(Mortality geeks only)
David Blake, Andrew Cairns and yours truly have just finished an article outlining a new(ish) mortality, model, CBDX. The purpose of the model is to offer a workhorse model that spans middle age as well as old age.
To recap: our original Cairns-Blake-Dowd (CBD) mortality model was specifically designed to capture the mortality behaviour of older people, e.g., people over 50. We were thinking of annuitants but equally it could apply to equity release borrowers, who must be at least 55.
Our original model had only two period (or passage of time) effects, the second of which enters the model through a coefficient that is a linear function of age. We then generalised then it to a CBD family consisting of 3 related models: M5, which is equivalent to a reconfigured CBD; M6, which is M5 plus a cohort (year of birth) effect; and M7, which is M6 plus a further period effect, which enters the model through a coefficient that is a quadratic function of age. More details on these models can be found here.
In subsequent work we discovered (to our surprise) that M7 performed robustly well across a number of different data sets. We had not expected a model with a quadratic function to perform as well as it did.
However, these models do not tend to perform well over age ranges that include younger ages. So one would not use the CBD family for, say, a model of a DC pension started at a youngish age. Andrew, David and I have long felt the need to remedy that limitation.
The solution, we felt, was to introduce an age-effect into a CBD-like model, but this turned out to be quite difficult to do. The programming dominated two successive summers’ work and I ended up writing a mortality econometrics toolbox to do the calculations.
One model that does have an age effect is Iain Currie’s age-period-cohort model: this model has one age effect, one period effect and one cohort effect. It is a good model in many respects, but a bit limited.
There is a second model, by Richard Plat, that adds further period effects. The Plat model comes in two versions, one with two period effects and the other with three period effects. The first Plat model is essentially the Currie model with an extra period effect, and the second period effect enters with a coefficient that is, like CBD-M5, linear in the age. The second Plat model is equal to the first Plat model with an extra, i.e., third, period effect. This third period effect has a coefficient that is a piecewise-linear function of age. We found that the first Plat model works fairly well but the second has identifiability problems and as far as I could reconstruct the model (which was no easy task) its performance was poor.
So what we did was replace the piecewise linear coefficient with a quadratic coefficient, which, remember, had worked well for M7.
And, lo and behold, our new CBDX model, or family of models:
CBDX1 = Currie APC model;
CBDX2 = 2 period-effect version of the Plat model; and
CBDX3 = CBDX2 + third period effect that enters the model via a coefficient that is a quadratic function of age.
It is this CBDX3 version that is our proposed workhorse model for the wider age ranges, i.e., the ones that start in middle age rather than old age.
All of which makes the process seem a whole lot more straightforward than it actually was.