Carter Model Research Articles

The role of domain wall junctions in Carter's pentahedral model

The role of domain wall junctions in Carter's pentahedral model is investigated both analytically and numerically. We perform, for the first time, field theory simulations of such model with various initial conditions. We confirm that there are very specific realizations of Carter's model corresponding to square lattice configurations with X-type junctions which could be stable. However, we show that more realistic realizations, consistent with causality constraints, do lead to a scaling domain wall network with Y-type junctions. We determine the network properties and discuss the corresponding cosmological implications, in particular for dark energy.

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.

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.

A case of motional stability of railway wheel pair

Motion of the railway wheel pair along a rectilinear segment of track was considered within the Carter model of wheel-rail interaction with regard for the nonlinear part of the creep forces. The linear parts of these forces were modified by the integral Volterra operator characterizing the visco-elastic characteristics of the material. Stability of motion of the wheel pair with zero conicity was studied using the integro-differential equations.

Modeling Mortality With Jumps: Applications to Mortality Securitization

AbstractIn this article, we incorporate a jump process into the original Lee–Carter model, and use it to forecast mortality rates and analyze mortality securitization. We explore alternative models with transitory versus permanent jump effects and find that modeling mortality via transitory jump effects may be more appropriate in mortality securitization. We use the Swiss Re mortality bond in 2003 as an example to show how to apply our model together with the distortion measure approach to value mortality‐linked securities. Pricing the Swiss Re mortality bond is challenging because the mortality index is correlated across countries and over time. Cox, Lin, and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries, but the problem of correlation over time remains unsolved. We show in this article how to account for the correlations of the mortality index over time by simulating the mortality index and changing the measure on paths.

Longevity Risk and Annuity Pricing with the Lee–Carter Model. Discussion Held at the Faculty of Actuaries

Longevity Risk and Annuity Pricing with the Lee–Carter Model. Discussion Held at the Faculty of Actuaries

A multivariate time series approach to projected life tables

AbstractThe method of mortality forecasting proposed by Lee and Carter describes a time series of age‐specific log‐death rates as a sum of an independent of time age‐specific component and a bilinear term in which one of the component is a time‐varying factor reflecting general change in mortality and the second one is an age‐specific parameter. Such a rigid model structure implies that on average the mortality improvements for different age groups should be proportional, regardless of the calendar period: a single time factor drives the future death rates. This paper investigates the use of multivariate time series techniques for forecasting age‐specific death rates. This approach allows for relative speed of decline in the log death rates specific to the different ages. The dynamic factor analysis and the Johansen cointegration methodology are successfully applied to project mortality. The inclusion of several time factors allows the model to capture the imperfect correlations in death rates from 1 year to the next. The benchmark Lee–Carter model appears as a special case of these approaches. An empirical study is conducted with the help of the Johansen cointegration methodology. A vector‐error correction model is fitted to Belgian general population death rates. A comparison is performed with the forecast of life expectancies obtained from the classical Lee–Carter model. Copyright © 2009 John Wiley & Sons, Ltd.

Hydraulic Fracture Propagation with 3-D Leak-off

Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are: (1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leak-off from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation.

A Quantitative Comparison of Stochastic Mortality Models Using Data From England and Wales and the United States

We compare quantitatively eight stochastic models explaining improvements in mortality rates in England and Wales and in the United States. On the basis of the Bayes Information Criterion (BIC), we find that, for higher ages, an extension of the Cairns-Blake-Dowd (CBD) model that incorporates a cohort effect fits the England and Wales males data best, while for U.S. males data, the Renshaw and Haberman (RH) extension to the Lee and Carter model that also allows for a cohort effect provides the best fit. However, we identify problems with the robustness of parameter estimates under the RH model, calling into question its suitability for forecasting. A different extension to the CBD model that allows not only for a cohort effect, but also for a quadratic age effect, while ranking below the other models in terms of the BIC, exhibits parameter stability across different time periods for both datasets. This model also shows, for both datasets, that there have been approximately linear improvements over time in mortality rates at all ages, but that the improvements have been greater at lower ages than at higher ages, and that there are significant cohort effects.

An index for longevity risk transfer

This paper discusses the choice of an appropriate longevity index to track the improvements in mortality in industrialized countries. Period life expectancies computed from national life tables turn out to be efficient in this context. A detailed analysis of the predictive distribution of this longevity index is performed in the Lee–Carter model where the period life expectancy is just a functional of the underlying time index.

Mechanobiological bone growth: comparative analysis of two biomechanical modeling approaches

Mechanobiological growth is the process whereby bone growth is modulated by mechanical loading. Analytical formulations of mechanobiological growth have been developed by Stokes et al. (J Orthop Res 17(5):646-653, 1990) and Carter et al. (J Orthop Res 6:804-816, 1988). The purpose of this study was to compare these two modeling approaches in a finite element model of a vertebra to investigate whether growth pattern induced by these models were equivalent. A finite element model of a thoracic vertebra, integrating a conceptual model of the growth plate, was developed and combined with the mechanobiological growth models. This model was further used to simulate vertebral growth modulation resulting from different physiological loading conditions. Different growth magnitudes were obtained under compression and combined tension/shear loading, whereas dissimilar growth patterns were triggered by shear forces and combined compression/shear. These two models represent mechanobiological bone growth under limited mechanical environment. Carter's model takes into account three-dimensional stress stimuli, but does not intrinsically incorporate the resulting growth orientation. Stokes' model adequately represents the mechanobiological contribution of axial stresses but does not take into account the contribution of non-axial stresses, which can occur in complex mechanical environment.

Mortality forecasting using neural networks and an application to cause‐specific data for insurance purposes

AbstractMortality forecasting is important for life insurance policies, as well as in other areas. Current techniques for forecasting mortality in the USA involve the use of the Lee–Carter model, which is primarily used without regard to cause. A method for forecasting morality is proposed which involves the use of neural networks. A comparative analysis is done between the Lee–Carter model, linear trend and the proposed method. The results confirm that the use of neural networks performs better than the Lee–Carter and linear trend model within 5% error. Furthermore, mortality rates and life expectancy were formulated for individuals with a specific cause based on prevalence data. The rates are broken down further into respective stages (cancer) based on the individual's diagnosis. Therefore, this approach allows life expectancy to be calculated based on an individual's state of health. Copyright © 2008 John Wiley & Sons, Ltd.

A parameterized approach to modeling and forecasting mortality

A new method is proposed of constructing mortality forecasts. This parameterized approach utilizes Generalized Linear Models (GLMs), based on heteroscedastic Poisson (non-additive) error structures, and using an orthonormal polynomial design matrix. Principal Component (PC) analysis is then applied to the cross-sectional fitted parameters. The produced model can be viewed either as a one-factor parameterized model where the time series are the fitted parameters, or as a principal component model, namely a log-bilinear hierarchical statistical association model of Goodman [Goodman, L.A., 1991. Measures, models, and graphical displays in the analysis of cross-classified data. J. Amer. Statist. Assoc. 86(416), 1085–1111] or equivalently as a generalized Lee–Carter model with p interaction terms. Mortality forecasts are obtained by applying dynamic linear regression models to the PCs. Two applications are presented: Sweden (1751–2006) and Greece (1957–2006).

Measurement of Longevity Risk Using Bootstrapping for Lee–Carter and Generalised Linear Poisson Models of Mortality

This paper provides a comparative investigation of simulation strategies for measuring the longevity risk associated with predictions of mortality rates and derived estimates of life expectancy. The study considers the Lee–Carter framework and a generalised linear Poisson model for representing the dynamics of mortality, as well as enhancements that allow for joint modelling of the dispersion and the effect of using a negative binomial rather than a Poisson assumption.

Mortality among Swedish insured

In this paper, the mortality among the Swedish voluntarily insured is described. It is based on calculations of the mortality among the Swedish insured from 2001 to 2005, and in the total Swedish population. The total population data has been used to compute the mortality trend with the Lee–Carter model.

Mortality, longevity and experiments with the Lee–Carter model

The paper reviews the Lee-Carter modelling framework, illustrated with an application, and then extends the framework through the development of a wider class of generalised, parametric, non-linear models. The choice of error distribution is also generalised. These extensions permit the modelling and extrapolation of age-specific cohort effects as well as the more familiar age-specific period effects: the age-period-cohort version of the model is discussed with a worked example. The paper also provides a comparative study of simulation strategies for assessing risk in mortality rate predictions and the associated forecast estimates of life expectancy and annuity values in both period and cohort perspectives.

Negative binomial version of the Lee–Carter model for mortality forecasting

AbstractMortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. For public policy, as well as for the management of financial institutions, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp(αx + βxκt). The log of the time series of age‐specific death rates is thus expressed as the sum of an age‐specific component αx that is independent of time and another component that is the product of a time‐varying parameter κt reflecting the general level of mortality, and an age‐specific component βx that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. The parameters are usually estimated via singular value decomposition or via maximum likelihood in a binomial or Poisson regression model. This paper demonstrates that it is possible to take into account the overdispersion present in the mortality data by estimating the parameter in a negative binomial regression model. Copyright © 2007 John Wiley & Sons, Ltd.

Association and heterogeneity of insured lifetimes in the Lee–Carter framework

This paper is devoted to the study of some unexpected consequences of the Lee–Carter model for mortality projection. The fact that survival probabilities are governed by a stochastic process induces some positive dependence between insured lifetimes (namely, association). This, in turn, has an impact on solvency capital (as measured by distortion risk measures, for instance). Failing to take this dependence into account, by assuming falsely that the lifetimes are independent, leads to systematic underestimations of the risk capital. The heterogeneity between the policy benefits and the insured lifetimes is also studied (with the help of majorisation, Schur-increasingness and a frailty model).

A Quantitative Comparison of Stochastic Mortality Models Using Data from England & Wales and the United States

We compare quantitatively eight stochastic models explaining improvements in mortality rates in England &Wales and in the US. On the basis of the Bayes Information Criterion (BIC), we find that an extension of the Cairns, Blake & Dowd (2006b) model that incorporates the cohort effect fits the England & Wales data best, while for US data, the Renshaw & Haberman (2006) extension to the Lee & Carter (1992) model that also allows for a cohort effect provides the best fit. However, we identify problems with the robustness of parameter estimates of these models over different time periods. A different extension to the Cairns, Blake & Dowd (2006b) model that allows not only for a cohort effect, but also for a quadratic age effect, while ranking below the other models in terms of the BIC, exhibits parameter stability across different time periods for both data sets. This model also shows, for both data sets, that there have been approximately linear improvements over time in mortality rates at all ages, but that the improvements have been greater at lower ages than at higher ages, and that there are significant cohort effects.

Robust forecasting of mortality and fertility rates: A functional data approach

A new method is proposed for forecasting age-specific mortality and fertility rates observed over time. This approach allows for smooth functions of age, is robust for outlying years due to wars and epidemics, and provides a modelling framework that is easily adapted to allow for constraints and other information. Ideas from functional data analysis, nonparametric smoothing and robust statistics are combined to form a methodology that is widely applicable to any functional time series data observed discretely and possibly with error. The model is a generalization of the Lee–Carter (LC) model commonly used in mortality and fertility forecasting. The methodology is applied to French mortality data and Australian fertility data, and the forecasts obtained are shown to be superior to those from the LC method and several of its variants.