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Browsing Mathematics & Statistics by Subject "Actuarial science"
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Item Open Access Multiple Risk Factors Dependence Structures With Applications to Actuarial Risk Management(2024-11-07) Su, Jianxi; Furman, EdwardActuarial and financial risk management is one of the most important innovations of the 20th century, and modelling dependent risks is one of its central issues. Traditional insurance models build on the assumption of independence of risks. Criticized as one of the main causes of the recent financial crisis, this assumption has facilitated the quantification of risks for decades, but it has often lead to under-estimation of the risks and as a result under-pricing. Hence importantly, one of the prime pillars of the novel concept of Enterprise Risk Management is the requirement that insurance companies have a clear understanding of the various interconnections that exist within risk portfolios. Modelling dependence is not an easy call. In fact, there is only one way to formulate independence, whereas the shapes of stochastic dependence are infinite. In this dissertation, we aim at developing interpretable practically and tractable technically probabilistic models of dependence that describe the adverse effects of multiple risk drivers on the risk portfolio of a generic insurer. To this end, we introduce a new class of Multiple Risk Factor (MRF) dependence structures. The MRF distributions are of importance to actuaries through their connections to the popular frailty models, as well as because of the capacity to describe dependent heavy-tailed risks. The new constructions are also linked to the factor models that lay in the very basis of the nowadays financial default measurement practice. Moreover, we use doubly stochastic Poisson processes to explore the class of copula functions that underlie the MRF models. Then, motivated by the asymmetric nature of these copulas, we propose and study a new notion of the paths of maximal dependence, which is consequently employed to measure tail dependence in copulas.Item Open Access Selected Computational Problems In Insurance(2025-04-10) Fleck, Andrew; Furman, EdwardThe 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.