Loss Modelling from First Principles - Report from the ASTIN Working Party

A common statistical modelling paradigm used in actuarial pricing is (a) assuming that the possible loss model can be chosen from a standard model dictionary; (b) selecting the model that provides the best trade-off between goodness of fit and complexity. Machine learning provides a rigorous framework for this selection and validation process.

An alternative modelling paradigm, common in the sciences, is to prove the adequacy of a statistical model from first principles: e.g., Planck’s distribution, which describes the spectral distribution of blackbody radiation empirically, was explained by Einstein by assuming that radiation is made of quantised harmonic oscillators (photons).

In this working party we have been exploring the extent to which loss models, too, can be derived from first principles. Claim count models traditionally used are the Poisson, negative binomial, and binomial distributions. They are used because they simplify the numerical calculation of the total loss distribution. We show how first-principle reasoning naturally leads to non-stationary Poisson processes, Levy processes, and multivariate Bernoulli processes depending on the context.

For modelling severities, we build on results from the paper by Parodi & Watson (2019) to show how graph (network) theory and epidemiological models can be used to model property-like losses. We note a tantalising relationship between the fire-spreading behaviour and whether the relevant exposure curve is in the Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac region of Bernegger’s MBBEFD curves. We show how the methodology can be extended to deal with business interruption/supply chain risks by considering networks with higher-order dependencies.

For liability business, we show the theoretical and practical limitations of traditional models such as the lognormal distribution (Benckert, 1962), and we consider the derivation of severity curves in territories where compensation tables are used and those where case law and the courts drive the compensation. We explore the question of where the power-law behaviour for some lines of business comes from, looking at criticality theory and universality laws.

This research is foundational in nature, but its results may prove useful to practitioners by guiding model selection and elucidating the relationship between the features of a risk and the model’s parameters.

Find the Q&A here: Q&A on 'Insurance Pricing and Maching Learning'