Pricing reinsurance and determining optimal retention based on the criterion of maximizing social expected utility

Purpose - The authors consider the mutual benefits of the ceding company and reinsurance company in the design of reinsurance contracts. Two objective functions to maximize social expected utilities are established, which are to maximize the sum of the expected utilities of both the ceding company and reinsurance company, and to maximize their products. The first objective function, additive, emphasizes the total gains of both parties, while the second, multiplicative, accounts for the degree of substitution of gains of one party through the loss of the other party. The optimal price and retention of reinsurance are found by a grid search method, and numerical analysis is conducted. The results indicate that the optimal solutions for two objective functions are quite different. However, optimal solutions are sensitive to the change of the means and volatilities of the claim loss for both objective functions. The results are potentially valuable to insurance regulators and government entities acting as reinsurers of last resort. Design/methodology/approach - In this paper, the authors apply relatively simple, but in the view significant, methods and models to discuss the optimization of excess loss reinsurance strategy. The authors only consider the influence of loss distribution on optimal retention and reinsurance price but neglect the investment factor. The authors also consider the benefits of both ceding company and reinsurance company to determine optimal premium and retention of reinsurance jointly based on maximizing social utility: the sum (or the product) of expected utilities of reinsurance company and ceding company. The authors solve for optimal solutions numerically, applying simulation. Findings - This paper establishes two optimization models of excess-of-loss reinsurance contract against catastrophic losses to determine optimal premium and retention. One model considers the sum of the expected utilities of a ceding company and a reinsurance company's expected utility; another considers the product of them. With an example, the authors find the optimal solutions of premium and retention of excess loss reinsurance. Finally, the authors carry out the sensitivity analysis. The results show that increasing the means and the volatilities of claim loss will increase the optimal retention and premium. For objective function I, increasing the coefficients of risk aversion of or reducing the coefficients of risk aversion of will make the optimal retention reduced but the optimal premium increased, and vice versa. However, for objective function 2, the change of coefficient of risk aversion has no effect on optimal solutions. Research limitations/implications - Utility of the two partners: The ceding company and the reinsurance company, may have different weights and different significance. The authors have not studied their relative significance. The simulation approach in numerical methods limits us to the probability distributions and stochastic processes the authors use, based on, generally speaking, lognormal models of rates of return. This may need to be generalized to other returns, including possible models of shocks through jump processes. Practical implications - In the recent two decades, reinsurance companies have played a great role in hedging mega-catastrophic losses. For example, reinsurance companies (and special loss sharing arrangements) paid as much as two-thirds of the insured losses for the September 11, 2001 tragedy. Furthermore, large catastrophic events have increased the role of governments and regulators as reinsurers of last resort. The authors hope that the authors provide guidance for possible balancing of the needs of two counterparties to reinsurance contracts. Social implications - Nearly all governments around the world are engaged in regulation of insurance and reinsurance, and some are reinsurers themselves. The authors provide guidance for them in these activities. Originality/value - The authors believe this paper to be a completely new and original contribution in the area, by providing models for balancing the utility to the ceding insurance company and the reinsurance company. 研究目的 - 我們探討分出公司和再保險公司在再保險合約的設計上、如何能達至互利互惠。研究確立了兩個目標函數，分別為把分出公司和再保險公司兩者之預期效用的總和最大化，以及把它們的產品最佳化。第一個目標函數是加法的，強調兩個參與方的總增益;而第二個目標函數則是乘法的，這個目標函數，闡釋參與方因另一方虧損而有所收益之取代度。再保險的最佳價格和自留額是利用網格搜索法找出的，數值分析也予以進行。研究結果顯示，兩個目標函數的最佳解決方案甚為不同。唯最佳解決方案會對就這兩個目標函數而言的追討損失的波動、以及其平均值之改變產生敏感反應。研究結果將會見其價值於作為在萬不得已的時候的再保險人的保險業規管機構和政府實體。 研究設計/方法/理念 - 在這學術論文裡，我們採用了相對簡單、但我們認為是重要的方法和模型，來探討超額賠款再保險策略的優化課題。我們只考慮虧損分佈對最佳自留額和再保險價格的影響，而不去檢視投資因素。我們亦考慮對分出公司和再保險公司兩者的利益，來釐定最佳保費和再保險的自留額，而這兩者則共同建基於把社會效益最大化之上：再保險公司和分出公司的預期效益的總和 (或其積數) 。 我們採用類比模仿方法、來解決尋求在數字上最佳解決方案的問題。 研究結果 - 本研究建立了就應對嚴重虧損而設的兩個超額賠款再保險合約的優化模型，來釐定最佳的保費和自留額。其中一個模型考慮了分出公司和再保險公司兩者各自的預期效益的總和。另外的一個模型則考慮了兩者的預期效益的積數。透過例子，我們找到了保費和超額虧損再保險自留額的最佳解決方案。最後，我們進行了敏感度分析。研究結果顯示、若增加追討損失的平均值和波動，則最佳自留額和保費也會隨之而增加。就第一個目標函數而言，若增加風險規避係數、或減少這個係數，則最佳自留額會隨之而減少，但最佳保費卻會隨之而增加，反之亦然。唯就第二個目標函數而言，風險規避係數的改變，對最佳解決方案是沒有影響的。 研究的局限/啟示 - – 有關的兩個夥伴之效用性：分出公司和再保險公司或有不同的份量和重要性。我們沒有探討兩者的相對重要性。– 我們以數值方法為核心的類比模仿研究法、使我們局限於機率分配和一般而言建基於投資報酬率對數常態模型之隨機過程的使用。我們或許需要調節研究法。以能概括其它回報收益，包括透過跳躍過程而可能達至之沖擊模型。 實務方面的啟示 - 在過去20年裡，再保險公司在控制極嚴重災難性的損失上曾扮演重要的角色。例如、再保險公司 (以及特殊的損失分擔安排) 為了2001年9月11日的災難事件而支付多至保險損失的三分之二的費用。而且，重大的災難性事件使政府及作為最後出路再保險人的調控者得扮演更重要的角色。我們希望研究結果能為再保險合約兩對手提供指導，以平衡雙方的需要。 社會方面的啟示 - 全球差不多每個政府都參與保險和再保險的管理工作，有部份更加本身就是再保險人。研究結果為他們的管理工作提供了指導。 研究的原創性/價值 - 我們相信本學術論文、提供了平衡分出保險公司和再保險公司效用性的模型，就此而言，本論文在相關的領域上作出了全新和獨創性的貢獻。