破产时刻和破产赤字对保险公司破产概率的影响研究——基于Erlang风险模型
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摘要
【目的】保险公司作为市场经济中重要活动主体,优胜劣汰是自由经济运行的必然结果,为了了解保险公司的生存现状,研究了当初始盈余值大于0时,个人索赔额服从Erlang(3)分布,索赔时间间隔服从Erlang(2)分布的Sparre Andersen风险模型。【方法】通过期望折现罚金函数,运用拉格朗日隐函数定理和拉普拉斯变换的技巧进行求解。【结果】得到破产时间和破产赤字的联合概率密度函数。【结论】根据中国上市保险公司的实际数据,对中国保险公司破产概率进行了实证研究,并证明该模型是有效的。
[Purposes]Although insurance companies play an important role in market economy,survival of the fittest is the inevitable result of free economy.In order to understand the real situation of insurance company survival,it studies the Sparre Andersen risk model with the individual claim amount obeying Erlang(3)distribution and the claim time interval obeying Erlang(2)distribution when the initial surplus value is greater than zero.[Methods]Applying the expected discounted penalty function.[Findings]The join density of time to ruin and the deficit at ruin is obtained by using Lagrange implicit function theorem and the Laplace transform.[Conclusions]Finally,according to the actual data of Chinese listed insurance companies,an empirical study on the ruin probability of Chinese insurance companies is carried out,and the model is proved to be effective.
[Purposes]Although insurance companies play an important role in market economy,survival of the fittest is the inevitable result of free economy.In order to understand the real situation of insurance company survival,it studies the Sparre Andersen risk model with the individual claim amount obeying Erlang(3)distribution and the claim time interval obeying Erlang(2)distribution when the initial surplus value is greater than zero.[Methods]Applying the expected discounted penalty function.[Findings]The join density of time to ruin and the deficit at ruin is obtained by using Lagrange implicit function theorem and the Laplace transform.[Conclusions]Finally,according to the actual data of Chinese listed insurance companies,an empirical study on the ruin probability of Chinese insurance companies is carried out,and the model is proved to be effective.
引文
[1]毕秀春,李荣,苏玉霞,等.一类风险模型的破产概率[J].曲阜师范大学学报(自然科学版),2010,36(2):35-38.BI X C,LI R,SU Y X,et al.Ruinprobability for a risk model[J].Journal of Qufu Normal University(Natural Science),2010,36(2):35-38.
[2]毕秀春,尹传存.更新风险模型中破产概率的一个局部结果[J].高校应用数学学报A辑(中文版),2005,20(1):29-36.BI X C,YIN C C.A partial result of ruin probability in the new risk model[J].Applied Mathematics A Journal(Chinese Version),2005,20(1):29-36.
[3]DICKSON D C M.Some explicit solutions for the joint desity of the time of ruin and the deficitat ruin[J].ASTIN Bulletin,2008,38(1):259-276.
[4]DICKSON D C M,WILLMOT G E.The density of the time to ruin in the classical Poisson risk model[J].ASTIN Bulletin,2005,35(1):45-60.
[5]郭晓莉,文丽壹.保险和金融风险相依的破产概率研究[J].重庆理工大学学报(自然科学),2016,30(5):135-140.GUO X L,WEN Y L.The ruin probability with dependent insurance and financial risks[J].Journal of Chongqing University of Technology(Natural Science),2016,30(5):135-140.
[6]李荣,毕秀春,张曙光.保费率-巨灾索赔相依的风险模型的破产概率[J].数学的实践与认识,2014,44(5):1-6.LI R,BI X C,ZHANG S G.Insurance rate-ruin probability of risk model for catastrophe claims[J].Practice and understanding of mathematics,2014,44(5):1-6.
[7]DAVID D C M,LI S.Finite time ruin problems for the Erlang(2)risk model[J].Insurance Mathematics and Economics,2009,46(1):12-18.
[8]DICKSOND C M,HIPP C.On the time to ruin for Erlang(2)risk processes[J].Insurance:Mathematics and Economics,2001,29(3):333-334.
[9]WILLMOT G E.On the discounted penalty function in the renewal risk model with general inter-claim times[J].Insurance:Mathematics and Economics,2007,41(1):17-31.
[10]SUN L J.The expected discounted penalty at ruin in the Erlang(2)risk process[J].Staristic and Probability Letters,2005,72(3):205-217.
[2]毕秀春,尹传存.更新风险模型中破产概率的一个局部结果[J].高校应用数学学报A辑(中文版),2005,20(1):29-36.BI X C,YIN C C.A partial result of ruin probability in the new risk model[J].Applied Mathematics A Journal(Chinese Version),2005,20(1):29-36.
[3]DICKSON D C M.Some explicit solutions for the joint desity of the time of ruin and the deficitat ruin[J].ASTIN Bulletin,2008,38(1):259-276.
[4]DICKSON D C M,WILLMOT G E.The density of the time to ruin in the classical Poisson risk model[J].ASTIN Bulletin,2005,35(1):45-60.
[5]郭晓莉,文丽壹.保险和金融风险相依的破产概率研究[J].重庆理工大学学报(自然科学),2016,30(5):135-140.GUO X L,WEN Y L.The ruin probability with dependent insurance and financial risks[J].Journal of Chongqing University of Technology(Natural Science),2016,30(5):135-140.
[6]李荣,毕秀春,张曙光.保费率-巨灾索赔相依的风险模型的破产概率[J].数学的实践与认识,2014,44(5):1-6.LI R,BI X C,ZHANG S G.Insurance rate-ruin probability of risk model for catastrophe claims[J].Practice and understanding of mathematics,2014,44(5):1-6.
[7]DAVID D C M,LI S.Finite time ruin problems for the Erlang(2)risk model[J].Insurance Mathematics and Economics,2009,46(1):12-18.
[8]DICKSOND C M,HIPP C.On the time to ruin for Erlang(2)risk processes[J].Insurance:Mathematics and Economics,2001,29(3):333-334.
[9]WILLMOT G E.On the discounted penalty function in the renewal risk model with general inter-claim times[J].Insurance:Mathematics and Economics,2007,41(1):17-31.
[10]SUN L J.The expected discounted penalty at ruin in the Erlang(2)risk process[J].Staristic and Probability Letters,2005,72(3):205-217.