α混合样本优化型CVaR估计的大样本性质
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摘要
在经济金融领域里,VaR是一个被广泛应用的风险度量,而且巴塞尔协议规定金融机构利用VaR来刻画金融风险和做相应的风险管理,但是在实际应用中,VaR却存在着一些不足之处,缺乏次可加性,测量的风险值相对大小不能完全反应真实风险状况.为了弥补VaR的不足,有学者提出条件风险价值CVaR(Conditional Value at Risk),而且Pflug(2000)指出可以将CVaR看成某一最优化问题的解,即损失变量X的置信水平为(1-α)%的CVaR可表示为：其中[α]+：=max{0,α}.设X1,X2,…,Xn为总体X的一组样本,A. Alexandra T(2007)等人给出了该优化形式下CVaR的估计并在独立同分布条件下讨论了该估计的相合性和渐近正态性,但没有给出这两个性质中任何一个的收敛速度.罗中德[47](2010)研究了ρ混合序列下CVaR估计的渐近性质.至于α混合样本的CVaR,该估计的渐进性还未见学者研究.本文将该优化估计推广到α混合样本条件下,并得出其收敛速度.一般地,金融、经济时间序列的样本并非独立,样本相依性则是它们固有的特性.特别地,a混合是金融数据中比较常见的混合形式.因此,研究在a混合样本条件下该估计的渐近性质有着重要的理论价值和应用价值.
本文在α混合序列下研究了上述CVaR估计的大样本性质,研究的主要内容和结果如下：
首先,讨论在样本为α混合序列的情形下CVaR估计的强相合性,并证明当a混合序列满足一定假设条件时,强相合的收敛速度为n-,其中：(i)当样本矩的阶数r≥2时,可以取任意的0≤κ<1/2；(ii)当1≤s≤r<2时,可以取κ=1-1/s.
其次,讨论在样本为α混合序列的情形下CVaR估计的一致渐近正态性,并且给出一致渐近正态的收敛速度,其收敛速度约为n-1/6.
再次,对一些a混合序列的CVaR进行随机模拟,并比较了优化估计方法与次序统计量方法所得估计的优劣,通过数值模拟得知,CVaR的优化型估计可以更好地处理α混合样本,而且相对于次序统计量方法而言,其误差更小、精度更高,特别是当样本容量较少时,优化方法的优越性更加显著.
最后,对我国股市的上证医药和上证材料的CVaR进行估计,由计算结果得出,在相同概率水平,上证医药的CVaR要小于上证材料的CVaR,即上证医药的风险要小于上证材料的风险.
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 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.
本文在α混合序列下研究了上述CVaR估计的大样本性质,研究的主要内容和结果如下：
首先,讨论在样本为α混合序列的情形下CVaR估计的强相合性,并证明当a混合序列满足一定假设条件时,强相合的收敛速度为n-,其中：(i)当样本矩的阶数r≥2时,可以取任意的0≤κ<1/2；(ii)当1≤s≤r<2时,可以取κ=1-1/s.
其次,讨论在样本为α混合序列的情形下CVaR估计的一致渐近正态性,并且给出一致渐近正态的收敛速度,其收敛速度约为n-1/6.
再次,对一些a混合序列的CVaR进行随机模拟,并比较了优化估计方法与次序统计量方法所得估计的优劣,通过数值模拟得知,CVaR的优化型估计可以更好地处理α混合样本,而且相对于次序统计量方法而言,其误差更小、精度更高,特别是当样本容量较少时,优化方法的优越性更加显著.
最后,对我国股市的上证医药和上证材料的CVaR进行估计,由计算结果得出,在相同概率水平,上证医药的CVaR要小于上证材料的CVaR,即上证医药的风险要小于上证材料的风险.
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
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.
引文
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[26]李燕华.我国证券市场的VaR与CVaR方法比较实证分析[J].西安山西财经大学学报,2007,29(1)：132-135.
[27]陈剑利,李胜宏CVaR风险度量模型在投资组合中的运用[J].运筹与管理,2004,13(1)：95-99.
[28]田新民,黄海平.基于条件VaR(CVaR)的投资组合优化模型及比较研究[J].数学的实践与认识,2004,34(7)：41-50.
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[2]A Alexandre Trindade, Stan Uryasev, Alexander Shapiro. Financial Prediction with Con-strained Tail Risk[J]. Journal of Banking and Finance,31(2007):3524-3538.
[3]Rockafella R T and S Uryasev. Optimization of Conditional Value-at-Risk[J]. The Journal Risk,2000,3(2):21-24.
[4]Uryasev S. Conditional Value-at-Risk:Optimization Algorithms and Applications[J]. Finan-cial Engineering News. can be downloaded from:www.gloriamundi.org.2000.
[5]Pownall R A,Koedijk K G. Capturing Downside Risk in Financial Markets:The Case of The Asian Crisis[J]. Journal of International Money and Finane,1999,1999(18):853-870.
[6]Acerbi C. Expected Shortfall:a natural coherent alternative to Value at Risk[J],2001,b(5).
[7]Rockafellar R T and S Uryasev. Conditional Value-at-Risk for General Loss Distributions [J]. Journal of Banking and Finance,2002, (26):1443-1471.
[8]Acerbi C. Spectral Measures of Risk:A Coherent Representation of Subjective Risk Aver-sion[J]. Journal of Banking and Finance,2002,26(7):1505-1518.
[9]Topaloglou N, Vladimirou H, Stavros A Zenios. CVaR models with selective hedging for international asset allocation[J]. Journal of Banking and Finance,2002,26(7):1535-1561.
[10]Acerbi C, Tasche D. On the Coherence of Expected Shortfall[J]. Journal ofBanking and Finance,2002,26(7).1487-1503.
[11]A LKibzun and E A Kuznetsov. Comparison of VaR and CVaR Criteria[J].Automation and Remote Control,2003,64(7):1154-1164.
[12]A LKibzun and E A Kuznetsov. Analysis of criteria VaR and CVaR[J]. Journal of Banking Finance,2006,30:779-796.
[13]Andersson F, Mausser H, Rosen D, et al. Credit Risk Optimization with Condi- tional Value-At-Risk Criterion[J]. Mathematical Programming,2000(B),89:273-291.
[14]Albert M. Conditional Value at Risk Optimization of a Credit Bond Portfolio[J]. can be downloaded from:www.gloriamundi.org,2004.
[15]Jonas Palmquist, Stanislav Uryasev and Pavlo Krokhmal. Portfolio Optimization with Con-ditional Value-at-Risk and Objective and Constraints[J]. The Journal of Risk,2002,4(2): 124-129.
[16]Alexander G, Baptista A. CVaR as a Measure of Risk:Implications for Portfolio Selection[J]. can be downloaded:www.gloriamundi.org.2003.
[17]David B. Brown. Large deviations bounds for estimating conditional value at-risk[J]. Oper-ations Research Letters,2007,35(2007):722-730.
[18]Jabr, Rabih A. Robust self-scheduling under price uncertainty using conditional value-at-risk[J]. IEEE Transactions on Power Systems,2005,20(4):1852-1858.
[19]Jobst N J, Mitra G, Zenios S A. Integrating market and credit risk:A simulation and opti-mization perspective[J]. Journal of Banking and Finance,2006,30(2):717-742.
[20]Mulvey J M, Erkan H G. Applying CVaR for decentralized risk management of financial companies[J]. Journal of Banking and Finance,2006,30(2):627-644.
[21]Webby R B, Adamson P T, Boland J, Howlett P G, Metcalfe A V, Piantadosi J. The Mekong-applications of value at risk (VaR) and conditional value at risk (CVaR) simulation to the benefits, costs and consequences of water resources development in a large river basin[J]. Ecological Modeling,2007,201(1):89-96.
[22]Zhou Y J, Chen X H, Wang Z R. Optimal ordering quantities for multi-products with stochas-tic demand:Return-CVaR model[J]. International Journal of Production Economics, In Press, Available online 28 July 2007.
[23]Meng Zhi-qing, Yu Xiao-fen, Jiang Min, Gao Hui. Risk Measure and Control Strategy of Investment Portfolio of Real Estate based on Dynamic CVaR Model[J]. SETP,2007,27(9): 69-76.
[24]王筱筱.上海股市VaR与CVaR的密度核估计[J].商业现代化,2009,7(下)：135-137.
[25]席金平,吴润衡CVaR在商业银行风险度量中的比较分析和应用[J].北京山西财经大学学报,2009,31(1)：193.
[26]李燕华.我国证券市场的VaR与CVaR方法比较实证分析[J].西安山西财经大学学报,2007,29(1)：132-135.
[27]陈剑利,李胜宏CVaR风险度量模型在投资组合中的运用[J].运筹与管理,2004,13(1)：95-99.
[28]田新民,黄海平.基于条件VaR(CVaR)的投资组合优化模型及比较研究[J].数学的实践与认识,2004,34(7)：41-50.
[29]蒋敏.条件风险值(CVaR)模型的理论研究[D].西安：西安电子科技大学博士学位论文,2005.
[30]吴军,汪寿阳CVaR与供应链风险管理[A].见：中国运筹学会第七届学术交流会论文集[C].2004：633-640.
[31]Chanda K. C.Strong Mixing Properties of Liner Stochastic Processess [J].Journal of Applied Probability,1974,11:401-408.
[32]Withers C S. Conditions for linear processes to be strong mixing.Z.Wahrsverw.Gebiete, 1981,57:477-480.
[33]Pham T D, Tran L T.Some Mixing Properties of Time series Models[J].S toch.Process. Appl.1985,19:297-303.
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