Statistics and Probability Letters 78 (2008) 2426–2432 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On a risk model with debit interest and dividend payments Kam-Chuen Yuen a,∗ , Ming Zhou b , Junyi Guo c a Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong b CIAS, Central University of Finance and Economics, Beijing 100081, PR China c School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China article info Article history: Received 3 June 2005 Received in revised form 21 January 2008 Accepted 14 February 2008 Available online 10 March 2008 MSC: 0167-6687 abstract We consider the compound Poisson risk model with debit interest and dividend payments. The model assumes that the company is allowed to borrow at some debit interest rate when the surplus turns negative, and that the premium incomes are paid out as dividends to shareholders when the surplus reaches a horizontal barrier of level b. We first derive integro-differential equations for the expected discounted value of all dividends until absolute ruin, V b (u), which is twice continuously differentiable. In the case of exponential claim amounts, we obtain explicit expressions for V b (u) and the optimal barrier b ∗ which maximizes V b (u). We then perform a similar study for the Gerber–Shiu expected discounted penalty function. Again, when claims are exponentially distributed, we are able to find explicit expressions for the joint distribution of the surplus just prior to absolute ruin and the deficit at absolute ruin, which is a special case of the Gerber–Shiu function. © 2008 Elsevier B.V. All rights reserved. 1. Introduction In the compound Poisson risk model, the surplus process of an insurance company is given by R(t) = u + ct − S(t) = u + ct − N(t)  k=1 X k , t ≥ 0, (1.1) where u ≥ 0 is the initial surplus, c > 0 is the rate of premium, N(t) is the Poisson claim-number process with intensity λ> 0, and {Z k , k = 1, 2,...}, independent of the claim-number process, are independent and identically distributed (i.i.d.) claim- size random variables with common distribution function F and probability density function f . Note that the compound Poisson risk model is also known as the classical risk model. In this paper, we consider the surplus process (1.1) with two additional features, namely debit interest and dividend payments. The feature of debit interest assumes that the company is allowed to borrow money at a debit interest rate β> 0 to pay claims when the surplus turns negative. As the company will pay the debts from its premium income, the negative surplus may return to a positive level. When the premium income is not enough to pay the debit interest (that is, the surplus falls below −c/β), absolute ruin is said to occur. In recent years, the issue of absolute ruin has received considerable attention in the actuarial literature. Some related results can be found in Dassios and Embrechts (1989), Embrechts and Schmidli (1994), Zhang and Wu (1999), Cai et al. (2006), Cai (2007) and Gerber and Yang (2007). For the feature of dividend payments, it is assumed that dividends are paid to shareholders according to the so-called barrier strategy. Under the barrier strategy, the premium incomes no longer go into the surplus process but are paid out ∗ Corresponding author. Tel.: +852 2859 1915; fax: +852 2858 9041. E-mail address: kcyuen@hku.hk (K.-C. Yuen). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.02.021