我国有限人口数据下长寿衍生产品定价的贝叶斯方法

鉴于我国人口死亡率数据匮乏对长寿风险定价的不利影响,文章采用贝叶斯MCMC方法来进行长寿衍生产品的定价.来自实际人口数据的研究结果表明,贝叶斯方法通过Gibbs抽样和MCMC模拟,更好地考虑了样本不足和样本质量问题,在死亡率建模的模型BIC值,残差方差值和预测稳健性上全面优越于传统方法,能有效提高死亡率预测的精度;同时,贝叶斯一体化框架结合最大熵原理能大幅减少定价过程中数据和参数风险的产生,累积和传导,提高长寿衍生产品定价结果的有效性和可靠性.其方法的优越性对保障我国有限人口数据下长寿衍生产品的成功开发具有积极的理论意义和现实价值.     

基于协整理论的中国人口死亡率预测

近年来,人类寿命明显延长.长寿风险对于国家养老金制度,保险公司寿险业务的影响日益凸现.长寿风险源于人口死亡率的非预期变动,精准预测人口死亡率是长寿风险研究的一项重要内容.文中提出了一种死亡率预测的新方法,将计量经济学中的协整理论引入死亡率预测,以弥补中国死亡率历史数据缺乏,并结合极值理论方法给出中国死亡率的预测.     

基于AE-LSTM改进模型的高龄人口死亡率预测

高龄人口死亡率预测是长寿风险度量和管理、养老金成本和债务评估的基础.基于高龄人口死亡率数据特征,本文建立一个AE-LSTM改进模型对高龄人口死亡率进行预测.首先利用AE模型从高龄人口死亡率数据提取潜在时间因子,把它作为LSTM模型的输入变量,然后通过解码得到高龄人口死亡率预测值.同时,选取我国大陆1994–2018年60–89岁高龄人口死亡率作为样本数据进行实证分析.研究结果表明,AE-LSTM改进模型较传统的人口死亡率CBD模型预测精度有显著提高,且预测结果呈现较强鲁棒性.     

多总体死亡率差异风险度量——以ILS债券为例

在回顾多总体动态死亡率预测模型研究成果的基础上,简要评述了已有模型的适应情况和假设条件,并依此构建了死亡率差异风险的度量模型.此后,并以ILS债券为例,利用HMD数据库中英国和美国人口死亡率数据,使用构建的死亡率差异风险度量模型,测量了ILS债券中的死亡率差异风险.定量分析结果显示:ILS为投资者设定了较高的安全阀值,保障了ILS的成功发行.     

基于Lee-Carter模型的互助养老年金研究

基于经典的双线性随机Lee-Carter模型,采用经济学的协整理论,对中国大陆男性人口死亡率进行预测,克服了ARIMA模型预测的局限性.在随机利率和Lee-Carter模型的基础上度量退休年金和生命年金的长寿风险,并为此提出应对策略,引入由消费者承担系统长寿风险、年金池承担个体长寿风险的群体自助养老年金(GSA),然后对其进行实证分析发现,与普通年金相比,GSA模型分担模式拥有较高的给付额.     

基于DEJD模型的人口寿命预测及SM债券定价

人口老龄化背景下的长寿风险,将会给国家养老保障体系带来极大的经济负担.如何度量和管理长寿风险,己成为近年来世界各国关注和研究的焦点.本文基于我国人口死亡率数据,在Lee-Carter模型的基础上,引入DEJD模型刻画时间序列因子的跳跃不对称性,并证实了 DEJD模型比Lee-Carter模型在拟合时间序列因子时更为有效...     

C-ROSS中长寿风险资本要求

为了应对长寿风险,保险公司需要对风险规模有清晰的认识.迄今为止,国内已有文献主要使用内部模型,针对中国保险公司的长寿风险做了度量.即将实施的C-ROSS为我国长寿风险度量提供了第一个标准模型,将是长寿风险度量的重要参考和强制标准.因此将在C-ROSS标准模型技术细节探讨的基础上,计算相应的长寿风险官方要求.根据文献梳理和究显示:长寿风险会给整个年金支付现值带来2~6%支付增加,其中由波动性长寿风险引发的支付增加为1.6~4%,其中(监管要求的)趋势性长寿风险引发的支付增加为1~3%.此外,与欧盟SolvencyⅡ相比,C-ROSS充分考虑了中国人口死亡率改善特点和未来发展趋势,在资本约束较强的背景下,设定了一个审慎、简洁的长寿风险资本要求.     

中国人口死亡率建模比较及长寿风险度量

以我国颁布的3套保险行业经验生命表为基础，结合1995-2017年国家统计局发布的《中国统计年鉴》中的死亡率数据，首先分析了中国全年龄人口数据死亡率动静态变动特点，其次比较了LC，CBD和APC 3种模型对中国死亡率数据的拟合优劣，最后采用最优APC模型度量了不同生命表下的长寿风险.死亡率的动态变化会导致以经验生命表为依据的年金产品定价出现偏差，增加养老金管理机构的承保风险.     

基于生命周期和交叠世代分析的养老风险管理文献综述及其启示

随着长寿风险的加剧以及人口死亡率的降低，养老风险管理的研究逐渐受到政府和学术界的广泛关注。本文系统梳理了养老风险相关驱动因素(死亡率和长寿风险)的研究，并介绍了生命周期框架与交叠世代模型(OLG)的建模过程，还对生命周期框架和OLG模型的应用研究进行了文献综述。从文献回顾来看，当前研究存在一些不足，未来的研究应基于我国人口生存特征去研究死亡率；同时，需要从生命周期框架和OLG模型研究研究应对养老风险的稳健的投资-消费决策，并在考虑对上一代老人赡养的情形下研究寿险决策。此外，随着养老风险逐步加剧，建议政府采取改进我国的多支柱养老保险体系、进一步放开生育、延迟退休及发展养老服务业等方式应对养老风险，保证我国养老金体系的可持续发展。     

基于Copula-AR(n)-LSV的死亡率建模及长寿风险度量

合理的死亡率模型是精准度量长寿风险的关键.考虑不同年龄组间死亡率的相依性以及各年龄组死亡率的自相关性和异方差结构,运用多元Copula和AR(n)-LSV模型构建了随机动态死亡率模型,并在此基础上进一步运用VaR、TVaR、GlueVaR对长寿风险进行测度.研究结果表明Copula-AR(n)-LSV模型比Lee-Ca...     

Longevity bond premiums: The extreme value approach and risk cubic pricing

The purpose of this study is to analyze the securitization of longevity risk with an emphasis on longevity risk modeling and longevity bond premium pricing. Various longevity derivatives have been proposed, and the capital market has experienced one unsuccessful attempt by the European Investment Bank (EIB) in 2004. After carefully analyzing the pros and cons of previous securitizations, we present our proposed longevity bonds, whose payoffs are structured as a series of put option spreads. We utilize a random walk model with drift to fit small variations of mortality improvements and employ extreme value theory to model rare longevity events. Our method is a new approach in longevity risk securitization, which has the advantage of both capturing mortality improvements within sample and extrapolating rare, out-of- sample longevity events. We demonstrate that the risk cubic model developed for pricing catastrophe bonds can be applied to mortality and longevity bond pricing and use the model to calculate risk premiums for longevity bonds.     

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.     

Heath–Jarrow–Morton modelling of longevity bonds and the risk minimization of life insurance portfolios

This paper has two parts. In the first, we apply the Heath–Jarrow–Morton (HJM) methodology to the modelling of longevity bond prices. The idea of using the HJM methodology is not new. We can cite Cairns et al. [Cairns A.J., Blake D., Dowd K, 2006. Pricing death: framework for the valuation and the securitization of mortality risk. Astin Bull., 36 (1), 79–120], Miltersen and Persson [Miltersen K.R., Persson S.A., 2005. Is mortality dead? Stochastic force of mortality determined by arbitrage? Working Paper, University of Bergen] and Bauer [Bauer D., 2006. An arbitrage-free family of longevity bonds. Working Paper, Ulm University]. Unfortunately, none of these papers properly defines the prices of the longevity bonds they are supposed to be studying. Accordingly, the main contribution of this section is to describe a coherent theoretical setting in which we can properly define these longevity bond prices. A second objective of this section is to describe a more realistic longevity bonds market model than in previous papers. In particular, we introduce an additional effect of the actual mortality on the longevity bond prices, that does not appear in the literature. We also study multiple term structures of longevity bonds instead of the usual single term structure. In this framework, we derive a no-arbitrage condition for the longevity bond financial market. We also discuss the links between such HJM based models and the intensity models for longevity bonds such as those of Dahl [Dahl M., 2004. Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts, Insurance: Math. Econom. 35 (1) 113–136], Biffis [Biffis E., 2005. Affine processes for dynamic mortality and actuarial valuations. Insurance: Math. Econom. 37, 443–468], Luciano and Vigna [Luciano E. and Vigna E., 2005. Non mean reverting affine processes for stochastic mortality. ICER working paper], Schrager [Schrager D.F., 2006. Affine stochastic mortality. Insurance: Math. Econom. 38, 81–97] and Hainaut and Devolder [Hainaut D., Devolder P., 2007. Mortality modelling with Lévy processes. Insurance: Math. Econom. (in press)], and suggest the standard pricing formula of these intensity models could be extended to more general settings.In the second part of this paper, we study the asset allocation problem of pure endowment and annuity portfolios. In order to solve this problem, we study the “risk-minimizing” strategies of such portfolios, when some but not all longevity bonds are available for trading. In this way, we introduce different basis risks.     

Securitization, structuring and pricing of longevity risk

Pricing and risk management for longevity risk have increasingly become major challenges for life insurers and pension funds around the world. Risk transfer to financial markets, with their major capacity for efficient risk pooling, is an area of significant development for a successful longevity product market. The structuring and pricing of longevity risk using modern securitization methods, common in financial markets, have yet to be successfully implemented for longevity risk management. There are many issues that remain unresolved for ensuring the successful development of a longevity risk market. This paper considers the securitization of longevity risk focusing on the structuring and pricing of a longevity bond using techniques developed for the financial markets, particularly for mortgages and credit risk. A model based on Australian mortality data and calibrated to insurance risk linked market data is used to assess the structure and market consistent pricing of a longevity bond. Age dependence in the securitized risks is shown to be a critical factor in structuring and pricing longevity linked securitizations.     

On the pricing of longevity-linked securities

For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products, so-called longevity derivatives, may be needed, even though a first attempt to issue a longevity bond in 2004 was not successful.While different methods of how to price such securities have been proposed in recent literature, no consensus has been reached. This paper reviews, compares and comments on these different approaches. In particular, we use data from the United Kingdom to derive prices for the proposed first longevity bond and an alternative security design based on the different methods.     

The role of longevity bonds in optimal portfolios

We study the optimal consumption and portfolio for an agent maximizing the expected utility of his intertemporal consumption in a financial market with: (i) a riskless asset, (ii) a stock, (iii) a bond as a derivative on the stochastic interest rate, and (iv) a longevity bond whose coupons are proportional to the population (stochastic) survival rate. With a force of mortality instantaneously uncorrelated with the interest rate (but not necessarily independent), we demonstrate that the wealth invested in the longevity bond must be taken from the ordinary bond and the riskless asset proportionally to the duration of the two bonds. This result is valid for both a complete and an incomplete financial market.     

Efficient versus inefficient hedging strategies in the presence of financial and longevity (value at) risk

This paper provides a closed-form Value-at-Risk (VaR) for the net exposure of an annuity provider, taking into account both mortality and interest-rate risk, on both assets and liabilities. It builds a classical risk-return frontier and shows that hedging strategies–such as the transfer of longevity risk–may increase the overall risk while decreasing expected returns, thus resulting in inefficient outcomes. Once calibrated to the 2010 UK longevity and bond market, the model gives conditions under which hedging policies become inefficient.     

On the robustness of longevity risk pricing

For longevity bond pricing, the most popular methods contain the risk-neutral method, the Wang transform and the Sharpe ratio rule. This paper studies robustness of these three methods and investigates connections and differences among them through theoretic analysis and numerical illustrations. We adopt the dynamic mortality models with jumps to capture the permanent effects caused by unexpected factors and allow the correlation between mortality and interest rate be nonzero. The analysis is based on four typical mortality models, including the mean-reverting models and the non mean-reverting ones. Our work may provide a guidance for participants on choice of pricing methods.     

A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions

We present a Bayesian approach to pricing longevity risk under the framework of the Lee-Carter methodology. Specifically, we propose a Bayesian method for pricing the survivor bond and the related survivor swap designed by Denuit et al. (2007). Our method is based on the risk neutralization of the predictive distribution of future survival rates using the entropy maximization principle discussed by Stutzer (1996). The method is illustrated by applying it to Japanese mortality rates.