All about the base(line)

When we first developed a technique for putting longevity trend risk into a 1-in-200 framework consistent with Solvency II, we sought to accommodate model risk by supporting a wide range of stochastic projection models. To this end, our VaR approach from 2012 supported both time-series and penalty projections across a range of model families, each with their own properties and approach. Since forecasters cannot know which model will produce results closest to the path of future mortality, a range of different models will best inform actuarial judgement.

The 2D P-spline family proposed by Currie, Durban and Eilers (2004) is a distinctive family of projection models. Both Age-Period and Age-Cohort variants use penalised smoothing as the intrinsic basis for the forecast and the optimisation of Age/Year smoothing factors and an overdispersion parameter occupies a substantial portion of the runtime. Such penalty models have specific forecasting properties. One notable characteristic of the 2D P-spline models is the often lengthy runtime usually provides for an superior fit to population data, far better than the fit you might expect from, say, models in the Lee-Carter family. However, this improved fit has a price: we can expect the models to be more sensitive to data changes year on year.

These features have repercussions in the automated value-at-risk approach to longevity trend risk, where each simulation projection uses a sample path from the baseline model to generate a hypothetical future-year of data to refit against. Even in the presence of parallel processing, slower projection runtime will necessarily make for a longer-running VaR. Further, projection sensitivity may sometimes stray across the border into volatility. Simply put, unattended reoptimisation of the overdispersion parameter and smoothing factors introduces the danger that too many instances of undersmoothing or other forecast problems might skew the VaR capital. In Richards, Currie and Ritchie (2012) we found that the 2D Age Period model had the highest capital requirement of all models tested - an ironic repayment for the fact it also had the longest VaR calculation runtime.

P-spline models benefit from an attentive analyst adjusting factors such as spline degree, penalty order, knot spacing and knot positioning to create a satisfactory baseline forecast with sensible smoothing and overdispersion values. Rather than jeopardise some of this investment by continual reoptimisation of smoothing and overdispersion within VaR, our latest paper Longevity trend risk over limited time horizons explores baseline-driven smoothing. In this approach, smoothing and overdispersion factors from the baseline model are imposed upon each and every VaR simulation. Aware that these baseline values would, by definition, be non-optimal for the simulated dataset, our work was exploratory; we simply hoped to evaluate the impact on capital and runtime while reducing model failure arising from extreme optimisation results. What we found was wholly positive, and is worth listing:

• For 2D Age Cohort model there was no impact on capital, but run time was reduced from thirty hours to less than three.
• For 2D Age Period capital was impacted with results clearly less volatile (and more sensible) than in the re-optimised case. Run time was reduced from seventeen hours to around one hour.
• Other smoothed models tested revealed no capital impact but performance improved by up to 50 percent.

With equivalent or objectively better results inside a shorter time-window, it appeared that we had stumbled over that most elusive of commodities: the free lunch. This feature increases the applicability of 2D models to use-cases where slow runtime performance had previously excluded them. And this is welcome: in an area where model risk is the only certainty, balancing the crowded field of ARIMA and drift models with a distinctive class of penalty projections provides a better foundation for actuarial judgement.