基于信息熵方法的非寿险定价研究
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摘要
保险业务分为寿险和非寿险,在它们的经营和管理过程中都涉及保险精算问题。寿险损失分布规律(生命表)比较稳定,寿险精算已经相当成熟和完备。由于非寿险精算涉及的随机因素更多、定量分析更困难,所以非寿险精算一直是研究的核心问题。其中非寿险定价是非寿险精算的一个主要问题,多年来受到研究者的广泛重视。在保险市场,针对非寿险定价问题,在不完全信息下进行保险风险分析,并且进行合理的保险定价是本论文研究的主要目的。
本文在研究中主要应用信息熵方法,根据熵的内涵、最大熵原理、最小叉熵原理及大偏差熵等,从不同的角度研究了非寿险定价所涉及的风险分析和保险定价问题。具体地,论文主要研究内容及所取得的研究成果包括以下方面:
1.风险分析
当获得损失分布的不完全信息时,把这些信息作为约束条件,通过最大熵原理在最不确定的情况下,得到最无偏的损失分布,并获得了损失分布的熵。基于熵度量不确定性的本质,把熵引入到风险度量上,与方差度量风险方法互相补充、共同决策风险大小。通过算例分析,探讨了熵参与风险度量的必要性。进而,把熵引入到实效保费原理中,对实效保费原理进行修正,建立方差—熵保费原理。具体结论是:(1)概率分布的熵是和高阶矩信息相联系的,是随矩信息的改变而改变的量。在不完全信息下,熵参与风险决策体现了更多损失变量的信息。(2)矩信息下的最大熵优化模型是易于求解的,所以新保费原理并没有增加计算上的难度,且由于熵的加入体现了更多关于保险定价的信息。
若已获得损失分布的先验信息,通过最小叉熵原理,建立了最小叉熵变换,变换的结果使高风险的比例增加了,体现了风险转移的本质。最小叉熵变换与Esscher变换有密切关系,并建立了它们之间的关系。由于Esscher变换在金融或保险领域通常被用来处理聚合风险的转移,这样基于两者之间的关系,得到用最小叉熵变换处理风险转移的方法。另外,当前保费价格中包含了关于潜在损失分布有用的信息,最小叉熵变换模型同时提供了合理提取这种信息的工具。具体结论是:(1)Esscher变换是特殊约束形式之下的最小叉熵变换,在约束条件中必须满足变换后的均值大于平均损失,以获得风险补偿。(2)用最小叉熵变换处理风险转移,更直观地体现了保险人对风险的态度,且使用更方便。
2.保险定价
在不完全的保险市场,基于金融风险中性定价方法,在满足风险中性约束和其他市场约束条件下,利用最小叉熵(最大熵)优化原理,以最无偏的方式选择唯一的风险中性密度,从而得到最小叉熵(最大熵)风险中性保费。通过算例分析,熵优化模型的应用是简单易行的,显示了更大的优势。具体结论是:(1)在所建的最小叉熵(最大熵)优化模型中,风险中性约束的形式是灵活的,且该模型是易于求解的,它的应用会带来很大的方便。(2)最小叉熵(最大熵)风险中性保费没有考虑风险附加,因此具有确定的形式,是一种非参数的保险定价方法。该保费对保险人和投保人均具有一定的指导意义。
在大偏差理论中非常重要的量是率函数,而相对经验概率分布的率函数则是叉熵。从Shannon熵、Jaynes最大熵和大偏差熵入手,揭示了大偏差率函数的信息论含义,从而率函数的求解可以通过熵优化原理简单得到。另外,从大偏差率函数的叉熵形式中推导出大偏差概率测度,在此基础上考察了基于大偏差概率测度上的保费。具体结论是:基于大偏差概率测度上的保费对不同数量的保单组合,可以估计出大偏差概率,显示了其他保费原理所没有的功效。
The insurance operation includes life and no-life insurance, in which both are relate to actuarial problems. The life actuarial science has been very perfect due to the stability of the loss distribution. The quantitative analysis on no-life actuarial science is more difficult than the one on life actuarial science because the former relates to more stochastic factors. So, the no-life actuarial science becomes the core problem that has being researched. On the other hand, the no-life insurance pricing problem, gotten the extensive recognition for many years, is one of no-life insurance actuarial science and is very important. The main object of this thesis is to analyze risk under incomplete information and give the reasonable price of insurance product to the question of no-life insurance pricing problem in insurance market.
The main tools are the information entropy methods in this thesis. Based on the essence of entropy, the maximum entropy principle, the minimum cross-entropy principle and the large deviation entropy, the no-life insurance pricing problems including risk analysis and insurance pricing are researched from the different aspects. The main investigations and achievements are composed of following portions.
1. Risk analysis
When the incomplete information of loss distribution is obtained, the estimate of the maximum entropy loss distribution and the value of entropy function are acquired by the maximum entropy principle under these information constraints. Based on the essence of uncertainty, entropy is introduced into the risk measure. Entropy and variance are reinforced each other and used to decide to the size of risk. The numerical example examines the necessity of entropy risk measure. Based on the research, entropy is added to actual effect premium principle and the new variance-entropy premium principle is presented. The result indicates that the entropy of probability distribution is related to higher moment information and it changes with the moment information. Entropy risk decision embodies the more information of loss variable. Furthermore, the solution of the maximum entropy optimization model under moment information is easily obtained. So, the calculation difficulty of the adjusted premium principle is not added and the new premium principle embodies more insurance pricing information because of the existence of entropy.
If the prior information on the loss distribution is obtained, the minimum cross-entropy transform is established by the minimum cross-entropy principle. The result of the minimum cross-entropy transform gives more weight to unfavorable events, which embodies the essence of risk change. The minimum cross-entropy transform is correlated to Esscher transform closely and their relationship is built. Based on the relationship, the new minimum cross-entropy transform way to deal with loss risk change is presented because Esscher transform usually used to deal with the risk change of aggregate risk. On the other hand, the useful information of the loss distribution contained in the current premium can be distilled by the minimum cross-entropy transform model. The results indicate that Esscher transform is the special minimum cross-entropy transform. The constraint, the transformed exception is larger than the mean loss, must be satisfied in order to obtain the risk compensate. The minimum cross-entropy transform embodies the insurer's attitude to loss risk more intuitively and is more convenient.
2. Insurance pricing
In the incomplete insurance market, based on the risk-neutral pricing ideal, the minimum cross-entropy or the maximum entropy optimization principle is used to select the only unprejudiced risk-neutral density under the risk-neutral constraint and other market constraints. The minimum cross-entropy or maximum entropy risk-neutral premium is established. Further, the numerical example shows that the application of minimum cross-entropy or maximum entropy optimization principle is simple and convenient, which make the new premium principle more advantaged. The results indicate that the form of the constraint of risk-neutral is flexible in the built minimum cross-entropy or maximum entropy optimization model and the solution of the model can be obtained easily, which must be convenient. Under the prior information on loss variable, the risk loading disappears in the minimum cross-entropy or maximum entropy risk-neutral premium. So, the method is no parameter. Otherwise, this premium principle has instructional significance to insurant and insurer.
The rate function, which is the cross-entropy to prior distribution, is very important in large deviation theory. Based on the Shannon entropy and Jaynes maximum entropy and large deviation entropy, the signification of information theory of large deviation rate function is explained. Thus, the rate function can be obtained easily by the entropy optimization principle. Otherwise, the large deviation probability measure is deduced from the cross-entropy form of large deviation rate function of probability distribution. Based on the large deviation probability measure, the insurance pricing problem is researched. The result indicates the premium based on the large deviation probability measure can estimate the large deviation probability for the different numbers insurance portfolio, which is the advantage that does not belong to other premium principle.
本文在研究中主要应用信息熵方法,根据熵的内涵、最大熵原理、最小叉熵原理及大偏差熵等,从不同的角度研究了非寿险定价所涉及的风险分析和保险定价问题。具体地,论文主要研究内容及所取得的研究成果包括以下方面:
1.风险分析
当获得损失分布的不完全信息时,把这些信息作为约束条件,通过最大熵原理在最不确定的情况下,得到最无偏的损失分布,并获得了损失分布的熵。基于熵度量不确定性的本质,把熵引入到风险度量上,与方差度量风险方法互相补充、共同决策风险大小。通过算例分析,探讨了熵参与风险度量的必要性。进而,把熵引入到实效保费原理中,对实效保费原理进行修正,建立方差—熵保费原理。具体结论是:(1)概率分布的熵是和高阶矩信息相联系的,是随矩信息的改变而改变的量。在不完全信息下,熵参与风险决策体现了更多损失变量的信息。(2)矩信息下的最大熵优化模型是易于求解的,所以新保费原理并没有增加计算上的难度,且由于熵的加入体现了更多关于保险定价的信息。
若已获得损失分布的先验信息,通过最小叉熵原理,建立了最小叉熵变换,变换的结果使高风险的比例增加了,体现了风险转移的本质。最小叉熵变换与Esscher变换有密切关系,并建立了它们之间的关系。由于Esscher变换在金融或保险领域通常被用来处理聚合风险的转移,这样基于两者之间的关系,得到用最小叉熵变换处理风险转移的方法。另外,当前保费价格中包含了关于潜在损失分布有用的信息,最小叉熵变换模型同时提供了合理提取这种信息的工具。具体结论是:(1)Esscher变换是特殊约束形式之下的最小叉熵变换,在约束条件中必须满足变换后的均值大于平均损失,以获得风险补偿。(2)用最小叉熵变换处理风险转移,更直观地体现了保险人对风险的态度,且使用更方便。
2.保险定价
在不完全的保险市场,基于金融风险中性定价方法,在满足风险中性约束和其他市场约束条件下,利用最小叉熵(最大熵)优化原理,以最无偏的方式选择唯一的风险中性密度,从而得到最小叉熵(最大熵)风险中性保费。通过算例分析,熵优化模型的应用是简单易行的,显示了更大的优势。具体结论是:(1)在所建的最小叉熵(最大熵)优化模型中,风险中性约束的形式是灵活的,且该模型是易于求解的,它的应用会带来很大的方便。(2)最小叉熵(最大熵)风险中性保费没有考虑风险附加,因此具有确定的形式,是一种非参数的保险定价方法。该保费对保险人和投保人均具有一定的指导意义。
在大偏差理论中非常重要的量是率函数,而相对经验概率分布的率函数则是叉熵。从Shannon熵、Jaynes最大熵和大偏差熵入手,揭示了大偏差率函数的信息论含义,从而率函数的求解可以通过熵优化原理简单得到。另外,从大偏差率函数的叉熵形式中推导出大偏差概率测度,在此基础上考察了基于大偏差概率测度上的保费。具体结论是:基于大偏差概率测度上的保费对不同数量的保单组合,可以估计出大偏差概率,显示了其他保费原理所没有的功效。
The insurance operation includes life and no-life insurance, in which both are relate to actuarial problems. The life actuarial science has been very perfect due to the stability of the loss distribution. The quantitative analysis on no-life actuarial science is more difficult than the one on life actuarial science because the former relates to more stochastic factors. So, the no-life actuarial science becomes the core problem that has being researched. On the other hand, the no-life insurance pricing problem, gotten the extensive recognition for many years, is one of no-life insurance actuarial science and is very important. The main object of this thesis is to analyze risk under incomplete information and give the reasonable price of insurance product to the question of no-life insurance pricing problem in insurance market.
The main tools are the information entropy methods in this thesis. Based on the essence of entropy, the maximum entropy principle, the minimum cross-entropy principle and the large deviation entropy, the no-life insurance pricing problems including risk analysis and insurance pricing are researched from the different aspects. The main investigations and achievements are composed of following portions.
1. Risk analysis
When the incomplete information of loss distribution is obtained, the estimate of the maximum entropy loss distribution and the value of entropy function are acquired by the maximum entropy principle under these information constraints. Based on the essence of uncertainty, entropy is introduced into the risk measure. Entropy and variance are reinforced each other and used to decide to the size of risk. The numerical example examines the necessity of entropy risk measure. Based on the research, entropy is added to actual effect premium principle and the new variance-entropy premium principle is presented. The result indicates that the entropy of probability distribution is related to higher moment information and it changes with the moment information. Entropy risk decision embodies the more information of loss variable. Furthermore, the solution of the maximum entropy optimization model under moment information is easily obtained. So, the calculation difficulty of the adjusted premium principle is not added and the new premium principle embodies more insurance pricing information because of the existence of entropy.
If the prior information on the loss distribution is obtained, the minimum cross-entropy transform is established by the minimum cross-entropy principle. The result of the minimum cross-entropy transform gives more weight to unfavorable events, which embodies the essence of risk change. The minimum cross-entropy transform is correlated to Esscher transform closely and their relationship is built. Based on the relationship, the new minimum cross-entropy transform way to deal with loss risk change is presented because Esscher transform usually used to deal with the risk change of aggregate risk. On the other hand, the useful information of the loss distribution contained in the current premium can be distilled by the minimum cross-entropy transform model. The results indicate that Esscher transform is the special minimum cross-entropy transform. The constraint, the transformed exception is larger than the mean loss, must be satisfied in order to obtain the risk compensate. The minimum cross-entropy transform embodies the insurer's attitude to loss risk more intuitively and is more convenient.
2. Insurance pricing
In the incomplete insurance market, based on the risk-neutral pricing ideal, the minimum cross-entropy or the maximum entropy optimization principle is used to select the only unprejudiced risk-neutral density under the risk-neutral constraint and other market constraints. The minimum cross-entropy or maximum entropy risk-neutral premium is established. Further, the numerical example shows that the application of minimum cross-entropy or maximum entropy optimization principle is simple and convenient, which make the new premium principle more advantaged. The results indicate that the form of the constraint of risk-neutral is flexible in the built minimum cross-entropy or maximum entropy optimization model and the solution of the model can be obtained easily, which must be convenient. Under the prior information on loss variable, the risk loading disappears in the minimum cross-entropy or maximum entropy risk-neutral premium. So, the method is no parameter. Otherwise, this premium principle has instructional significance to insurant and insurer.
The rate function, which is the cross-entropy to prior distribution, is very important in large deviation theory. Based on the Shannon entropy and Jaynes maximum entropy and large deviation entropy, the signification of information theory of large deviation rate function is explained. Thus, the rate function can be obtained easily by the entropy optimization principle. Otherwise, the large deviation probability measure is deduced from the cross-entropy form of large deviation rate function of probability distribution. Based on the large deviation probability measure, the insurance pricing problem is researched. The result indicates the premium based on the large deviation probability measure can estimate the large deviation probability for the different numbers insurance portfolio, which is the advantage that does not belong to other premium principle.
引文
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