Copula函数在联合寿险精算中的应用

发布时间：2018-03-27 07:34

本文选题：联合寿险　切入点：Markov模型　出处：《安徽工程大学》2013年硕士论文

【摘要】：随着社会的发展,一个人投保已经不能满足整个家庭的需要,而为家庭内的每个成员均投保又会增加家庭的负担.家庭联合寿险将家庭内的两个(或两个以上)成员作为联合被保险人,该险种大大降低了保费额,同时又能让参保的每个家庭成员享受保障.但在对联合寿险的保单进行定价的时候,大多学者采用了简单叠加的方法去处理各成员之间的关系.但实际上我们知道,一个家庭中各成员之间一定不会是简单的独立存在,各个体之间有着密不可分的关系,简单的叠加处理会带来较大的误差.那么应当如何正确处理这种关系,Copula函数作为一种连接函数,成为了学者们处理联合寿险中个体间相关性的重要工具.Copula函数将一个联合分布与该联合分布的各个边缘分布连接在一起,可以有效的处理随机变量相依性结构问题. 本文首先在家庭联合寿险中考虑家庭因素,将一对夫妻生存状况的未来发展构造成Markov模型,考虑该模型内各状态的各种转移方式,得到其转移概率矩阵.再利用转移强度与转移概率的关系,得到该矩阵中各转移概率的计算方法.然后用转移概率表示出联合生存状态和最后生存者状态的多生命精算函数,由此得到家庭联合寿险均衡年保费精算现值.但在此过程中,因为对个体间的关系简单的采用了相互独立的方法去处理,导致所得保费定价偏低.为了更符合实际情况,本文采用了改进的多生命函数符号,将单生命与二元生命精算符号的条件统 .再结合二元条件Copula函数,对条件阿基米德Copula函数进行改进,得到在统一条件下的条件阿基米德Copula函数及其生成元.再采用条件阿基米德Copula函数处理联合寿险中个体间的相关性,对随机利率下的家庭联合寿险模型作出研究,得到其均衡年保费的精算现值.并给出了用不同方法处理个体相关性所得的均衡年保费精算现值的实例应用. 本文做了如下工作： 1、将家庭联合寿险中成员的未来生存状态构造成一个Markov模型,并将其作为家庭因素引入家庭联合寿险精算模型中； 2、对传统的多生命精算符号进行了改进,统一了单生命精算符号与多生命精算符号的条件；并在统一的条件下引入条件阿基米德Copula函数刻画家庭联合寿险中成员的相依关系； 3、针对上述两种情况分别给出了家庭联合寿险均衡年保费的定价方式,并通过实例分别与普通的家庭联合寿险做了对比分析.
[Abstract]:With the development of society, one person's insurance can no longer meet the needs of the whole family. And insurance for every member of the family increases the burden on the family. The family union life insurance takes two (or more) members of the family as co-insured, which greatly reduces the amount of the premium. But when it comes to pricing a joint life insurance policy, most scholars use a simple superposition method to deal with the relationship between the members, but in fact we know that, The members of a family must not be simply independent of each other. Each individual is inextricably connected. A simple superposition process can lead to large errors. So how to correctly handle this relationship Copula function as a connection function, Copula function is an important tool to deal with the correlation between individuals in joint life insurance. The Copula function connects a joint distribution with each edge distribution of the joint distribution, which can effectively deal with the dependence structure of random variables. In this paper, the family factor is considered in the family joint life insurance, and the future development of a couple's living condition is constructed into a Markov model, and the various transfer modes of each state in the model are considered. The transfer probability matrix is obtained, and then the calculation method of each transition probability in the matrix is obtained by using the relation between the transfer intensity and the transition probability. Then, the multi-life actuarial function of the combined survival state and the last survivor state is expressed by the transfer probability. Thus, the actuarial present value of the equilibrium annual premium of the family joint life insurance is obtained. However, in the process, the relationship between individuals is simply treated by independent methods, which leads to the underpricing of the premium. In order to conform to the actual situation, In this paper, an improved multi-life function symbol is used to unify the conditions of the actuarial symbol of single life and binary life. Combining with the binary conditional Copula function, the conditional Archimedes Copula function is improved. The conditional Archimedes Copula function and its generator are obtained under unified conditions. Then the conditional Archimedes Copula function is used to deal with the correlation between individuals in joint life insurance, and the family joint life insurance model with random interest rate is studied. The actuarial present value of the equilibrium annual premium is obtained, and the application of the actuarial present value of the equilibrium annual premium obtained by different methods to deal with the individual correlation is given. This paper has done the following work:. 1. Construct a Markov model for the future survival status of the members in family joint life insurance, and introduce it into the actuarial model of family joint life insurance as a family factor; (2) the traditional multi-life actuarial symbol is improved to unify the conditions of single-life actuarial symbol and multi-life actuarial symbol, and the conditional Archimedes Copula function is introduced to describe the dependent relationship of the members in the family joint life insurance under the unified condition. 3. In view of the above two cases, the pricing method of the equilibrium annual premium of the family joint life insurance is given, and the comparison with the common family joint life insurance is made through the examples.
【学位授予单位】：安徽工程大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F840.4;F224

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