On the dual risk model with tax payments

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Abstract

In this paper, we study the dual risk process in ruin theory (see e.g. Cramér, H. 1955. Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes. Ab Nordiska Bokhandeln, Stockholm, Takacs, L. 1967. Combinatorial methods in the Theory of Stochastic Processes. Wiley, New York and Avanzi, B., Gerber, H.U., Shiu, E.S.W., 2007. Optimal dividends in the dual model. Insurance: Math. Econom. 41, 111–123) in the presence of tax payments according to a loss-carry forward system. For arbitrary inter-innovation time distributions and exponentially distributed innovation sizes, an expression for the ruin probability with tax is obtained in terms of the ruin probability without taxation. Furthermore, expressions for the Laplace transform of the time to ruin and arbitrary moments of discounted tax payments in terms of passage times of the risk process are determined. Under the assumption that the inter-innovation times are (mixtures of) exponentials, explicit expressions are obtained. Finally, we determine the critical surplus level at which it is optimal for the tax authority to start collecting tax payments.

Introduction

The classical risk model describes the surplus process {U(t),t0} of an insurance company as U(t)=u+ctS(t), where u is the initial surplus in the portfolio, c is the premium rate and {S(t),t0} 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 {R(t),t0} by R(t)=uct+S(t), where u is the company’s initial surplus, c is now the constant rate at which expenses are paid out and {S(t),t0} is the aggregate revenue process (Takacs, 1967).

In this paper, we assume that {S(t),t0} is a pure jump process defined as S(t)=i=1N(t)Yi where

  • the innovation number process {N(t),t0} is a renewal process with independent and identically distributed (i.i.d.) inter-innovation times T1,T2, with cumulative distribution function K.

  • the random variable (r.v.) Yj corresponds to the revenue associated to the j-th innovation (j=1,2,). The r.v.’s {Yj}j=1 form a sequence of i.i.d. exponentially distributed r.v.’s with d.f. p(y)=βexp{βy} (y0).

We also assume that the innovation sizes {Yj}j=1 and the inter-innovation times {Tj}j=1 are mutually independent. For convenience, we define the sequence of innovation times {Wj}j=1 by W0=0 and Wj=T1++Tj for jN+.

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 γ (0<γ<1) 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 Wj (j=1,2,). 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 σ0=0 and define σn=infkN{k>σn1:j=σn1+1k(YjcTj)>0}, to be the number of innovations up to the time of the n-th record high. Let J0=u and Jn=Jn1+(1γ)j=σn1+1σn(YjcTj), be the value of the n-th record high. The resulting surplus process in the dual model with tax is given by Rγ(t)=JΞ(t)j=σΞ(t)+1N(t)(cTjYj)c(tWN(t))=JΞ(t)c(tWσΞ(t))+j=σΞ(t)+1N(t)Yj, where Ξ(t)=sup{nN:Wσnt}. For practical considerations, we assume that the net profit condition c<E[Y1]/E[T1] is satisfied, i.e. the drift of the (before-tax) surplus process (2) is positive.

The time to ruin τγ of the surplus process (4) is defined as τγ=inf{t0,Rγ(t)=0} (with the convention τγ= if Rγ(t)>0 for all t0) and its Laplace transform is denoted by ργ,δ(u)=E[eδτγ1{τγ<}|Rγ(0)=u], where δ0 can also be interpreted as a discount rate and 1A denotes the indicator of the event A. An important special case of (6) is the ruin probability ψγ(u)=ργ,0(u).

We point out that the surplus process R0(t) (i.e. γ=0) is the dual equivalent of the Sparre Andersen risk model in ruin theory and can also be viewed as a GI/M/1 queueing system (see e.g. Cohen (1982) and Prabhu (1998)). Thus, the time to ruin in the surplus process (4) with γ=0 can be interpreted as the length of the first busy period in the GI/M/1 queue. Here, the positive security loading condition (5) translates into a traffic intensity ρ>1 in the queueing system GI/M/1 (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 ρ<1, so that we will derive some results for the congested queue in the Appendix.

For the limit γ=1, it is immediate that the surplus process {R1(t),t0} 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 γ 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 Rγ(t) 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 ϕγ(u)=1ψγ(u). Indeed, starting with an initial surplus u, the surplus process {Rγ(t),t0} shall (up-) cross level u at least once in order to avoid ruin. Let ξu=inf{t>0:Rγ(t)u}, be the time of the first up-crossing of Rγ(t) above a level exceeding its initial value u.

Laplace transform of the time to ruin τγ

Let us now consider the Laplace transform ργ,δ(u) of the time to ruin τγ. Starting with an initial surplus u, the surplus process {Rγ(t),t0} can either reach a new record high at time ξu avoiding ruin en route, or reach level 0 before any visit to levels greater than u. For the latter, the Laplace transform hδ(u) of the corresponding passage time νu is defined as hδ(u)=E[eδνu1{Zu}|Rγ(0)=u], where Z=sup{Rγ(t):0t<νu<} corresponds to the maximum surplus level before ruin.

Thus, by conditioning

Discounted tax payments

Let Dγ,δ(u) denote the discounted tax payments before ruin in the surplus process (4) defined as Dγ,δ(u)γ1γn=1eδWσn(JnJn1)1{τγ>Wσn}. In this section, we will analyze the nth moment of Dγ,δ(u), namely Mn(u)=E[(Dγ,δ(u))n], for n=1,2,. By conditioning on the first upper exit time ξu of the surplus process (4), one finds Mn(u)=gnδ(u)0β1γeβ1γxE[(Dγ,δ(u+x)+γ1γx)n]dx=gnδ(u)uβ1γeβ1γ(xu)E[(Dγ,δ(x)+γ1γ(xu))n]dx. Differentiating (35) w.r.t. to u gives Mn(u)=(ddulngnδ(u)+β1γ(1gnδ(u)

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 b(b>u). Let vb(u) denote the resulting expected discounted tax payments. By a probabilistic argument, one easily shows vb(u)=Bδ(u,b)[γβ+0β1γeβ1γxM1(b+x)dx], where Bδ(u,b) is the Laplace transform of the first passage from level u to any level above b avoiding ruin en route. Furthermore, an expression for Bδ(u,b) can be obtained in terms of gδ(x) for uxb

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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