Distortion Risk Measures The Concentration Of Polymerization Risk

This thesis is devoted to risk measures and approximation properties of risk measures. Risk measure is the foundation of modern risk measures, which provides information of risk to guide behavior of economic agents. We get the equivalent characterizations of risk measures by researching the first-order asymptotic properties of risk measures.The first chapter of this thesis introduce some basic properties, such as the definition and the properties of distortion risk measure, coherent risk measure exhaustive distortion risk measure, as well as some common risk measures and how these common risk measure are expressed as distortion risk measures by using different distortion functions.The tail distortion risk measure at level p was first introduced in Zhu and Li(2012),where the parameter p ∈(0, 1) indicates the confidence level. They established first-order asymptotics for this risk measure, as p ↑ 1, for Fréchet case. The first order approximation is only a crude way to understand the tail behavior of risks, and the second order regular variation(2RV) provides a tractable tool for studying second order properties of risk measures.The second chapter mainly tells us some properties of 2RV and based on the risk measure of risk concentration the second-order expansions under the theory of 2RV and second-order asymptotics. We have known the first-order, second-order property of VaR and the risk concentration based on CTE. The main result of this paper is studying the asymptotics property of risk risk concentration based on ES.