Bayesian Stochastic Mortality Modelling for Two Populations

The paper introduces a new framework for modelling the joint development over time of mortality rates in a pair of related populations by combining a number of recent and novel developments in stochastic mortality modelling. First, we develop an underlying stochastic model which incorporates a mean-reverting stochastic spread that allows for different trends in mortality improvement rates in the short-run, but parallel improvements in the long run in line with the principles of biological reasonableness. Second, we fit the model using a Bayesian framework that allows us to combine estimation of the unobservable state variables and the parameters of the stochastic processes driving them into a single procedure. This procedure employs Markov chain Monte Carlo (MCMC) techniques, permitting us to analyse uncertainty in the estimates of the historical age, period and cohort effects, and this helps us to smooth out noise in parameter estimates attributable to small populations. Mortality rates arising from this framework provide consistent forecasts for the two populations. Further, estimated correlations based on the simulated mortality improvement factors for two populations are consistent with historical data. The framework is illustrated using two-population extensions of the Age-PeriodCohort and Lee-Carter models on the following populations: England & Wales national and CMI assured lives males and females, and US and California males. The approach is designed for large populations coupled with a small sub-population, but is easily adaptable to other combinations. A key application of the modelling framework would be to allow longevity risk hedgers to model the basis risk that exists between mortality rates in two populations in the case where the hedger wishes to hedge the risk in one population using a hdging instrument based on the second population.