A Bayesian approach to modelling mortality, with applications to insurance

The purpose of this research was to use Bayesian statistics to develop flexible mortality models that could be used to forecast human mortality rates. Several models were developed as extensions to existing mortality models, in particular the Lee-Carter mortality model and the age-period-cohort model, by including some of the following features: age-period and age-cohort interactions, random effects on mortality, measurement errors in population count and smoothing of the mortality rate surface. One expects mortality rates to change in a relatively smooth manner between neighbouring ages or between neighbouring years or neighbouring cohorts. The inclusion of random effects in some of the models captures additional fluctuations in these effects. This smoothing is incorporated in the models by ensuring that the age, period and cohort parameters of the models have a relatively smooth sequence which is achieved through the choice of the prior distribution of the parameters. Three different smoothing priors were employed: a random walk, a random walk on first differences of the parameters and an autoregressive model of order one on the first differences of the parameters. In any model only one form of smoothing was used. The choice of smoothing prior not only imposes different patterns of smoothing on the parameters but is seen to be very influential when making mortality forecasts. The mortality models were fitted, using Bayesian methods, to population data for males and females from England and Wales. The fits of the models were analysed and compared using analysis of residuals, posterior predictive intervals for both in-sample and out-of-sample data and the Deviance Information Criterion. The models fitted the data better than did both the Lee-Carter model and the age-period-cohort model. From the analysis undertaken, for any given age and calendar year, the preferred model based on the Deviance Information Criterion score, for male and female death counts was a Poisson model with the mean parameter equal to the number of lives exposed to risk of dying for that age in that calendar year multiplied by a mortality parameter. The logit of this mortality parameter was a function of the age, year (period) and cohort with additional interactions between the age and period parameters and between the age and cohort parameters. The form of parameter smoothing that suited the males was an autoregressive model of order one on the first differences of the parameters and that for the females was a random walk. Moreover, it was found useful to add Gaussian random effects to account for overdispersion caused by unobserved heterogeneity in the population mortality. The research concluded by the application of a selection of these models to the provision of forecasts of period and cohort life expectancies as well as the numbers of centenarians for males and females in England and Wales. In addition, the thesis illustrated how Bayesian mortality models could be used to consider the impact of the new European Union solvency regulations for insurers (Solvency II) for longevity risk. This research underlined the important role that Bayesian stochastic mortality models can play in considering longevity risk