【作者】 周小兴；

【导师】 夏亚峰；

【作者基本信息】 兰州理工大学 ， 应用数学， 2008， 硕士

【摘要】 风险理论是现代精算和数学界研究的热点,风险结构是风险理论的重要内容。在假设风险相互独立的情况下,经典风险理论主要处理保险事务中的随机风险模型,讨论有限时间内的生存概率以及最终破产概率问题。对随机风险模型,研究较多的是连续时间模型,且大都集中于复合Poisson过程的风险模型,而对离散时间模型则研究的较少。而作为连续时间的离散化,离散时间风险模型意义直观,在实践中更易于应用,关于离散时间风险模型,讨论最多的是复合二项风险模型,二项风险模型的特点是其理赔次数的均值大于其方差,特别适用于同质性保单组合的理赔次数模型,而当保单组合的理赔次数观察分布的样本方差大于其均值时,显然用复合二项风险模型不再合适、本文研究离散的复合负二项风险模型,利用鞅理论,在理赔次数的方差大于均值条件下,证明了该模型的最终破产概率,证明方法有较大的改进。本研究中,将每张保单的保费及单位时间内收取保费次数都推广为随机变量,建立了保费完全随机的复合负二项风险模型,使之更加贴近保险公司经营实际,并讨论该模型的性质,利用离散鞅理论得到它的最终破产概率及Lundberg不等式。上述各种风险模型适用于单一的险种,且保费收入过程与理赔过程相互独立,现实情况是保险公司经营多种险种,各险种的风险不一定严格符合现有的风险模型,但还想了解各组合资产风险,此时经典风险理论不再适用。本文将利用风险价值VaR(Value at risk)对组合资产风险进行研究。当风险超出自身的承受范围,那么保险商就会考虑进行再保险。已有学者研究了单个资产的风险值、停止损失保费及满足一些特定关系的风险组合的风险值,本文研究了理赔损失分布在一般情况下封闭式集合风险模型的风险值及停止损失保费问题,给出了具体的计算方法。最后运用非负随机变量函数凸序研究随机资产的风险值(VaR),证明了任意资产组合的风险不会超过其各个资产风险值之和,给出了组合资产风险值的上界。风险结构问题是风险理论研究的重要内容,对金融企业控制风险有重要参考意义。本文研究了更加符合保险公司经营现实的风险模型及资产组合风险的VaR,对风险结构及相关问题进行探讨,着重从理论上研究如何预防破产的发生,所获得研究结果为金融企业控制风险提供理论依据。更多还原

【Abstract】 Risk theory is a hot spot in modern actuarial science and risk structure is an important content in risk theory.On the assumption that risk is mutually independent, Classical risk theories give priority to stochastic risk models of insurance and survival or ruin probability within a limited time.In stochastic risk models,we consider mostly continual time risk models,especially compound Poisson procession risk models, discrete risk models are seldom treated,but discrete risk models are easier to perceive and more applicable.When treating discrete risk models,we often mention a compound binomial risk model whose characteristic is that the mean of claim numbers is greater than the variance of claim numbers,a compound binomial risk model is suitable for homogeneous policies assemblages,has been discussed.Actually policies assemblages have more or less non-homogeneity,a compound negative binomial risk model is more suitable for non-homogeneous policies assemblages than a compound binomial risk model and the property is that the variance of claim numbers is greater than the mean of claim numbers.This thesis Studies a compound negative binomial risk model whose formula of ultimate ruin probability has been proved by applying discrete martingale theory,the method is different form others references.We construct a compound negative binomial risk model with a completely stochastic premium where the premium of every policy and the number of insure charges at per unit time are random variables.This paper discusses some properties of a compound negative binomial risk model with a completely stochastic premium.By applying discrete martingale theory,this article proves the formula of ultimate ruin probability and the Lundberg inequality.When only considering a policy whose premium and claim are independent each other,we can apply all sorts of risk models.But every insurer has a lot of policies whose premium and claim don’t accord with existing risk models,this results that we can not apply the classics risk theories and use value at risk(VaR)to study a portfolio. Insures reinsure a policy whose loss exceeds the insurer’s anticipation.Some scholars have studied an asset’s VaR and stop-loss premium and some special portfolio’s VaR, this thesis has discussed arbitrary asset portfolio’s VaR and stop-loss premium, analytical expressions for risk value and stop-loss premiums of sums of independent random variables have been given.At finial,function convex order of non—negative random variables is introduced for random assets’ VaR,the conclusion is that portfolio assets VaR can not exceed the sum of individual asset VaR,and the upper of portfolio is also given.Risk structure problem is an important aspect of risk theory and is important significance for controlling risk of financial enterprises.This thesis has studied a new risk model which is more in line with the reality of insurers and arbitrary assets’ portfolio’s VaR,we study risk structure problem to guard against the occurrence of bankruptcy and our results are the gist for controlling risk of financial enterprises.更多还原