Optimal Dividend And Risk Control Strategies Under The Influences Of Transition Costs And Terminal Value

From the viewpoint of corporate finance,the value of a corporation can be defined by the expected discounted sum of dividend payments until the time of bankruptcy.During the last twenty years,dividend strategy has become a hot topic in actuarial science.In order to maximize the value of the company,people focus on searching for the optimal dividend strategy.In addition,since reinsurance has long being an important method for the insurance company to reduce risk,researchers further consider the combined opti-mal dividend and reinsurance strategy under the objective of maximizing the company’s value.But for the purpose of mathematic convenience,most of the research is based on the hypothesis of expected value principle and using proportional reinsurance or excess of loss reinsurance.Besides,when an insurance company is on the edge of ruin,it can choose to declare bankrupt and claim the terminal value or to inject caption which will accordingly cause the cost of injection.Then company’s value can be measured by the expected discounted dividend minus the discounted cost of capital injection before ruin.So researchers study the combined optimal dividend,capital injection and reinsurance problem under the new objective function.But very few papers have consider the case when the terminal value is non-positive.In the real financial world,the influences of the transaction costs cannot be neglected since it can affect the management strategy of company.For convenience,the transaction costs can be divided into two parts:the proportional cost which is relevant to the force of the control(such as tax and commission cost)and the fixed cost which is relevant to the frequency of the control(such as advisory and consulting cost).Generally speaking,the proportional cost will have effect on the parameters of the optimal strategy but cannot change the style of the optimal strategy.But the fixed cost will make the optimal problem much more complicated and can usually change the type of the control strategy.In this article,we further study the combined optimal dividend,capital injection and reinsurance problem in the framework of the diffusion risk model.Under the objec-tive of maximizing the value of the company,we use the expectation premium principle,variance premium principle and exponential premium principle to calculate the premium,respectively.We analyse the influences of the transaction costs and arbitrary terminal value to the optimal strategy and value function as well.We find that when there is no fixed transaction cost and the dividend rates are not bounded,barrier dividend strategy is optimal.That’s to say,if only when the company’s surplus exceeds some barrier u,the excess should be paid out immediately as dividends.When there’s no fixed cost and the dividend rates are restricted,threshold dividend strategy is optimal.That is,dividends are paid at the maximum admissible rate as soon as the surplus exceeds a certain thresh-old d.And when there exists fixed transaction cost and the dividend rate is unrestricted,the impulse dividend strategy is optimal.It means that when the surplus exceeds some barrier b2,it is optimal to distribute dividends immediately and make the surplus jump to a lower level b1.When the company is on the edge of bankruptcy,it is profitable to inject capital only in the case when the company’s profitability is relatively high and in the meanwhile the transaction costs are relatively low.Or else,the company should choose to bankrupt and declare the terminal value.The insurance company should reduce the purchase of the reinsurance as the increase of the surplus and buy no insurance when the surplus is large enough.In addition to the above conclusions,we also give some numerical calculations and some economic explanations as well.In the end,our study promote the research of the combined optimal dividend,capital injection and reinsurance problem and expand the existing results.