A. E. Renshaw

Lee–Carter mortality forecasting with age-specific enhancement

Abstract We investigate the feasibility of constructing mortality forecasts on the basis of the first two sets of single value decomposition vectors, rather than just on the first such set of vectors, as in the established Lee–Carter (Gaussian) approach to mortality forecasting. Three applications are presented and the resulting forecasts compared with those constructed using two similar approaches based on generalised linear and bilinear models with Poisson error structures.

Lee–Carter mortality forecasting: a parallel generalized linear modelling approach for England and Wales mortality projections

The paper presents a reinterpretation of the model underpinning the Lee-Carter methodology for forecasting mortality (and other vital) rates. A parallel methodology based on generalized linear modelling is introduced. The use of residual plots is proposed for both methods to aid the assessment of the goodness of fit. The two methods are compared in terms of structure and assumptions. They are then compared through an analysis of the gender- and age-specific mortality rates for England and Wales over the period 1950-1998 and through a consideration of the forecasts generated by the two methods. The paper also compares different approaches to the forecasting of life expectancy and considers the effectiveness of the Coale-Guo method for extrapolating mortality rates to the oldest ages. Copyright 2003 Royal Statistical Society.

On the forecasting of mortality reduction factors

Abstract Ways in which the so-called Lee–Carter time series approach to forecasting mortality patterns can be modified to forecast the possible future behaviour of mortality reduction factors (RFs) are described. A comparison is drawn with an alternative regression type approach to the forecasting of mortality RFs, based on the same model structure. Case studies, illustrating different aspects of the methodology, making use of both the more recent mortality experience of UK male pensioner lives and the historical mortality experience of UK male annuitants, are presented.

Generalized linear models and actuarial science

The authors review the applications of generalized linear models to actuarial problems. This rich class of statistical model has been successfully applied in recent years to a wide range of problems, involving mortality, multiple-state models, lapses, premium rating and reserving. Selective examples of these applications are presented.

MODELLING THE CLAIMS PROCESS IN THE PRESENCE OF COVARIATES

An overview of the potential of Generalized Linear Models as a means of modelling the salient features of the claims process in the presence of rating factors is presented. Specific attention is focused on the rich variety of modelling distributions which can be implemented in this context. The claims process in non-life insurance comprises two components, claim frequency and claim serverity, in which the product of the underlying expected claim rate and expected claim severity defines the pure or risk premium. Specifically, considerable attention is given to the probabalistic modelling of various aspects of a single batch of claims, often focusing on the aggregate claims accruing in a time period of fixed duration, typically one year, under a variety of assumptions imposed on the claim frequency and claim severity mechanisms. In this paper, attention is refocused on the considerable potential of generalized linear models (GLMs) as a comprehensive modelling tool for the study of the claims process in the presence of covariates. Section 2 contains a brief summary of the main features of GLMs which are of potential interest in modelling various aspects of the claims process. Particular attention is drawn to the rich variety of modelling distributions which are available and to the parameter estimation and model fitting techniques based on the concepts of quasi-likelihood and extended quasi-likelihood. Sections 3 and 4 focus respectively on the modelling of the claim frequency and claim severity components of the process in the presence of covariates. An overview of the potential of GLMs as a means of modelling these two aspects of the claims process is discussed. Relevant published applications are referenced, although an exhaustive search of the literature has not been conducted. A number of the suggested modelling techniques are illustrated in Section 5.

ACTUARIAL GRADUATION PRACTICE AND GENERALISED LINEAR AND NON-LINEAR MODELS

It is demonstrated how existing actuarial graduation practice, used in the construction of life tables, can be extended to considerable effect by formulating the techniques within the generalised linear and non-linear modelling framework.

STATISTICAL ANALYSIS OF LIFE ASSURANCE LAPSES

Data have been generously supplied by the Faculty of Actuaries Withdrawals Research Group. These data cover the lapse or withdrawal experience for the calendar year 1976 of seven Scottish life offices. An extensive analysis has been published by the Research Group although, for reasons we shall discuss later, we believe that the approach outlined here is better able to describe the structure of the data than the detailed (and somewhat pedestrian) tabulations of this earlier paper.

On the graduations associated with a multiple state model for permanent health insurance

Abstract The paper puts forward a comprehensive methodology for graduating the transition intensities in a multiple state model for permanent health insurance applications. The approach is based on generalized linear models (GLM) and utilises the data collected and analysed by the U.K. Continuous Mortality Investigation (CMI) Bureau (under the auspices of the Institute and Faculty of Actuaries) in respect of male standard experience of individual PHI policies for the period 1975–1978. The comprehensive and versatile nature of this approach is shown to be applicable to three sets of transition intensities for which data are available: for sickness recovery (as functions of age at sickness onset, x , and duration of sickness, z ), for death as sick (also a bivariate function of x and z ) and for sickness inception (a function of x only). The full potential of the GLM methodology means that approximations to normality which would lead to complex, iterative methods of fitting can be avoided and that the presence of duplicate policies can be allowed for. A novel feature of the graduation proposed is the introduction of break-point predictors (similar to splines) into the graduation formula, and the method of location of the number and optimum positions of the underlying break-points (or knots).

Longevity-indexed life annuities

Abstract This paper addresses the problem of the sharing of longevity risk between an annuity provider and a group of annuitants. An appropriate longevity index is designed in order to adapt the amount of the periodic payments in life annuity contracts. This accounts for unexpected longevity improvements experienced by a given reference population. The approach described in the present paper is in contrast with group self-annuitization, where annuitants bear their own risk. Here the annuitants bear only the nondiversifiable risk that the future mortality trend departs from that of the reference forecast. In that respect, the life annuities discussed in this paper are substitutes for reinsurance and securitization of longevity risk.

JOINT MODELLING FOR ACTUARIAL GRADUATION AND DUPLICATE POLICIES

In this paper it is demonstrated how recently developed statistical techniques designed to facilitate the joint modelling of the mean and dispersion are well suited to model the presence of duplicate policies in graduation.