Extreme Value Theory and Extremal Dependence with Application to Real-Life Insurance Data

Specialeforsvar: Camilla Lundtoft Kristensen

Titel: Extreme Value Theory and Extremal Dependence with Application to Real-Life Insurance Data

Abstract:  This thesis investigates fire insurance claims from a Danish insurance company using extreme value theory in one and two dimensions. The insurance data includes two components: building and content losses. The primary objective is to analyze the tail-behavior of each component individually and subsequently explore the extremal dependence between the two components. First, we present some primary results from univariate extreme value theory. Analyzing building and content losses individually, with methods from the univariate theory, we explore their heavy-tailed behaviour, finding them to be in the maximum domain of attraction of the Fréchet distribution. Under this assumption, we use various estimation methods to estimate the tail-distribution and high quantiles of the two components.
Extending the study to paired losses, we utilize bivariate extreme value theory, emphasizing the importance of the spectral measure in assessing asymptotic dependence. We transform data with tail-estimates from the univariate analysis to ensure the two components share the same tail-distribution. Estimating tail-indices, spectral measures, and probabilities of different failure
sets, we learn that the paired observations with only one component equal to 0 dominate the dependence structure completely. This results in asymptotic independence within the full data. Due to this dominating behaviour, we analyse the data where both paired components are greater than 0, resulting in a higher degree of asymptotic dependence. We also learn that these claims are not as heavy-tailed as first anticipated. Lastly, we estimate the coefficient of upper tail dependence.

Vejleder: Thomas Mikosch
Censor:   Mette Havning