For a recent survey, Barnett Waddingham gathered information on Solvency II capital requirements for longevity from eight internal model firms.
We found significant variations in assumptions. For males ages 65, the increase in life expectancy between best-estimate and a 1-in-200 event was over twice as large for some companies as for others.
While some of this difference can be explained by reasonable variation in companies’ perceptions of longevity risk, we are concerned that some of the difference may be due to weaknesses of particular models.
"It is important for users of longevity models to understand which models have spurious parameters, and to be aware of the weaknesses of those models."
Most longevity models express mortality using parameters indexed by age, period (calendar year) and cohort (birth year). It is natural to assume that these parameters reflect corresponding historical effects – e.g. period parameters reflect patterns of year-by-year mortality rates. But for some models this is not the case.
For example, the structure of the Age-Period-Cohort (APC) model is not detailed enough to directly reflect patterns of mortality improvement by age. Instead the APC model has to ‘cheat’ and make use of period and cohort parameters to mimic age affects. This means that the period parameters do not purely reflect period effects; they are ‘spurious’.
The standard approach to projecting the APC model is to assume a steady linear trend in the period parameters. If the period parameters purely reflected period effects, then this would reflect an assumption of steady period mortality improvements. However, because the spurious period parameters also reflect age effects, our research  shows that the standard approach implicitly assumes that period improvements will increase indefinitely without limit. This seems implausible, and we would not expect any companies to make such an assumption explicitly.
This finding affects several, but not all, models; and it can explain differences in their projections. For example, the best-estimate projections from the APC model are more extreme than the 1-in-200 stress from the Cairns-Blake-Dowd (CBD) model . In the past this had been considered as 'model risk'. However our research shows that this difference instead stems from a weakness of the APC model.
Impact on capital requirements
It is important for users of longevity models to understand which models have spurious parameters, and to be aware of the weaknesses of those models. These concerns also affect the PRA’s view of risk.
Our survey shows that the PRA’s view of capital requirements is significantly higher than the view of many companies. The PRA has described its use of 'quantitative indicators'  as one part of its review of a company’s longevity risk. Over a one-year horizon, the PRA considers two distinct components of risk:
- 'data risk' – risk emerging from that year’s mortality data; and
- 'event risk' – risk emerging from new information that is not that year’s mortality data.
The PRA uses model risk as a proxy for event risk. It compares best-estimate outcomes from four 'model families'. These include the APC model, so unreliable projections from that model can lead to an unreliable estimate of event risk. It is possible that some of the other models used by the PRA may also suffer from a similar problem, but this is not clear from the PRA’s brief description of the models.
The PRA’s approach means that it is important for companies to have a clear understanding of not only their preferred longevity model, but also those models considered by the PRA. Understanding the weaknesses of these models will allow companies to robustly defend their views. Barnett Waddingham’s research means that we are in a strong position to help insurers in their negotiations with the PRA to ensure capital is being held as efficiently as possible.
 When is a cohort not a cohort? Spurious parameters in stochastic longevity models - presented at the International Mortality and Longevity Symposium, 8 September 2016.
 This was noted in discussion of A Value-at-Risk framework for longevity trend risk.