dividend strategies.

Keywords: Dividends, Brownian motion, Ornstein-Uhlenbeck process, Mean reverting

JEL Classification: G35, G32, G22, C44

Suggested Citation

Avanzi, Benjamin and Wong, Bernard, On a Mean Reverting Dividend Strategy with Brownian Motion (November 19, 2009). UNSW Australian School of Business Research Paper No. 2009ACTL12. Available at SSRN: https://ssrn.com/abstract=1504401 or http://dx.doi.org/10.2139/ssrn.1504401

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