The Large Sample Properties Of CVaR Estimator Under α Mixing Series

VaR is a risk measure which is widely used in the economic financial field, and the Basel Accord requires financial institutions must to use VaR to characterize the financial risks and make the corresponding risk managements, however, the VaR there are some shortcomings in practical applications,Lack of Subadditivity, measuring risk is not really response risk situation. To make up for the lack of the VaR, some scholars gave the CVaR (Conditional Value at Risk) which is a new risk measure, and Pflug(2000) pointed out that the CVaR can be viewed as the solution of an optimization problem, namely, the CVaR of the loss variable X with the confidence level (1-α)% can be defined as where [a]+:= max{0, a}. Notes X1, X2,…, Xn are a set of samples of a population X, A. Alexandre T(2007) and the other scholars, who gave the optimal estimate of the CVaR At the same time, they had discussed the consistency and asymptotic normality of the estimator under the independent and identically distributed samples, but they didn't give their convergence rate of above properties. Luozhongde(2010)has researched the consistency and asymptotic normality of the estimator under the p mixing. However, this estimator has not been researched by few scholars under the a mixing.In the paper,I have discussed the consistency and asymptotic normality of the estimator under the a mixing and get the consistency and asymptotic normality of convergence rate.As generally, Financial and economic time series samples are not independent, and the sample dependence is their inherent characteristics. Particular, the a mixing is the more common mixed form in the financial data. Therefore, It has important theoretical value and appli-cation value to study the asymptotic properties of this estimator under the a mixing sequences.In this paper, we have study the large sample property of the above CVaR estimator under the a mixing 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 cases where samples are a mixing random sequences. And the convergence rate of the strong consis-tency is n-κwhen the a mixing random sequences are satisfying certain assumptions, where:(ⅰ) When sample moments r≥2, we can take any 0≤κ<1/2; (ⅱ) when 1≤s<r<2, we can takeκ=1-1/s.Secondly, the paper discusses the uniformly asymptotic normality of CVaR estimator in cases where samples are a mixing random sequences, and the convergence rate of uniformly asymptotic normality is given,that the convergence rate of the uniformly asymptotic normality is about n-1/6.Thirdly, the CVaR of some a mixing sequences are random simulated in the paper, and the pros 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 with a mixing data effectively while we are calculating CVaR, but also the error of the optimal method is smaller than the order statistics method, and higher accuracy, particularly, the optimal method there are more significant advantages when the sample size is fewer.Finally, the CVaR of the Shanghai Medicine Index and the CVaR of the Shanghai Materials Index on China's stock market are estimated. From the calculating results we know that the CVaR of the Shanghai Medicine Index is less than the CVaR of the Shanghai Materials Index under the same probability level, namely, the risk of the Shanghai Medicine Index is less than the risk of the Shanghai Materials Index.