THE TAX IDENTITY IN RISK THEORY - A SIMPLE PROOF AND AN EXTENSION HANSJ ¨ ORG ALBRECHER * , SEM BORST, ONNO BOXMA, AND JACQUES RESING Abstract. By linking queueing concepts with risk theory, we give a simple and insight- ful proof of the tax identity in the Cram´ er-Lundberg model that was recently derived in Albrecher & Hipp (2007), and extend the identity to arbitrary surplus-dependent tax rates. Keywords: Compound Poisson Model, Insurance Risk, Survival Probability, Maximum Workload, Tax Payments 1. Introduction Consider the surplus process (1) R t = u + ct − Nt  i=1 X i of the classical Cram´ er-Lundberg model in risk theory that describes the surplus at time t of an insurance portfolio, where u is the initial capital, c is a constant premium intensity and {X i } i≥1 is a sequence of independent and identically distributed positive random variables. The claim number process N t is assumed to be a homogeneous Poisson process with intensity λ, which is independent of {X i } i≥1 . A crucial quantity in risk theory is the infinite-time survival probability φ 0 (u)= P(R t ≥ 0 ∀ t ≥ 0 | R 0 = u), which will be positive under the net profit condition c>λ E(X i ) (see Asmussen [2] for a survey on risk theory). In a recent paper, Albrecher & Hipp [1] studied the Cram´ er-Lundberg model under the additional assumption that tax payments are deducted from the premium income (with a constant proportion γ< 1), whenever the free surplus R t is at a running maximum, i.e. in a profitable situation (cf. Figure 1). This leads to a modified risk process R (γ) t . Using conditioning techniques and product identities, Theorem 1 in [1] established the remarkably simple formula (2) φ γ (u)=(φ 0 (u)) 1/(1−γ) , where φ γ (u) denotes the survival probability of the process R (γ) t with tax at rate 0 <γ< 1. In this note we provide a short alternative proof of Identity (2), which shows more clearly the reason for the simplicity of the identity (Section 2). The approach also allows us to generalize the tax identity to the case of a tax rate γ (s) that depends on the current surplus level s (Section 3). * Supported by the Austrian Science Foundation Project P18392. 1