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 model's 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.     

Coherent modeling of mortality patterns for age-specific subgroups

Decisions in Economics and Finance - The recent actuarial literature has shown that mortality patterns and trajectories in closely related populations are similar in some respects and that small...     

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.     

The Poisson Log-Bilinear Lee-Carter Model

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.