Research On Measures Of Coherent Variability And Risk

Along with the development of human civilization,the form of competition be-tween countries is quiet changed.Today,financial markets have become more and more important between the two countries5 competition.With the researches on fi-nancial markets,people have proposed a series of theoretical systems on financial ac-tivities.Among the theoretical systems,risk management,as an issue that cannot be avoided by financial institutions,has been paid more and more attention by investors.Risk measure,as an effective tool to quantify risk,plays a very important role in the risk management process.Based on the theory of risk measurement,this paper investigated the cumulative residual entropy(CRE)and the distortion distance risk measure family,and used the combination of the variability measure and the conherent risk measure,to deduce the Shortfall risk measures of the two volatility measures.With the actual finan-cial data,we estimate the investigation results of paper,and compare the performance of risk measures.At the same time,we also study the extreme-aggregation measure in this paper,and combine the extreme-aggregation measure with the rank-dependent expected utility model.In Chapter 1,we introduce the research background and development perspective of risk measure,as well as the related studies and innovations of this dissertation.Chapter 2 is mainly about the literature review,reviewing the development pro-cess of risk measure and listing many predecessors' work results.Many of them are related to the content of this dissertation.Also we add our own ideas about the review,leading to the following introduction to the results of the study.In Chapter 3,we begin to show our main research about variability measure,choosing the cumulative residual entropy(CRE)as the origin of the research.First-ly,The Chapter focuses on the definition of cumulative residual entropy and prove that it can be written in the form of Choquet integral,which means that it is a coherent volatility measure with comonotonically additive.Next,the research turn the direction to the tail of distribution function,because in the real world,the extreme case is more concerned.Unfortunately,the tail cumulative residual entropy TCRE does not have subadditive.Therefore,refering to the predecessor's idea,we combine the coherent risk measure with the tail cumulative residual entropy to get the Shortfall measure,and adjust the parameters to make it have excellent statistical properties,also give a simple expression of the Shortfall measure and extend it to the general random variable at the same time.After the study on the cumulative residual entropy,we pay our attention to the dis-torted distance risk measure family in Chapter 4,under the coherent risk measure axiom,give the necessary parameter threshold to meet the coherent risk measure.Furthermore,by selecting special distortion functions,we list several risk measures included in the distorted distance risk measure family.The Chapter 5 applies the investigated results to practics,using stock index data to make an estimate of the variability risk measures included in Chapter 3 and Chapter 4.The cumulative residual entropy Shortfall measure,and the special distorted distance Shortfall measures.We compare the performance of the cumulative residual entropy Shortfall measure under different stock indices,as well as the figures of special distor-tion distance Shortfall measures under different parameters,and study the influence of parameters on these measures.In Chapter 6,the extreme-aggregation measures become research object,a risk measure used to characterize superadditive.Based on the work of others,the extreme-aggregate measures combine with the rank-dependent expected utility model.In this chapter,we give the explicit forms of the extreme-aggregation measure induced by the rank-dependent expected utility model and the Shortfall measure of this extreme ag-gregation measure,These two explicit forms make the extreme-aggregation measure forward to practical applications.Finally,in Chapter 7,we summarize the work of the dissertation,and also point put the problems found during investigation process.These will be the direction of future work and research plan.