Dynamic mortality factor model with conditional heteroskedasticity

In most methods for modeling mortality rates, the idiosyncratic shocks are assumed to be homoskedastic. This study investigates the conditional heteroskedasticity of mortality in terms of statistical time series. We start from testing the conditional heteroskedasticity of the period effect in the naïve Lee–Carter model for some mortality data. Then we introduce the Generalized Dynamic Factor method and the multivariate BEKK GARCH model to describe mortality dynamics and the conditional heteroskedasticity of mortality. After specifying the number of static factors and dynamic factors by several variants of information criterion, we compare our model with other two models, namely, the Lee–Carter model and the state space model. Based on several error-based measures of performance, our results indicate that if the number of static factors and dynamic factors is properly determined, the method proposed dominates other methods. Finally, we use our method combined with Kalman filter to forecast the mortality rates of Iceland and period life expectancies of Denmark, Finland, Italy and Netherlands.     

Estimating the term structure of mortality

In modeling and forecasting mortality the Lee-Carter approach is the benchmark methodology. In many empirical applications the Lee-Carter approach results in a model that describes the log central death rates by means of linear trends. However, due to the volatility in (past) mortality data, the estimation of these trends, and, thus, the forecasts based on them, might be rather sensitive to the sample period employed. We allow for time-varying trends, depending on a few underlying factors, to make the estimates of the future trends less sensitive to the sampling period. We formulate our model in a state-space framework, and use the Kalman filtering technique to estimate it. We illustrate our model using Dutch mortality data.     

Comonotonic approximations to quantiles of life annuity conditional expected present value

In large portfolios, the risk borne by annuity providers (insurance companies or pension funds) is basically driven by the randomness in the future mortality rates. To fix the ideas, we adopt here the standard Lee-Carter framework, where the future forces of mortality are decomposed in a log-bilinear way. This paper aims to provide accurate approximations for the quantiles of the conditional expected present value of the payments to the annuity provider, given the future path of the Lee-Carter time index. Mortality is stochastic while the discount factors are derived from a zero-coupon yield curve and are assumed to be deterministic. Numerical illustrations based on Belgian mortality (general population and insurance market statistics) show that the accuracy of the approximations proposed in this paper is remarkable, with relative difference less than 1% for most probability levels.     

Is the Home Equity Conversion Mortgage in the United States sustainable? Evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional Esscher transform

The purpose of this paper is to build a modeling and pricing framework to investigate the sustainability of the Home Equity Conversion Mortgage (HECM) program in the United States under realistic economic scenarios, i.e., whether the premium payments cover the fair premiums for the inherent risks in the HECM program. We note that earlier HECM models use static mortality tables, neglecting the dynamics of mortality rates and extreme mortality jumps. The earlier models also assume housing prices follow a geometric Brownian motion, which contradicts the fact that housing prices exhibit strong autocorrelation and varying volatility over time. To solve these problems, we propose a generalized Lee-Carter model with asymmetric jump effects to fit the mortality data, and model the house price index via an ARIMA-GARCH process. We then employ the conditional Esscher transform to price the non-recourse provision of reverse mortgages and compare it with the calculated mortgage insurance premiums. The HECM program turns out to be sustainable based on our model setup and parameter settings.     

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...     

An application of vector GARCH model in semiconductor demand planning

Demand planning has been the key to supply chain management in semiconductor industry. With an appropriate weight assignment scheme, the planning accuracy resulting from combinational forecasts can be improved by merging several individual candidate methods. In this paper we discuss the applicability of vector generalized autoregressive conditional heteroskedasticity (GARCH) model to determine the optimal combinational weights of component forecasts, where the conditional variances and correlations of forecast errors from candidate methods are represented and estimated by a maximum-likelihood procedure. The asymptotical properties of parameter estimators for GARCH model are investigated by simulation experiments. An example of the proposed method to real time series of electronic products demonstrates that this weight-varying combinational method produces less prediction errors, compared to other commonly used forecasting approaches that are based on single model selection criteria or fixed weights.     

中国城市人口死亡率的预测

死亡率是随时间变动的具有不确定性的变量,基本养老保险的养老金给付必须考虑动态死亡率的影响,因此需要对中国城市人口的未来死亡率变动进行预测。针对部分年的中国城市分性别人口死亡率数据缺失的实际状况,本文运用死亡人数服从Poisson分布的Lee-Carter模型进行了预测,结果表明该模型的拟合较好。由上述预测得出,随时间的延续,中国城市人口的预期寿命将明显增加,为基本养老保险的支付带来严重的风险,该风险导致基本养老保险个人账户的收入远不足以支付未来的养老金,必须引起重视。本文就如何规避这一风险给出了一些政策建议。     

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 longevity risks using a principal component approach: A comparison with existing stochastic mortality models

This research proposes a mortality model with an age shift to project future mortality using principal component analysis (PCA). Comparisons of the proposed PCA model with the well-known models—the Lee-Carter model, the age-period-cohort model (Renshaw and Haberman, 2006), and the Cairns, Blake, and Dowd model—employ empirical studies of mortality data from six countries, two each from Asia, Europe, and North America. The mortality data come from the human mortality database and span the period 1970-2005. The proposed PCA model produces smaller prediction errors for almost all illustrated countries in its mean absolute percentage error. To demonstrate longevity risk in annuity pricing, we use the proposed PCA model to project future mortality rates and analyze the underestimated ratio of annuity price for whole life annuity and deferred whole life annuity product respectively. The effect of model risk on annuity pricing is also investigated by comparing the results from the proposed PCA model with those from the LC model. The findings can benefit actuaries in their efforts to deal with longevity risk in pricing and valuation.     

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.     

Time-simultaneous prediction bands: A new look at the uncertainty involved in forecasting mortality

Conventionally, isolated (point-wise) prediction intervals are used to quantify the uncertainty in future mortality rates and other demographic quantities such as life expectancy. A pointwise interval reflects uncertainty in a variable at a single time point, but it does not account for any dynamic property of the time-series. As a result, in situations when the path or trajectory of future mortality rates is important, a band of pointwise intervals might lead to an invalid inference. To improve the communication of uncertainty, a simultaneous prediction band can be used. The primary objective of this paper is to demonstrate how simultaneous prediction bands can be created for prevalent stochastic models, including the Cairns-Blake-Dowd and Lee-Carter models. The illustrations in this paper are based on mortality data from the general population of England and Wales.     

A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates

With the decline in the mortality level of populations, national social security systems and insurance companies of most developed countries are reconsidering their mortality tables taking into account the longevity risk. The Lee and Carter model is the first discrete-time stochastic model to consider the increased life expectancy trends in mortality rates and is still broadly used today. In this paper, we propose an alternative to the Lee-Carter model: an AR(1)-ARCH(1) model. More specifically, we compare the performance of these two models with respect to forecasting age-specific mortality in Italy. We fit the two models, with Gaussian and t-student innovations, for the matrix of Italian death rates from 1960 to 2003. We compare the forecast ability of the two approaches in out-of-sample analysis for the period 2004-2006 and find that the AR(1)-ARCH(1) model with t-student innovations provides the best fit among the models studied in this paper.     

Composite quantile regression estimation for P-GARCH processes

We consider the periodic generalized autoregressive conditional heteroskedasticity(P-GARCH) process and propose a robust estimator by composite quantile regression. We study some useful properties about the P-GARCH model. Under some mild conditions, we establish the asymptotic results of proposed estimator.The Monte Carlo simulation is presented to assess the performance of proposed estimator. Numerical study results show that our proposed estimation outperforms other existing methods for heavy tailed distributions.The proposed methodology is also illustrated by Va R on stock price data.     

A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities

Dynamic life tables arise as an alternative to the standard (static) life table, with the aim of incorporating the evolution of mortality over time. The parametric model introduced by Lee and Carter in 1992 for projected mortality rates in the US is one of the most outstanding and has been used a great deal since then. Different versions of the model have been developed but all of them, together with other parametric models, consider the observed mortality rates as independent observations. This is a difficult hypothesis to justify when looking at the graph of the residuals obtained with any of these methods.Methods of adjustment and prediction based on geostatistical techniques which exploit the dependence structure existing among the residuals are an alternative to classical methods. Dynamic life tables can be considered as two-way tables on a grid equally spaced in either the vertical (age) or horizontal (year) direction, and the data can be decomposed into a deterministic large-scale variation (trend) plus a stochastic small-scale variation (residuals).Our contribution consists of applying geostatistical techniques for estimating the dependence structure of the mortality data and for prediction purposes, also including the influence of the year of birth (cohort). We compare the performance of this new approach with different versions of the Lee-Carter model. Additionally, we obtain bootstrap confidence intervals for predicted q_xt resulting from applying both methodologies, and we study their influence on the predictions of e_65t and a_65t.     

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.     

Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity

Compared to the conditional mean or median, conditional quantiles provide a more comprehensive picture of a variable in various scenarios. A semi-parametric quantile estimation method for a double threshold auto-regression with exogenous regressors and heteroskedasticity is considered, allowing representation of both asymmetry and volatility clustering. As such, GARCH dynamics with nonlinearity are added to a nonlinear time series regression model. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between the quantile loss function and the asymmetric-Laplace distribution, is employed for estimation and inference, simultaneously estimating and accounting for nonlinear heteroskedasticity plus unknown threshold limits and delay lags. A simulation study illustrates sampling properties of the method. Two data sets are considered in the empirical applications: modelling daily maximum temperatures in Melbourne, Australia; and exploring dynamic linkages between financial markets in the US and Hong Kong.     

A class of random field memory models for mortality forecasting

This article proposes a parsimonious alternative approach for modeling the stochastic dynamics of mortality rates. Instead of the commonly used factor-based decomposition framework, we consider modeling mortality improvements using a random field specification with a given causal structure. Such a class of models introduces dependencies among adjacent cohorts aiming at capturing, among others, the cohort effects and cross generations correlations. It also describes the conditional heteroskedasticity of mortality. The proposed model is a generalization of the now widely used AR-ARCH models for random processes. For such a class of models, we propose an estimation procedure for the parameters. Formally, we use the quasi-maximum likelihood estimator (QMLE) and show its statistical consistency and the asymptotic normality of the estimated parameters. The framework being general, we investigate and illustrate a simple variant, called the three-level memory model, in order to fully understand and assess the effectiveness of the approach for modeling mortality dynamics.     

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.     

Forecasting mortality rate improvements with a high-dimensional VAR

Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial–temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the differences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main factor-based mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyper-parameters. The variance–covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort and the period effects which are implicitly captured by our model. Third, our approach can be extended to model the several populations in long run perspectives, without raising issue in the estimation process. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort and period effects for these populations.