Bayesian Stochastic Mortality Modelling for Two Populations

Andrew Cairns, David Blake, Kevin Dowd, Guy Coughlan and Marwa Khalaf-Allah

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 rea- sonableness. 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 uncer- tainty 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 im- provement factors for two populations are consistent with
historical data. The framework is illustrated using two-population extensions of the
Age-Period- Cohort 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
hedging instrument based on the second population.

Keywords: Stochastic mortality, two populations, small sub-populations, mortality
spreads, age effect, period effect, cohort effect, basis risk, Markov chain Monte Carlo,
parameter uncertainty.

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