The Asymptotic Properties Of CVaR Estimator Under ρ Mixing Sequences

VaR is a risk measure which is widely used in the financial field, and the Basel Accordrequires financial institutions must to use VaR to characterize the financial risks and make thecorresponding risk managements, however, the VaR there are some shortcomings in practicalapplications. To make up for the lack of the VaR, some scholars gave the CVaR (ConditionalValue-at-Risk) which is a new risk measure, and P?ug [1] (2000) pointed out that the CVaR can beviewed as the solution of an optimization problem. namely, the CVaR of the loss variable X withthe confidence level (1 ?α)% can be defined aswhere [a]+ := max{0,a}. Notes X1,X2,···,Xn are a set of samples of a population X, A.Alexandre Trindade [2](2007) and the other scholars, who gave the optimal estimate of the CVaRAt the same time, they had discussed the consistency and asymptotic normality of the estimatorunder the independent and identically distributed samples,but they didn't give their convergencerate of above properties. However, this estimator has not been researched by any scholars underthe mixing samples. As generally, Financial and economic time series samples are not indepen-dent, and the sample dependence is their inherent characteristics. Particular, theρmixing is themore common mixed form in the financial data. Therefore, It has important theoretical value and application value to study the asymptotic properties of this estimator under theρmixingsequences.In this paper, we have study the asymptotic property of the above CVaR estimator under theρmixing random sequences, the main research contents and results are as follows:First of all, the paper discusses the strong consistency property of CVaR estimator in caseswhere samples areρmixing random sequences. And the convergence rate of the strong consis-tency is n?κwhen theρmixing random sequences are satisfying certain assumptions, where: (i)When sample moments r≥2, we can take any 0≤κ< 1/2; (ii) when 1≤r < 2, we can takeκ= 1 ? 1/r.Secondly, the paper discusses the uniformly asymptotic normality of CVaR estimator in caseswhere samples areρmixing random sequences, and the convergence rate of uniformly asymptoticnormality is given,that the convergence rate of the uniformly asymptotic normality is about n?1/6.Finally, the CVaR of someρmixing sequences are random simulated in the paper, and thepros and cons of this optimal estimation method are compared with the order statistics method.We know, through the numerical simulation, that not only this method can deal withρmixing dataeffectively while we are calculating CVaR, but also the error of the optimal method is smaller thanthe order statistics method, and higher accuracy, particularly, the optimal method there are moresignificant advantages when the sample size is fewer. As subsequently, the CVaR of the ShanghaiComposite Index and the CVaR of the Shenzhen Component Index on China's stock market areestimated. From the calculating results we know that the CVaR of the Shanghai Composite Indexis less than the CVaR of the Shenzhen Component Index under the same probability level, namely,the risk of the Shanghai Composite Index is less than the risk of the Shenzhen Component Index.