Second-order Expansions Of Risk Concentrations For Aggregated Risk Variables

Risk measure is the premise and foundation of modern risk management, which provides quantitative information of risk to guide behavior of economic agents. Asymptotic study gives many uncovered performance and relationships between risk measures by their first order approximations, but 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.This thesis is devoted to the investigation of further properties of2RV, risk measures and risk concentration under the conditions of the second order regular variation.The main content and innovation points of the thesis contains are as follows.1. We investigate further properties of2RV. These properties include the preservation properties of2RV under the composition operation and the general-ized inverse transform, among others.2. we derive second-order expansions of the tail probabilities of convolutions of non-iid heavily tailed random variables, and establish second-order expansions of risk concentration under mild assumptions. The main results extend some ones in Barbe and McCormick (2005) and Degen et al.(2010) from the iid case to non-iid case.3. Based on the risk measure of the conditional tail expectation and the tail distortion risk measure, we establish second-order expansions of risk concentration as confidence level p↑1for a portfolio of n iid loss random variables with a common survival function possessing the2RV property.4. According to the max-domains of the Frechet, Weibull and Gumbel distri- butions, respectively we get the first-order asymptotics of spectral risk measures (SRMs). We also investigate the second-order expansions of operational risk SRMs under the theory of2RV and the second-order subexponentiality.