金融资产的离散过程动态风险度量研究
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摘要
随着金融市场的波动性不断加剧,金融衍生工具所蕴涵的风险结构越来越复杂,金融风险的危害性不但引起了各国政府和金融界对金融风险管理的密切关注,也使得其自身越来越成为现代企业和金融机构面临的主要风险之一,金融风险管理也因此成为金融工程与现代金融理论的核心内容。
风险度量作为金融风险管理的核心内容,由于风险的复杂性、不确定性以及巨大的社会危害性,必然成为金融风险管理领域的重点和难点。风险度量的目标是针对金融数据的时间序列进行研究,定量测量金融资产在未来一定时期内的风险大小,以便更好地完成金融风险的预测、监管、控制和规避。目前金融界主流的风险度量方法仍是建立在静态风险度量框架下的VaR及其衍生方法,风险度量的公理系统也是以静态框架下的一致性风险度量为主,未能实现动态框架下风险度量公理系统、建模及应用研究,导致迄今为止还没有真正意义上的动态风险度量公理系统和动态风险度量方法得以实现。针对此种现状,本文在风险度量方法及公理系统发展历史综述的基础上,指出动态风险度量方法的缺失首先应归因于动态风险度量框架的缺失,建立动态风险度量框架是解决目前动态风险度量方法缺失的根本途径,更是不应该被忽视的风险度量领域重要问题。根据静态风险度量框架的构造特点和属性,本文建立了广义概率空间下的动态风险度量框架,提出了适应该风险度量框架的动态风险度量方法DVaR,并完成了相关实证研究。
首先本文在风险多种理解的基础上界定风险内涵,从数学角度引入了金融风险度量的基本公理,针对本文主要采用的两种风险度量模型——VaR和CVaR的定义、计算过程和性质进行研究,并介绍谱风险度量的思想。风险内涵的确立是本文研究的风险度量框架及风险度量方法的立题之本。基于期望理论的风险定义,严格区别于基于效用理论的定义,这就决定本文研究的风险是基于损失考虑,而不是基于波动性或不确定性的。基于投资者心理的风险内涵界定和风险度量方法是风险度量理论的基础性问题。
其次本文研究了静态框架下的风险度量公理系统。静态框架下的风险度量公理系统以目前的一致性风险度量和凸性风险度量为代表,本文在此基础上提出了可行静态风险度量框架。可行静态风险度量框架作为静态风险度量框架研究成果的发展,为静态框架下的风险度量方法提供验证标准。针对有限概率空间下风险度量公理系统存在的问题,本文也对广义概率空间下静态风险度量进行了增量研究。
再次本文建立动态风险度量框架。动态风险度量框架以动态风险度量属性研究为核心,它解决了静态风险度量框架无法实现多时期风险度量的问题。动态风险度量属性中,各属性之间相互关系的推导为动态风险度量框架的独立性和充分性提供了保证;针对现有风险度量方法的动态属性证伪为新的风险度量方法建立提供了必要性准备。
然后本文提出动态风险度量方法DVaR。动态风险度量方法是本研究的最终目的,在动态风险度量框架下建立了基于离散过程的动态风险度量方法DVaR,解决了多时期风险度量方法缺失问题;对提出的动态风险度量方法进行了风险度量属性证明,保证了动态风险度量方法对于动态风险度量框架的适用性;针对动态风险度量方法DVaR给出了动态风险度量方法的模型求解算法,实现了模型假设、建立和求解的全过程。
最后本文对所建立的动态风险度量方法进行实证研究。实证研究是保证动态风险度量方法DVaR可行性的重要步骤。实证研究中选取了我国两大股市的两个代表性指数在相当长时期内的收盘数据,运用多种软件对VaR、CVaR和DVaR进行了比较计算,并对结果进行了详细比较分析,为风险度量方法DVaR的合理性提供支持。
本文将风险度量公理系统和风险度量方法紧密结合,在多时期风险度量研究领域内进行探索性研究,旨在解决风险度量研究领域内的动态风险度量框架和动态风险度量方法共同缺失的局面。
With the strengthening trend of economy globalization and finance conglomeration, the risk structure within the financial derivative tools was more complicated. All these changes caused some potential disasters throughout the world, which have been paid special attentions to among every country, and made the financial risk to be one of the key risks for modern enterprises and financial institutes. The financial risk management has also become the core of both finance engineering and modern finance theory.
Because of its complexity, uncertainty and great harmfulness, risk measure which was the key problem of financial risk management, has become into the important problem in the financial risk management fields. The destination of risk measure was to make deep research in the historical time series of financial data, to measure the risk of financial series in the future quantitatively, and to finish the job of forecasting, supervising, controlling and escaping towards financial risk. At present, the methods of risk measure applied in the finance fields were mainly concentrated on the Value-at-Risk and its derivatives under static risk measures framework, while the axiom of risk measure was represented by coherent measure of risk under static risk measures framework. Both of them couldn’t realize the research of risk measure axiom under dynamic settings, modeling and application. Utile now, there were not any axiom and model of dynamic risk measures available, which created the departure from theory and practice. To solve all of the problems above, we concluded the development of models and axioms of risk measure. We showed that to establish the dynamic risk measures framework is the only way to solve the shortage of dynamic risk measures models, which can’t be neglected by the theoretical scholars. According to the characters and properties of static risk measures framework, we established dynamic risk measures framework on general probability space, and proposed dynamic risk measures model—DVaR under dynamic risk measures framework. The empirical research has also been completed.
Firstly, some definitions of risk and basic axioms of financial risk measures have been introduced. The advantage and disadvantage of VaR and CVaR have been demonstrated, including spectral measure of risk. The expression of risk is the base of risk measure axioms and models in this paper. The definition of risk based on the Prospect Theory, was quite different from the definition based on Utility Theory. The definition showed the research in this paper was based on the loss, while not on the fluctuation and uncertainty. The definition followed the ideas of behavior finance, and also followed the psychology of investors. The risk definition and model based on the psychology of investors will not only accelerate the development of risk measure theory, but also the development of behavior finance theory.
Secondly, we optimized the risk measures axioms under the static framework. The risk measure axioms under the static framework were represented by coherent measure of risk and convex risk measure, based on which we proposed the acceptability static risk measures framework. The acceptability static risk measures framework was the summary of research under static framework. It provided the standard to verify the static risk measures models. Because of the problem existing on the finite probability space, we extended the research into the general probability space, which made the static risk measure axioms perfect.
Thirdly, we established the dynamic risk measures framework. Dynamic risk measures framework concentrated on the properties of dynamic risk measures, which realized the multi-period risk measures rather than static risk measures framework. Among all the dynamic risk measures properties, the relationship between them guaranteed the independence and sufficiency for the dynamic risk measures framework. The proof of dynamic risk measures properties of present risk measure models provided the necessary prepare for the new risk measure model.
Fourthly, we proposed the dynamic risk measures model DVaR. The dynamic risk measures model was the final goal in this paper. Under the dynamic risk measures framework, we established the dynamic risk measures model DVaR based on discrete-time process, which solved the problem of multi-period risk measures. The coherence between DVaR and risk measure properties has been proved, which guaranteed the efficiency of dynamic risk measures model under dynamic risk measures framework. We provided the programs of dynamic risk measures model DVaR. All of the above have realized the whole process from assumption, establishment to settlement.
At last, the empirical research of dynamic risk measures model has been carried out. The empirical research is the necessary step to assure the acceptability of dynamic risk measures model DVaR. Two of the representative stock indices in our country have been chosen as the empirical objects. After calculation of VaR, CVaR and DVaR by some software, the result was analyzed detailedly, which supported the acceptability of DVaR.
In this paper, the axioms and models of risk measure have been combined closely. We tried to make some exploratory research in the fields of multi-period risk measures. We were intended to solve the problem of shortage of dynamic risk measures framework and model. The achievement in the paper was supposed to contribute to the research of theory and practice of dynamic risk measures, which would also promote the research of dynamic risk measures.
风险度量作为金融风险管理的核心内容,由于风险的复杂性、不确定性以及巨大的社会危害性,必然成为金融风险管理领域的重点和难点。风险度量的目标是针对金融数据的时间序列进行研究,定量测量金融资产在未来一定时期内的风险大小,以便更好地完成金融风险的预测、监管、控制和规避。目前金融界主流的风险度量方法仍是建立在静态风险度量框架下的VaR及其衍生方法,风险度量的公理系统也是以静态框架下的一致性风险度量为主,未能实现动态框架下风险度量公理系统、建模及应用研究,导致迄今为止还没有真正意义上的动态风险度量公理系统和动态风险度量方法得以实现。针对此种现状,本文在风险度量方法及公理系统发展历史综述的基础上,指出动态风险度量方法的缺失首先应归因于动态风险度量框架的缺失,建立动态风险度量框架是解决目前动态风险度量方法缺失的根本途径,更是不应该被忽视的风险度量领域重要问题。根据静态风险度量框架的构造特点和属性,本文建立了广义概率空间下的动态风险度量框架,提出了适应该风险度量框架的动态风险度量方法DVaR,并完成了相关实证研究。
首先本文在风险多种理解的基础上界定风险内涵,从数学角度引入了金融风险度量的基本公理,针对本文主要采用的两种风险度量模型——VaR和CVaR的定义、计算过程和性质进行研究,并介绍谱风险度量的思想。风险内涵的确立是本文研究的风险度量框架及风险度量方法的立题之本。基于期望理论的风险定义,严格区别于基于效用理论的定义,这就决定本文研究的风险是基于损失考虑,而不是基于波动性或不确定性的。基于投资者心理的风险内涵界定和风险度量方法是风险度量理论的基础性问题。
其次本文研究了静态框架下的风险度量公理系统。静态框架下的风险度量公理系统以目前的一致性风险度量和凸性风险度量为代表,本文在此基础上提出了可行静态风险度量框架。可行静态风险度量框架作为静态风险度量框架研究成果的发展,为静态框架下的风险度量方法提供验证标准。针对有限概率空间下风险度量公理系统存在的问题,本文也对广义概率空间下静态风险度量进行了增量研究。
再次本文建立动态风险度量框架。动态风险度量框架以动态风险度量属性研究为核心,它解决了静态风险度量框架无法实现多时期风险度量的问题。动态风险度量属性中,各属性之间相互关系的推导为动态风险度量框架的独立性和充分性提供了保证;针对现有风险度量方法的动态属性证伪为新的风险度量方法建立提供了必要性准备。
然后本文提出动态风险度量方法DVaR。动态风险度量方法是本研究的最终目的,在动态风险度量框架下建立了基于离散过程的动态风险度量方法DVaR,解决了多时期风险度量方法缺失问题;对提出的动态风险度量方法进行了风险度量属性证明,保证了动态风险度量方法对于动态风险度量框架的适用性;针对动态风险度量方法DVaR给出了动态风险度量方法的模型求解算法,实现了模型假设、建立和求解的全过程。
最后本文对所建立的动态风险度量方法进行实证研究。实证研究是保证动态风险度量方法DVaR可行性的重要步骤。实证研究中选取了我国两大股市的两个代表性指数在相当长时期内的收盘数据,运用多种软件对VaR、CVaR和DVaR进行了比较计算,并对结果进行了详细比较分析,为风险度量方法DVaR的合理性提供支持。
本文将风险度量公理系统和风险度量方法紧密结合,在多时期风险度量研究领域内进行探索性研究,旨在解决风险度量研究领域内的动态风险度量框架和动态风险度量方法共同缺失的局面。
With the strengthening trend of economy globalization and finance conglomeration, the risk structure within the financial derivative tools was more complicated. All these changes caused some potential disasters throughout the world, which have been paid special attentions to among every country, and made the financial risk to be one of the key risks for modern enterprises and financial institutes. The financial risk management has also become the core of both finance engineering and modern finance theory.
Because of its complexity, uncertainty and great harmfulness, risk measure which was the key problem of financial risk management, has become into the important problem in the financial risk management fields. The destination of risk measure was to make deep research in the historical time series of financial data, to measure the risk of financial series in the future quantitatively, and to finish the job of forecasting, supervising, controlling and escaping towards financial risk. At present, the methods of risk measure applied in the finance fields were mainly concentrated on the Value-at-Risk and its derivatives under static risk measures framework, while the axiom of risk measure was represented by coherent measure of risk under static risk measures framework. Both of them couldn’t realize the research of risk measure axiom under dynamic settings, modeling and application. Utile now, there were not any axiom and model of dynamic risk measures available, which created the departure from theory and practice. To solve all of the problems above, we concluded the development of models and axioms of risk measure. We showed that to establish the dynamic risk measures framework is the only way to solve the shortage of dynamic risk measures models, which can’t be neglected by the theoretical scholars. According to the characters and properties of static risk measures framework, we established dynamic risk measures framework on general probability space, and proposed dynamic risk measures model—DVaR under dynamic risk measures framework. The empirical research has also been completed.
Firstly, some definitions of risk and basic axioms of financial risk measures have been introduced. The advantage and disadvantage of VaR and CVaR have been demonstrated, including spectral measure of risk. The expression of risk is the base of risk measure axioms and models in this paper. The definition of risk based on the Prospect Theory, was quite different from the definition based on Utility Theory. The definition showed the research in this paper was based on the loss, while not on the fluctuation and uncertainty. The definition followed the ideas of behavior finance, and also followed the psychology of investors. The risk definition and model based on the psychology of investors will not only accelerate the development of risk measure theory, but also the development of behavior finance theory.
Secondly, we optimized the risk measures axioms under the static framework. The risk measure axioms under the static framework were represented by coherent measure of risk and convex risk measure, based on which we proposed the acceptability static risk measures framework. The acceptability static risk measures framework was the summary of research under static framework. It provided the standard to verify the static risk measures models. Because of the problem existing on the finite probability space, we extended the research into the general probability space, which made the static risk measure axioms perfect.
Thirdly, we established the dynamic risk measures framework. Dynamic risk measures framework concentrated on the properties of dynamic risk measures, which realized the multi-period risk measures rather than static risk measures framework. Among all the dynamic risk measures properties, the relationship between them guaranteed the independence and sufficiency for the dynamic risk measures framework. The proof of dynamic risk measures properties of present risk measure models provided the necessary prepare for the new risk measure model.
Fourthly, we proposed the dynamic risk measures model DVaR. The dynamic risk measures model was the final goal in this paper. Under the dynamic risk measures framework, we established the dynamic risk measures model DVaR based on discrete-time process, which solved the problem of multi-period risk measures. The coherence between DVaR and risk measure properties has been proved, which guaranteed the efficiency of dynamic risk measures model under dynamic risk measures framework. We provided the programs of dynamic risk measures model DVaR. All of the above have realized the whole process from assumption, establishment to settlement.
At last, the empirical research of dynamic risk measures model has been carried out. The empirical research is the necessary step to assure the acceptability of dynamic risk measures model DVaR. Two of the representative stock indices in our country have been chosen as the empirical objects. After calculation of VaR, CVaR and DVaR by some software, the result was analyzed detailedly, which supported the acceptability of DVaR.
In this paper, the axioms and models of risk measure have been combined closely. We tried to make some exploratory research in the fields of multi-period risk measures. We were intended to solve the problem of shortage of dynamic risk measures framework and model. The achievement in the paper was supposed to contribute to the research of theory and practice of dynamic risk measures, which would also promote the research of dynamic risk measures.
引文
1 Markowitz Harry M. Portfolio Selection. Journal of Finance. 1952,7(1): 77~79
2 Dhaene J., Goovaerts M. J., Kaas R. Economic Capital Allocation Derived from Risk Measures. North American Actuarial Journal. 2003,7(2):44~56
3 Kahneman D., Tversky A. Prospect theory: An analysis of decision under risk. Econometrica. 1979, 47(2): 263~291
4 Artzner, F. Delbaen, J.-M. Eber and D. Heath. Thinking Coherently. Risk. 1997,4: 10~11
5 Konno H, Yamazaki H. Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market. Management Science. 1991,(37): 519~531
6 Roy A. D. Safety First and the Holding of Assets. Econometrica. 1952, 20(3): 431~449
7 Markowitz H. M. Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. 1959: 89~102
8 Harlow W. V. Asset Allocation in a Downside Risk Framework. Salomon Brothers. New York. 1991: 57~68
9 Bawa Vijay S. Optimal Rules For Ordering Uncertain Prospects. Journal of Financial Economics. 1975, 2(1): 95~121
10 Nawrocki David. A Brief History of Downside Risk Measures. Journal of Investing. 1999, 8(3): 9~25
11 Baumol W. J. An Expected Gain-Confidence Limit Criterion for Portfolio Selection. Management Science. 1963, 10(2): 174~182
12 J. P. Morgan. Risk Metrics-technical Document. U.S.: J.P. Morgan Inc. 1994: 32~56
13 Danielson J. Extreme Returns, Tail Estimation and Value-at-Risk. In: The 52th ESEM conference. Toulouse. 1997: 89~93
14 Vlaar, P. Value at Risk Models for Dutch Bond Portfolios. Journal of Banking and Finance. 2000,24: 1131~1154
15 Venkataraman Subu. Value at Risk for a Mixture of Normal Distribution:The Use of Quasi-Bayesian Estimation Techniques. Economic Perspectives(Federal Reserve Bank of Chicago). 1997,3/4: 24~31
16 Bulter J. S., Barry Schachter. Estimating Value at Risk with a Precision Measure by Combining Kernel Estimation with Historical Simulation. Review of Derivatives Research. 1998,1: 371~390
17 Jon Danielsson, Casper G. de Vries. Tail Index and Quantile Estimation with Very High Finance Data. Journal of Empirical Finance. 1997,4: 241~257
18 Fmhrechts Paul, Sidney Resnick and Gennady Samorodnidky. Extreme Value Theory as a Risk Management Tool. North American Actuarial Journal. 1999,3(1): 30~41
19 Iacono, Frank and David Skeie. Translating VaR Using Square Root of T. Derivatives Week. 1996,8(10): 25~33
20 Jamshidian F., Zhu Y. Scenario Simulation Model for Risk Management. Capital Market Strategy. 1996,12: 17~21
21 Jauri, Osmo and P. Toivonen. Risk Management Information Systems: Value-at-Risk Analysis with Stochastic with Vision. Proceedings of the First Simulation in International Mathematica Symposium, Computational Mechanics Publications. 1996
22 Kwiatkowski. Current Issues in Value-at-Risk for Portfolios of Derivatives. Derivatives Use, Tranding & Regulation. 1997,3
23 Simko David. Applying Value-at-Risk Measures to Derivatives in Portfolio Management. Charlottesville Derivatives: AIMR. 1998
24 Kupiec Paul. Techniques for Verifying the Accuracy Risk of Measurement Models. Journal of Derivatives. 1995,3(Winter)
25 Matthew Pritsker. Evaluating Value at Risk Methodologies: Accuracy Versus Computational Time. Journal of Financial Services Research. 1996,12
26 Hendricks Darryll, Beverly Hirtle. Bank Capital Requirements for Market Risk: The Internal Models Approach. Federal Reserve Rank of New York Economic Policy Review. 1997,3(12)
27 Artzner P., F. Delbaen, J-M. Eber and D. Heath. Coherent Measures of Risk. Mathematical Finance. 1999,3: 203~228
28 Harry H. Panjer. Measurement of Risk, Solvency Requirements and Allocation of Capital within Financial Conglomerates (Working paper).Society of Actuaries. 2001
29 Acerbi C, Tasche D. On the Coherence of Expected Shortfall. Journal of Banking & Finance. 2002, 26(7): 1487~1503
30 Yamai Yasuhiro, Yoshiba Toshinao. Value-at-risk Versus Expected Shortfall: A Practical Perspective. Journal of Banking & Finance. 2005,4(29): 997~1015
31 Acerbi C. Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion. Journal of Banking and Finance. 2002, 26(7): 1505~1518
32 田新民,黄海平. 基于条件 VaR 的投资组合优化模型及比较研究. 数学的实践与认识. 2004,7:39~49
33 冯春山,蒋馥,吴家肴. 应用半参数方法计算市场风险的受险价值. 系统工程理论方法应用. 2005,14(4): 379~381
34 H. Follmer, A. Schied. Convex Measures of Risk and Trading Constraints. Finance and Stochastics. 2002,6: 429~447
35 Delbaen F. Coherent Risk Measures on General Probability Spaces. Essays in Honour of Dieter Sondermann. Springer-Verlag, 2002
36 Cheridito P., Delbaen F., Kupper M. Coherent and Convex Monetary Risk Measures for Unbounded Cadlag Processes. Finance and Stochastics. 2005,9(3): 369~387
37 Cheridito Patrick, Freddy Delbaen, Michael Kupper. Coherent and Convex Risk Measures for Unbounded Cadlag Processes. Finance and Stochastics. 2004,9: 349~367
38 A. Ruszczynski, A. Shapiro. Optimization of Convex Risk Functions. E-print, available at: http://www.optimization-online.org. 2004
39 Pelessoni Renato, Vicig Paolo. Convex Imprecise Previsions for Risk Measurement. Workingpaper. 2003
40 Susanne Kloppel, Martin Schweizer. Dynamic Utility Indifference Valuation via Convex Risk Measures. to appear in Mathematical Finance. 2005
41 Jorn Dunkel, Stefan Weber. Efficient Monte Carlo Methods for Convex Risk Measures in Portfolio Credit Risk Models. Workingpaper. 2005
42 Follmer H., A. Schied. Robust Representation of Convex Measures of Risk. in Advances in Finance and Stochastics. Essays in Honour of DieterSondermann, Springer-Verlag. 2002b: 39~56
43 Fritelli M., E. Rosazza Gianin. Putting Order in Risk Measures. Journal of Banking and Finance. 2002, 26: 1473~1486
44 A. Jobert, L.C.G. R. Pricing Operators and Dynamic Convex Risk Measures. Preliminary and incomplete. University of Cambridge. 2005,3
45 Kai Detlefsen, Giacomo Scandolo. Conditional and Dynamic Convex Risk Measures. SFB 649 Discussion Paper, 2005
46 Roorda B., Schumacher J.M., Engwerda J.C. Coherent Acceptability Measures in Multiperiod Models. Mathematical Finance. 2005,15: 589~612
47 Kang Boda, Jerzy A. Filar. Time Consistent Dynamic Risk Measures. Journal of Mathematical Methods of Operations Research, 2006,63(1): 169~186
48 Roorda B., J.M. Schumacher. Time Consistency Conditions for Acceptability Measures, with an Application to Tail Value at Risk. Insurance: Mathematical and Economics, 2006, 40(2): 209~230
49 Artzner P., Delbaen F., Eber J.M., Heath D., Ku H. Coherent Multiperiod Risk Adjusted Values and Bellman's Principle. Forthcoming in Ann. Oper. Res. 2005
50 Coquet F., Hu Y., Mémin J., Peng S. Filtration Consistent Nonlinear Expectations and Related g-expectations. Probability Theory and Related Fields. 2002,123: 1~27
51 De Giorgi E. Reward-risk Portfolio Selection and Stochastic Dominance. Journal of Banking and Finance. 2005,29: 895~926
52 杨晓光, 马超群, 文风华. VaR 之下厚尾分布的最优资产组合的收敛性. 管理科学学报. 2002,1(5): 65~69
53 陈剑利, 李胜宏. CVaR 风险度量模型在投资组合中的运用. 运筹与管理. 2004,1(13): 95~99
54 文凤华, 马超群, 陈牡妙, 兰秋军, 杨晓光. 一致性风险价值及其算法与实证研究. 系统工程理论与实践. 2004,(10): 15~21
55 梅雪晖. CVaR 风险度量模型在带有交易费用的投资组合中的运用. 乌鲁木齐职业大学学报. 2005,1(14): 28~31
56 唐湘晋, 童仕宽. CVaR 有界限制下的风险资本配置. 武汉理工大学学报. 2005,2(29): 292~295
57 司继文, 张明佳, 龚朴. 基于 MonteCarlo 模拟和混合整数规划的 CVaR(VaR)投资组合优化. 武汉理工大学学报(交通科学与工程版). 2005,29(3): 411~414
58 曲圣宁, 田新时. 投资组合风险管理中 VaR 模型的缺陷以及 CVaR 模型研究. 统计与决策. 2005,(5): 18~20
59 李纲, 杨辉耀, 郭海燕. 基于极值理论的深市 VaR 和 Expected Shortfall度量. 财经科学. 2002, 增刊: 77~79
60 陈学华, 杨辉耀. 基于 Expected Shortfall 的投资组合优化模型. 管理科学. 2003,16(4): 55~59
61 唐湘晋, 李楚霖. 关于风险度量: 期望亏空、最坏条件期望和尾部条件期望的等价定理. 工程数学学报. 2003,(12): 55~59
62 张蒙, 杨国孝. 期望不足量在信用风险中的应用. 管理科学. 2004,5(17): 75~80
63 彭飞, 史本山, 吴炯. 基于 E-Sh 风险度量的实物限量资本投资组合模型. 技术经济与管理研究. 2004,(1): 29~31
64 吴世农, 陈斌. 风险度量方法与金融资产配置模型的理论和实证研究. 经济研究. 1999,(9): 30~38
65 韦廷权. 风险度量和投资组合构造的进一步实证. 南开经济研究. 2001,(2): 3~7
66 朱奉云, 邱菀华, 刘善存. 投资组合均值-方差模型和极小极大模型的实证比较. 中国管理科学. 2002,10(6): 13~17
67 徐绪松, 王频, 侯成琪. 基于不同风险度量的投资组合模型的实证比较. 武汉大学学报(理学版). 2004, 50(3): 311~314
68 李军农, 陈彦斌. 中国股票市场卖空约束研究. 中国软科学. 2004,(9): 63~66
69 周德才, 李志雄. 现代风险技术(ES)对我国债券市场的实证研究. 西安邮电学院学报. 2005,10(2): 50~52
70 陈德胜, 冯宗宪. 资产组合信用风险度量技术比较研究——基于 VAR. 财经问题研究. 2005,(2): 38~43
71 朱世武. 基于 Copula 函数度量违约相关性. 统计研究. 2005,(4): 61~64
72 舒建平, 胡培, 范钛. 长记忆下的风险度量及中国市场投资期限效应的实证分析. 系统工程. 2005,23(2): 39~44
73 李萍, 李楚霖. 标准化风险度量与投资决策的双目标优化. 应用数学. 2005, 18(1): 167~173
74 李华, 李兴斯. 均值-叉熵证券投资组合优化模型. 数学的实践与认识. 2005, 35(5): 65~70
75 胡支军, 黄登仕. 一个非对称风险度量模型及组合证券投资分析. 中国管理科学. 2005,13(2): 8~14
76 孟生旺. 论金融风险度量方法的一致性要求. 现代财经. 2004,24(9): 18~21
77 何琳洁, 文凤华, 马超群. 基于一致性风险价值的投资组合优化模型研究. 湖南大学学报(自然科学版). 2005,32(2): 125~128
78 王爱民, 何信. 金融风险统计度量标准研究. 统计研究. 2005,(2): 67~72
79 Knight Frank H. Risk, Uncertainty, and Profit. Houghton Mifflin, New York. 1921,126~165
80 王春峰. 金融市场风险管理. 天津: 天津大学出版社,2001,418~426
81 李毳,欧阳昌民. 不确定性、信息不对称与风险投资契约工具选择. 中央财经大学学报. 2007,(6):92~96
82 A. H. Willett. The Economic Theory of Risk and Insurance. New York: Columbia University Press, 1901: 10
83 雍迥敏,刘道百. 数学金融学. 上海:上海人民出版社,2002:142~152
84 Fishburn P. C. Foundations of Risk Measurement. Management Science. 1984,(30): 396~496
85 Fishburn P. C. Foundations of Decision Analysis: Along the Way. Management Science. 1989,(35): 387~405
86 Fishburn P. C, P Wakker. The Invention of the Independence Condition for Preferences. Management Science. 1995,(41): 1130~1144
87 Coquet F., Hu Y., Memin J., Peng S. Filtration Consistent Nonlinear Expectations and Related g-expectations. Probability Theory and Related Fields. 2002, 123 (11): 1~27
88 Ramsay C M. Loading Gross Premiums for Risk without Using Utility Theory with Discussions. Transactions of the Society of Actuaries. 1994,(XLV): 305~349
89 Tasche D. Expected Shortfall and Beyond. Journal of Banking and Finance.2002,26(7): 1519~1533
90 Szego G. Measures of Risk. Journal of Banking and Finance. 2002, 26(7): 1505~1518
91 Kusuoka S. On Law Invariant Coherent Risk Measures. in: Advances in Mathematical Economics. Springer, Tokyo. 2001, (3): 83~95
92 何信. 金融风险度量及其应用研究. 天津大学博士学位论文. 天津, 天津大学, 2004: 13~37
93 Basel committee. Capital Requirement and Bank Behavior: Infection of Basel Accord. 1994
94 Basak S., Shapiro A. Value-at-risk Management: Optimal Policies and Asset Prices. Review of Financial Studies. 2001, (14): 371~405
95 Frey R., McNeil A. J. VaR and Expected Shortfall in Portfolios of Dependent Credit Risks: Conceptual and Practical Insights. Journal of Banking and Finance. 2002,26(7): 1317~1334
96 Fischer T. Risk Capital Allocation by Coherent Risk Measures Based on One-sided Moments. Insurance: Mathematics and Economics. 2003,32(1): 135~146
97 Follmer H., Schied A. Robust Preferences and Convex Measures of Risk. in: Advances in Finance and Stochastics (K. Sandmann and P.J. Sch?onbucher eds.) Springer-Verlag, 2002: 39~56
98 Duffie D., Pan J. An Overview of Value at Risk. The Journal of Derivatives. 1997,4(1): 7~49
99 Frittelli M., Rosazza Gianin E. Putting order in risk measures. Journal of Banking and Finance. 2002,26(7):1473~1486
100 宋逢明. 金融工程原理——无套利均衡分析. 第一版. 北京, 清华大学出版社, 1999: 26~53
101 Wang T. A Characterization of Dynamic Risk Measures. Working Paper. University of British Columbia, 1996
102 Weber S. Distribution-invariant Risk Measures, Information, and Dynamic Consistency. Mathematical Finance. 2006,(16): 419~441
103 Simon Benninga, Zvi Wiener. Value-at-Risk. Mathematica in Education and Research. 1998, 7(4): 1~8
104 Rockafellar R T, Uryasev S. Optimization of Conditional Value-at-risk.Journal of Risk. 2000,(2):21~42
105 Rockafellar R T, Uryasev S. Conditional Value-at-risk for General Loss Distributions. Journal of Bank and Finance. 2002,(26):1443~1471
106 Schweizer M. Variance-optimal Hedging in Discrete Time. Mathematics of Operations Research. 1995, (20):1~31
107 Kusuoka S. On Law Invariant Coherent Risk Measures. Advances in Mathematical Economics. 2001,(3): 83~95
108 菲利浦·乔瑞. 风险价值 VaR(第二版). 北京: 中信出版社, 2006:208~209
109 黄雄艳. 上证指数的 VaR 风险测量及有效性分析. 长沙大学学报. 2005, 19(6): 38~41
110 舒建平. 证券风险度量及其在中国股市投资价值分析中的应用. 西南交通大学博士学位论文. 2006,8: 78~84
111 胡晓. 武汉房地产投资风险分析——风险价值 VaR 的应用. 中南财经政法大学研究生学报. 2006,(5): 28~33
112 薛宏刚,徐成贤,李二平,苗宝山. 金融风险管理的 VaR 方法及实证分析. 工程数学学报. 2004,21(6): 941~946
2 Dhaene J., Goovaerts M. J., Kaas R. Economic Capital Allocation Derived from Risk Measures. North American Actuarial Journal. 2003,7(2):44~56
3 Kahneman D., Tversky A. Prospect theory: An analysis of decision under risk. Econometrica. 1979, 47(2): 263~291
4 Artzner, F. Delbaen, J.-M. Eber and D. Heath. Thinking Coherently. Risk. 1997,4: 10~11
5 Konno H, Yamazaki H. Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market. Management Science. 1991,(37): 519~531
6 Roy A. D. Safety First and the Holding of Assets. Econometrica. 1952, 20(3): 431~449
7 Markowitz H. M. Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. 1959: 89~102
8 Harlow W. V. Asset Allocation in a Downside Risk Framework. Salomon Brothers. New York. 1991: 57~68
9 Bawa Vijay S. Optimal Rules For Ordering Uncertain Prospects. Journal of Financial Economics. 1975, 2(1): 95~121
10 Nawrocki David. A Brief History of Downside Risk Measures. Journal of Investing. 1999, 8(3): 9~25
11 Baumol W. J. An Expected Gain-Confidence Limit Criterion for Portfolio Selection. Management Science. 1963, 10(2): 174~182
12 J. P. Morgan. Risk Metrics-technical Document. U.S.: J.P. Morgan Inc. 1994: 32~56
13 Danielson J. Extreme Returns, Tail Estimation and Value-at-Risk. In: The 52th ESEM conference. Toulouse. 1997: 89~93
14 Vlaar, P. Value at Risk Models for Dutch Bond Portfolios. Journal of Banking and Finance. 2000,24: 1131~1154
15 Venkataraman Subu. Value at Risk for a Mixture of Normal Distribution:The Use of Quasi-Bayesian Estimation Techniques. Economic Perspectives(Federal Reserve Bank of Chicago). 1997,3/4: 24~31
16 Bulter J. S., Barry Schachter. Estimating Value at Risk with a Precision Measure by Combining Kernel Estimation with Historical Simulation. Review of Derivatives Research. 1998,1: 371~390
17 Jon Danielsson, Casper G. de Vries. Tail Index and Quantile Estimation with Very High Finance Data. Journal of Empirical Finance. 1997,4: 241~257
18 Fmhrechts Paul, Sidney Resnick and Gennady Samorodnidky. Extreme Value Theory as a Risk Management Tool. North American Actuarial Journal. 1999,3(1): 30~41
19 Iacono, Frank and David Skeie. Translating VaR Using Square Root of T. Derivatives Week. 1996,8(10): 25~33
20 Jamshidian F., Zhu Y. Scenario Simulation Model for Risk Management. Capital Market Strategy. 1996,12: 17~21
21 Jauri, Osmo and P. Toivonen. Risk Management Information Systems: Value-at-Risk Analysis with Stochastic with Vision. Proceedings of the First Simulation in International Mathematica Symposium, Computational Mechanics Publications. 1996
22 Kwiatkowski. Current Issues in Value-at-Risk for Portfolios of Derivatives. Derivatives Use, Tranding & Regulation. 1997,3
23 Simko David. Applying Value-at-Risk Measures to Derivatives in Portfolio Management. Charlottesville Derivatives: AIMR. 1998
24 Kupiec Paul. Techniques for Verifying the Accuracy Risk of Measurement Models. Journal of Derivatives. 1995,3(Winter)
25 Matthew Pritsker. Evaluating Value at Risk Methodologies: Accuracy Versus Computational Time. Journal of Financial Services Research. 1996,12
26 Hendricks Darryll, Beverly Hirtle. Bank Capital Requirements for Market Risk: The Internal Models Approach. Federal Reserve Rank of New York Economic Policy Review. 1997,3(12)
27 Artzner P., F. Delbaen, J-M. Eber and D. Heath. Coherent Measures of Risk. Mathematical Finance. 1999,3: 203~228
28 Harry H. Panjer. Measurement of Risk, Solvency Requirements and Allocation of Capital within Financial Conglomerates (Working paper).Society of Actuaries. 2001
29 Acerbi C, Tasche D. On the Coherence of Expected Shortfall. Journal of Banking & Finance. 2002, 26(7): 1487~1503
30 Yamai Yasuhiro, Yoshiba Toshinao. Value-at-risk Versus Expected Shortfall: A Practical Perspective. Journal of Banking & Finance. 2005,4(29): 997~1015
31 Acerbi C. Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion. Journal of Banking and Finance. 2002, 26(7): 1505~1518
32 田新民,黄海平. 基于条件 VaR 的投资组合优化模型及比较研究. 数学的实践与认识. 2004,7:39~49
33 冯春山,蒋馥,吴家肴. 应用半参数方法计算市场风险的受险价值. 系统工程理论方法应用. 2005,14(4): 379~381
34 H. Follmer, A. Schied. Convex Measures of Risk and Trading Constraints. Finance and Stochastics. 2002,6: 429~447
35 Delbaen F. Coherent Risk Measures on General Probability Spaces. Essays in Honour of Dieter Sondermann. Springer-Verlag, 2002
36 Cheridito P., Delbaen F., Kupper M. Coherent and Convex Monetary Risk Measures for Unbounded Cadlag Processes. Finance and Stochastics. 2005,9(3): 369~387
37 Cheridito Patrick, Freddy Delbaen, Michael Kupper. Coherent and Convex Risk Measures for Unbounded Cadlag Processes. Finance and Stochastics. 2004,9: 349~367
38 A. Ruszczynski, A. Shapiro. Optimization of Convex Risk Functions. E-print, available at: http://www.optimization-online.org. 2004
39 Pelessoni Renato, Vicig Paolo. Convex Imprecise Previsions for Risk Measurement. Workingpaper. 2003
40 Susanne Kloppel, Martin Schweizer. Dynamic Utility Indifference Valuation via Convex Risk Measures. to appear in Mathematical Finance. 2005
41 Jorn Dunkel, Stefan Weber. Efficient Monte Carlo Methods for Convex Risk Measures in Portfolio Credit Risk Models. Workingpaper. 2005
42 Follmer H., A. Schied. Robust Representation of Convex Measures of Risk. in Advances in Finance and Stochastics. Essays in Honour of DieterSondermann, Springer-Verlag. 2002b: 39~56
43 Fritelli M., E. Rosazza Gianin. Putting Order in Risk Measures. Journal of Banking and Finance. 2002, 26: 1473~1486
44 A. Jobert, L.C.G. R. Pricing Operators and Dynamic Convex Risk Measures. Preliminary and incomplete. University of Cambridge. 2005,3
45 Kai Detlefsen, Giacomo Scandolo. Conditional and Dynamic Convex Risk Measures. SFB 649 Discussion Paper, 2005
46 Roorda B., Schumacher J.M., Engwerda J.C. Coherent Acceptability Measures in Multiperiod Models. Mathematical Finance. 2005,15: 589~612
47 Kang Boda, Jerzy A. Filar. Time Consistent Dynamic Risk Measures. Journal of Mathematical Methods of Operations Research, 2006,63(1): 169~186
48 Roorda B., J.M. Schumacher. Time Consistency Conditions for Acceptability Measures, with an Application to Tail Value at Risk. Insurance: Mathematical and Economics, 2006, 40(2): 209~230
49 Artzner P., Delbaen F., Eber J.M., Heath D., Ku H. Coherent Multiperiod Risk Adjusted Values and Bellman's Principle. Forthcoming in Ann. Oper. Res. 2005
50 Coquet F., Hu Y., Mémin J., Peng S. Filtration Consistent Nonlinear Expectations and Related g-expectations. Probability Theory and Related Fields. 2002,123: 1~27
51 De Giorgi E. Reward-risk Portfolio Selection and Stochastic Dominance. Journal of Banking and Finance. 2005,29: 895~926
52 杨晓光, 马超群, 文风华. VaR 之下厚尾分布的最优资产组合的收敛性. 管理科学学报. 2002,1(5): 65~69
53 陈剑利, 李胜宏. CVaR 风险度量模型在投资组合中的运用. 运筹与管理. 2004,1(13): 95~99
54 文凤华, 马超群, 陈牡妙, 兰秋军, 杨晓光. 一致性风险价值及其算法与实证研究. 系统工程理论与实践. 2004,(10): 15~21
55 梅雪晖. CVaR 风险度量模型在带有交易费用的投资组合中的运用. 乌鲁木齐职业大学学报. 2005,1(14): 28~31
56 唐湘晋, 童仕宽. CVaR 有界限制下的风险资本配置. 武汉理工大学学报. 2005,2(29): 292~295
57 司继文, 张明佳, 龚朴. 基于 MonteCarlo 模拟和混合整数规划的 CVaR(VaR)投资组合优化. 武汉理工大学学报(交通科学与工程版). 2005,29(3): 411~414
58 曲圣宁, 田新时. 投资组合风险管理中 VaR 模型的缺陷以及 CVaR 模型研究. 统计与决策. 2005,(5): 18~20
59 李纲, 杨辉耀, 郭海燕. 基于极值理论的深市 VaR 和 Expected Shortfall度量. 财经科学. 2002, 增刊: 77~79
60 陈学华, 杨辉耀. 基于 Expected Shortfall 的投资组合优化模型. 管理科学. 2003,16(4): 55~59
61 唐湘晋, 李楚霖. 关于风险度量: 期望亏空、最坏条件期望和尾部条件期望的等价定理. 工程数学学报. 2003,(12): 55~59
62 张蒙, 杨国孝. 期望不足量在信用风险中的应用. 管理科学. 2004,5(17): 75~80
63 彭飞, 史本山, 吴炯. 基于 E-Sh 风险度量的实物限量资本投资组合模型. 技术经济与管理研究. 2004,(1): 29~31
64 吴世农, 陈斌. 风险度量方法与金融资产配置模型的理论和实证研究. 经济研究. 1999,(9): 30~38
65 韦廷权. 风险度量和投资组合构造的进一步实证. 南开经济研究. 2001,(2): 3~7
66 朱奉云, 邱菀华, 刘善存. 投资组合均值-方差模型和极小极大模型的实证比较. 中国管理科学. 2002,10(6): 13~17
67 徐绪松, 王频, 侯成琪. 基于不同风险度量的投资组合模型的实证比较. 武汉大学学报(理学版). 2004, 50(3): 311~314
68 李军农, 陈彦斌. 中国股票市场卖空约束研究. 中国软科学. 2004,(9): 63~66
69 周德才, 李志雄. 现代风险技术(ES)对我国债券市场的实证研究. 西安邮电学院学报. 2005,10(2): 50~52
70 陈德胜, 冯宗宪. 资产组合信用风险度量技术比较研究——基于 VAR. 财经问题研究. 2005,(2): 38~43
71 朱世武. 基于 Copula 函数度量违约相关性. 统计研究. 2005,(4): 61~64
72 舒建平, 胡培, 范钛. 长记忆下的风险度量及中国市场投资期限效应的实证分析. 系统工程. 2005,23(2): 39~44
73 李萍, 李楚霖. 标准化风险度量与投资决策的双目标优化. 应用数学. 2005, 18(1): 167~173
74 李华, 李兴斯. 均值-叉熵证券投资组合优化模型. 数学的实践与认识. 2005, 35(5): 65~70
75 胡支军, 黄登仕. 一个非对称风险度量模型及组合证券投资分析. 中国管理科学. 2005,13(2): 8~14
76 孟生旺. 论金融风险度量方法的一致性要求. 现代财经. 2004,24(9): 18~21
77 何琳洁, 文凤华, 马超群. 基于一致性风险价值的投资组合优化模型研究. 湖南大学学报(自然科学版). 2005,32(2): 125~128
78 王爱民, 何信. 金融风险统计度量标准研究. 统计研究. 2005,(2): 67~72
79 Knight Frank H. Risk, Uncertainty, and Profit. Houghton Mifflin, New York. 1921,126~165
80 王春峰. 金融市场风险管理. 天津: 天津大学出版社,2001,418~426
81 李毳,欧阳昌民. 不确定性、信息不对称与风险投资契约工具选择. 中央财经大学学报. 2007,(6):92~96
82 A. H. Willett. The Economic Theory of Risk and Insurance. New York: Columbia University Press, 1901: 10
83 雍迥敏,刘道百. 数学金融学. 上海:上海人民出版社,2002:142~152
84 Fishburn P. C. Foundations of Risk Measurement. Management Science. 1984,(30): 396~496
85 Fishburn P. C. Foundations of Decision Analysis: Along the Way. Management Science. 1989,(35): 387~405
86 Fishburn P. C, P Wakker. The Invention of the Independence Condition for Preferences. Management Science. 1995,(41): 1130~1144
87 Coquet F., Hu Y., Memin J., Peng S. Filtration Consistent Nonlinear Expectations and Related g-expectations. Probability Theory and Related Fields. 2002, 123 (11): 1~27
88 Ramsay C M. Loading Gross Premiums for Risk without Using Utility Theory with Discussions. Transactions of the Society of Actuaries. 1994,(XLV): 305~349
89 Tasche D. Expected Shortfall and Beyond. Journal of Banking and Finance.2002,26(7): 1519~1533
90 Szego G. Measures of Risk. Journal of Banking and Finance. 2002, 26(7): 1505~1518
91 Kusuoka S. On Law Invariant Coherent Risk Measures. in: Advances in Mathematical Economics. Springer, Tokyo. 2001, (3): 83~95
92 何信. 金融风险度量及其应用研究. 天津大学博士学位论文. 天津, 天津大学, 2004: 13~37
93 Basel committee. Capital Requirement and Bank Behavior: Infection of Basel Accord. 1994
94 Basak S., Shapiro A. Value-at-risk Management: Optimal Policies and Asset Prices. Review of Financial Studies. 2001, (14): 371~405
95 Frey R., McNeil A. J. VaR and Expected Shortfall in Portfolios of Dependent Credit Risks: Conceptual and Practical Insights. Journal of Banking and Finance. 2002,26(7): 1317~1334
96 Fischer T. Risk Capital Allocation by Coherent Risk Measures Based on One-sided Moments. Insurance: Mathematics and Economics. 2003,32(1): 135~146
97 Follmer H., Schied A. Robust Preferences and Convex Measures of Risk. in: Advances in Finance and Stochastics (K. Sandmann and P.J. Sch?onbucher eds.) Springer-Verlag, 2002: 39~56
98 Duffie D., Pan J. An Overview of Value at Risk. The Journal of Derivatives. 1997,4(1): 7~49
99 Frittelli M., Rosazza Gianin E. Putting order in risk measures. Journal of Banking and Finance. 2002,26(7):1473~1486
100 宋逢明. 金融工程原理——无套利均衡分析. 第一版. 北京, 清华大学出版社, 1999: 26~53
101 Wang T. A Characterization of Dynamic Risk Measures. Working Paper. University of British Columbia, 1996
102 Weber S. Distribution-invariant Risk Measures, Information, and Dynamic Consistency. Mathematical Finance. 2006,(16): 419~441
103 Simon Benninga, Zvi Wiener. Value-at-Risk. Mathematica in Education and Research. 1998, 7(4): 1~8
104 Rockafellar R T, Uryasev S. Optimization of Conditional Value-at-risk.Journal of Risk. 2000,(2):21~42
105 Rockafellar R T, Uryasev S. Conditional Value-at-risk for General Loss Distributions. Journal of Bank and Finance. 2002,(26):1443~1471
106 Schweizer M. Variance-optimal Hedging in Discrete Time. Mathematics of Operations Research. 1995, (20):1~31
107 Kusuoka S. On Law Invariant Coherent Risk Measures. Advances in Mathematical Economics. 2001,(3): 83~95
108 菲利浦·乔瑞. 风险价值 VaR(第二版). 北京: 中信出版社, 2006:208~209
109 黄雄艳. 上证指数的 VaR 风险测量及有效性分析. 长沙大学学报. 2005, 19(6): 38~41
110 舒建平. 证券风险度量及其在中国股市投资价值分析中的应用. 西南交通大学博士学位论文. 2006,8: 78~84
111 胡晓. 武汉房地产投资风险分析——风险价值 VaR 的应用. 中南财经政法大学研究生学报. 2006,(5): 28~33
112 薛宏刚,徐成贤,李二平,苗宝山. 金融风险管理的 VaR 方法及实证分析. 工程数学学报. 2004,21(6): 941~946