Classic Risk Model With Tax And Dividends

In recent years, the issue of dividends and taxes has received remarkable attention in the actuarial ?elds, while few people combine them to study. In this paper, we discuss a compound Poisson risk model with taxes and dividends. More precisely, we assume taxes are paid according to a loss-carry-forward system and dividends are paid under a threshold strategy. The main problems we discussed are the expected discounted penalty function,the expected total discounted tax payment and the Laplace-Stieltjes transform of dividends which are not paid between two consecutive taxation periods.First of all, according to the dividend level b, the Gerber-Shiu functions of the initial surplus above and below the barrier b are divided into ΦUand ΦL. We ?rst solve the GerberShiu function ΦU. Conditioning on the time and the amount of the ?rst claim, we have(1) the ?rst claim causes ruin.(2) the ?rst claim does not cause ruin and the surplus is below the level b without taxes and dividends.(3) the ?rst claim does not cause ruin, the surplus which exceeds the level b can be divided into two cases. One is that it reaches its running maximum before ruin while does not. A system of integro-differential equations and the boundary conditions satis?ed by the expected discounted penalty function are derived.Then the closed-form expression of the expected discounted penalty function ΦUis obtained.Due to the expression and method of ΦU, we can conclude the closed-form expression of ΦL.Then, we get the range of the expected accumulated discounted tax payments according to its de?nition which is given beforehand. Conditioning on the time and the amount of the ?rst claim, the closed-form expressions for the expected accumulated discounted tax payments are deduced.At last, in order to get the expression for the Laplace-Stieltjes transform of the total duration of without paying dividends, we review the preparation knowledge. The sample paths of the surplus process we discussed coincide with those of the process during the periods when the insurer is not in a pro?table situation. Due to the law of total expectation, one can determine the probability function of the number of excursions below the level b. Denoting the total duration of no dividends over the time of no taxes, we derive the Laplace-Stieltjes transform of the dividends which are not paid between two consecutive taxation periods.