Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example

Bjarne Hø Jgaard and Michael Taksar

Mathematical Finance, 1999, vol. 9, issue 2, 153-182

Abstract: The paper represents a model for financial valuation of a firm which has control of the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. This model extends the classical Miller–Modigliani theory of firm valuation to the situation of controllable business activities in a stochastic environment. We associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). The example we consider is a large corporation, such as an insurance company, whose liquid assets in the absence of control fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, and drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure while simultaneously reducing the potential profit—for example, by using proportional reinsurance with another carrier for an insurance company. Management of a company controls the dividends paid out to the shareholders, and the objective is to find a policy that maximizes the expected total discounted dividends paid out until the time of bankruptcy. Two cases are considered: one in which the rate of dividend payout is bounded by some positive constant M, and one in which there is no restriction on the rate of dividend payout. We use recently developed techniques of mathematical finance to obtain an easy understandable closed form solution. We show that there are two levels u0 and u1 with u0≤u1. As a function of currently available reserve, the risk exposure monotonically increases on (0,u0) from 0 to the maximum possible. When the reserve exceeds u1 the dividends are paid at the maximal rate in the first case and in the second case every excess above u1 is distributed as dividend. We also show that for M small enough u0=u1 and the optimal risk exposure is always less than the maximal.

Date: 1999
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