基于风险立方方法的长寿风险债券定价研究

面对世界范围内长寿风险越来越严峻的趋势,长寿风险管理成为全世界面临的共同难题。近年来死亡率风险证券化引起人们的广泛关注,长寿债券作为死亡率风险证券化中最常用的一种方法,可以有效地将长寿风险转移至资本市场。本文通过对国外经典死亡率债券的比较,在离散型死亡率模型假设条件下,设计一支可调整上触碰点的触发型长寿债券,运用带永久跳跃的APC模型和风险立方方法对长寿债券进行定价。实证结果显示风险溢价的结果比较稳定,设置不同的初始上触碰点,风险溢价差异较大。     

长寿债券的运行机制与定价模型

通过分析长寿债券的市场发展以及连续型和触发型两类长寿债券的运行机制,采用风险中性定价方法推导出当死亡率服从双指数跳跃(DEJD)分布时,长寿债券的定价解析式,研究发现,无论从理论还是实践看,设计并发行触发型长寿债券是一种应对长寿风险更为明智的选择。     

长寿风险证券化的理论研究动态

国际寿险业在资本市场上尝试进行长寿风险证券化的同时,学术界也不断在探索长寿风险资本市场的创新性解决方案,并已取得了丰硕的理论成果。本文阐述了关于连续型和触发型长寿债券的设计机制及其定价模型的研究成果;分析了长寿互换的设计机制和定价模型的研究进展;梳理了其他长寿风险金融衍生工具的设计机制和定价模型的研究动态。     

我国养老保险公司的长寿风险证券化——基于百分位分层定价法

长寿风险证券化可以把长寿风险转移到资本市场,有效管理养老保险公司的长寿风险。本文主要研究利息率与死亡率挂钩的逆生存债券,研究内容分为死亡率预测与债券定价两个部分。由于投资者对风险的偏好不同,本文利用百分位分层定价法对逆生存债券进行定价。数值分析展示了债券价格随利率变化的情况,共分四个层次,不同的分层对应不同的价格和风险,投资者可以根据自身的风险偏好选择不同的分层债券。最后就养老保险公司在长寿风险证券化方面存在的问题给出相关建议。     

我国随机死亡率的长寿风险建模和衍生品定价

长寿风险近年来对各国保险业、养老金体系、社会保障体系造成大规模影响,成为保险和风险管理学术界关注和研究的重点。采用国际前沿的研究方法,系统深入地采用中国数据研究这一问题。在Lee-Carter模型的基础上,通过双指数跳跃扩散模型对Lee-Carter模型中的时间序列因子进行拟合,较好地刻画了中国人口死亡率的长寿跳跃和死亡跳跃;引用Swiss Re死亡债券度量长寿风险的市场价格,预估未来中国人口死亡率,并得出了寿险衍生品Q型远期的中国定价。     

基于套利理论与ICIR模型的债券市场发行定价偏离研究

基于套利定价理论与利率期限结构理论,运用Tobit多元线性回归模型,得出债券发行定价的主要影响因素为债券无风险利率、债券期限溢价、债项信用评级、债券主体信用评级和债券赎回风险溢价,在此基础上再通过改进的CIR定价模型(ICIR)对2006～2010年各债券定价偏离现象进行研究的结果表明,在1%的显著性水平上,ICIR模型测算的债券理论价格通过了二级市场的定价检验,ICIR模型对债券发行定价偏离进行检验具有较强的合理性;同时,从发行年份来看,近五年来,债券定价偏离总体呈逐年下降趋势,债券发行定价与ICIR定价与二级市场定价逐步接轨,市场化程度越来越高。     

基于金融衍生工具视角的长寿风险管理

长寿风险已成为养老保障发展所面临的重要风险,而作为养老保障产品供给者的政府、年金和寿险公司等机构难以持续、有效地管理长寿风险。本文在分析长寿风险发展态势和现有管理方案的缺陷后,研究了最近的长寿风险管理工具创新及其发展动向,即死亡率巨灾债券、EIB/BNP长寿债券和远期等,并在此基础上分析了基于资本市场的长寿/死亡率风险相关衍生品设计与交易,包括长寿债券、死亡率互换、死亡率期货和死亡率期权,最后是长寿/死亡率衍生品交易市场建设的启示。     

优先级债券与次级债券风险溢价差的理论分析——基于期权定价理论的应用

国内运用期权定价理论分析债券的定价主要集中于对可转换债券、单一零息票债券和违约风险债券的定价分析,但对于同时发行优先级债券和次级债券二者的收益率差研究较少。在国外已有的次级债券期权定价方法和结论基础上,文章具体论证出次级债券相对于无风险国债的风险溢价与发行主体资产价值之间呈非单调关系的结论,并得到次级债券与优先级债券收益率差的定价公式。     

极端死亡率风险证券化的理论研究综述

国际寿险业和理论界在不断地研究探索极端死亡率风险资本市场的创新性解决方案,已取得丰硕的理论成果。本文阐述了本金赔付非累积阈值型和累积阈值型两类极端死亡率债券的设计机制及其定价方面的研究成果;分析了极端死亡率互换的设计机制和定价方面的研究动态;概述了极端死亡率风险衍生工具q远期合约的设计机制和定价方面的研究进展。     

新型农村社会养老保险中的逆向选择问题研究

国际寿险业和理论界在不断地研究探索极端死亡率风险资本市场的创新性解决方案,已取得丰硕的理论成果。本文阐述了本金赔付非累积阈值型和累积阈值型两类极端死亡率债券的设计机制及其定价方面的研究成果;分析了极端死亡率互换的设计机制和定价方面的研究动态;概述了极端死亡率风险衍生工具q远期合约的设计机制和定价方面的研究进展。     

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.     

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.     

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.     

Longevity/Mortality Risk Modeling and Securities Pricing

Securitizing longevity/mortality risk can transfer longevity/mortality risk to capital markets. Modeling and forecasting mortality rate is key to pricing mortality‐linked securities. Catastrophic mortality and longevity jumps occur in historical data and have an important impact on security pricing. This article introduces a stochastic diffusion model with a double‐exponential jump diffusion process that captures both asymmetric rate jumps up and down and also cohort effect in mortality trends. The model exhibits calibration advantages and mathematical tractability while better fitting the data. The model provides a closed‐form pricing solution for J.P. Morgan’s q‐forward contract usable as a building block for hedging.     

An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach

This article investigates the natural hedging strategy to deal with longevity risks for life insurance companies. We propose an immunization model that incorporates a stochastic mortality dynamic to calculate the optimal life insurance–annuity product mix ratio to hedge against longevity risks. We model the dynamic of the changes in future mortality using the well‐known Lee–Carter model and discuss the model risk issue by comparing the results between the Lee–Carter and Cairns–Blake–Dowd models. On the basis of the mortality experience and insurance products in the United States, we demonstrate that the proposed model can lead to an optimal product mix and effectively reduce longevity risks for life insurance companies.     

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.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

Forecasting Longevity Gains for a Population with Short Time Series Using a Structural SUTSE Model: An Application to Brazilian Annuity Plans

In this article, a multivariate structural time series model with common stochastic trends is proposed to forecast longevity gains of a population with a short time series of observed mortality rates, using the information of a related population for which longer mortality time series exist. The state space model proposed here makes use of the seemingly unrelated time series equation and applies the concepts of related series and common trends to construct a proper model to predict the future mortality rates of a population with little available information. This common trends approach works by assuming the two populations’ mortality rates are affected by common factors. Further, we show how this model can be used by insurers and pension funds to forecast mortality rates of policyholders and beneficiaries. We apply the proposed model to Brazilian annuity plans where life expectancies and their temporal evolution are predicted using the forecast longevity gains. Finally, to demonstrate how the model can be used in actuarial practice, the best estimate of the liabilities and the capital based on underwriting risk are estimated by means of Monte Carlo simulation. The idiosyncratic risk effect in the process of calculating an amount of underwriting capital is also illustrated using that simulation.