商业养老保险定价的风险因素与精算模型研究
摘要
随着老龄化社会的到来,基本养老保险及企业补充养老保险已不能满足个人日益增长的养老需求,商业养老保险是个人自主实现养老需求的重要保障。基于此,论文重点研究了商业养老保险的风险因素及相关精算定价模型,在此基础上给出了国内商业养老保险产品的研发方向,为保险公司定价提供科学有效的指导方法。
对于商业养老保险定价的风险因素,主要表现为死亡率风险及利率风险。论文死亡率风险的研究为商业养老险死亡率改善因素的及时准确预测提供了新方法。利率风险的研究发现波动因素对不同定价利率的影响程度不同。对于商业养老保险定价的精算模型,论文从市场需求、客户效用、公司偿付能力等角度予以定价,论证了市场可接受费率、逆选择情形及市场需求均衡下的费率。在此基础上,利用经验数据,考虑影响因素的相关性,给出利润目标导向的费率。
论文首先分析了商业养老保险定价的风险因素,具体如下：
第一,关于死亡率风险评估。论文首先给出了个体死亡率随时间推移的变化规律。分别从国内寿险业生命表演变及年度国民人口统计数据角度得出了较一致的预测结论,给出了分年龄、性别的参数估计,克服了使用行业生命表的时滞缺陷,创造性地解决了死亡率随时间变化难以及时准确预计的问题。在此基础上,结合国内某大型寿险公司的养老险数据,得出定价所用的经验死亡率。其次论证了逆选择因素的存在。利用推广后的单一时间段效用最大化模型,阐释了高收入、低死亡率人群更愿意购买年金的现象,该现象在低利率时代尤为明显。将特定的效用函数拓展为一般效用函数,并考虑利率因素,通过多时间段的消费-投资模型,论证了低死亡率人群减少当期消费额度的倾向。通过对养老数据的实证分析,发现购买保单数量大的保单经验死亡率也较低。接着论证了综合人群的死亡率风险,首次将特定的风险分布函数拓展为一般风险分布函数,发现整体人群的综合死亡效力及一阶导数低于标准个体死亡效力,同时给出了实证分析。
第二,关于利率风险评估。在Vasicek及CIR连续利率效力模型的基础上,研究了一般化的CKLS模型,并将计算结果与Vasicek模型作对比,发现期望及模拟结果所得结论均比较一致。对于低固定利率的贴现结果,随机利率贴现结果与固定利率贴现结果是相似的。而高固定利率的贴现结果,随机利率贴现结果会明显高于固定利率贴现结果,且差异随着利率贴现时间的延长越加明显。说明高固定利率贴现风险较大,尤其对于长期养老险。结合公司实际资产状况,利用已有利率模型,创造性给出了分年度资产预期收益率,为随机利率下以利润为导向的定价创造重要条件。
第三,关于风险因素相关性的评估。通过实证分析,对于死亡率与退保率的关系,表明不同年份的死亡率与退保因素的正负相关性存在差异；对于死亡率与利率的关系,表明死亡率与利率几乎不相关,定价时可假设死亡率与利率是独立的,解决了长期难以确定的死亡率与经济环境的关系。对于退保率与市场利率,存在同向关系。
第四,关于长寿风险及防范措施的探讨。首先利用年度国民人口统计数据,通过不同阶数的泰勒展开式,考虑死亡率随时间递减因子的相关性,计算生存概率,给出年金精算现值的上、下限,实证表明两者预测差异较小,并从可信度角度证明了预测结果的充足性,及时客观估计了未来生存概率。关于长寿风险,给出了发行生存债券防范方式下的额外边际及风险的市场价格,为长寿风险的防范提供了思路。
接着分析了精算定价模型,具体如下：
第一,市场需求模型。针对目前国内外从费率与风险方差最小化的客户需求角度或公司偿付能力角度单方面核算费率且未考虑费用因素的问题,给出考虑费用因素的客户需求角度定价、公司偿付能力角度定价的等价条件。研究结果表明,无论使用何种等价指标,最小化要求下的两者费率值是相同的。且从优化角度,其费用率应为0,反映了市场对保险公司的低费用附加的严格要求,因此降低保险公司附加费用是市场需求的直接体现。随着投保人群的增大,费率会随之变小。同时结合国内商业保险养老数据,给出了费率与购买人数平衡后的费率调整数据。
第二,客户效用模型。国外文献假设高、低死亡率购买年金险的概率是相同的,此假设与实际承保状况可能存在差异。论文给出了低死亡率客户的各种购买可能性假设下的情形分析,且低死亡率水平假设对生存率调整因子的影响较大。测算结果表明,生存率调整因子均大于1,也证明了逆选择因素的存在。第三,破产概率模型。给定公司最低资本要求,已有文献给出了静态固定死亡率下的偿付能力结果。论文考虑了死亡率随时间变化的因素、年度分红因素等对公司偿付能力的影响,给出了一定盈余目标下的费率要求,发现生命表改善因素对公司破产概率的影响较大。
第四,以利润为导向的定价模型。在对承保数据分析基础上,分析相关风险因素间的相互影响关系。利用相关风险因素间的变量关系,结合保险公司分红资产的利率估计,从保险公司利润目标角度,首次给出了各影响因素相互作用下的综合定价模型。
最后指出了未来养老产品发展方向。第一,变额年金。该类产品不保证回报率和本金安全,保险公司不承担利率风险,但需承担死差与费差。第二,指数年金产品。通过投资期权实现保底保障,将投资成果与股票指数挂钩,给出了产品示例。第三,住房养老保险产品。对于国内反向抵押贷款的养老保险进行了实证研究,此研究结果对该业务的开展具有实践指导意义。
With the advent of an aging society, pension insurance and enterprise supplementary pension insurance can't meet the individual increasing needs of enjoying good retired life. Annuity is personal independent realization to ensure retirement life demand. This article focuses on annuity risk factors and related actuarial pricing model. Moreover, this study provides domestic annuity products development directions. This paper aims at offering scientific and effective guidance pricing method for insurance companies.
As to the risk factors of annuity pricing, the main risks include mortality risk and interest rate risk. Mortality risk means mortality improvement factor varied with time. This study provides a new method for timely and accurate forecast. Interest rate risk study shows that fluctuation factors have different effects on different pricing interest rates. For annuity pricing actuarial models, this article provides corresponding models from the market demand, customer utility and company solvency etc. It demonstrates market acceptable rates, adverse selection conditions and market demand equilibrium rates. Furthermore, based on the use of empirical data, this study gives the profit-oriented rate considering the factors correlation.
This paper first analyses annuity pricing risk factors, as follows:
First, mortality risk assessment. This article presents individual mortality rate changes over time. From the domestic historical life tables and the annual nation demographic data, a consistent forecast conclusion and parameter estimation about age and sex were obtained, and the time lag problem of insurance industry life tables was effectively resolved. With the annuity data from a large nationwide life insurance company, we get the mortality experience results. On the other hand, the existence of adverse selection factors was verified. It illustrates that people with high income and low mortality are more willing to buy annuity by using a single period utility maximization model. The phenomenon is particularly evident in the era of low interest rates. By using a generic utility functions instead of specific one as well as taking into account interest factor, it demonstrates that low mortality populations have tendency to reduce the current consumption. Through empirical analysis of the annuity data, we find out the larger amount policy is accompanied with lower mortality. Moreover, comprensive mortality force and its first order derivative are both lower than individual one. It's remarkable that this study uses distribution function of the General risk other than a specific risk for the first time. At the same time, the paper gives the empirical analysis.
Secondly, interest rate risks assessment. Considering continuous interest rate model, such as Vasicek, CIR and the generalized CKLS model, this article makes contrasts Vasicek model with CKLS model. We find out the expectations and simulation results are relatively consistent. For a low fixed discount rate, stochastic interest rate discounted results and fixed interest rate discounted results are similar. But, high fixed discount rate results are much lower than the corresponding stochastic interest rate discount results. It shows higher fixed discount rate is more risky than smaller one, especially for long-term annuity. Combined with the actual situation of the insurance company assets, this study gives the expected annual assets return rate creatively through the existing interest rate models. It offers a good basis to use profit-oriented comprehensive pricing model.
Thirdly, the assessment of the risk factors correlation. Through empirical analysis, for the mortality rate and surrender rate, the mortality rate and surrender rate have different correlation factors in different years. Factors are positive in some years, but may be negative in other years; for mortality and interest rates, mortality rates and the interest rate are almost irrelevant, pricing assumption can be assumed that mortality rate is independent of economic environment. The assumption has been a debate for many years. For surrender rate and market interest rate, they change in the same direction.
Lastly, the longevity risk and its preventive measures. With the annual national demographic data, considering the correlation between the decreasing mortality factors over time, this study gives the upper and lower estimation of survival probability through different orders of Taylor expansions. The difference between the upper and lower one is relatively small. The forecast result is adequate from the credibility theory. It offers a useful way to forecast survival probability timely and objectively in the future. With regard to longevity risk, issuing survival bonds can hedge it. This article gives the additional margin and the price of market risk about survival bonds.
Then analyzes the actuarial pricing model, as follows:
First, market demand model. Currently, whether from customer demand requirement or company solvency, rates were calculated unilaterally and the expense factors were not included. Given the expense factors, the equivalent conditions are deduced between customer demand pricing and company solvency pricing. The results of the study show that both rates are the same with minimization requirement under equivalent conditions. From the optimization view, its optimized expense factors should be 0, which reflects the market has much stricter requirement on expense loading. Reducing insurance company additional expenses is a direct requirement of market demand. The larger the number of insured people, the smaller the rates will be. At the same time, the study gives the balance process example between the rate and the purchase number adjustment.
Secondly, customer utility model. It was reported that the probability to purchase annuity is the same regardless mortality rate level. The hypothesis may be unfit for actual underwriting condition. We successfully separate low mortality rate from the purchase possibility and then give results under scenario analysis. The result shows that both assumptions have greater impact on survival level adjustment factor. The survival level adjustment factor is greater than 1 and the study shows the adverse selection factors exist.
Thirdly, bankruptcy probability model. Given the minimum capital requirement for a company, the reported solvency results were obtained from static mortality rate. Considering the mortality factors changed with time as well as the annual dividend factors, the present work studies their effects on the solvency of the company. The improvement in the life table has greater impact on the company's bankruptcy probability.
Fourthly, the profit-oriented pricing model. Based on data analysis, the interaction between associated risk factors was introduced to this model. Using the relevant risk factors, combined with the expected asset return rate of profit sharing product, the rate under profit target was deduced for the first time.
As the extension of the study, we point out the annuity development directions in the future. First, the variable annuity products. These products don't guarantee return rate and principal safety. Insurance companies will not be liable for interest rate risk, but to mortality risk and expense risk. Secondly, the equity-indexed annuity products. The minimum guarantee was achieved by options purchase and potential ideal return was realized by stocks investment. Thirdly, reverse mortgage products. Empirical research results have much significance to the practice.
对于商业养老保险定价的风险因素,主要表现为死亡率风险及利率风险。论文死亡率风险的研究为商业养老险死亡率改善因素的及时准确预测提供了新方法。利率风险的研究发现波动因素对不同定价利率的影响程度不同。对于商业养老保险定价的精算模型,论文从市场需求、客户效用、公司偿付能力等角度予以定价,论证了市场可接受费率、逆选择情形及市场需求均衡下的费率。在此基础上,利用经验数据,考虑影响因素的相关性,给出利润目标导向的费率。
论文首先分析了商业养老保险定价的风险因素,具体如下：
第一,关于死亡率风险评估。论文首先给出了个体死亡率随时间推移的变化规律。分别从国内寿险业生命表演变及年度国民人口统计数据角度得出了较一致的预测结论,给出了分年龄、性别的参数估计,克服了使用行业生命表的时滞缺陷,创造性地解决了死亡率随时间变化难以及时准确预计的问题。在此基础上,结合国内某大型寿险公司的养老险数据,得出定价所用的经验死亡率。其次论证了逆选择因素的存在。利用推广后的单一时间段效用最大化模型,阐释了高收入、低死亡率人群更愿意购买年金的现象,该现象在低利率时代尤为明显。将特定的效用函数拓展为一般效用函数,并考虑利率因素,通过多时间段的消费-投资模型,论证了低死亡率人群减少当期消费额度的倾向。通过对养老数据的实证分析,发现购买保单数量大的保单经验死亡率也较低。接着论证了综合人群的死亡率风险,首次将特定的风险分布函数拓展为一般风险分布函数,发现整体人群的综合死亡效力及一阶导数低于标准个体死亡效力,同时给出了实证分析。
第二,关于利率风险评估。在Vasicek及CIR连续利率效力模型的基础上,研究了一般化的CKLS模型,并将计算结果与Vasicek模型作对比,发现期望及模拟结果所得结论均比较一致。对于低固定利率的贴现结果,随机利率贴现结果与固定利率贴现结果是相似的。而高固定利率的贴现结果,随机利率贴现结果会明显高于固定利率贴现结果,且差异随着利率贴现时间的延长越加明显。说明高固定利率贴现风险较大,尤其对于长期养老险。结合公司实际资产状况,利用已有利率模型,创造性给出了分年度资产预期收益率,为随机利率下以利润为导向的定价创造重要条件。
第三,关于风险因素相关性的评估。通过实证分析,对于死亡率与退保率的关系,表明不同年份的死亡率与退保因素的正负相关性存在差异；对于死亡率与利率的关系,表明死亡率与利率几乎不相关,定价时可假设死亡率与利率是独立的,解决了长期难以确定的死亡率与经济环境的关系。对于退保率与市场利率,存在同向关系。
第四,关于长寿风险及防范措施的探讨。首先利用年度国民人口统计数据,通过不同阶数的泰勒展开式,考虑死亡率随时间递减因子的相关性,计算生存概率,给出年金精算现值的上、下限,实证表明两者预测差异较小,并从可信度角度证明了预测结果的充足性,及时客观估计了未来生存概率。关于长寿风险,给出了发行生存债券防范方式下的额外边际及风险的市场价格,为长寿风险的防范提供了思路。
接着分析了精算定价模型,具体如下：
第一,市场需求模型。针对目前国内外从费率与风险方差最小化的客户需求角度或公司偿付能力角度单方面核算费率且未考虑费用因素的问题,给出考虑费用因素的客户需求角度定价、公司偿付能力角度定价的等价条件。研究结果表明,无论使用何种等价指标,最小化要求下的两者费率值是相同的。且从优化角度,其费用率应为0,反映了市场对保险公司的低费用附加的严格要求,因此降低保险公司附加费用是市场需求的直接体现。随着投保人群的增大,费率会随之变小。同时结合国内商业保险养老数据,给出了费率与购买人数平衡后的费率调整数据。
第二,客户效用模型。国外文献假设高、低死亡率购买年金险的概率是相同的,此假设与实际承保状况可能存在差异。论文给出了低死亡率客户的各种购买可能性假设下的情形分析,且低死亡率水平假设对生存率调整因子的影响较大。测算结果表明,生存率调整因子均大于1,也证明了逆选择因素的存在。第三,破产概率模型。给定公司最低资本要求,已有文献给出了静态固定死亡率下的偿付能力结果。论文考虑了死亡率随时间变化的因素、年度分红因素等对公司偿付能力的影响,给出了一定盈余目标下的费率要求,发现生命表改善因素对公司破产概率的影响较大。
第四,以利润为导向的定价模型。在对承保数据分析基础上,分析相关风险因素间的相互影响关系。利用相关风险因素间的变量关系,结合保险公司分红资产的利率估计,从保险公司利润目标角度,首次给出了各影响因素相互作用下的综合定价模型。
最后指出了未来养老产品发展方向。第一,变额年金。该类产品不保证回报率和本金安全,保险公司不承担利率风险,但需承担死差与费差。第二,指数年金产品。通过投资期权实现保底保障,将投资成果与股票指数挂钩,给出了产品示例。第三,住房养老保险产品。对于国内反向抵押贷款的养老保险进行了实证研究,此研究结果对该业务的开展具有实践指导意义。
With the advent of an aging society, pension insurance and enterprise supplementary pension insurance can't meet the individual increasing needs of enjoying good retired life. Annuity is personal independent realization to ensure retirement life demand. This article focuses on annuity risk factors and related actuarial pricing model. Moreover, this study provides domestic annuity products development directions. This paper aims at offering scientific and effective guidance pricing method for insurance companies.
As to the risk factors of annuity pricing, the main risks include mortality risk and interest rate risk. Mortality risk means mortality improvement factor varied with time. This study provides a new method for timely and accurate forecast. Interest rate risk study shows that fluctuation factors have different effects on different pricing interest rates. For annuity pricing actuarial models, this article provides corresponding models from the market demand, customer utility and company solvency etc. It demonstrates market acceptable rates, adverse selection conditions and market demand equilibrium rates. Furthermore, based on the use of empirical data, this study gives the profit-oriented rate considering the factors correlation.
This paper first analyses annuity pricing risk factors, as follows:
First, mortality risk assessment. This article presents individual mortality rate changes over time. From the domestic historical life tables and the annual nation demographic data, a consistent forecast conclusion and parameter estimation about age and sex were obtained, and the time lag problem of insurance industry life tables was effectively resolved. With the annuity data from a large nationwide life insurance company, we get the mortality experience results. On the other hand, the existence of adverse selection factors was verified. It illustrates that people with high income and low mortality are more willing to buy annuity by using a single period utility maximization model. The phenomenon is particularly evident in the era of low interest rates. By using a generic utility functions instead of specific one as well as taking into account interest factor, it demonstrates that low mortality populations have tendency to reduce the current consumption. Through empirical analysis of the annuity data, we find out the larger amount policy is accompanied with lower mortality. Moreover, comprensive mortality force and its first order derivative are both lower than individual one. It's remarkable that this study uses distribution function of the General risk other than a specific risk for the first time. At the same time, the paper gives the empirical analysis.
Secondly, interest rate risks assessment. Considering continuous interest rate model, such as Vasicek, CIR and the generalized CKLS model, this article makes contrasts Vasicek model with CKLS model. We find out the expectations and simulation results are relatively consistent. For a low fixed discount rate, stochastic interest rate discounted results and fixed interest rate discounted results are similar. But, high fixed discount rate results are much lower than the corresponding stochastic interest rate discount results. It shows higher fixed discount rate is more risky than smaller one, especially for long-term annuity. Combined with the actual situation of the insurance company assets, this study gives the expected annual assets return rate creatively through the existing interest rate models. It offers a good basis to use profit-oriented comprehensive pricing model.
Thirdly, the assessment of the risk factors correlation. Through empirical analysis, for the mortality rate and surrender rate, the mortality rate and surrender rate have different correlation factors in different years. Factors are positive in some years, but may be negative in other years; for mortality and interest rates, mortality rates and the interest rate are almost irrelevant, pricing assumption can be assumed that mortality rate is independent of economic environment. The assumption has been a debate for many years. For surrender rate and market interest rate, they change in the same direction.
Lastly, the longevity risk and its preventive measures. With the annual national demographic data, considering the correlation between the decreasing mortality factors over time, this study gives the upper and lower estimation of survival probability through different orders of Taylor expansions. The difference between the upper and lower one is relatively small. The forecast result is adequate from the credibility theory. It offers a useful way to forecast survival probability timely and objectively in the future. With regard to longevity risk, issuing survival bonds can hedge it. This article gives the additional margin and the price of market risk about survival bonds.
Then analyzes the actuarial pricing model, as follows:
First, market demand model. Currently, whether from customer demand requirement or company solvency, rates were calculated unilaterally and the expense factors were not included. Given the expense factors, the equivalent conditions are deduced between customer demand pricing and company solvency pricing. The results of the study show that both rates are the same with minimization requirement under equivalent conditions. From the optimization view, its optimized expense factors should be 0, which reflects the market has much stricter requirement on expense loading. Reducing insurance company additional expenses is a direct requirement of market demand. The larger the number of insured people, the smaller the rates will be. At the same time, the study gives the balance process example between the rate and the purchase number adjustment.
Secondly, customer utility model. It was reported that the probability to purchase annuity is the same regardless mortality rate level. The hypothesis may be unfit for actual underwriting condition. We successfully separate low mortality rate from the purchase possibility and then give results under scenario analysis. The result shows that both assumptions have greater impact on survival level adjustment factor. The survival level adjustment factor is greater than 1 and the study shows the adverse selection factors exist.
Thirdly, bankruptcy probability model. Given the minimum capital requirement for a company, the reported solvency results were obtained from static mortality rate. Considering the mortality factors changed with time as well as the annual dividend factors, the present work studies their effects on the solvency of the company. The improvement in the life table has greater impact on the company's bankruptcy probability.
Fourthly, the profit-oriented pricing model. Based on data analysis, the interaction between associated risk factors was introduced to this model. Using the relevant risk factors, combined with the expected asset return rate of profit sharing product, the rate under profit target was deduced for the first time.
As the extension of the study, we point out the annuity development directions in the future. First, the variable annuity products. These products don't guarantee return rate and principal safety. Insurance companies will not be liable for interest rate risk, but to mortality risk and expense risk. Secondly, the equity-indexed annuity products. The minimum guarantee was achieved by options purchase and potential ideal return was realized by stocks investment. Thirdly, reverse mortgage products. Empirical research results have much significance to the practice.
引文
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3. 曾庆五,变额年金与变额寿险给付额的核算[J].经济数学.1994,1,70-75.
4. 杜鹃,长寿风险与年金保险研究[J].金融发展研究.2008,6,70-73.
5. 段正敏、何欣,中国寿险生命表演变规律及其未来中国人口预测的应用[J].保险研究(增刊),2008,第2期.
6. 范子文.以房养老一住房反向抵押贷款的国际经验与我国的现实选择[M].北京：中国金融出版社,2006.
7. 方越峦, 《个人年金保险定价中若干问题的探讨》,硕士论文,2005,25.
8. 高建伟,《养老保险精算定价模型研究及其应用》,博士论文,2002,43-44,84-100.
9. 李冰清,王默航,基于极大似然估计的中国拆借市场短期利率模型实证研究[J].保险研究(增刊).2008，2,64-69.
10.卢仿先,尹莎,Lee-Carter方法在预测中国人口死亡率中的应用[J].保险职业学院学报,2005,6,9-11.
11.陆健瑜,杨步青,梁子君,修正生命表的信度理论方法[J].精算通讯2008,第2期
12.吕兆友,基于单因子利率期限结构模型的中国银行间短期利率行为的实证研究[J].国际商务-对外经济贸易大学学报,2007,1,35-38.
13.潘冠中,邵斌,单因子利率模型的极大似然估计-对中国利率的实证分析[J].财经研究.2004,10，62-69.
14.饶瑞明,浅谈资产份额在寿险保单定价上的应用[J].精算通讯,1998年5月.
15.石玉凤,王立杰,动态资产份额定价理论模型[J].数学的实践与认识,2005,3,8-13.
16.万里霜,李琼,寿险业债券化有效性的实证分析[J].经济评论,2007,2,100-103.
17.王彬,一类随机利率模型下的年金精算现值[J].应用数学与计算数学学报,2007,21(1),77-81.
18.王修文,规避寿险公司利率风险[J].上海企业,2003,5,43-46.
19.王修文,钱林义,关于2000-2003新生命表出台对寿险业的影响分析[J].应用概率统计.2008,24(1),98-106.
20.魏迎宁,中国寿险业发展的又一里程碑-中国人寿保险业经验生命表(2000-2003)编制成功[J].中国金融,2006,2,9-10.
21.魏迎宁主编,中国人寿保险业经验生命表[M].2006,137-143.
22.余伟强,长寿风险的证券化探索[J].复旦大学学报(自然科学版).2006,45(5),664-669.
23.郑海涛,徐楠楠,马佳,任若恩,基于CBS模型的分红保险负债评估研究[J].保险研究(增刊).2008,2,87-93.
24.朱世武,陈健恒,利用均衡利率模型对浮动利率债券定价[J].世界经济,2005,2,48-59.
25. Balzer,1. and Benjamin, S. Dynamic response of insurance systems with delayed P rofit/Loss Feedback to mortality and survival model, Journal of the Institute of Actuaries.1980,107,513-528.
26. Becker, D.N. Pricing for Profitability in ART, Best's Review (Life/Health Insurance Edition) 85(September).1984,26.
27. Beekman, J. A ruin function approximation and several survival models. Trans. Of the Sue. Of Actuaries.1969,21,41-48.
28. Blake, D. and Bufrrows, W. Survivor Bonds:Helping to Hedge Mortality Risk. The Journal of Risk and Insurance.2001,68(2),339-348.
29. Blake, D., Caims, A., Dowd, K. and Mac Minn, R. Longevity Bonds:Financial Engineering, Valuation and Hedging. The Journal of Risk and Insurance.2006,73(4),647-672.
30. Blake, D., Caims. A. and Dowd. K. Living with Mortality:Longevity Bonds and Other Mortality-linked Securities. British Actuarial Journal.2006,12(1),153-197.
31. Blanchet-Scalliet, Christophette etc. Dynamic Asset Pricing Theory with Uncertain Time-Horizon. Journal of Economic Dynamics and Control.2005,29,1737-1764.
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35. Bruce, L. Jones. A model for analyzing the impact of selective lap sation on mortality, north American actuarial Joural,1998,2,79-86.
36. Brugiavini, A. Uncertainty Resolution and the Timing of Annuity Purchases, Journal of Public Economics.1993,50,31-62.
37. Cantor, MS., Cole, J.B., Sandor, R.L. Insurance Derivatives:A New Asset Class for the Capital Markets and a New Hedging%o 1 for the Insurance Industry. Journal of Applied Corporate Finance. 1997,10(3),69-83.
38. Chan, K.C., Andrew, Karolyi G., Francis A. Long staff and Anthony B. Sanders. An Emprical Comparison of Alternative Models of the Term Structure of Interest Rates. The Journal of Finance 1992,47,1209-1228.
39. Constantinides George M., and Jonathan E., Ingersoll, Optimal bond trading with personal Taxes. Journal of Finance Economics.1984,13,299-335.
40. Cowley, A., Cummins J. D., Securitization of Life Insurance Assets and Liabilities, Journal of Risk and Insurance.2005,72,193-226.
41. Cox John, C., Jonathan, E. Ingersoll, Jr. and Stephen, A. Ross. A Theory of the Term Structure of Interest Rates. Econometrica.1985,53,385-407.
42. Cox, S., Pedersen, H. Catastrophe Risk Bonds, North American Actuarial Journal,2000,4,56-82.
43. Cox, S., Pedersen, H., Fairchild, J. Economic Aspects of Securitization of Risk, ASTIN Bulletin,2000, 30,157-193.
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