Analysis of Risk Measures and Multi-dimensional Risk Dependence

In this thesis, we try to provide a broadeconometric analysis of a class of risk measures, distortion risk measures (DRM). With carefully selected functional form, theValue-at-Risk (VaR) and Tail-VaR (TVaR) are special cases of DRMs. Besides, the DRM also admits interpretation in the sense ofnon-expected utility type of preferences. We first provide a unified statistical framework for the nonparametric estimators ofthe DRMs in a univariate case. The asymptotic properties of both theDRMs and their sensitivities with respect to the parameters representing risk aversion and/or pessimism are derived. Moreover,the relationships between the VaR and TVaR are also investigated in detail, which, we hope, can shed new lights on the way passing one risk measure to another. Then, the analysis of DRMs are extended to a multi-dimensional framework, where the DRM is computed for a portfolio consisting of many primitive assets. Analogous to themean-variance frontier analysis, we study the efficient portfolio frontier when both objective and constraint are replaced by theDRMs. We call this the DRM-DRM framework. Under a nonparametric setting, we propose three asymptotic test statistics for evaluating the efficiency of a given portfolio. Finally, we discuss thecriteria used for evaluating models used to forecast the VaRs. More precisely, we propose a criterion which takes into account the loss levels beyond the VaRs.