The Discrete-Time Risk Models With Dividends And The Applications Of Transition Probabilities For Markov Chains To Them

The risk theory has been developed in a so long period that the theory about the classical risk model has almost been perfected, especially, about the classical continuous time risk model. Beyond all doubt, the theory is playing am important role for an insurer to devise new products and to manage them. Some dividend problems based on the continuous time risk model have attracted significant interest in some recent literature. Dividend strategies for insurance risk models were firstly proposed by De Finetti in a paper presented to the International Congress of Actuaries in New York in 1957. Obviously, dividend strategies can more realistically reflect the surplus cash flows in an insurance portfolio and the theories about them axe very valuable in the devising and managing of products with dividends. But, our colleagues only have been concerned to the dividend problems in some continuous time models until now, while the discrete time risk models with dividends are also worthy of our discussion owing to not only their independent interest but also their use as an approximation to the continuous time models with dividends. Therefore, in this paper, we are interested in the discrete time risk models with dividends.In this paper we consider two discrete time risk models with dividends and call them as Dividend Model (I) and Dividend Model (II) respectively. Two methods are applied when we discuss the two models, i.e., Gerber-Shiu (Expected Discounted) Penalty Function and Markov Transition Matrix suggested by myself. The Gerber-Shiu Penalty Function was proposed by Hans U. Gerber and Elias S.W. Shiu in 1997, and was widely applied to the continuous time models with excellent behaves. But we almost can't find the method in the discrete time models owing to the difficulty of its applications in some such models. In this paper, we try to apply the method and obtain some satisfied results. As far as the Markov Transition Matrix, it is difficult to find it not only in discrete time models but also in continuous time models. The cause may be the great quantity of calculation and that it can't directly be applied to continuous time models. I think the Markov Transition Matrix should be paid attention to and be generalized because of the developed technique of computer at present. This is a motive for me to write this paper.The Dividend Model (I) is a discrete time model with randomized decisions on paying dividends, i.e., a compound binomial model modified by the inclusion of dividends. The insurer pays a dividend of 1 to the insured or the stock holders with a probability q₀ when the surplus is greater or equal to a non-negative integer x called as a dividend bound or threshold. We will discuss the model in the chapter 2, 3 and 4. In the chapter 2, we will find that the values of Gerber-Shiu penalty functionΦ(0),Φ(1),…,Φ(x) satisfy a set of linear equation(s), and will prove that there exists an unique solution of the set of equations under the assumption of a positive security loading by some knowledge of matrix theory. When the surplus of the insurer u >x, we find that the penalty functionΦ(u) satisfies two recursion formulas. Furthermore, we derive an asymptotic estimate for the penalty function. The above formulas and equations are derived with a hard process, and the method of derivation is completely different from the method for continuous time models. According to the penalty function, we give the linear equation sets, recursion formulas and asymptotic estimates for some important ruin quantities such as the ruin probability, the distribution function of the deficit at ruin, the generating function of the deficit at ruin, and the mass probability function of the surplus prior to ruin. In the end of the chapter, there is a numerical illustration and we can find from the value tables that the above formulas have fairly good behaves.In the chapter 3, we derive some matrix expressions for some important ruin quantities for the Dividend Model (I) by another method, i.e., the Markov Transition Matrix. It is worth mentioning that these expressions are explicit. The values computed by the matrix expressions completely coincide with the values in the chapter 2. There are following results in the chapter. Under some assumptions, the surplus process of the insurance company is a homogeneous discrete time Markov chain with the initial distribution Pr[U(0) = u] = 1. If we kill the process at the stopping time T (the time of ruin), we can obtain a killed process U(t∧T), which is still a homogeneous discrete time Markov chain. Applying the one-step transition probability matrix of the killed process, we derive the explicit expression for the joint probability function of the time of ruin, the surplus prior to ruin, and the deficit at ruin, which can lead to some marginal distributions such as the finite-time ruin probability, infinite-time ruin probability, the distribution function of the deficit at ruin, and the conditional distribution function of the deficit at ruin given ruin (or the time of ruin). As mentioned in above, the quantities of computation are remarkably greater by the formulas in the chapter 3 than the ones in the chapter 2. But, it is very satisfying that some ruin quantities can be obtained which can't be gotten in chapter 2, for example, the finite-time ruin probability and the conditional distribution function of the deficit at ruin.In the chapter 4, we consider the duration of negative surplus for the Dividend Model (I) under the condition of a positive security loading, i.e., the time of surplus temporarily staying below the zero level if ruin occurs. Due to the assumptions presented, the surplus will go to infinity with probability 1. So, once ruin has occurred at time T and the surplus (sooner or later) will reach the zero level. Afterwards, the second ruin may occur, the third ruin may occur, and so on. We find that there are two sorts of distribution for N (the number of ruin): (1) N obeys a geometric distribution if the initial surplus u = 0; (2) N obeys the following distribution if u≥1:Distinguishing the two cases, we derive the moments, the generating functions and the distribution functions (expressed by matrix) for the first duration of negative surplus, the second (or the third, or the forth,…) duration of negative surplus, and the (stochastic) sum of the durations respectively. Note that the duration of negative surplus matters with the time of first passage at level y when the deficit is y at ruin. There is a sort of mature method for us to derive the distribution of the time of first passage. It is very lucky that this method can successfully be applied to the Dividend Model (I), which is a key for writing this chapter.There is an obvious defect about the familiar compound binomial model recommended by Gerber in 1988. In Gerber (1988), there was such an assumption that the premiums for each period are one, which implies that every claim is only an integral multiple of the premium rate. Obviously, this doesn't accord with the universality. Therefore, in chapter 5, we modify Gerber's compound binomial model and consider the model with general premium rate. But the Gerber-Shiu penalty function is derived more difficultly. It is very lucky that we still find a linear equation about the penalty function, an upper bound, a lower bound and a not perfect recursion formula. Although we can't accurately compute the values of the penalty function by the above formulas, we can offer an. estimate for the penalty function. The formulas mentioned in above are also of another use, i.e., we can apply them to estimate the errors of approximate values for some ruin quantities with matrix expressions in the chapter 6. Though the formulas with matrix expressions in the chapter 6 are exact formulas for some ruin quantities, we can only get their approximate values because the orders of the matrixes are infinite. So, the formulas in the chapter 5 are some useful supplements for the chapter 6.In the chapter 6, we propose a new model, namely, the Dividend Model (II). It is a discrete time risk model based on the compound binomial model with general premium rate. The insurer pays a dividend of 1 when the surplus is greater than or equal to a non-negative integer dividend bound x. The dividends are not paid stochastically, which is different from the Dividend Model (I). The Dividend Model (II) is a sort of generalization of the compound binomial model. Hence our results in this chapter include the corresponding results of the classic risk model. The main method applied in this chapter is the Markov Transition Matrix (similar to the chapter 3), and is more efficient than the Penalty Function in the chapter 5. I think the method may be widely applied to other risk models. Recommending the method is a main purpose of this chapter.Our new method, i.e., the Markov Transition Matrix, is worthy generalizing owing to not only its strong efficiency when deal with some discrete time risk model, but also its use on approximate calculation of the ruin quantities in some continuous time models. As an example, we consider the Sparre Andersen model in the chapter 7. We successfully derive an approximate formula, an upper bound and a lower bound for the ruin probability. The approximate formula has a great advantage: the error is controllable theoretically.Prom this paper, we know that the Gerber-Shiu penalty function has a special virtue in its application, i.e., a small computational number. But the derivations of the formulas for it are usually difficult in the discrete time risk models. Solving these problems needs the studious work of more colleagues. In comparison, the Markov Transition Matrix has a bright prospect. Only two models with dividends are proposed in this paper. But we can still devise more and much more complex models with dividends. Furthermore, we may consider some other problems in the discrete time models with dividends except for ruin quantities, for example, the optimal dividend pay-out.