Selected Computational Problems In Insurance

The coming together of digital data sets, computational power and rigorous probability theory has transformed finance and insurance in the last century. Once the purview of heuristics and an almost artisanal knowledge, these fields have increasingly taken on a scientific sophistication in technique. Modern regulations even require firms to retain the mathematical skill necessary to perform complex risk analysis. Mechanical heuristics originally developed in the absence of probabilistic assumptions have been demystified and reworked for novel applications. Complex structured products can be simulated and statistical learning algorithms can be applied to gain insights where none existed before. This dissertation is concerned with such problems.

In the realm of property and casualty insurance, this thesis addresses challenges in risk estimation, quantification and allocation when the risk can be modelled by multivariate Stable distributions, which we will argue provide a suitable null model in the case of heavy-tailed losses. Traditionally the lack of means and distribution functions has rendered these distributions difficult to work with. We will sidestep this issue in estimation by using an integral transformation-based method of estimation. For risk quantification, we develop computationally simple and efficient representations of commonly used risk measures. Allocation then follows from our choice of dependence structure.

In the area of life contingencies, we will study a relatively new product, the fixed index annuity (FIA). The variety of annuity parameters and the complexity of the underlying index make FIA comparisons very challenging. While still an insurance product, FIAs require sophisticated models of equity indices to analyze. We elect to use machine learning techniques to reproduce FIA-linked equity indices. In order to understand our often surprising results, we make use of a few stochastic volatility models.