On the dual risk model with tax payments
Introduction
The classical risk model describes the surplus process of an insurance company as where is the initial surplus in the portfolio, is the premium rate and represents the aggregate claim amount process that is assumed to be a compound Poisson process (see e.g. Bühlmann (1970),Grandell (1991) or Rolski et al. (1999)).
As pointed out by e.g. Avanzi et al. (2007), its dual process may also be relevant for companies whose inherent business involves a constant flow of expenses while revenues arrive occassionally due to some contingent events (e.g. discoveries, sales). For instance, pharmaceutical or petroleum companies are prime examples of companies for which it is reasonable to model their surplus process by where is the company’s initial surplus, is now the constant rate at which expenses are paid out and is the aggregate revenue process (Takacs, 1967).
In this paper, we assume that is a pure jump process defined as where
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the innovation number process is a renewal process with independent and identically distributed (i.i.d.) inter-innovation times with cumulative distribution function .
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the random variable (r.v.) corresponds to the revenue associated to the -th innovation (). The r.v.’s form a sequence of i.i.d. exponentially distributed r.v.’s with d.f. ().
We also assume that the innovation sizes and the inter-innovation times are mutually independent. For convenience, we define the sequence of innovation times by and for .
In Albrecher and Hipp (2007), the effect of tax payments under a loss-carry forward system in the Cramér–Lundberg model was studied and a remarkably simple relationship between the ruin probability of the surplus process with and without tax has been established. In addition, the authors found a simple criterion to determine the optimal surplus level at which taxation should start, subject to the maximization of the expected discounted tax payments before ruin. It is natural to question whether similar relations hold in the dual model (2) which is of comparable complexity to the Cramér–Lundberg model and is in several ways closely related.
In this paper, we address the above questions by introducing a tax component of loss-carry forward type in the dual surplus process (2) with general inter-innovation times and exponential innovation sizes. Hence the company pays tax at rate () on the excess of each new record high of the surplus over the previous one. Due to the structure of the process, a new record high can only be achieved by an innovation which implies that tax payments only occur at the innovation times (). In this paper, we show that whereas the relationship for the ruin probability with and without tax is slightly more complicated in the dual model, the criterion for identifying the optimal starting taxation level is identical to the one known from the Cramér–Lundberg risk model.
Let and define to be the number of innovations up to the time of the -th record high. Let and be the value of the -th record high. The resulting surplus process in the dual model with tax is given by where . For practical considerations, we assume that the net profit condition is satisfied, i.e. the drift of the (before-tax) surplus process (2) is positive.
The time to ruin of the surplus process (4) is defined as (with the convention if for all ) and its Laplace transform is denoted by where can also be interpreted as a discount rate and denotes the indicator of the event . An important special case of (6) is the ruin probability .
We point out that the surplus process (i.e. ) is the dual equivalent of the Sparre Andersen risk model in ruin theory and can also be viewed as a queueing system (see e.g. Cohen (1982) and Prabhu (1998)). Thus, the time to ruin in the surplus process (4) with can be interpreted as the length of the first busy period in the queue. Here, the positive security loading condition (5) translates into a traffic intensity in the queueing system (congested queue). However, it seems that most of the explicit results in queueing theory that are relevant for the present purpose are based on the assumption , so that we will derive some results for the congested queue in the Appendix.
For the limit , it is immediate that the surplus process corresponds to a dual model with a horizontal barrier strategy, where the initial surplus level is at the barrier (see Avanzi et al. (2007) for a detailed study of that case). This gives rise to an alternative interpretation of tax payments, as they can also be viewed as dividend payments to shareholders who ask for a proportion of each new profit.
The paper is organized as follows: in Section 2, we derive a relation between the ruin probability with and without tax payments in this dual model and give some more explicit expressions for (mixtures of) exponential inter-innovation times. In Section 3, the Laplace transform of the time to ruin for the surplus process in (4) is studied and an explicit expression is obtained under exponential inter-innovation times. Section 4 is then devoted to the analysis of the moments of the discounted tax payments before ruin. Finally, in Section 5, we determine the critical surplus level at which it is optimal for a tax authority to start collecting taxes.
Section snippets
Ruin probability
First, we consider the impact of the defined tax system on the ruin probability of the surplus process of the dual model (4). The analysis will be carried out by a study of its complement, namely the non-ruin probability . Indeed, starting with an initial surplus , the surplus process shall (up-) cross level at least once in order to avoid ruin. Let be the time of the first up-crossing of above a level exceeding its initial value .
Laplace transform of the time to ruin
Let us now consider the Laplace transform of the time to ruin . Starting with an initial surplus , the surplus process can either reach a new record high at time avoiding ruin en route, or reach level 0 before any visit to levels greater than . For the latter, the Laplace transform of the corresponding passage time is defined as where corresponds to the maximum surplus level before ruin.
Thus, by conditioning
Discounted tax payments
Let denote the discounted tax payments before ruin in the surplus process (4) defined as In this section, we will analyze the th moment of , namely for . By conditioning on the first upper exit time of the surplus process (4), one finds Differentiating (35) w.r.t. to gives
Delayed start of tax payments
In this section we consider a variant of the tax system where tax payments start only after the surplus is greater than a threshold level . Let denote the resulting expected discounted tax payments. By a probabilistic argument, one easily shows where is the Laplace transform of the first passage from level to any level above avoiding ruin en route. Furthermore, an expression for can be obtained in terms of for
Acknowledgements
Andrei Badescu and David Landriault gratefully acknowledge financial support received from the Natural Sciences and Engineering Research Council of Canada (NSERC). Hansjörg Albrecher was supported by the Austrian Science Fund Project P18392.
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