The risk assessment is an important topic in risk management. The parametric VaR models are widely used in risk measurement. However, they are subject to large errors of model misspecification. In order to avoid the defect of parametric models, two nonparametric models for estimating VaR were proposed, which are SQ-ARCH and Nop-Quantile models. These two models are not restricted by their own specific structures and have great flexibility and stability in use. By the robust quantile regression method, we derived respectively the calculative steps and obtained the closed expressions of VaRs based on the two models. Monte Carlo simulation confirms that the nonparametric VaR models are more robust than the type of parametric ARCH VaR models, regardless of the correct or wrong setting of models. In addition, the two robust nonparametric VaR models are applied to qualify the risk of Chinese stock market by using the composite index data of Shanghai. It is founded that the returns of sample are non-normal and fat-tailed distribution. The technique of backtesting is used to examine the statistical properties of the nonparametric models and the ARCH models. The test results show that the robust nonparametric models outperform the type of non-robust parametric ARCH models in measuring VaR. The estimated risk values of ARCH are quite variable relative to the nonparametric models. Furthermore, the SQ-ARCH and Nop-Quantile models can yield more accurate VaR estimates than the ARCH models. The suggested models provided two effective methods for risk measurement.
[1] Mandelbrot B. The variation of certain speculative prices [J]. The Journal of Business, 1963, 36(4): 394-419.
[2] Fama E. The behavior of stock market prices [J]. Journal of Business, 1965, 38(1): 34-105.
[3] Hagerman R. More evidence on the distribution of security returns [J]. Journal of Finance, 1978, 33(4): 1213-1221.
[4] McDonald J B, Newey W K. Partially adaptive estimation of regression models via the generalized t distribution [J]. Econometric Theory, 1988, 4(3): 428-457.
[5] Hansen B. Autoregressive conditional density estimation [J]. International Economic Review, 1994, 35(3): 705-730.
[6] Theodossiou P. Financial data and the skewed generalized t distribution [J]. Management Science, 1998, 44(12): 1650-1661.
[7] Cappuccio N, Lubian D, Raggi D. MCMC Bayesian estimation of a skew-GED stochastic volatility model [J]. Studies in Nonlinear Dynamics and Econometrics, 2004, 8(2): 1558-3708.
[8] Engle R F. Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation [J]. Econometrica, 1982, 50(4): 987-1007.
[9] Bollerslev T. Generalized autoregressive conditional heteroscedasticity [J]. Journal of Econometrics, 1986, 31(3): 307-327.
[10] Giot P, Laurent S. Modelling daily Value-at-Risk using realized volatility and ARCH type models [J]. Journal of Empirical Finance, 2004, 11(3): 379-398.
[11] Kuester K, Mittnik S, Paolella M S. Value-at-Risk prediction: A comparison of alternative strategies [J].Journal of Financial Econometrics, 2006, 4(1): 53-89.
[12] 刘向丽, 成思危, 汪寿阳, 等. 期现货市场间信息溢出效应研究 [J]. 管理科学学报, 2008, 3(11): 125-139.
[13] 魏宇. 股票市场的极值风险测度及后验分析研究 [J]. 管理科学学报, 2008, 11(1): 78-88.
[14] 林宇, 卫贵武, 魏宇, 等. 基于Skew-t-FIAPARCH的金融市场动态风险VaR测度研究 [J]. 中国管理科学, 2009,17(6): 17-24.
[15] Lee C F, Su J B. Alternative statistical distributions for estimating value-at-risk: Theory and evidence [J]. Review of Quantitative Finance and Accounting, 2012, 39(3): 309-331.
[16] 王鹏, 魏宇, 王鸿. 沪深300股指期货的风险测度模型研究 [J]. 数理统计与管理, 2014, 33(4): 724-733.
[17] 谢尚宇, 姚宏伟, 周勇. 基于ARCH-Expectile方法的VaR和ES尾部风险测量 [J]. 中国管理科学, 2014, 22(9): 1-9.
[18] Dowd K. Estimating VaR with order statistics [J]. The Journal of Derivatives, 2001, 8(3): 23-30.
[19] Scaillet O. Nonparametric estimation and sensitivity analysis of expected shortfall [J]. Mathematical Finance, 2004, 14(1): 115-129.
[20] Chen Songxi, Tang Chengyong. Nonparametric inference of value at risk for dependent financial returns [J]. Journal of Financial Econometrics, 2005, 3(2): 227-255.
[21] 叶五一, 缪柏其, 吴振翔. 基于收益率修正分布的VaR估计 [J]. 数理统计与管理, 2007, 26(5): 867-874.
[22] Cai Zongwu, Wang Xian.Nonparametric estimation of conditional var and expected shortfall [J]. Journal of Econometrics, 2008, 147(1): 120-130.
[23] 赵晓玲, 陈雪蓉, 周勇. 金融风暴中基于非参估计VaR和ES方法的风险度量 [J]. 数理统计与管理, 2012, 31(3): 381-383.
[24] Koenker R, Zhao Quanshui. Conditional quantile estimation and inference of ARCH models [J]. Econometric Theory, 1996, 12(5): 793-813.
[25] Wu Guojun, Xiao Zhijie. An analysis of risk measures [J]. Journal of Risk, 2002, 4(4): 53-75.
[26] Fan Jianqing, Gijbels I. Data-driven bandwidth selection in local ploynomial fitting: variable bandwidth and spatial adaption [J]. Journal of the Royal Statistical Sotiety, Series B, 1995, 57(2): 371-394.
[27] Yu Keming, Jones M C. Local linear quantile regression [J]. Journal of the American Statistical Association, 1998, 93(441): 228-237.
[28] Peter C. Elements of financial risk management, second edition [M].Massachusetts,US: Academic Press, 2011.
[29] Hendricks D. Evaluation of Value-at-Risk models using historical data [J]. Economic Policy Review, 1996, 2(4): 39-70.
[30] Kupiec P H. Techniques for verifying the accuracy of risk measurement models [J]. Journal of Derivatives, 1995, 3(2): 73-84.
