Asymptotics for Risk Measures of Extreme Risks

This thesis focuses on measuring extreme risks in insurance business. We mainly use extreme value theory to develop asymptotics for risk measures. We also study the characterization of upper comonotonicity for multiple extreme risks.

Firstly, we conduct asymptotics for the Haezendonck--Goovaerts (HG) risk measure of extreme risks at high confidence levels, which serves as an alternative way to statistical simulations. We split the study of this problem into two steps. In the first step, we concentrate on the HG risk measure with a power Young function, which yields certain explicitness. Then we derive asymptotics for a risk variable with a distribution function that belongs to one of the three max-domains of attraction separately. We extend our asymptotic study to the HG risk measure with a general Young function in the second step. We study this problem using different approaches and overcome a lot of technical difficulties. The risk variable is assumed to follow a distribution function that belongs to the max-domain of attraction of the generalized extreme value distribution and we show a unified proof for all three max-domains of attraction.

Secondly, we study the first- and second-order asymptotics for the tail distortion risk measure of extreme risks. Similarly as in the first part, we develop the first-order asymptotics for the tail distortion risk measure of a risk variable that follows a distribution function belonging to the max-domain of attraction of the generalized extreme value distribution. In order to improve the accuracy of the first-order asymptotics, we further develop the second-order asymptotics for the tail distortion risk measure. Numerical examples are carried out to show the accuracy of both asymptotics and the great improvements of the second-order asymptotics.

Lastly, we characterize the upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the HG risk measure of the sum of upper comonotonic random variables with exponential marginal distributions.

The methodology developed in this thesis is expected to work with the same efficiency for generalized quantiles (such as expectile, L^p-quantiles, ML-quantiles and Orlicz quantiles), quantile based risk measures or risk measures which focus on the tail areas, and also work well on capital allocation problems.