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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, wasexplained 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. Loss 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
reasoning from first principles naturally leads to non-stationary
Poisson processes, Lévy 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 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. We explore
the question of where the ubiquitous power-law behaviour comes
from, finding a natural explanation in random growth models. We
also address the derivation of severity curves in territories where
compensation tables are
used.
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.
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