Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy

Abstract

This paper represents a model for risk management in a firm which exercises control of its risk as well as potential profit by choosing different business activities among those available to it. Furthermore, the firm has an option of investing its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is associated with potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is described by a classical Black–Scholes model. The management of the company also controls the dividend pay-outs to shareholders. The objective is to find a policy, consisting of investment strategy, risk control and dividend distribution scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy. We apply the theory of controlled diffusions to solve the problem and show that there is a level u 1>0 such that the optimal action is to distribute all the reserve in excess of u 1 as dividends. Furthermore, there exists a constant x 0, with x 0<u 1, such that the risk exposure monotonically increases on (0, x 0) from zero to the maximum possible. The optimal choice of investments depends on the market price of risk , where r 0, r 1 denotes the mean rate of return of bond and stock respectively and σP denotes the volatility of the stock price. We get the following results. (1) m p≤0: invest everything in bond. (2) m p is large: invest everything in stock. (3) m p is small: there exists x 0<x 1<u 1, such that the optimal fraction of the reserve invested in stock is constant when the current reserve x is less than x 0 and it is an increasing function of x on [x 0, x 1] with all the reserve to be invested in stock whenever the reserve level exceeds x 1.