Optimal risk and dividend control for a company with a debt liability

Abstract

We consider a model of a corporation, which can choose a production/business policy among an available set of control policies with different expected profits and associated risks. In addition, there is a choice of the amount of dividends to be paid out to the shareholders. Notwithstanding any policy decision there is a constant payment of a corporate debt, such as bond liability or loan amortization. The objective is to find the policy which maximizes the expected total discounted dividend pay-outs until the time of bankruptcy. We model the dynamics of the corporate assets as a diffusion process with the diffusion and drift coefficients being affine functions of the risk control variable. In other words, the potential profit in our model is proportional to risk. The cumulative dividends are modeled by an increasing process. The resulting problem becomes a mixed regular-singular control problem for diffusion processes. We show that there exists an optimal level b 1 such that the optimal dividend policy is to keep the company's wealth below b 1 and pay out as dividends all the amounts in excess of this level. On the other hand, the profit/risk control policy depends on the ratio of the maximal possible expected profit to the liability payments. When this ratio is small, the optimal policy is always to undertake the maximal risk. When this ratio is large, the risk depends on the current amount of the company's wealth. The risk as a function of the wealth is monotone and increasing, reaching its maximum at some b 0 < b 1.