Maria Russolillo

Extending the Lee–Carter model: a three-way decomposition

In this paper, we focus on a Multi-dimensional Data Analysis approach to the Lee–Carter (LC) model of mortality trends. In particular, we extend the bilinear LC model and specify a new model based on a three-way structure, which incorporates a further component in the decomposition of the log-mortality rates. A multi-way component analysis is performed using the Tucker3 model. The suggested methodology allows us to obtain combined estimates for the three modes: (1) time, (2) age groups and (3) different populations. From the results obtained by the Tucker3 decomposition, we can jointly compare, in both a numerical and graphical way, the relationships among all three modes and obtain a time-series component as a leading indicator of the mortality trend for a group of populations. Further, we carry out a correlation analysis of the estimated trends in order to assess the reliability of the results of the three-way decomposition. The models goodness of fit is assessed using an analysis of the residuals. Finally, we discuss how the synthesised mortality index can be used to build concise projected life tables for a group of populations. An application which compares 10 European countries is used to illustrate the approach and provide a deeper insight into the model and its implementation.

The Poisson Log-Bilinear Lee-Carter Model: Applications Of efficient bootstrap methods to annuity analyses

Abstract Life insurance companies deal with two fundamental types of risks when issuing annuity contracts: financial risk and demographic risk. Recent work on the latter has focused on modeling the trend in mortality as a stochastic process. A popular method for modeling death rates is the Lee-Carter model. This methodology has become widely used, and various extensions and modifications have been proposed to obtain a broader interpretation and to capture the main features of the dynamics of mortality rates. In order to improve the measurement of uncertainty in survival probability estimates, in particular for older ages, the paper proposes an extension based on simulation procedures and on the bootstrap methodology. It aims to obtain more reliable and accurate mortality projections, based on the idea of obtaining an acceptable accuracy of the estimate by means of variance reducing techniques. In this way the forecasting procedure becomes more efficient. The longevity question constitutes a critical element in the solvency appraisal of pension annuities. The demographic models used for the cash flow distributions in a portfolio impact on the mathematical reserve and surplus calculations and affect the risk management choices for a pension plan. The paper extends the investigation of the impact of survival uncertainty for life annuity portfolios and for a guaranteed annuity option in the case where interest rates are stochastic. In a framework in which insurance companies need to use internal models for risk management purposes and for determining their solvency capital requirement, the authors consider the surplus value, calculated as the ratio between the market value of the projected assets to that of the liabilities, as a meaningful measure of the company’s financial position, expressing the degree to which the liabilities are covered by the assets.

The mortality of the Italian population: Smoothing techniques on the Lee–Carter model

Several approaches have been developed for forecasting mortality using the stochastic model. In particular, the Lee-Carter model has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity. Hyndman-Ullah show a particular version of the Lee-Carter methodology, the so-called Functional Demographic Model, which is one of the most accurate approaches as regards some mortality data, particularly for longer forecast horizons where the benefit of a damped trend forecast is greater. The paper objective is properly to single out the most suitable model between the basic Lee-Carter and the Functional Demographic Model to the Italian mortality data. A comparative assessment is made and the empirical results are presented using a range of graphical analyses.

Detecting Common Longevity Trends by a Multiple Population Approach

Recently the interest in the development of country and longevity risk models has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare, and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups or the dependence structure across time in a single population setting—a sort of intradependence structure. On the other hand, the dependence across multiple populations, which we describe as interdependence, can be explored for capturing common long-run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee-Carter (LC) framework with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intragroup error correlation structure are preserved. We introduce a model which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current article, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee-Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.

Computational framework for longevity risk management

Longevity risk threatens the financial stability of private and government sponsored defined benefit pension systems as well as social security schemes, in an environment already characterized by persistent low interest rates and heightened financial uncertainty. The mortality experience of countries in the industrialized world would suggest a substantial age-time interaction, with the two dominant trends affecting different age groups at different times. From a statistical point of view, this indicates a dependence structure. It is observed that mortality improvements are similar for individuals of contiguous ages (Wills and Sherris, Integrating financial and demographic longevity risk models: an Australian model for financial applications, Discussion Paper PI-0817, 2008). Moreover, considering the dataset by single ages, the correlations between the residuals for adjacent age groups tend to be high (as noted in Denton et al., J Population Econ 18:203–227, 2005). This suggests that there is value in exploring the dependence structure, also across time, in other words the inter-period correlation. In this research, we focus on the projections of mortality rates, contravening the most commonly encountered dependence property which is the “lack of dependence” (Denuit et al., Actuarial theory for dependent risks: measures. Orders and models, Wiley, New York, 2005). By taking into account the presence of dependence across age and time which leads to systematic over-estimation or under-estimation of uncertainty in the estimates (Liu and Braun, J Probability Stat, 813583:15, 2010), the paper analyzes a tailor-made bootstrap methodology for capturing the spatial dependence in deriving confidence intervals for mortality projection rates. We propose a method which leads to a prudent measure of longevity risk, avoiding the structural incompleteness of the ordinary simulation bootstrap methodology which involves the assumption of independence.

Adaptive Neuro-Fuzzy Inference Systems vs. Stochastic Models for Mortality Data

A comparative analysis is done between stochastic models and Adaptive Neuro–Fuzzy Inference System applied to the projection of the longevity trend. The stochastic models provides the heuristic rule for obtaining projections. In the context of ANFIS models, the fuzzy logic allows for determining the learning algorithm on the basis of the relationship between inputs and outputs. In other words the rule is here deducted by the actual mortality data, because this allows for fuzzy systems to learn from the data they are modelling. This is possible by computing the membership function parameters that best allow the associated fuzzy inference system to track the input/output data. The literature indicates that the self-predicting model of ANFIS is better than other models in a lot of fields. Shortcomings and advantages of both approaches are here highlighted.

Lee-Carter error matrix simulation: heteroschedasticity impact on actuarial valuations

Recently a number of approaches have been developed for forecasting mortality. In this paper, we consider the Lee-Carter model and we investigate in particular the hypothesis about the error structure implicitly assumed in the model specification, i.e., the errors are homoschedastic. The homoschedasticity assumption is quite unrealistic, because of the observed pattern of the mortality rates showing a different variability at old ages than younger ages. Therefore, the opportunity to analyse the robustness of estimated parameter is emerging. To this aim, we propose an experimental strategy in order to assess the robustness of the Lee-Carter model by inducing the errors to satisfy the homoschedasticity hypothesis. Moreover, we apply it to a matrix of Italian mortality rates. Finally, we highlight the results through an application to a pension annuity portfolio.

Comparing Mortality Trends via Lee-Carter Method in the Framework of Multidimensional Data Analysis

In the framework of demographic processes, different approaches can be identified. According to their aims, these approaches can be differentiated in extrapolative and structural methods. The first focus on the homogeneity of trends in order to obtain projection. The second are based on structural models relating demographic variables to other kinds of variables (geographical, social, economical, etc.). Nowadays, this distinction is not so clear and the joint use of explorative and explanatory approaches is increasing. In this paper, we focus on the extrapolative features of the Lee-Carter model (1992) and propose a reading of such method in the framework of the Multidimensional Data Analysis. Our aim is to propose a data analysis strategy exploiting the analytical and geometrical properties of the Lee-Carter method.

Empirical Evidence from the Three-Way LC Model

The three-way Lee-Carter (LC) model was proposed as an extension of the original LC model when a three-mode data structure is available. It provides an alternative for modelling mortality differentials. This variant of the LC model adds a subpopulation parameter that deals with different drifts in mortality. Making use of several tools of exploratory data analysis, it allows giving a new perspective to the demographic analysis supporting the analytical results with a geometrical interpretation and a graphical representation. When facing with a three-way data structure, several choices on data pre-treatment will affect the whole data modelling. The first step of three-way mortality data investigation should be addressed by exploring the different source of variations and highlighting the significant ones. In this contribution, we consider the three-way LC model investigated by means of a three-way analysis of variance with fixed effects, where the three main effects, the three two-way interactions and one three-way interaction are analyzed. Aim of the paper is to highlight the technical-applicative infrastructure behind the methodology.

Three-Way Data Analysis Applied to Cause Specific Mortality Trends

The costs of the social security public systems, in almost all developed countries, are affected by two phenomena: an increasing survival in higher ages and a smaller number of births. The combination of these two aspects largely impacts on the societies dealing with the rising pension and healthcare costs. In spite of the common trend given by the ageing population and the growing longevity, the mortality rates are also influenced by gender, countries, ethnicity, income, wealth, causes of death and so on. According to the WHO a “right” recognition of the causes of death is important for forecasting more accurately mortality. In this framework we intend to investigate the main causes of death impacting on the upcoming human survival, throughout a Multi-dimensional Data Analysis approach to the Lee Carter model of mortality trends. In a previous paper, we stated that the crude mortality data can be considered according to several criteria. In this contribution we take into account a three way array holding mortality data structured by time, age-group and causes of death. The model decomposition we propose is a modified version of the classical Lee Carter model allowing for three-way data treatment, analysis of residuals, graphical representation of the different components. A case study based on actual data will be discussed.