基于盈余过程模型下短期健康险业务的定价研究
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摘要
随着医疗条件和经济条件的改善,人们购买健康险产品的意识逐渐增强,各保险公司不断开发各类健康险产品以满足市场的需求。如何厘定合理的保费,谨慎地提取准备金,是保险公司开发产品面临的首要问题。健康险业务因为其自身特点(主要作为附加险产品销售,主要为一年期的短期险,保险给付方式主要有定额给付和实际发生费用补偿给付两种类型),其定价模型较为简单。目前主要有：一元回归分析法、趋势平均数法和正态分布法。对于实际发生费用给付型产品,其理赔次数和理赔额更具有随机性,精算师在使用上述三种模型评估未来风险损失时,需要大量的相关经验数据。然而健康险业务在中国的保险市场中还处于发展的初期阶段,对于缺少相关业务经验数据的公司,很多短期健康险产品都由于低估风险造成了巨大的亏损。
本文就保险公司缺少经验数据造成的定价困难问题展开讨论,提出一套新的短期健康险业务定价方案——假设公司只经营一种产品,通过模拟产品上市后单位保单的资本盈余过程,利用控制破产概率和隐含收益率的思想,调整产品的保费。具体实施步骤为：第一步、假设单位保单的初始资本U0和初始保费P;第二步、从公司数据库中按性别和年龄段分类提取每月单位保单的实际赔付支出数据,分析其隐含的信息和规律,估计未来的理赔损失分布：第三步、模拟未来资本盈余过程,利用控制破产概率和隐含收益率来调整产品的初始资本和保费。实务中不同的保险产品有不同的投资组合与之匹配,为了研究的方便,本文仅考虑一种投资,并使用GRACH模型描述其波动规律。
本文分为三个部分,第一部分(第一、二章)介绍实务中短期健康险产品的定价模型以及本文提出的定价方案；第二部分(第三章)介绍定价模型中投资回报率和风险损失分布研究的理论基础,及其应用于实际中得到的结果；第三部分(第四章)应用软件模拟产品上市后的盈余过程,并结合定价方案模拟保费和初始资本对破产概率的影响,讨论其在实务中的意义,最后对本文进行了总结。
本文主要的创新点有：一是通过模拟短期健康险产品上市后单位保单的资本盈余过程,利用控制破产概率和隐含收益率的思想,调整产品的保费和初始资本；二是改进盈余过程,讨论每月的实际赔付。考虑到每月实际赔付随时间的相依结构,将其分解成当月发生的理赔和之前发生理赔的线性组合；三是利用非参数方法——变换核估计的迭代算法估计理赔额的概率密度函数。
With the improvement of medical and economic conditions, the awareness of buying health insurance products has increased over the years. Many insurance companies are continually developing various health insurance products to meet the market demand.The major topics for insurance companies are to price these products reasonably and draw ad-equate reserves.Because of the features of health insurance business, the premium pricing models are relatively simple.There are mainly three models to consider at present:simple regression analysis, aver age trend method and normal distribution method.For actual cost payment products, their claim number and claims are more random than other kinds of products, so actuaries usually need a great deal of historical data to value the risk.However. China's health insurance business is still in the early stage. Many short-term health prod-ucts are in the state of poor performance due to lack of historical data.
This paper proposes a new model to deal with short-term health products. This model considers the situation that an insurance company only manages one product. We predict the future capital surplus process per policy and determine the premium by controlling the ruin probability and IRR.There are three implementation steps in this model. The first is to set initial capital and premium. Then we draw the data of claim paid per policy from the database, find out its implicit rule and estimate the loss distribution. At last, we simulate the future capital surplus process and determine the initial capital and premium. In practice, every insurance product has its own investment portfolio. In this paper, we simply choose GRACH model to fit the investment return of this product.
This paper consists of three parts. The first part (Chapter one and two) introduces the pricing models used in practice and proposed in the paper. The second part (chapter 3) describes the theories of both the investment return and the distribution of loss. The last part (chapter 4) illustrates the effect of the premium and initial surplus on the probability of bankrupt by simulating future capital surplus process.
There are three major contributions of this article. The first is to propose a new model for short-term health products pricing. The second is to treat the actual claim paid per month as a liner combination of the claim occurred this month and the past months. The third is to use transformed kernel estimation to estimate the claim's density function.
本文就保险公司缺少经验数据造成的定价困难问题展开讨论,提出一套新的短期健康险业务定价方案——假设公司只经营一种产品,通过模拟产品上市后单位保单的资本盈余过程,利用控制破产概率和隐含收益率的思想,调整产品的保费。具体实施步骤为：第一步、假设单位保单的初始资本U0和初始保费P;第二步、从公司数据库中按性别和年龄段分类提取每月单位保单的实际赔付支出数据,分析其隐含的信息和规律,估计未来的理赔损失分布：第三步、模拟未来资本盈余过程,利用控制破产概率和隐含收益率来调整产品的初始资本和保费。实务中不同的保险产品有不同的投资组合与之匹配,为了研究的方便,本文仅考虑一种投资,并使用GRACH模型描述其波动规律。
本文分为三个部分,第一部分(第一、二章)介绍实务中短期健康险产品的定价模型以及本文提出的定价方案；第二部分(第三章)介绍定价模型中投资回报率和风险损失分布研究的理论基础,及其应用于实际中得到的结果；第三部分(第四章)应用软件模拟产品上市后的盈余过程,并结合定价方案模拟保费和初始资本对破产概率的影响,讨论其在实务中的意义,最后对本文进行了总结。
本文主要的创新点有：一是通过模拟短期健康险产品上市后单位保单的资本盈余过程,利用控制破产概率和隐含收益率的思想,调整产品的保费和初始资本；二是改进盈余过程,讨论每月的实际赔付。考虑到每月实际赔付随时间的相依结构,将其分解成当月发生的理赔和之前发生理赔的线性组合；三是利用非参数方法——变换核估计的迭代算法估计理赔额的概率密度函数。
With the improvement of medical and economic conditions, the awareness of buying health insurance products has increased over the years. Many insurance companies are continually developing various health insurance products to meet the market demand.The major topics for insurance companies are to price these products reasonably and draw ad-equate reserves.Because of the features of health insurance business, the premium pricing models are relatively simple.There are mainly three models to consider at present:simple regression analysis, aver age trend method and normal distribution method.For actual cost payment products, their claim number and claims are more random than other kinds of products, so actuaries usually need a great deal of historical data to value the risk.However. China's health insurance business is still in the early stage. Many short-term health prod-ucts are in the state of poor performance due to lack of historical data.
This paper proposes a new model to deal with short-term health products. This model considers the situation that an insurance company only manages one product. We predict the future capital surplus process per policy and determine the premium by controlling the ruin probability and IRR.There are three implementation steps in this model. The first is to set initial capital and premium. Then we draw the data of claim paid per policy from the database, find out its implicit rule and estimate the loss distribution. At last, we simulate the future capital surplus process and determine the initial capital and premium. In practice, every insurance product has its own investment portfolio. In this paper, we simply choose GRACH model to fit the investment return of this product.
This paper consists of three parts. The first part (Chapter one and two) introduces the pricing models used in practice and proposed in the paper. The second part (chapter 3) describes the theories of both the investment return and the distribution of loss. The last part (chapter 4) illustrates the effect of the premium and initial surplus on the probability of bankrupt by simulating future capital surplus process.
There are three major contributions of this article. The first is to propose a new model for short-term health products pricing. The second is to treat the actual claim paid per month as a liner combination of the claim occurred this month and the past months. The third is to use transformed kernel estimation to estimate the claim's density function.
引文
[1]潘红宇,金融时间序列模型,北京,对外经济贸易大学出版社,2008.
[2]孙山泽,非参数统计讲义,北京,北京大学出版社,2000.
[3]王静龙,非寿险精算,北京,中国人民大学出版社,,2004.
[4]王沁,时间序列分析及其应用,成都,西南交通大学出版社,2008.
[5]王星,非参数统计,北京,清华大学出版社,2009.
[6]Brockman, M.J., Wright, T.S., Statistical motor rating:making effective use of your data, J. Inst. Actuaries 119,457-543.,1992.
[7]Bolshev, L.N. & Smirnov, N.V., Tables of Mathematical Statistics, Nauka, Moscow, 1965.
[8]Bowman, A.W., An alternative method of cross-validation for the smoothing of den-sity estimates, Biometrika 71(2),353-360,1984.
[9]Brockwell, P.J. & Davis, R.A., Time series: theory and methods, Springer, New York,1987.
[10]Castellana, J.V. & Leadbetter,M.R., On smoothed probability density estimation for stationary processes, Stochastic Processes and their Applications 21,179-193,1986.
[11]Conover W J, Practical Nonparametric Statistics (2nd ed). New York:John Wiley k Sons,1980.
[12]Devroye, L. & Gyorfi, L., Nonparametric density estimation. Wiley New York,1985.
[13]Haberman, S., Renshaw. A.E., Generalized linear models and actuarial science, The Statistician 45,407-436.,1996.
[14]Hall, P., Lahiri. S.N. and Truong, Y.K., On bandwidth choice for density estimation based on dependent data, Annals of Statistics,23(6).2241-2263,1995.
[15]Hall, P. & Weissman, I., On the estimation of extreme tail probabilities, Annals of Statistics,25(3),1311-1326,1997.
[16]Hall, P. & Welsh, A.H., Adaptive estimates of parameters regular variation. Annals of Statistics,13.331-341.1985.
[17]Hart, J.D. & Vieu, P., Data-driven bandwidth choice for density estimation based on dependent data, Annals of Statistics 13,743-756,1985.
[18]Horst, M. M., Radar sea clutter model IEE Conf. Public, No.169,28-30, Nov., 1978.
[19]McCullagh, P., Nelder, J.A., Generalized Linear Models, Chapman and Hall, New York,1989.
[20]Nadine Gatzert, Combining fair pricing and capital requirements for non-life insur-ance companies, Journal of Banking& Finance,2008.
[21]Natalia Markovich, Nonparametric Analysis of Univariate Heavy Tailed Data:Re-search and Practice, Wiley-Interscience,2007.
[22]Parzen, E, On estimation of a probability density function and mode Annals of Mathematical Statistics 33(3),1065-1076,1962.
[23]Renshaw, A.E., Modeling the claims process in the presence of covariates, ASTIN Bull.24,265-285.,1994.
[24]Rosenblatt, M, Remarks on some nonparametric estimates of density function,27(3), 832-837,1956.
[25]Rudemo, M, Empirical choice of histogram and kernel density estimators, Scandi-navian Journal of Statistics 9.65-78,1982.
[26]Silverman, B.W, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability,1986.
[27]Stone, C.J., Asymptotically optimal window selection rule for kernel density esti-mates,Annals of Statistics 12,1285-1297,1984.
[28]Stone, M., Cross-validation choice and assessment of statistical predictions (with discussion), Journal of Royal Statistical Society B 36,111-147,1974.
[29]Tsai, C.C.L., Willmot, G.E., On the moments of the surplus process perturbed by diffusion, Insurance:Mathematics and Economics Vol.31, pp 327-350,2002.
[30]Yuen. K.C., Guo, J.Y., Ng. K.W., On ultimate ruin in a delayed-claims risk model, Journal of Applied Probability,42,163-174,2005.
[2]孙山泽,非参数统计讲义,北京,北京大学出版社,2000.
[3]王静龙,非寿险精算,北京,中国人民大学出版社,,2004.
[4]王沁,时间序列分析及其应用,成都,西南交通大学出版社,2008.
[5]王星,非参数统计,北京,清华大学出版社,2009.
[6]Brockman, M.J., Wright, T.S., Statistical motor rating:making effective use of your data, J. Inst. Actuaries 119,457-543.,1992.
[7]Bolshev, L.N. & Smirnov, N.V., Tables of Mathematical Statistics, Nauka, Moscow, 1965.
[8]Bowman, A.W., An alternative method of cross-validation for the smoothing of den-sity estimates, Biometrika 71(2),353-360,1984.
[9]Brockwell, P.J. & Davis, R.A., Time series: theory and methods, Springer, New York,1987.
[10]Castellana, J.V. & Leadbetter,M.R., On smoothed probability density estimation for stationary processes, Stochastic Processes and their Applications 21,179-193,1986.
[11]Conover W J, Practical Nonparametric Statistics (2nd ed). New York:John Wiley k Sons,1980.
[12]Devroye, L. & Gyorfi, L., Nonparametric density estimation. Wiley New York,1985.
[13]Haberman, S., Renshaw. A.E., Generalized linear models and actuarial science, The Statistician 45,407-436.,1996.
[14]Hall, P., Lahiri. S.N. and Truong, Y.K., On bandwidth choice for density estimation based on dependent data, Annals of Statistics,23(6).2241-2263,1995.
[15]Hall, P. & Weissman, I., On the estimation of extreme tail probabilities, Annals of Statistics,25(3),1311-1326,1997.
[16]Hall, P. & Welsh, A.H., Adaptive estimates of parameters regular variation. Annals of Statistics,13.331-341.1985.
[17]Hart, J.D. & Vieu, P., Data-driven bandwidth choice for density estimation based on dependent data, Annals of Statistics 13,743-756,1985.
[18]Horst, M. M., Radar sea clutter model IEE Conf. Public, No.169,28-30, Nov., 1978.
[19]McCullagh, P., Nelder, J.A., Generalized Linear Models, Chapman and Hall, New York,1989.
[20]Nadine Gatzert, Combining fair pricing and capital requirements for non-life insur-ance companies, Journal of Banking& Finance,2008.
[21]Natalia Markovich, Nonparametric Analysis of Univariate Heavy Tailed Data:Re-search and Practice, Wiley-Interscience,2007.
[22]Parzen, E, On estimation of a probability density function and mode Annals of Mathematical Statistics 33(3),1065-1076,1962.
[23]Renshaw, A.E., Modeling the claims process in the presence of covariates, ASTIN Bull.24,265-285.,1994.
[24]Rosenblatt, M, Remarks on some nonparametric estimates of density function,27(3), 832-837,1956.
[25]Rudemo, M, Empirical choice of histogram and kernel density estimators, Scandi-navian Journal of Statistics 9.65-78,1982.
[26]Silverman, B.W, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability,1986.
[27]Stone, C.J., Asymptotically optimal window selection rule for kernel density esti-mates,Annals of Statistics 12,1285-1297,1984.
[28]Stone, M., Cross-validation choice and assessment of statistical predictions (with discussion), Journal of Royal Statistical Society B 36,111-147,1974.
[29]Tsai, C.C.L., Willmot, G.E., On the moments of the surplus process perturbed by diffusion, Insurance:Mathematics and Economics Vol.31, pp 327-350,2002.
[30]Yuen. K.C., Guo, J.Y., Ng. K.W., On ultimate ruin in a delayed-claims risk model, Journal of Applied Probability,42,163-174,2005.