Numerical algorithms for Panjer recursion by applying Bernstein approximation

In actuarial science, Panjer recursion (1981) is used in insurance to compute the loss distribution of the compound risk models. When the severity distribution is continuous with density function, numerical calculation for the compound distribution by applying Panjer recursion will involve an approximation of the integration. In order to simplify the numerical algorithms, we apply Bernstein approximation for the continuous severity distribution function and obtain approximated recursive equations, which are used for computing the approximated values of the compound distribution. The theoretical error bound for the approximation is also obtained. Numerical results show that our algorithm provides reliable results.     

VaR Criteria for optimal limited change-loss and truncated change-loss reinsurance

Reinsurance can provide an effective way for insurer to manage its risk exposure. In this paper, we further analyze the optimal reinsurance models recently proposed by J. Cai and K. S. Tan [Astin Bulletin, 2007, 37(1): 93-112]. With the criteria of minimizing the value-at-risk (VaR) risk measure of insurer’s total loss exposure, we derive the optimal values of sharing proportion a, retention d, and layer l of two reinsurance treaties: the limited changeloss f(x) = a{(x - d)+ - (x - l)+} and the truncated change-loss f(x) = a(x-d)₊I_(x≤l). Both of the reinsurance plans have been considered to be more realistic and practical in the real business. Our solutions have several appealing features: (i) there is only one condition to verify for the existence of optimal limited change-loss reinsurance while there always exists an optimal truncated change-loss reinsurance, (ii) the resulting optimal parameters have simple analytic forms which depend only on assumed loss distribution, reinsurer’s safety loading, and insurer’s risk tolerance, (iii) the optimal retention d for limited change-loss reinsurance is the same as that by Cai and Tan while the optimal value is smaller for truncated change-loss, (iv) the optimal sharing proportion and layer are always the same for both reinsurance plans, (v) minimized VaR are strictly lower than the values derived by Cai and Tan, (vi) the optimization results reveal possible drawbacks of VaR-based risk management that a heavy tail risk exposure may be expressed by lower VaR.     

含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

均衡的再保险市场

本文讨论了再保险市场达到均衡状态的定义、性质以及与 Pareto最优的关系 .     

离散时间比例再保险模型的破产概率

研究具有相依结构的离散时间比例再保险模型的破产概率.在模型中假设随机利率和索赔间隔时间是相依的.利用更新递归技巧,首先得到了破产概率满足的递归方程.然后,根据该递归方程得到了破产概率的上下界估计.     

最小化破产概率的保险人鲁棒投资再保险策略研究

在模型不确定条件下,研究以破产概率最小化为目标的模糊厌恶型保险公司的最优投资再保险问题. 假设保险公司可投资于一种风险资产,也可购买比例再保险. 分别考虑风险资产的价格过程服从随机波动率模型和非随机波动率模型的两种情况,根据动态规划原理建立相应的HJB方程,得到保险公司的最优鲁棒投资再保险策略和价值函数的解析解. 最后,通过数值模拟分析了各模型参数对最优策略和价值函数的影响.     

The optimal reinsurance strategy — the individual claim case

This paper is concerned with the optimal form of reinsurance when the cedent seeks to maximize the adjustment coefficient of the retained risk (related to the probability of ultimate ruin)-which we prove to be equivalent to maximizing the expected utility of wealth, with respect to an exponential utility with a certain coefficient of risk aversion-and restricts the reinsurance strategies to functions of the individual claims, which is the case for most nonproportional treaties placed in the market.Assuming that the premium calculation principle is a convex functional we prove the existence and uniqueness of solutions and provide a necessary optimality condition (via needle-like perturbations, widely known in optimal control). These results are used to find the optimal reinsurance policy when the reinsurance loading is increasing with the variance. The optimal contract is described by a nonlinear function, of a similar form than in the aggregate case.     

最优比例与超额损失组合再保险下的破产概率

本文站在保险人的立场上,讨论了保险公司的最优组合再保险问题.通过纯粹比例再保险,纯粹超额损失再保险,或者这两类再保险的组合方式,把保险公司的部分风险分担出去.在最大化调节系数的最优准则下,我们得出了布朗运动模型和复合Poisson模型中最优值的显示表达,并且给出了复合Poisson模型中最优策略下破产概率的最小指数上界.我们还得出结论:在一定的条件下,总存在一种纯粹超额损失再保险策略比任何一类组合再保险策略都要好.最后,通过一些数例和图表来进一步说明我们在文中所获得的结论.     

带干扰复合泊松模型下对调整系数的新研究(英文)

本文研究了带干扰复合泊松模型中采用成数再保与超额损失再保险混合策略时作为自留额水平函数的调整系数.我们按照原始条款计算成数再保费,按照期望值保费原则计算超额损失再保费,这样得到了调整系数是超额损失自留额极限的单峰函数的结论.本文最后部分给出了有限时间破产概率的上界.     

Optimal multivariate quota-share reinsurance: A nonparametric mean-CVaR framework

In this paper, the Conditional Value-at-Risk (CVaR) is adopted to measure the total loss of multiple lines of insurance business and two nonparametric estimation methods are introduced to explore the optimal multivariate quota-share reinsurance under a mean-CVaR framework. While almost all the existing literature on optimal reinsurance are based on a probabilistic derivation, the present paper relies on a statistical analysis. The proposed optimal reinsurance models are directly formulated on empirical data and no explicit distributional assumption on the underlying risk vector is required. The resulting nonparametric reinsurance models are convex and computationally amenable, circumventing the difficulty of computing CVaR of the sum of a generally dependent random vector. Statistical consistency of the resulting estimators for the best CVaR is established for both nonparametric models, allowing empirical data to be generated from any stationary process satisfying strong mixing conditions. Finally, numerical experiments are presented to show that a routine bootstrap procedure can capture the distributions of the resulting risk measures well for independent data.