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This paper proposes a discrete-time risk model that has a certain type of correlation between premiums and claim amounts. It is motivated by the well-known bonus-malus system (also known as the no claims discount) in the car insurance industry. Such a system penalises policyholders at fault in accidents by surcharges, and rewards claim-free years by discounts. For simplicity, only up to three levels of premium are considered in this paper and recursive formulae are derived to calculate the ultimate ruin probabilities. Explicit expressions of ruin probabilities are obtained in a simplified case. The impact of the proposed correlation between premiums and claims on ruin probabilities is examined through numerical examples. In the end, the joint probability of ruin and deficit at ruin is also considered.

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In this paper, we study the joint Laplace transform of the occupation times in disjoint intervals until ruin in a delayed Sparre Andersen risk model with general inter-claim times and exponential claims. We extend the transformation method in the literature and apply the theoretical fluctuation techniques to derive an explicit expression of the joint Laplace transform under consideration. Further, with the presence of a constant dividend barrier, we derive explicit expressions for the Laplace transforms of the time of ruin and the non-dividend paying duration, namely the total length of non-dividend paying periods prior to ruin. This quantity is of practical interest but has not been studied in the literature to date. Within this paper, all of the Laplace transforms are expressed in terms of scale functions associated with the given spectrally negative Lévy process. Numerical examples are also provided at the end of this paper regarding the Laplace transform of the non-dividend paying duration to illustrate how the distribution of this occupation time behaves in response to varying parameters and the impact of delay on the occupation times comparing with an ordinary Sparre Andersen risk model.

Abstract. The leniency bias is one of the most common biases in performance evaluation. It has an adverse effect on the organization's performance evaluation. Based on the definition of performance appraisal bias, this paper discusses the common performance appraisal biases, summarizes the previous research on the influencing factors of leniency effect, and based on the performance appraisal system proposed by Landy and Farr Category summary, formed a systematic framework. Finally, the paper puts forward the insufficiency and suggestion of the research on the influence factors of the leniency bias.