调整未成年人死亡概率及实证分析

由于我国人口普查存在着漏报现象,致使各年龄的人口总数与实际存在一定程度的不吻合,尤其是未成年人(0-14岁),人口数量存在的问题更为严重,从而由此推算的未成年人的年龄别死亡概率qx,c与实际不符。针对此问题,本文建立了一个修正后的Gompertz生存模型,并利用这一修正后的生存模型对我国普查所得的年龄别死亡概率进行调整,为准确估计未成年人年龄别死亡概率提供一个新的方法。     

基于极值理论高龄死亡率模型的比较研究

基于极值理论模型,对中国与日本高龄人口死亡率进行拟合和预测,克服了其他死亡率参数外推模型的主观性.在极值理论高龄死亡率模型的基础上使用加权最小二乘法,通过反复试验方式选择最优门限年龄和模型参数估计值,并且预测中国与日本人口最高年龄以及最高年龄区间估计.此研究为我国经验生命表的编制工作提供借鉴.     

几种结构非概率可靠性模型的比较研究

相对概率可靠性模型和模糊可靠性模型,基于区间分析的结构非概率可靠性模型对数据的要求低,因此在实际工程中对非概率可靠性模型的研究越来越重要.近年来,非概率可靠性理论得到了很好的发展和完善.文中综述了已有的4种主要的非概率可靠性模型,针对线性结构功能函数,分别从度量原理、可靠性指标物理意义、适用范围和结果精度等方面对各可靠性模型进行比较与总结;针对非线性结构功能函数,对各可靠性模型的适用性进行了初步的讨论,从而对非概率可靠性模型有更加全面和深刻的理解,为实际工程中非概率可靠性模型的选取提供重要的理论依据.     

具有相依结构离散时间模型破产概率的上界

研究两类具有相依结构的离散时间风险模型的破产概率问题.其中,索赔和利率过程假设为2个不同的自回归移动平均模型.利用更新递归技巧,首先得到了该模型下破产概率所满足的递归方程.然后,根据该递归方程得到了破产概率的上界估计.最后对两类风险模型的破产概率的上界进行了比较.     

相依索赔的二项风险模型的破产问题

考虑一类相依索赔的二项风险模型,根据索赔额的大小随机产生一副索赔.通过引入辅助模型,研究相应模型生存概率的母函数,对任意的初始值u,得到了有限时间内生存概率的递推解,并结合保险实例进行了数值模拟.在某些特殊情形下得到有限时间生存概率和最终破产概率的明确表达式.     

极端损失条件下分组群体的主观概率累积偏差仿真研究

为了研究个体死亡这种极端损失条件下分组群体的主观概率累积偏差的形成与演化规律,构建不同分组方式下的群体主观概率元胞自动机模型并运用Netlogo仿真平台进行实验。在对仿真结果进行图形分析的基础上,进一步对实验数据进行了回归分析和案例验证分析。结果表明:分组群体的主观概率演化一般会经历从急速变化到趋于稳定的过程;分组作业方式能够提高安全冗余,但不利于个体事故经验积累;个体死亡等极端损失会导致群体的主观概率低估和分化,但分组措施能够降低此作用;客观事故概率的减少会增大群体内部主观概率的差异,但分组措施能够降低此作用;事故率与死亡率之间存在互补效应,分组措施对其存在双向调节作用;群体人数增加有助于减少主观概率偏差和差异。     

具有因病死亡且输人为Berverton-Holt函数的离散SIS传染病模型分析

引入相应的概率建立了考虑因病死亡且输入为Berverton-Holt的离散SIS传染病模型,确定了决定其动力性态的阈值,在阈值之下模型仅存在无病平衡点,且无病平衡点是全局渐近稳定的;在阈值之上模型是一致持续的,有唯一的地方病平衡点存在,且可以猜想地方病平衡点是全局渐近稳定的.     

静态迁徙下依赖于年龄的随机环境中分枝过程

引进了静态迁徙下依赖于年龄的随机环境中分枝过程的模型,给出了该模型的条件母函数的更新方程并考虑了特殊情形下的随机Kolmogorov方程.与此同时,通过研究更新方程得到了分枝过程的各阶矩,考虑了简单情形下的灭绝概率.最后给出了一个开问题.     

一类推广的带干扰风险模型的条件破产概率

本文首先利用随机时刻变换推广了一类带干扰的风险模型,然后讨论这类风险模型的条件破产概率.研究表明条件破产概率相对于无条件破产概率能提供更多有用信息,这对保险公司及时调整投资和管理策略是很有帮助的.     

缺失数据下线性模型的半参数估计

通过比较参数方法和非参数方法对选择概率建模的优缺点,基于充分降维的思想提出了一种利用单指标模型对选择概率建模的半参数方法.基于逆概率加权方法和半参数方法,研究了缺失数据下线性模型的统计推断问题.建立的逆概率加权估计方程可以处理不同的数据缺失情形,给出了线性模型中兴趣参数的估计,并证明了它的渐近正态性.最后通过模拟研究说明提出的方法具有较好的有限样本性质.     

Reviews on the Research of Dynamic Mortality Models

Mortality forecasting is the basis of population forecasting. In recent years, new progress has been made in mortality models. From the earliest static mortality models, mortality models have been developed into dynamic forecasting models including time terms, such as Lee-Carter model family, CBD model family and so on. This paper reviews and sorts out relevant literature on mortality forecasting models. With the development of dynamic models, some scholars have developed a series of mortality improvement models based on the level of mortality improvement. In addition, with the progress of mortality research, multi-population mortality modeling attracted the attention of researchers, and the multi-population forecasting models have been constantly developed and improved, which play an important role in the mortality forecasting. With the continuous enrichment and innovation of mortality model research methods, new statistical methods (such as machine learning) have been applied in mortality modeling, and the accuracy of fitting and prediction has been improved. In addition to the extension of classical modeling methods, issues such as small-area population or missing data of the population, the elderly population, the related population mortality modeling are still worth studying.     

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.     

Modeling stochastic mortality with O-U type processes

Modeling log-mortality rates on O-U type processes and forecasting life expectancies are explored using U.S. data. In the classic Lee-Carter model of mortality, the time trend and the age-specific pattern of mortality over age group are linear, this is not the feature of mortality model. To avoid this disadvantage, O-U type processes will be used to model the log-mortality in this paper. In fact, this model is an AR(1) process, but with a nonlinear time drift term. Based on the mortality data of America from Human Mortality database (HMD), mortality projection consistently indicates a preference for mortality with O-U type processes over those with the classical Lee-Carter model. By means of this model, the low bounds of mortality rates at every age are given. Therefore, lengthening of maximum life expectancies span is estimated in this paper.     

Hedging pure endowments with mortality derivatives

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. (2009). The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.     

Stochastic portfolio specific mortality and the quantification of mortality basis risk

In the last decade a vast literature on stochastic mortality models has been developed. However, these models are often not directly applicable to insurance portfolios because:

(a) For insurers and pension funds it is more relevant to model mortality rates measured in insured amounts instead of measured in the number of policies.

(b) Often there is not enough insurance portfolio specific mortality data available to fit such stochastic mortality models reliably.

Therefore, in this paper a stochastic model is proposed for portfolio specific mortality experience. Combining this stochastic process with a stochastic country population mortality process leads to stochastic portfolio specific mortality rates, measured in insured amounts. The proposed stochastic process is applied to two insurance portfolios, and the impact on the Value at Risk for longevity risk is quantified. Furthermore, the model can be used to quantify the basis risk that remains when hedging portfolio specific mortality risk with instruments of which the payoff depends on population mortality rates.     

Characterization of between-group inequality of longevity in European Union countries

Comparisons of differential survival by country are useful in many domains. In the area of public policy, they help policymakers and analysts assess how much various groups benefit from public programs, such as social security and health care. In financial markets and especially for actuaries, they are important for designing annuities and life insurance products. This paper presents a method for clustering information about differential mortality by country. The approach is then used to group mortality surfaces for European Union (EU) countries. The aim of this paper is to measure between-group inequality in mortality experience in EU countries through a range of mortality indicators. Additionally, the indicators permit the characterization of each group. It is important to take into account characteristics such as sex; therefore, this study differentiates between males and females in order to detect whether their patterns and characterizations are different. It is concluded that there are clear differences in mortality between the east and west of the EU that are more important than the traditional south–north division, with a significant disadvantage for Eastern Europe, and especially for males in Baltic countries. We find that the mortality indicators have evolved in all countries in such a way that the gap between groups has been maintained, both in terms of the differences in mortality levels and variability.     

修匀法的案例分析

在人寿保险中，预定死亡率是压定纯保费率的一个重要因素之一．本文通过一个具体的实例，介绍预定死亡率的修匀方法．     

A forecast reconciliation approach to cause-of-death mortality modeling

Life expectancy has been increasing sharply around the globe since the second half of the 20th century. Mortality modeling and forecasting have therefore attracted increasing attention from various areas, such as the public pension systems, commercial insurance sectors, as well as actuarial, demographic and epidemiological research. Compared to the aggregate mortality experience, cause-specific mortality rates contain more detailed information, and can help us better understand the ongoing mortality improvements. However, when conducting cause-of-death mortality modeling, it is important to ensure coherence in the forecasts. That is, the forecasts of cause-specific mortality rates should add up to the forecasts of the aggregate mortality rates. In this paper, we propose a novel forecast reconciliation approach to achieve this goal. We use the age-specific mortality experience in the U.S. during 1970–2015 as a case study. Seven major causes of death are considered in this paper. By incorporating both the disaggregate cause-specific data and the aggregate total-level data, we achieve better forecasting results at both levels and coherence across forecasts. Moreover, we perform a cluster analysis on the cause-specific mortality data. It is shown that combining mortality experience from causes with similar mortality patterns can provide additional useful information, and thus further improve forecast accuracy. Finally, based on the proposed reconciliation approach, we conduct a scenario-based analysis to project future mortality rates under the assumption of certain causes being eliminated.     

Forecasting mortality for small populations by mixing mortality data

In this paper we address the problem of projecting mortality when data are severely affected by random fluctuations, due in particular to a small sample size, or when data are scanty. Such situations may emerge when dealing with small populations, such as small countries (possibly previously part of a larger country), a specific geographic area of a (large) country, a life annuity portfolio or a pension fund, or when the investigation is restricted to the oldest ages. The critical issues arising from the volatility of data due to the small sample size (especially at the highest ages) may be made worse by missing records; this is the case, for example, of a small country previously part of a larger country, or a specific geographic area of a country, given that in some periods mortality data could have been collected just at an aggregate level.We suggest to ‘replicate’ the mortality of the small population by mixing appropriately the mortality data obtained from other populations. We design a two-step procedure. First, we obtain the average mortality of ‘neighboring’ populations. Three alternative approaches are tested for the assessment of the average mortality; conversely, the identification and the weight of the neighboring populations are obtained through (standard) optimization techniques. Then, following a sort of credibility approach, we mix the original mortality data of the small population with the average mortality of the neighboring populations.In principle, the approach described in the paper could be adopted for any population, whatever is its size, aiming at improving mortality projections through information collected from other groups. Through backtesting, we show that the procedure we suggest is convenient for small populations, but not necessarily for large populations, nor for populations not showing noticeable erratic effects in data. This finding can be explained as follows: while the replication of the original data implies the increase of the size of the sample, it also involves a smoothing of data, with a possible loss of specific information relating to the group referred to. In the case of small populations showing major erratic movements in mortality data, the advantages gained from the larger sample size overcome the disadvantages of the smoothing effect.     

Semi-parametric accelerated hazard relational models with applications to mortality projections

In this paper, we propose new relational models linking some specific mortality experience to a reference life table. Compared to existing relational models which distort the forces of mortality, we work here on the age scale. Precisely, age is distorted making individuals younger or older before performing the computations with the reference life table. This is in line with standard actuarial practice, specifically with the so-called Rueff’s adjustments. It is shown that the statistical inference can be conducted with the help of a suitably modified version of the standard IRWLS algorithm in a Poisson GLM/GAM setting. A dynamic version of this model is proposed to produce mortality projections. Numerical illustrations are performed on Belgian mortality statistics.