长寿风险对寿险和年金产品定价的对冲效应研究
|
摘要
作为老龄社会的重要风险,长寿风险专题研究是近20年来公共养老金领域、保险公司关注的热点。长寿风险引发的保险公司寿险产品定价高估和年金产品定价低估之间存在潜在的自然对冲效应。为了量化这种对冲效应的长期影响,本文基于构建的同时涵盖低龄、高龄和超高龄在内的整个生命跨度的全年龄人口动态死亡率模型,采用对冲弹性量化终身寿险与终身年金、两全保险与定期年金、递延寿险与递延年金三类保障型寿险产品和养老型年金产品对冲效应的动态演变,并通过敏感性分析扩展探讨利率变化对对冲效应的长期影响。研究发现,从单位寿险和年金产品组合的净对冲效应来看,由于保险公司的产品定价区分了性别差异,使得女性的对冲效应更明显,因而女性对应的产品组合中的长寿风险对保险公司的影响更不显著。作为系统性风险,利率风险和长寿风险也存在对冲,利率上升能抵消或对冲长寿风险的影响,低利率下长寿风险更显著。
As an important risk in the aging society,longevity risks have become an active topic in both the field of public pension plans and for life insurance companies during the last two decades. There is a potential natural hedging effect between the overestimated pricing of life insurance products and underestimated pricing of annuity products resulted from longevity risk. In order to quantify the long term relative influences of this hedging effect,we used hedging elasticity to quantify the dynamic evolutions of hedging effects of three types of life insurance and pension annuity portfolios,i.e.,whole life insurance and whole life annuity,endowment insurance and term annuity,and deferred life insurance and deferred annuity,by using the proposed dynamic mortality rate models for the entire life spans which cover all ages,with young,old,and advanced old inclusive. Furthermore,we discussed the long term impacts of interest rate changes on the hedging effect through sensitivity analysis. The main conclusions were as follows. From the net hedging effect of unit life insurance and annuity product portfolio,the hedging effect of female insured was more obvious because insurance companies distinguished the gender differences in products pricing,so the impact of longevity risk in female insured's corresponding product portfolio was less significant on insurance companies. Both interest rate risk and longevity risk were systemic risks,and there was also a hedge to some extent between them. The rise of interest rate could offset or hedge the impact of longevity risk,and longevity risk was more significant under lower interest rates.
As an important risk in the aging society,longevity risks have become an active topic in both the field of public pension plans and for life insurance companies during the last two decades. There is a potential natural hedging effect between the overestimated pricing of life insurance products and underestimated pricing of annuity products resulted from longevity risk. In order to quantify the long term relative influences of this hedging effect,we used hedging elasticity to quantify the dynamic evolutions of hedging effects of three types of life insurance and pension annuity portfolios,i.e.,whole life insurance and whole life annuity,endowment insurance and term annuity,and deferred life insurance and deferred annuity,by using the proposed dynamic mortality rate models for the entire life spans which cover all ages,with young,old,and advanced old inclusive. Furthermore,we discussed the long term impacts of interest rate changes on the hedging effect through sensitivity analysis. The main conclusions were as follows. From the net hedging effect of unit life insurance and annuity product portfolio,the hedging effect of female insured was more obvious because insurance companies distinguished the gender differences in products pricing,so the impact of longevity risk in female insured's corresponding product portfolio was less significant on insurance companies. Both interest rate risk and longevity risk were systemic risks,and there was also a hedge to some extent between them. The rise of interest rate could offset or hedge the impact of longevity risk,and longevity risk was more significant under lower interest rates.
引文
[1] 段白鸽,石磊.中国高龄人口死亡率的动态演变——基于年份、城镇乡、性别的分层建模视角[J].人口研究,2015,39(4):3-18.
[2] 段白鸽,孙佳美.极值理论在高龄死亡率建模中的应用[J].数量经济技术经济研究,2012,29(7):120-133.
[3] 黄顺林,王晓军.基于VaR方法的长寿风险自然对冲模型[J].统计与信息论坛,2011,26(2):48-51.
[4] 金博轶.随机利率条件下保险公司长寿风险自然对冲策略研究[J].保险研究,2013,(5):31-38.
[5] 魏华林,宋平凡.随机利率下的长寿风险自然对冲研究[J].保险研究,2014,(3):3-10.
[6] 曾燕,曾庆邹,康志林.基于价格调整的长寿风险自然对冲策略[J].中国管理科学,2015,23(12):11-19.
[7] Bensusan H,EI Karoui N,Loisel S,Salhi Y.Partial Splitting of Longevity and Financial Risks:The Longevity Nominal Choosing Swaptions[J].Insurance:Mathematics and Economics,2016,68(1):61-72.
[8] Blake D,Burrows W.Survivor Bonds:Helping to Hedge Mortality Risk[J].Journal of Risk and Insurance,2001,68(2):339-348.
[9] Cairns A J G,Blake D,Dowd K.A Two-factor Model for Stochastic Mortality with Parameter Uncertainty:Theory and Calibration[J].Journal of Risk and Insurance,2006,73(4):687-718.
[10] Chan K C,Karolyi G A,Longstate F A,Sanders A B.An Empirical Comparison of Alternative Models of the Short-term Interest Rate[J].The Journal of Finance,1992,47(3):1209-1227.
[11] Chan W S,Li J S-H,Li J.The CBD Mortality Indexes:Modeling and Applications[J].North American Actuarial Journal,2014,18(1):38-58.
[12] Chen Hua,Cummins J D.Longevity Bond Premiums:The Extreme Value Approach and Risk Cubic Pricing[J].Insurance:Mathematics and Economics,2010,46(1):150-161.
[13] Cowley A,Cummins D.Securitization of Life Insurance Assets and Liabilities[J].Journal of Risk and Insurance,2005,72(1):193-226.
[14] Lee R D,Carter L R.Modeling and Forecasting U.S.Mortality[J].Journal of the American Statistical Association,1992,87(419):659-671.
[15] Li J,Haberman S.On the Effectiveness of Natural Hedging for Insurance Companies and Pension Plans[J].Insurance:Mathematics and Economics,2015,61(2):286-297.
[16] Li J S-H,Hardy M R,Tan K S,Threshold Life Tables and Their Applications[J].North American Actuarial Journal,2008,12(1):99-115.
[17] Luciano E,Regis L.Efficient Versus Inefficient Hedging Strategies in the Presence of Financial and Longevity (Value at) Risk[J].Insurance:Mathematics and Economics,2014,55(1):68-77.
[18] Milevsky M A,Promislow S D.Mortality Derivatives and the Option to Annuities[J].Insurance:Mathematics and Economics,2001,29(3):299-318.
[19] Ngai A,Sherris M.Longevity Risk Management for Life and Variable Annuities:The Effectiveness of Static Hedging Using Longevity Bonds and Derivatives[J].Insurance:Mathematics and Economics,2011,49(1):100-114.
[20] Tan C I,Li J,Li J S-H,Balasooriya U.Parametric Mortality Indexes:From Index Construction to Hedging Strategies[J].Insurance:Mathematics and Economics,2014,59(2):285-299.
[21] Wang C W,Huang H C,Hong D C.A Feasible Natural Hedging Strategy for Insurance Companies[J].Insurance:Mathematics and Economics,2013,52(3):532-541.
① 长寿风险可细分为个体长寿风险和聚合长寿风险。其中,个体长寿风险是可以分散的非系统性风险或特定风险,而聚合长寿风险则是无法根据大数法则进行分散的系统性风险。在实际中,我们并不畏惧个体长寿风险,而担心或害怕聚合长寿风险对经济社会发展的影响。如无特别说明,本文中的长寿风险均指聚合长寿风险。
② 由于对冲弹性是个相对概念,不受死亡率样本数据的影响,本文后续部分进一步对基于普查及抽样调查数据、寿险业参保人群经验数据计算的对冲弹性的可比性进行解释说明。年金三类产品组合的对冲弹性的动态演变规律,并通过利率敏感性分析扩展考虑利率变化对对冲弹性的影响。这些研究基于性别、投保年龄、保险期限、递延期限和利率假设等多层次视角,对中国寿险业承担的长寿风险进行了较全面的量化分析。本文的主要贡献在于,第一,在长寿风险量化与管理中,找到了基于人口普查及抽样调查死亡数据与寿险业参保人群死亡经验数据可比性的方法,通过两类数据计算的对冲弹性具有可比性。对冲弹性的值越负,对冲效应越明显,从而长寿风险越不显著。第二,重新界定了长寿风险对寿险和年金产品定价的双重影响,丰富了国内学者对长寿风险的性别差异的研究。研究表明,由于保险公司在产品定价时区分了死亡率的性别差异,使得女性的对冲效应更明显,从而女性对应的产品组合中的长寿风险对保险公司的影响更不显著。然而,基本养老保险体系在缴费筹资模式和待遇领取模式的设计中都未严格区分死亡率的性别差异,且没有相应的保障型寿险产品与养老型年金产品对冲,通常面临的女性参保人群的长寿风险更显著。第三,通过对寿险和年金产品进行分组,得出终身寿险与终身年金组的对冲效应最强,两全保险与定期年金组的对冲效应次之,递延寿险与递延年金组的对冲效应最弱。这为保险公司提供了一套利用不同产品组合内部的对冲效应量化自身长寿风险的方法。第四,通过考虑利率变化对对冲效应的净影响,诠释了利率风险和长寿风险两大系统性风险的相互作用,即利率上升能抵消或对冲长寿风险的影响。
(1)门限年龄为建模中考虑的超高龄人口年龄的起点。较早研究中年龄分段点是人为选取的,为了克服分段点选取的主观性,近年研究中则设定了多个可选择的分段点,也称为门限年龄(Threshold Age),在估计模型参数过程中,通过反复试验方法(Tatonnement Approach)来选取合适的门限年龄(即最优门限年龄)以确保最优分段形式的模型结构,保证在最优门限年龄下模型整体拟合效果最好。本文沿用近年研究中的做法,即假设门限年龄是可选择的,而最优门限年龄则可以通过统计方法唯一确定。
(2)除0岁外,通常假设分数年龄服从均匀分布,即α=0.5,而α的取值通常位于[0.06,0.3],故可将α视为已知量。
(3)这里,超参数包括固定效应参数q、随机效应变量的分布参数mx,t和qx,t、模型误差项的分布参数μx,t和x。这些超参数通过极大似然估计或相关的优化技术来估计。
(4)本文在构建参数t的ARIMA模型时,采用的是参数θt的随机效应估计值θt-μθ组成的时间序列,在最后的预测结果中,再加上参数qx,t的固定效应估计值x∈[0,ωt]。
(5)本文后续部分将进一步介绍对冲弹性的利率敏感性分析。
(6)为了评价最优模型对样本外数据的预期精度,本文通过比较最优模型预测的2011～2013年各年龄的条件死亡概率与利用2012～2014年《中国人口和就业统计年鉴》中的死亡数据计算的0～89岁的条件死亡概率,得出最优模型的短期预测能力相当好。
(7)实际上,保险产品的价格由净保费和附加保费两部分组成,其中附加保费包括费用附加和利润附加。由于附加保费因公司而异,故本文在量化长寿风险对保险产品定价的影响中暂时不考虑附加保费,直接采用对净保费的影响来代替对定价的影响。
(8)如式(6)～式(8)所示,这里绘制的对冲弹性的动态演变是指1995～2030年寿险与年金产品净保费相对于基期1994年净保费的变化率之比随时间的变化趋势。由于目前保险行业仍采用固定死亡率精算假设为各种保险产品进行定价,这种相对于基期的量化方法比相对于上一期的量化方法更适宜刻画实际中寿险产品与年金产品的对冲效应。故这里采用相对于基期的量化方法。
(9)该结论也可以从第二部分对式(8)的解释中得到验证。
[2] 段白鸽,孙佳美.极值理论在高龄死亡率建模中的应用[J].数量经济技术经济研究,2012,29(7):120-133.
[3] 黄顺林,王晓军.基于VaR方法的长寿风险自然对冲模型[J].统计与信息论坛,2011,26(2):48-51.
[4] 金博轶.随机利率条件下保险公司长寿风险自然对冲策略研究[J].保险研究,2013,(5):31-38.
[5] 魏华林,宋平凡.随机利率下的长寿风险自然对冲研究[J].保险研究,2014,(3):3-10.
[6] 曾燕,曾庆邹,康志林.基于价格调整的长寿风险自然对冲策略[J].中国管理科学,2015,23(12):11-19.
[7] Bensusan H,EI Karoui N,Loisel S,Salhi Y.Partial Splitting of Longevity and Financial Risks:The Longevity Nominal Choosing Swaptions[J].Insurance:Mathematics and Economics,2016,68(1):61-72.
[8] Blake D,Burrows W.Survivor Bonds:Helping to Hedge Mortality Risk[J].Journal of Risk and Insurance,2001,68(2):339-348.
[9] Cairns A J G,Blake D,Dowd K.A Two-factor Model for Stochastic Mortality with Parameter Uncertainty:Theory and Calibration[J].Journal of Risk and Insurance,2006,73(4):687-718.
[10] Chan K C,Karolyi G A,Longstate F A,Sanders A B.An Empirical Comparison of Alternative Models of the Short-term Interest Rate[J].The Journal of Finance,1992,47(3):1209-1227.
[11] Chan W S,Li J S-H,Li J.The CBD Mortality Indexes:Modeling and Applications[J].North American Actuarial Journal,2014,18(1):38-58.
[12] Chen Hua,Cummins J D.Longevity Bond Premiums:The Extreme Value Approach and Risk Cubic Pricing[J].Insurance:Mathematics and Economics,2010,46(1):150-161.
[13] Cowley A,Cummins D.Securitization of Life Insurance Assets and Liabilities[J].Journal of Risk and Insurance,2005,72(1):193-226.
[14] Lee R D,Carter L R.Modeling and Forecasting U.S.Mortality[J].Journal of the American Statistical Association,1992,87(419):659-671.
[15] Li J,Haberman S.On the Effectiveness of Natural Hedging for Insurance Companies and Pension Plans[J].Insurance:Mathematics and Economics,2015,61(2):286-297.
[16] Li J S-H,Hardy M R,Tan K S,Threshold Life Tables and Their Applications[J].North American Actuarial Journal,2008,12(1):99-115.
[17] Luciano E,Regis L.Efficient Versus Inefficient Hedging Strategies in the Presence of Financial and Longevity (Value at) Risk[J].Insurance:Mathematics and Economics,2014,55(1):68-77.
[18] Milevsky M A,Promislow S D.Mortality Derivatives and the Option to Annuities[J].Insurance:Mathematics and Economics,2001,29(3):299-318.
[19] Ngai A,Sherris M.Longevity Risk Management for Life and Variable Annuities:The Effectiveness of Static Hedging Using Longevity Bonds and Derivatives[J].Insurance:Mathematics and Economics,2011,49(1):100-114.
[20] Tan C I,Li J,Li J S-H,Balasooriya U.Parametric Mortality Indexes:From Index Construction to Hedging Strategies[J].Insurance:Mathematics and Economics,2014,59(2):285-299.
[21] Wang C W,Huang H C,Hong D C.A Feasible Natural Hedging Strategy for Insurance Companies[J].Insurance:Mathematics and Economics,2013,52(3):532-541.
① 长寿风险可细分为个体长寿风险和聚合长寿风险。其中,个体长寿风险是可以分散的非系统性风险或特定风险,而聚合长寿风险则是无法根据大数法则进行分散的系统性风险。在实际中,我们并不畏惧个体长寿风险,而担心或害怕聚合长寿风险对经济社会发展的影响。如无特别说明,本文中的长寿风险均指聚合长寿风险。
② 由于对冲弹性是个相对概念,不受死亡率样本数据的影响,本文后续部分进一步对基于普查及抽样调查数据、寿险业参保人群经验数据计算的对冲弹性的可比性进行解释说明。年金三类产品组合的对冲弹性的动态演变规律,并通过利率敏感性分析扩展考虑利率变化对对冲弹性的影响。这些研究基于性别、投保年龄、保险期限、递延期限和利率假设等多层次视角,对中国寿险业承担的长寿风险进行了较全面的量化分析。本文的主要贡献在于,第一,在长寿风险量化与管理中,找到了基于人口普查及抽样调查死亡数据与寿险业参保人群死亡经验数据可比性的方法,通过两类数据计算的对冲弹性具有可比性。对冲弹性的值越负,对冲效应越明显,从而长寿风险越不显著。第二,重新界定了长寿风险对寿险和年金产品定价的双重影响,丰富了国内学者对长寿风险的性别差异的研究。研究表明,由于保险公司在产品定价时区分了死亡率的性别差异,使得女性的对冲效应更明显,从而女性对应的产品组合中的长寿风险对保险公司的影响更不显著。然而,基本养老保险体系在缴费筹资模式和待遇领取模式的设计中都未严格区分死亡率的性别差异,且没有相应的保障型寿险产品与养老型年金产品对冲,通常面临的女性参保人群的长寿风险更显著。第三,通过对寿险和年金产品进行分组,得出终身寿险与终身年金组的对冲效应最强,两全保险与定期年金组的对冲效应次之,递延寿险与递延年金组的对冲效应最弱。这为保险公司提供了一套利用不同产品组合内部的对冲效应量化自身长寿风险的方法。第四,通过考虑利率变化对对冲效应的净影响,诠释了利率风险和长寿风险两大系统性风险的相互作用,即利率上升能抵消或对冲长寿风险的影响。
(1)门限年龄为建模中考虑的超高龄人口年龄的起点。较早研究中年龄分段点是人为选取的,为了克服分段点选取的主观性,近年研究中则设定了多个可选择的分段点,也称为门限年龄(Threshold Age),在估计模型参数过程中,通过反复试验方法(Tatonnement Approach)来选取合适的门限年龄(即最优门限年龄)以确保最优分段形式的模型结构,保证在最优门限年龄下模型整体拟合效果最好。本文沿用近年研究中的做法,即假设门限年龄是可选择的,而最优门限年龄则可以通过统计方法唯一确定。
(2)除0岁外,通常假设分数年龄服从均匀分布,即α=0.5,而α的取值通常位于[0.06,0.3],故可将α视为已知量。
(3)这里,超参数包括固定效应参数q、随机效应变量的分布参数mx,t和qx,t、模型误差项的分布参数μx,t和x。这些超参数通过极大似然估计或相关的优化技术来估计。
(4)本文在构建参数t的ARIMA模型时,采用的是参数θt的随机效应估计值θt-μθ组成的时间序列,在最后的预测结果中,再加上参数qx,t的固定效应估计值x∈[0,ωt]。
(5)本文后续部分将进一步介绍对冲弹性的利率敏感性分析。
(6)为了评价最优模型对样本外数据的预期精度,本文通过比较最优模型预测的2011～2013年各年龄的条件死亡概率与利用2012～2014年《中国人口和就业统计年鉴》中的死亡数据计算的0～89岁的条件死亡概率,得出最优模型的短期预测能力相当好。
(7)实际上,保险产品的价格由净保费和附加保费两部分组成,其中附加保费包括费用附加和利润附加。由于附加保费因公司而异,故本文在量化长寿风险对保险产品定价的影响中暂时不考虑附加保费,直接采用对净保费的影响来代替对定价的影响。
(8)如式(6)～式(8)所示,这里绘制的对冲弹性的动态演变是指1995～2030年寿险与年金产品净保费相对于基期1994年净保费的变化率之比随时间的变化趋势。由于目前保险行业仍采用固定死亡率精算假设为各种保险产品进行定价,这种相对于基期的量化方法比相对于上一期的量化方法更适宜刻画实际中寿险产品与年金产品的对冲效应。故这里采用相对于基期的量化方法。
(9)该结论也可以从第二部分对式(8)的解释中得到验证。