Gaussian Process Models for Actuarial Science

We have written in collaboration Professor Michael Ludkowski and Ph.D candidate James Risk of U.C. Santa Barbara, a paper introducing Gaussian Processes (GPs) to the actuarial community.  Our paper has focused on a practical implementation of GPs in mortality analysis.  A pre-publication version of the paper can be found here:

GPs are an extremely powerful tool that have broad applicability across many areas of actuarial science. The paper shows how GPs can be used as a means of graduating mortality rates across multiple dimensions and that they provide uncertainty quantification around mortality improvement rates and mortality predictions.  The actuarial community is by and large still employing trusted techniques such as the Whittaker-Henderson graduation method, developed almost 100 years ago (see for example the RP2014/RP2015 tables)..  We show that GPs have a number of significant advantages over traditional actuarial techniques, including:

  • GPs estimate the uncertainty of the mortality curves rather than merely providing point estimates. This becomes increasingly important as we look at improvements in mortality rates.  Noisy estimates of the underlying mortality are amplified when assessing improvement rates.  Such noise amplification is not observable when analyzing a mortality table graduated using standard actuarial techniques, but is immediately understood when using the GP method.
  • GPs are less susceptible to “edge” issues. In the SOA’s pension tables, data from the last two years of mortality experience is partly ignored because the Whittaker-Henderson technique is not reliable at the “edge” of the data.  Such dropping of information is not necessary with GPs which have self-calibrating credible intervals.

GPs can be naturally updated with new emerging data.  This provides a means of efficiently keeping models up-to-date.Beyond the technical aspects of GPs, our paper clearly shows some interesting aspects of mortality that we will be discussing in more detail over time on this blog. In particular, there has been a significant decline in rates of mortality improvement over recent years.  There has been much discussion in the press (see Washington Post for example) about the worsening of mortality particularly with white, female, middle aged and more rural people in the US.  Our paper shows that the deterioration in mortality is in fact quite wide-spread and reaches into the older ages for both males and females.  This deterioration  is not just a female, middle aged issue! Further there is a clear divergence in the rate of mortality improvement as exemplified by the Society of Actuaries recently published tables (see MP2014) and the actual mortality improvement occurring in the US.  This is shown in the following chart, extracted from the paper:

Mortality Improvement 2014

The red line shows the raw mortality improvement rates for males observed in the CDC data, the green line is the Society of Actuaries estimate of improvement, based on the CDC experience and the blue line is our GP estimate, again based on the CDC experience.

It is clear that the SOA rates show a reasonable representation of population mortality improvement in 2000, but by 2014, there appears to be a material divergence. This divergence has broad implications, particularly since the SOA mortality improvement scales are almost universally used by pension funds to value their future obligations.  Lower mortality improvement assumptions could potentially reduce pension obligations for many companies.  Interestingly, in the UK, recent mortality investigations are also showing a pause in mortality improvement, with one UK actuarial study stating “both the one-year and four-year mortality improvements to 2015 are lower than at any other time in the 41-year data period used to calibrate the Mode”.

GPs can be used in many areas beyond graduation.   As a topical and entertaining application of GPs, we have shown how GPs can be used to track the presidential elections (see here and here).  GPs are particularly valuable when analyzing non-linear systems. Over the coming weeks, we will extend our discussion to other ways in which actuarial analysis can be significantly improved using the GP approach.