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Posts feedConstraints: a lot of fuss about nothing?
Our paper, "A stochastic implementation of the APCI model for mortality projections", was presented at the Institute and Faculty of Actuaries in October 2017. There was quite a discussion of the role of constraints in the fitting and forecasting of models of mortality. This got me wondering if constraints weren't in fact a red herring. This blog is a short introduction to the results of my investigation into the role, or indeed the non-role, of constraints in modelling and forecasting mortality.
Introducing the Product Integral
Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:
\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]
Fathoming the changes to the Lee-Carter model
Ancient Greek philosophers had a paradox called "The Ship of Theseus"; if pieces of a ship are replaced over time as they wear out until every one of the original components is gone, is it still the same ship? At this point you could be forgiven for thinking (a) that this couldn't possibly be further removed from mortality modelling, and (b) that I had consumed something a lot more potent than tea at breakfast.
Solid progress
From small steps to big results
In survival-model work there is a fundamental relationship between the \(t\)-year survival probability from age \(x\), \({}_tp_x\), and the force of mortality, \(\mu_x\):
\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right).\qquad(1)\]
Occupational hazard
Everything points to Poisson
One recurring theme in our forthcoming book, Modelling Mortality with Actuarial Applications, is the all-pervading role of likelihoods that suggest the lurking presence of a Poisson distribution. A popular assumption in modelling hazard rates is that the number of deaths observed at any given age is a Poisson random variable, so perhaps that might explain it?
Mortality by the book
Risk transfer...and transfer risk
Age rating
Back in the days before personal computers, actuaries relied solely on published tables for their calculations. These were not just the mortality tables, but monetary functions of these tables known as commutation factors. My old student tables from 1980 list commutation and other factors at discount rates of 4%, 6% and 8% (the latter rate seems almost comically high by current standards).