基于死亡率免疫理论的自然对冲有效性评估

为应对长寿风险对年金产品的影响,本文提出分段对冲策略,并以死亡率免疫和死亡率久期规则为理论基础探讨该策略的有效性问题。为避免传统久期匹配方法中参数估计误差的累积和传导,借助WinBUGS软件和贝叶斯Markov Chain Monte Carlo方法,在统一的计算框架下完成了死亡率预测、死亡率久期计算和对冲效果的数值模拟;并以4种分段组合准备金数据的三维图、方差缩减比(VRR)和VaR值为指标进行长寿风险对冲有效性的对比,结果表明低年龄寿险保单和高年龄年金保单组合具有最平滑的三维图,最小的VRR和VaR值,可明显提高长寿风险自然对冲的有效性。     

基于贝叶斯MCMC方法的年金长寿风险度量研究

长寿风险的准确度量是年金长寿风险管理的基础和前提，具有一定的学术和实际意义。通过利用贝叶斯MCMC（马尔科夫链蒙特卡洛模拟）算法统筹死亡率预测的Lee-Carter模型和利率预测的CIR模型，将长寿风险的度量转化成长寿期权的定价问题，考查了年金中长寿风险的变动规律和影响因素。MCMC抽样和数值模拟的结果表明，年金中的长寿风险与预测年份和年金持有人年龄成正向关系，其在年金中所占的比重随着年金持有人年龄的增加而增加，60岁的终身生存年金中长寿风险的占比高达10.11%；同时长寿风险与利率成反向关系，当前的低利率环境将会给年金发行人造成更高的长寿风险压力。     

个人年金产品中蕴含的长寿风险研究

本文基于随机死亡率预测,并将年金合同定价问题与年金保单组的破产概率相结合,对我国年金业务中蕴含的长寿风险进行了实证研究。一方面,在死亡率预测的基础上,研究了即期年金保单组未来现金流的分布特征,探讨了保单规模和性别对长寿风险的影响;另一方面,在考虑长寿风险条件下,测算了即期年金保单组的未来现金流,讨论了长寿风险对保单组破产概率和破产时间的影响,以及对冲长寿风险时对资产回报率要求。     

基于双因素Wang转换方法的长寿风险债券定价研究

在比较国外经典债券设计的基础上,基于离散型死亡率模型假设,设计一种可调整上触碰点的触发型长寿债券,运用带永久跳跃的APC模型和双因素Wang转换定价方法对长寿债券进行定价,实证结果表明:在不同的参数组合下的风险溢价均处在一个合理的范围,由于模型参数多、可用死亡率数据年限短,风险溢价的结果对无风险利率等参数敏感性较高.     

高龄人口死亡率预测模型的比较与选择

高龄人口死亡率预测模型是人口预测、养老金成本和债务评估以及长寿风险度量与管理的基础。我国大陆地区高龄人口死亡数据量少、数据波动性大,如何选择适合我国高龄数据特点的死亡率预测模型,是重要的研究课题。本文在归纳总结死亡率预测模型研究进展的基础上,先采用数据较为充分的台湾地区高龄死亡数据,选用Lee-Carter、CBD、贝叶斯分层模型等八种死亡率模型,对模型的拟合效果、预测效果和稳健性做出比较。在此基础上,基于修正和平滑后的我国大陆人口死亡数据,采用CBD模型和贝叶斯分层模型建模和预测。结果显示:贝叶斯分层模型能捕捉我国大陆高龄死亡率数据的历史波动,预测区间能够涵盖全部死亡率的真实值,但预测区间过宽,生存曲线不收敛;相比之下,CBD模型对我国大陆地区高龄死亡率的拟合和预测较好,预测区间和生存曲线合理。在长寿风险度量中,建议采用CBD模型。     

长寿风险与年金保险研究

随着经济发展、科学技术进步、医疗卫生水平提升和生活方式的转变,人类的寿命延长已成为一种必然趋势。长寿对人类固然是件好事,但它对各种养老保险产品产生的影响却不容小视,尤其是对年金产品造成了巨大的偿付压力。本文借鉴国际上相关领域的优秀研究成果,运用Lee-Carter模型和年金精算模型对我国面临的长寿风险进行了定量分析,进而提出现阶段我国保险业应对长寿风险的最佳路径。     

动态死亡率下个人年金的长寿风险分析

传统的精算定价方法假定死亡率是静态的,实际上死亡率是随时间而变动的具有动态不确定性的变量。在动态死亡率的框架下定量分析长寿风险对于个人年金产品定价的影响:引入Wang转换的风险定价方法度量长寿风险的市场价格,并运用模拟分析的方法分析长寿风险对个人年金定价的影响。最后,基于分析结果,就保险公司如何管理这一风险给出建议。     

长寿风险对寿险和年金产品定价的对冲效应研究

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。     

死亡率分解模型及其在长寿分级基金构建中的应用

论文提出了一种有别于传统死亡率模型的新的长寿风险度量模型,叫做死亡率分解模型。基于该模型,论文对美国的死亡率数据进行了分析和对比,同时对中国的死亡率进行了两个角度的分析,并给出了相对更有说服力的结果。同时,论文借助模型的分解结果,提出构建多层次的长寿风险基金,用以应对中国社会日益严重的养老问题。     

基于投资约束条件的企业年金最优投资组合研究

利用马科维兹的均值-方差模型,在目前我国企业年金投资约束条件的前提下测算四种主要投资工具在不同投资组合下的收益率和风险。研究表明:在现有年金投资约束条件下,从企业年金投资组合的安全性和收益率出发,国债和企业债券两者之间企业债券更优;两种以上投资工具的组合更能分散风险;投资组合中必须要选择股票投资;股票投资占比越高,企业年金投资组合的收益率越大。     

The CBD Mortality Indexes: Modeling and Applications

Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.     

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly. A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.     

Biometric Solvency Risk for Portfolios of General Life Contracts. I. The Single-Life Multiple Decrement Case

Abstract

Solvency II splits life insurance risk into seven risk classes consisting of three biometric risks (mortality risk, longevity risk, and disability/morbidity risk) and four nonbiometric risks (lapse risk, expense risk, revision risk, and catastrophe risk). The best estimate liabilities for the biometric risks are valued with biometric life tables (mortality and disability tables), while those of the nonbiometric risks require alternative valuation methods. The present study is restricted to biometric risks encountered in traditional single-life insurance contracts with multiple causes of decrement. Based on the results of quantitative impact studies, process risk was deemed to be not significant enough to warrant an explicit calculation. It was therefore assumed to be implicitly included in the systematic/parameter risk, resulting in a less complex standard formula. For the purpose of internal models and improved risk management, it appears important to capture separately or simultaneously all risk components of biometric risks. Besides its being of interest for its own sake, this leads to a better understanding of the standard approach and its application extent. Based on a total balance sheet approach we express the liability risk solvency capital of an insurance portfolio as value-at-risk and conditional value-at-risk of the prospective liability risk understood as random present value of future cash flows at a given time. The proposed approach is then applied to determine the biometric solvency capital for a portfolio of general life contracts. Using the conditional mean and variance of a portfolio’s prospective liability risk and a gamma distribution approximation we obtain simple solvency capital formulas as well as corresponding solvency capital ratios. To account for the possibility of systematic/parameter risk, we propose either to shift the biometric life tables or to apply a stochastic biometric model, which allows for random biometric rates. A numerical illustration for a cohort of immediate life annuities in arrears reveals the importance of process risk in the assessment of longevity risk solvency capital.     

Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.     

A partial internal model for longevity risk

This paper proposes a simple partial internal model for longevity risk within the Solvency 2 framework. The model is closely linked to the mechanisms associated with the so-called Danish longevity benchmark, where the underlying mortality intensity and the trend is estimated yearly based on mortality experience from the Danish life and pension insurance sector, and on current data from the entire Danish population. Within this model, we derive an estimate for the 99.5% percentile for longevity risk, which differs from the longevity stress of 20% from the standard model. The new stress explicitly reflects the risk associated with unexpected changes in the underlying population mortality intensity on a one-year horizon and with a 99.5% confidence level. In addition, the model contains a component, which quantifies the unsystematic longevity risk associated with a given insurance portfolio. This last component depends on the size of the specific portfolio.     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

Canonical Valuation of Mortality‐Linked Securities

A fundamental question in the study of mortality‐linked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This article develops a framework for pricing mortality‐linked securities on the basis of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortality‐linked security, and the results are compared against those derived from other methods.     

Lee-Carter模型在中国城市人口死亡率预测中的应用与改进

由死亡率下降带来的长寿风险给社会、政治以及经济带来了新的挑战。为了更加准确地对长寿风险进行评估和管理,需要对未来死亡率趋势进行预测。本文针对我国死亡率数据样本量小以及数据存在缺失的实际情况,对Lee-Carter模型进行了改进,通过一个双随机过程对Lee-Carter模型中的时间项进行建模。在模型中考虑了样本量不足对预测结果造成的影响,使得改进后的Lee-Carter模型更加适合目前中国的人口死亡率预测。