Optimal Retention Level for Infinite Time Horizons under MADM

Başak Bulut Karageyik, Şule Şahin
2016 Risks  
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more » ... von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract: In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum benefit criteria: released capital, expected profit and exponential-fractional-logarithmic utility from the insurer's point of view. Numerical examples are presented to explain how the optimal initial surplus and retention level are changed according to the individual claim amounts, loading factors and weights of the criteria. In the decision making process, we use The Analytical Hierarchy Process (AHP) and The Technique for Order of Preference by Similarity to ideal Solution (TOPSIS) methods as the Multi-Attribute Decision Making methods (MADM) and compare our results considering different combinations of loading factors for both exponential and Pareto individual claims. Risks 2017, 5, 1 2 of 24 probability of survival of the insurer. Kaluszka [7] proposes the optimal reinsurance which minimizes the ruin probability for the truncated stop loss reinsurance based on different pricing rules, such as the economic principle, generalized zero-utility principle, Esscher principle and mean-variance principle. Dickson and Waters [8] focus on a dynamic reinsurance strategy to minimize the ruin probability. They derive a formula for the finite time ruin probability for discrete and continuous time by using the Bellman optimality principle. Moreover, they show how the optimal strategies are determined by approximating the compound Poisson aggregate claims distributions by translated gamma distributions and by approximating the compound Poisson process by a translated gamma process, respectively. Kaishev and Dimitrova [9] generalize a joint survival optimal reinsurance model for the excess of loss reinsurance under the assumption that the individual claim amounts are modeled by continuous dependent random variables with a joint distribution. The optimal retention levels that maximize both the joint survival function and the premium income are determined. Nie et al. [10] propose a new kind of reinsurance arrangement, for which the reinsurer's payments are bounded above by a fixed level. In this reinsurance type, whenever the insurer's surplus falls between zero and this fixed level, the reinsurance company makes an additional payment called capital injections. The optimal pair of initial surplus and the fixed reinsurance level is determined to make the ultimate ruin probability minimum. Centeno [11], Aase [12], Ignatov et al. [6], Balbas et al. [13] and Centeno and Simoes [14] summarize the research techniques that are used in optimal reinsurance and provide further references about optimal reinsurance studies. Briefly, the findings of these studies indicate that optimal reinsurance levels are mostly determined by using a single criterion (e.g., minimizing a ruin probability). Furthermore, there are few studies in the literature that focus on determining the optimal reinsurance level under different constraints. Dimitrova and Kaishev [15] and Hürlimann [16] have studied optimal reinsurance by considering different risks from the point of both the insurer and the reinsurer. Karageyik and Dickson [17] suggest optimal reinsurance criteria as the released capital, expected profit and expected utility of resulting wealth. They aim to find the pair of initial surplus and reinsurance level that maximizes the output of these three quantities under the minimum finite time ruin probability by using the translated gamma process to approximate the compound Poisson process. In order to obtain the optimal reinsurance, they take the advantage of the decision theory and use the TOPSIS method with the Mahalanobis distance. Based on the approach introduced in Karageyik and Dickson [17], the purpose of this paper is to determine the optimal initial surplus and retention level that maximize the optimal reinsurance criteria under the minimum ultimate ruin probability constraint. Different from Karageyik and Dickson [17], we investigate the optimal reinsurance level by using three utility functions: exponential, fractional and logarithmic, besides the expected profit and released capital criteria. Although, Karageyik and Dickson [17] examine optimal reinsurance under the finite time ruin probability, we prefer to use the ultimate ruin probability constraint. In addition, we use two multi-attribute decision making methods: AHP and TOPSIS with four normalization and two distance measure techniques in the decision analysis part. We have obtained and compared the optimal initial surplus and retention level for the combinations of different loading factors. The rest of the paper is structured as follows: Section 2 describes the classical risk model. Section 3 briefly introduces the ultimate ruin probability under the assumption of the aggregate claims amount approximated by the translated gamma process. Section 4 explains the optimal reinsurance criteria: released capital, expected profit, exponential, fractional and logarithmic utility functions. Section 5 presents two multi-attribute decision making methods: AHP and TOPSIS. Section 6 focuses on the application to determine the optimal initial surplus and retention level for the exponential and Pareto claims. Section 7 concludes the paper. Risks 2017, 5, 1 3 of 24 Classical Risk Model The insurer's surplus process at time t, t ≥ 0, is: where u ≥ 0 is the initial surplus, c is the constant premium rate with c > 0 and S(t) is the aggregate claim amounts up to time t. The aggregate claim amount up to time t, S(t), is:
doi:10.3390/risks5010001 fatcat:meeexav42zdpxaneup5t2ucjca