Optimal Dividends

Abstract

In the absence of dividends, the surplus of a company is modeled by a Wiener process (or Brownian motion) with positive drift. Now dividends are paid according to a barrier strategy: Whenever the (modified) surplus attains the level b, the “overflow” is paid as dividends to shareholders. An explicit expression for the moment-generating function of the time of ruin is given. Let D denote the sum of the discounted dividends until ruin. Explicit expressions for the expectation and the moment-generating function of D are given; furthermore, the limiting distribution of D is determined when the variance parameter of the surplus process tends toward infinity. It is shown that the sum of the (undiscounted) dividends until ruin is a compound geometric random variable with exponentially distributed summands. The optimal level b* is the value of b for which the expectation of D is maximal. It is shown that b* is an increasing function of the variance parameter; as the variance parameter tends toward infinity, b* tends toward the ratio of the drift parameter and the valuation force of interest, which can be interpreted as the present value of a perpetuity. The leverage ratio is the expectation of D divided by the initial surplus invested; it is observed that this leverage ratio is a decreasing function of the initial surplus. For b = b*, the expectation of D, considered as a function of the initial surplus, has the properties of a risk-averse utility function, as long as the initial surplus is less than b*. The expected utility of D is calculated for quadratic and exponential utility functions. In the appendix, the original discrete model of De Finetti (1957) is explained and a probabilistic identity is derived.