The problem

One aspect of the CMI’s mortality and morbidity experience reports is that they do not account for the variability in experience between insurers. Significant variation exists because insurers differ in the customers they target and how they underwrite risks and manage claims. As a consequence, differences in the mix of contributors may distort the analysis of the pooled data and lead to results that might not reflect the “true” underlying pattern and trends in experience.

A statistical model and computer algorithms were required to analyse this heterogeneous pooled data and ultimately produce estimates that are adjusted to compensate for variations in the mix of contributors. Meeting the confidentiality requirement of the CMI, and making the techniques involved re-applicable to future data input and types of analysis, are some additional criteria that made the project even more challenging.

“This internship was a wonderful opportunity for me to learn and apply statistics to solve a real problem and gain experience in the financial services industry and it has had an immediate and beneficial impact on my career. I have learnt and improved many interpersonal skills while working alongside Jon, Neil and his colleagues on this challenging project and I am so grateful to them. I am also proud having used “Bayesian” statistics to such a level, considering that my PhD was more in the “frequentist” side of statistics. I can now say that I hunger for more knowledge and projects after this exceptional experience.” said intern Pascal Ah-Kine, PhD student, University of Southampton.

The approach

First, it was essential to understand the nature of the CMI’s pooled data and the heterogeneity within it. There are publishable explanatory factors, such as year, age and gender (among others), and there is a non-publishable explanatory factor which is the relative effect of each of the contributing insurers. Second, a response that would be determined using those explanatory factors and that would relate to the observations in the data was required. The ratio of actual values to expected events (where events could be deaths, claims, etc) was chosen for reporting purposes. Once it was understood how each explanatory factor could affect the response, a well-developed and parameterised model was built.

A Bayesian hierarchical model was developed where variability within and between insurers was modelled at two different levels of the hierarchy. Hierarchical models have previously been used in Health Services and Education where the hierarchy involves, for example, variability at student, and at school level. Computer-intensive Markov Chain Monte Carlo methods were used to estimate the parameters of the model, and quantify the associated uncertainty. Those estimates could then be used to provide mortality ‘tables’ produced under the model and to compare them with actual experience.

In addition to a model that can be used to calculate the ratio of actual to expected events for different types of investigations, probability intervals for those ratios were also made readily available as a spin-off of the computer simulations. Input and output scripts that are flexible to changes and addition of future data have also been developed for the benefit of future CMI reporting.

“Pascal has brought his skills in statistical modelling to an important application for the CMI, and his energy and drive have overcome demanding technical challenges. The support offered by Prof. Jon Forster, whose depth of knowledge in Bayesian methods has been crucial, has been first class. The project has delivered models of immediate value to the CMI and enough knowledge for us to continue the work.” said Neil Robjohns, Industrial Supervisor, Barnett Waddingham, for the CMI.

References

[1] Gelman, A., Hill, J.: Data Analysis Using Regression and Multilevel/Hierarchical Models, 2007, Cambridge.
[2] Robert, C. P., Casella, G., Monte Carlo Statistical Methods, 2nd edition, 2004, Springer.