Mortality crossover

In a previous blog I discussed the importance of mortality convergence to actuaries, i.e. how mortality differentials narrow with age and how this interacts with discounting of cashflows. The traditional approach to allowing for mortality convergence is to have two parameters, one for the main effect of a risk factor and a second to allow the impact to vary by age. This second parameter is known as an interaction. However, one unwelcome side-effect of this is mortality crossover, an example of which is shown in Figure 1:

Figure 1. Gompertz model by age and pension size, showing mortality convergence but also crossover between ages 85 and 90. Source: Macdonald et al (2018).

As pointed out in Richards (2019), there are five drawbacks to traditional interactions with age in mortality modelling:

1. The age interaction causes crossover. While there is usually clear evidence of a mortality differential up to age 90, there is usually no evidence of a differential above age 95. The crossover is therefore an artefact of the model that is unsupported by the data.

2. Model distortion. Since the data do not support crossover, the model fit tries to balance that fact against the mortality differential at younger ages. The model thus ends up under-stating the size of the differential at younger ages.

3. The crossover is not actuarially prudent at all ages. In Figure 1 those with larger pensions have lower mortality below age 85, which is fine. However, it is not actuarially prudent to assume that they have higher mortality above age 90, as forced by the model structure in Figure 1.

4. Estimation of the age interaction is tricky. It is one thing to have enough data to estimate a difference between two sub-groups. However, more data are required to detect the subtler change of that risk factor with age. Not every portfolio has enough data for this.

5. Redundancy. If you know that the mortality differential will largely vanish by age 90–95, the age-interaction parameter is in one sense redundant — one could derive its value from knowledge of the main effect, \(\alpha\) say, and the assumption that the age interaction must be such that the effect is zero at age 95. This would mean that the interaction was \(-\alpha/95\), and that no second parameter would need to be estimated.

How do we get rid of these five drawbacks? When modelling post-retirement mortality we would ideally have the following:

1. No redundant parameters.

2. Automatic convergence with age with a single parameter per risk factor.

3. No crossover.

How do we achieve this? As it happens, this is what we get from the Hermite-spline approach to mortality modelling, the technical details of which are in Richards (2019). The Hermite family offers parsimonious mortality models with automatic convergence and guaranteed no crossover.