An application of comonotonicity theory in a stochastic life annuity framework

A life annuity contract is an insurance instrument which pays pre-scheduled living benefits conditional on the survival of the annuitant. In order to manage the risk borne by annuity providers, one needs to take into account all sources of uncertainty that affect the value of future obligations under the contract. In this paper, we define the concept of annuity rate as the conditional expected present value random variable of future payments of the annuity, given the future dynamics of its risk factors. The annuity rate deals with the non-diversifiable systematic risk contained in the life annuity contract, and it involves mortality risk as well as investment risk. While it is plausible to assume that there is no correlation between the two risks, each affects the annuity rate through a combination of dependent random variables. In order to understand the probabilistic profile of the annuity rate, we apply comonotonicity theory to approximate its quantile function. We also derive accurate upper and lower bounds for prediction intervals for annuity rates. We use the Lee-Carter model for mortality risk and the Vasicek model for the term structure of interest rates with an annually renewable fixed-income investment policy. Different investment strategies can be handled using this framework.     

Modeling stochastic mortality for joint lives through subordinators

There is a burgeoning literature on mortality models for joint lives. In this paper, we propose a new model in which we use time-changed Brownian motion with dependent subordinators to describe the mortality of joint lives. We then employ this model to estimate the mortality rate of joint lives in a well-known Canadian insurance data set. Specifically, we first depict an individual’s death time as the stopping time when the value of the hazard rate process first reaches or exceeds an exponential random variable, and then introduce the dependence through dependent subordinators. Compared with existing mortality models, this model better interprets the correlation of death between joint lives, and allows more flexibility in the evolution of the hazard rate process. Empirical results show that this model yields highly accurate estimations of mortality compared to the baseline non-parametric (Dabrowska) estimation.     

Analysis of survivorship life insurance portfolios with stochastic rates of return

A general portfolio of survivorship life insurance contracts is studied in a stochastic rate of return environment with a dependent mortality model. Two methods are used to derive the first two moments of the prospective loss random variable. The first one is based on the individual loss random variables while the second one studies annual stochastic cash flows. The distribution function of the present value of future losses at a given valuation time is derived. For illustrative purposes, an AR(1) process is used to model the stochastic rates of return, and the future lifetimes of a couple are assumed to follow a copula model. The effects of the mortality dependence, the portfolio size and the policy type, as well as the impact of investment strategies on the riskiness of portfolios of survivorship life insurance policies are analyzed by means of moments and probability distributions.     

关于2000-2003新生命表出台对寿险业的影响分析

随着2000-2003新生命表的出台, 寿险业对生命表的关注程度日益加强, 本文第一部分介绍了研究背景, 第二部分对死亡效力(mortality force)进行模拟, 并进行了可靠性检验. 第三部分结合中国人寿保险业1990-1993生命表、2000-2003生命表, 给出了时间推移下同年龄死亡效力之间的关系. 基于此, 引入了布朗运动的随机变量, 将死亡效力随机化, 并进行模拟, 优化了可靠性检验结果.第四部分预测了生命表改善对年金保险(annuity)净费率的影响, 分析了延期承保的费率影响趋势, 指出了长寿风险. 最后给出了相关评价及未来预测思路.     

Applications of a multi-state risk factor/mortality model in life insurance

Mortality rates are known to depend on socio-economic and behavioral risk factors, and actuarial calculations for life insurance policies usually reflect this. It is typically assumed, however, that these risk factors are observed only at policy issue, and the impact of changes that occur later is not considered. In this paper, we present a discrete-time, multi-state model for risk factor changes and mortality. It allows one to more accurately describe mortality dynamics and quantify variability in mortality. This model is extended to reflect health status and then used to analyze the impact of selective lapsation of life insurance policies and to predict mortality under reentry term insurance.     

An EOQ model for deteriorating inventory with alternating demand rates

The present paper deals with an economic order quantity model for items deteriorating at some constant rate with demand changing at a known and at a random point of time in the fixed production cycle.     

Applications of a multi-state risk factor/mortality model in life insurance

Mortality rates are known to depend on socio-economic and behavioral risk factors, and actuarial calculations for life insurance policies usually reflect this. It is typically assumed, however, that these risk factors are observed only at policy issue, and the impact of changes that occur later is not considered. In this paper, we present a discrete-time, multi-state model for risk factor changes and mortality. It allows one to more accurately describe mortality dynamics and quantify variability in mortality. This model is extended to reflect health status and then used to analyze the impact of selective lapsation of life insurance policies and to predict mortality under reentry term insurance.     

模糊随机环境下的连续型变额生命年金

采用模糊随机理论,构建连续支付型变额生命年金模型.假定利率为三角模糊数,死亡率为随机变量.结合精算理论,给出了连续支付型变额生命年金精算现值的期望、方差以及分布函数和分位数的模糊表达式.最后,通过实证分析计算出一个在养老保险中常见的生命年金的相关值,验证模型的可行性.     

带有随机订货点的两阶段报童模型

销售商可通过二次订货以达到降低风险、增加利润的目的,然而在实际中由于生厂商供货能力不足等不可控因素,销售商往往无法确定何时能进行第二次订货。针对这种现象,本文研究二次订货时间不确定的报童问题,假设随机订货时间点和需求率均服从均匀分布,建立了带有随机订货点的两阶段报童模型,给出了两阶段最优期望总订货量,使得零售商在整个销售期内的期望利润达到最大值。最后通过数值算例,对比分析了本文的二次订购模型与传统一次订购模型,研究结果指出在整个销售期内二次订货可以提高零售商的期望利润。     

Using fuzzy random variables in life annuities pricing

This paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates. We show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered. So, we firstly describe fuzzy random variables and define some associated measures: the mathematical expectation, the variance, distribution function and quantiles. Secondly, we show several ways to estimate the discount rates to price annuities. Subsequently, the present value of life annuities is modelled with fuzzy random variables. We finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables.     

不确定环境下定期人寿保险的破产概率区间的计算

针对实际问题存在的不确定因素,研究了含不确定参数的定期人寿保险的破产模型,其中死亡率和净年保单数分别用区间数和随机参数刻画.推导了破产概率区间的计算公式,且用泊松分布近似时得到其近似计算方法.该模型的建立既考虑了初始准备金的利息积累和任何时刻的新投保人的加入,并采用了新的分组方式,又考虑了实际问题中的不确定因素,因而能够更加真实地刻画了实际过程,比传统模型更具实用性.     

Predicting daily exchange rate with singular spectrum analysis

This paper uses univariate and multivariate singular spectrum analysis for predicting the value and the direction of changes in the daily pound/dollar exchange rate. In prediction of daily pound/dollar rate, we use the rescaled and bootstrapped daily euro/dollar rate as a guidepost for the singular spectrum analysis method. We use the random walk model as a benchmark to evaluate performances of the singular spectrum analysis as a prediction method. Empirical results show that the forecast based on the multivariate singular spectrum analysis compares favorably to the forecast of the random walk model both for predicting the value and the direction of changes in the daily pound/dollar exchange rate. We compared the prediction results based on an error correction model in the context of a restricted vector autoregressive model and compared them with the prediction results by a random walk as well as by those of singular spectrum and multiple singular spectrum models and found that the VEC results are inferior.     

Mean-variance optimization problems for an accumulation phase in a defined benefit plan

In this paper we deal with contribution rate and asset allocation strategies in a pre-retirement accumulation phase. We consider a single cohort of workers and investigate a retirement plan of a defined benefit type in which an accumulated fund is converted into a life annuity. Due to the random evolution of a mortality intensity, the future price of an annuity, and as a result, the liability of the fund, is uncertain. A manager has control over a contribution rate and an investment strategy and is concerned with covering the random claim. We consider two mean-variance optimization problems, which are quadratic control problems with an additional constraint on the expected value of the terminal surplus of the fund. This functional objectives can be related to the well-established financial theory of claim hedging. The financial market consists of a risk-free asset with a constant force of interest and a risky asset whose price is driven by a Lévy noise, whereas the evolution of a mortality intensity is described by a stochastic differential equation driven by a Brownian motion. Techniques from the stochastic control theory are applied in order to find optimal strategies.     

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.     

On cumulative jump random variables

Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This paper provides some tools for working with such processes. We develop a sample path formula for the cumulative jump height over a given time interval. From this formula an expression for the expected value of the cumulative jump random variable is developed under reasonable conditions. The results are applied to finding the expected number of failures in the separate maintenance model over a stated time interval and to the expected number of occurrences of a regenerative event over a stated time interval.     

The impact of the determinants of mortality on life insurance and annuities

Extended risk classification has become an important issue recently in life insurance and annuity markets. Various risk factors have been explored and identified by past research. Using those risk factors, one can construct various risk classes. This enables insurers to provide more equitable life insurance and annuity benefits for individuals in different risk classes and to manage mortality/longevity risk more efficiently. The challenge of modeling mortality using various risk factors is to reflect complicated mortality dynamics in a model while maintaining statistical significance. This paper discusses the development of a mortality model that reflects the impact of various risk factors on mortality. Longitudinal survey data from the Canadian National Population Health Survey was used to determine the significant risk factors and quantify their effect on mortality. The model is used to illustrate how the various risk factors influence actuarial present values of life insurance and annuity benefits.     

居民家庭耐用消费品的消费时机选择:实物期权的方法

针对当前居民家庭消费的特点，为了合理地选择耐用消费品的最佳购买时机，本分析耐用品的价格和性能具有实物期权的属性，建立了一个关于“性价比”变量的随机微分方程，并且求出了购买时机临界值的解析公式。最后对日常生活中发生的购买时间现象给予了解释。     

Full predictivistic modeling of stock market data: Application to change point problems

In change point problems in general we should answer three questions: how many changes are there? Where are they? And, what is the distribution of the data within the blocks? In this paper, we develop a new full predictivistic approach for modeling observations within the same block of observation and consider the product partition model (PPM) for treating the change point problem. The PPM brings more flexibility into the change point problem because it considers the number of changes and the instants when the changes occurred as random variables. A full predictivistic characterization of the model can provide a more tractable way to elicit the prior distribution of the parameters of interest, once prior opinions will be required only about observable quantities. We also present an application to the problem of identifying multiple change points in the mean and variance of a stock market return time series.     

周期与季节性因素影响下的短生命周期产品预测优化模型

短生命周期产品因为需求的随机性和产品价值的瞬间变化性,对预测准确性提出了更高的要求.然而许多企业在使用多种预测模型后发现其预测准确率并没有得到显著提升.以短生命周期产品需求特点为背景,在需求预测影响的BASS模型基础上,建立受生命周期和季节性因素影响的需求预测优化模型,最后通过一个产品的实例证实了验证了模型的合理性.     

Pitfalls and merits of cointegration-based mortality models

In recent years, joint modelling of the mortality of related populations has received a surge of attention. Several of these models employ cointegration techniques to link underlying factors with the aim of producing coherent projections, i.e. projections with non-diverging mortality rates. Often, however, the factors being analysed are not fully identifiable and arbitrary identification constraints are (inadvertently) allowed to influence the analysis thereby compromising its validity. Taking the widely used Lee–Carter model as an example, we point out the limitations and pitfalls of cointegration analysis when applied to semi-identifiable factors. On the other hand, when properly applied cointegration theory offers a rigorous framework for identifying and testing long-run relations between populations. Although widely used as a model building block, cointegration as an inferential tool is often overlooked in mortality analysis. Our aim with this paper is to raise awareness of the inferential strength of cointegration and to identify the time series models and hypotheses most suitable for mortality analysis. The concluding application to UK mortality shows by example the insights that can be obtained from a full cointegration analysis.