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。     

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。     

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。     

我国个人年金长寿风险的资本要求度量

随着人口平均寿命的延长,系统性的长寿风险正在逐步成为理论界和保险业关注的热点。然而,以往研究大多止步于对死亡率的期望估计,很难满足实践中对长寿风险资本需求的度量。本文采用Bootstrap方法将研究拓展到死亡率的分布上,并以此为基础计算年金保单组现值的分布,度量年金保单组长寿风险的风险价值及其资本要求。结果表明,虽然经营年金业务保险公司为了应对长寿风险需要额外的资本准备,但中短期内这种资本要求并不高,保险公司可以通过增收保费和提高资本市场收益来达到相应的要求。     

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。     

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。     

随机利率下的长寿风险自然对冲研究

死亡率降低和居民寿命的提高是社会发展的一大进步,但是超出预期水平的死亡率改善也带来了一定的负面影响,如养老金机构可能会因为死亡率的降低而收支不抵最终破产等,我们称之为长寿风险。因此,长寿风险也越来越引起学者们的关注。关于长寿风险的基本管理方法有两种,一种是通过死亡率相关的衍生品将长寿风险向资本市场分散,另外一种,是利用寿险产品和年金产品性质上的差异来进行对冲的策略,我们称为自然对冲。文章采用随机死亡率的模型,并分别在固定利率和随机利率的框架下,分析我国的养老金组织将来面临的长寿风险并给出相应自然对冲策略。     

动态死亡率建模与长寿风险量化研究评述

首先,系统梳理了国内外已有的动态死亡率建模方法存在的问题,提出了改进的思路,总结了量化长寿风险的各种模型与方法。其次,结合国外长寿风险管理工具的应用及发展,主要包括保险公司年金产品与寿险产品的自然对冲、再保险、长寿风险证券化的理论研究与实践经验,探讨其在我国实施的可行性及实施中可能存在的问题。该研究可作为我国长寿风险量化与管理的基础研究,有望为我国长寿风险定量评估体系的构建提供参考与借鉴。     

动态死亡率下寿险公司的准备金风险分析

寿险产品定价方法一般采用静态的死亡率,没有考虑死亡率改善对保障类产品或年金类产品的影响,这样可能导致寿险公司的准备金估计过高或更低,进而影响寿险公司的经营。本文首先借助Monte Carlo方法模拟静态死亡率和动态死亡率下寿险公司的责任准备金分布,然后比较两种情况下寿险公司责任准备金风险的变化情况。结果表明,死亡率改善对寿险公司责任准备金的影响显著,建议寿险公司在产品设计时考虑死亡率改善因素。     

The Political Economy of Government-Issued Longevity Bonds

This article explores the trade‐offs associated with government issuance of longevity bonds as a way of stimulating private annuity supply in the presence of aggregate mortality risk. We provide new calculations suggesting a 5 percent chance that aggregate mortality risk could ex post raise annuity costs for private insurers by as much as 5–10 percentage points, with the most likely effect based on historical patterns toward the lower end of that range. While we suspect that aggregate mortality risk does exert some upward pressure on annuity prices, evidence from private market pricing suggests that, to the extent that private insurers are accurately pricing this risk, the effect is less than 5 percentage points. We discuss ways that the private market can spread this risk, while emphasizing that the government has the unique ability to spread aggregate risk across generations. We note factors that might hamper such an efficient allocation of risk, including potential political incentives for the government to shift more than the optimal amount of risk onto future generations, and the possibility that government fiscal policy might allocate risk less efficiently within each generation than would private markets. We also discuss how large‐scale longevity bond issuance might affect government borrowing costs, as well as political economy aspects of how the proceeds from such a bond issuance might be used.     

Basis risk in static versus dynamic longevity-risk hedging

This paper provides a tractable, parsimonious model for assessing basis risk in longevity and its effect on the hedging strategies of Pension Funds and annuity providers. Basis risk is captured by a single parameter, that measures the co-movement between the portfolio and the reference population’s longevity. The paper sets out the static, full and customized swap-hedge for an annuity, and compares it with a dynamic, partial, and index-based hedge. We calibrate our model to the UK and Scottish populations. The effectiveness of static versus dynamic strategies depends on the rebalancing frequency of the second, on the relative costs, and on basis risk, which does not affect fully-customized, static hedges. We show that appropriately calibrated dynamic hedging strategies can still be reasonably effective, even at low rebalancing frequencies.     

“A Direct Approach to the Discounted Penalty Function”, Hansjörg Albrecher,Hans U. Gerber,and Hailiang Yang,Volume 14, No. 4, 2010

Abstract

Mortality improvements, especially of the elderly, have been a common phenomenon since the end of World War II. The longevity risk becomes a major concern in many countries because of underestimating the scale and speed of prolonged life. In this study we explore the increasing life expectancy by examining the basic properties of survival curves. Specifically, we check if there are signs of mortality compression (i.e., rectangularization of the survival curve) and evaluate what it means to designing annuity products. Based on the raw mortality rates, we propose an approach to verify if there is mortality compression. We then apply the proposed method to the mortality rates of Japan, Sweden, and the United States, using the Human Mortality Database. Unlike previous results using the graduated mortality rates, we found no obvious signs that mortality improvements are slowing down. This indicates that human longevity is likely to increase, and longevity risk should be seriously considered in pricing annuity products.     

A General Semi-Markov Model for Coupled Lifetimes

Joint-life annuities with a high last survivor benefit play an important role in the optimal annuity portfolio for a retired couple. The dependence between coupled lifetimes is crucial for valuing joint-life annuities. Existing bivariate modeling of coupled lifetimes is based on outdated data with limited observation periods and does not take into account mortality improvement. In this article, we propose a transparent and dynamic framework for modeling coupled lifetime dependence caused by both marital status and common mortality improvement factors. Dependence due to marital status is captured by a semi-Markov joint life model. Dependence due to common mortality improvement, which represents the correlation between mortality improvement patterns of coupled lives, is incorporated by a two-population mortality improvement model. The proposed model is applied to pricing the longevity risk in last survivor annuities sold in the United States and the United Kingdom.     

Mortality regimes and longevity risk in a life annuity portfolio

This paper explores the presence of changes of trends or jumps in French mortality from 1947 to 2007, and assesses their implications on the longevity risk management of a life annuity portfolio. We accomplish this by extending the Poisson log-bilinear regression developed by Brouhns et al. (2002) with a regime-switching model. Estimation results show that French mortality is characterized by two distinct regimes. One refers to a strong uncertainty state, which corresponds to the longevity conditions observed during the decade following World War II. The second regime is related to the low volatility of longevity improvements observed during the last 30 years. We use these results to analyze the impact of mortality regimes on the longevity risk management of a life annuity portfolio. Simulation results suggest that the changes of trends in the mortality process have some implications for longevity risk management.     

Author’s Reply: Statistical Independence and Fractional Age Assumptions - Discussion by Cecil Nesbitt; Elias S. W. Shiu; Serena Tiong,January 1997

Abstract

This paper addresses the problem of the sharing of longevity risk between an annuity provider and a group of annuitants. An appropriate longevity index is designed in order to adapt the amount of the periodic payments in life annuity contracts. This accounts for unexpected longevity improvements experienced by a given reference population. The approach described in the present paper is in contrast with group self-annuitization, where annuitants bear their own risk. Here the annuitants bear only the nondiversifiable risk that the future mortality trend departs from that of the reference forecast. In that respect, the life annuities discussed in this paper are substitutes for reinsurance and securitization of longevity risk.     

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.     

Sharing Longevity Risk: Why Governments Should Issue Longevity Bonds

Government-issued longevity bonds would allow longevity risk to be shared efficiently and fairly between generations. In exchange for paying a longevity risk premium, the current generation of retirees can look to future generations to hedge their systematic longevity risk. Longevity bonds will lead to a more secure pension savings market, together with a more efficient annuity market. By issuing longevity bonds, governments can aid the establishment of reliable longevity indices and key price points on the longevity risk term structure and help the emerging capital market in longevity-linked instruments to build on this term structure with liquid longevity derivatives.     

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.     

Application of Relational Models in Mortality Immunization

The prediction of future mortality rates by any existing mortality models is hardly exact, which causes an exposure to mortality (longevity) risk for life insurers (annuity providers). Since a change in mortality rates has opposite impacts on the surpluses of life insurance and annuity, hedging strategies of mortality and longevity risks can be implemented by creating an insurance portfolio of both life insurance and annuity products. In this article, we apply relational models to capture the mortality movements by assuming that the realized mortality sequence is a proportional change and/or a constant shift of the expected one, and the size of the changes varies in the length of the sequences. Then we create a variety of non-size-free matching strategies to determine the weights of life insurance and annuity products in an insurance portfolio for mortality immunization, where the weights depend on the sizes of the proportional and/or constant changes. Comparing the hedging performances of four non-size-free matching strategies with corresponding size-free ones proposed by Lin and Tsai, we demonstrate with simulation illustrations that the non-size-free matching strategies can hedge against mortality and longevity risks more effectively than the size-free ones.