Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data

Value-at-Risk (VaR) is a popular measure of market risk. To convey information regarding potential exceedances beyond the VaR, Expected Shortfall (ES) has become the risk measure for trading book bank regulation. However, the estimation of VaR and ES is challenging, as it requires the estimation of the tail behaviour of daily returns. In this paper, we take advantage of recent research that develops joint scoring functions for VaR and ES. Using these functions, we present a novel approach to estimating the two risk measures based on intraday data. We focus on the intraday range, which is the difference between the highest and lowest intraday log prices. In contrast to intraday observations, the intraday low and high are widely available for many financial assets. To alleviate the challenge of modelling extreme risk measures, we propose the use of the intraday low series. We draw on a theoretical result for Brownian motion to show that a quantile of the daily returns can be estimated as the product of a constant term and a less extreme quantile of the intraday low returns, which we define as the difference between the lowest log price of the day and the log closing price of the previous day. In view of this, we use estimates of the VaR and ES of the intraday low returns to estimate the VaR and ES of the daily returns. We provide empirical support for the new proposals using data for five stock indices and five individual stocks.     

证券市场指数的尖峰厚尾特征与风险量估计

在对DOW,Nasdaq,S&P500和FTSE100等四个证券市场指数进行实证分析基础上,展示了证券市场指数的对数收益率具有尖峰厚尾的分布特征,并利用Logistic分布得到了很好的拟合,同时给出了基于Logistic分布的风险量VaR和CVaR的估计公式,以此计算证券市场指数的对数收益率的风险量VaR和CVaR的估计值.     

基于正则逆Gamma分布和广义极值分布的VaR计算

股指收益率的分布和风险价值(VaR)的计算是证券市场研究的热点问题.本文对来自上证指数和深证成指日收益率采用正则逆Gamma分布和偏T分布(SST)分别进行拟合,对极值序列(周、月极大值和极小值)建立广义极值分布函数。并由此计算VaR值,度量这几种序列的风险价值.结果表明正则逆Gamma分布能更好地拟合日收益率的分布,以及采用周极值收益率的广义极值分布计算VaR值来估计风险较为合理.     

Portfolio value-at-risk estimation in energy futures markets with time-varying copula-GARCH model

This paper combines copula functions with GARCH-type models to construct the conditional joint distribution, which is used to estimate Value-at-Risk (VaR) of an equally weighted portfolio comprising crude oil futures and natural gas futures in energy market. Both constant and time-varying copulas are applied to fit the dependence structure of the two assets returns. The findings show that the constant Student t copula is a good compromise for effectively fitting the dependence structure between crude oil futures and natural gas futures. Moreover, the skewed Student t distribution has a better fit than Normal and Student t distribution to the marginal distribution of each asset. Asymmetries and excess kurtosis are found in marginal distributions as well as in dependence. We estimate VaR of the underlying portfolio to be 95% and 99%, by using the Monte Carlo simulation. Then using backtesting, we compare the out-of-sample forecasting performances of VaR estimated by different models.     

应用EWMA-GARCH(1,1)-M模型对沪深300股指期货最小VaR套期保值效果研究

VaR是目前国际上应用最广泛的度量金融风险的指标之一,其核心在于波动率,也就是方差的参数估计.采用EWMA模型估计方差,并且结合风险溢价特征的GARCH(1,1)-M模型计算出沪深300股指及其期货的最优衰减因子为0.933 25,摒弃了以往采用0.940 0作为衰减因子的一贯做法,并且运用Cornish-Fisher方程对正态分布的分位数进行了修正,得到修正后的套期保值比率以及资产组合的VaR,与传统的套期保值模型相比,该模型的风险价值VaR降低的程度明显,并且对投资组合未来的VaR具有很好的预测效果,表明EWMA-GARCH(1,1)-M模型对沪深300股指期货的套期保值效果较好.     

Estimation of Value at Risk by Extreme Value Methods

Value at Risk (VaR) is defined as a low quantile in the distribution of financial profits and losses. It is the most commonly used measure of market risk in the financial industry. The methods currently used for estimation of VaR have various short comings as they are not aimed specifically at modeling the tails of the distribution of profits and losses; extreme value methods may prove valuable towards improving the current estimation methods. In this paper we give an overview of the current state of the art in applying extreme value methods to financial data and the problems encountered when doing so. We compare the performance of methods currently used for estimation of VaR to the performance of various extreme value methods and outline advantages and drawbacks of the different methods.     

Analyzing return asymmetry and quantiles through stochastic volatility models using asymmetric Laplace error via uniform scale mixtures

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.     

我国股市收益的双长记忆性检验——基于VaR估计的ARFIMA-HYGARcH-skt模型

针对股市收益分布的"尖峰肥尾"特征,引入了偏t分布作为新息分布。基于VaR方法,从风险估计的角度,利用ARFIMA(2,d_1,0)-HYGARCH(1,d_2,1)-skt模型对1996年12月17日至2007年7月5日期间的沪深股市收益进行了实证分析.实证结果显示:沪深股市具有显著的双长记忆特征;上海股市的日收益率和波动率的长记忆性均比深圳股市强;ARFIMA(2,d_1,0)- HYGARCH(1,d_2,1)-skt模型对我国股市收益具有较强的风险估计和预测能力。     

投资者情绪和市场利率对证券市场指数收益影响特点的动态刻画

本文基于2006年10月到2015年6月市场层面的投资者情绪和上证综指收益率,刻画了投资者情绪和市场利率对证券市场指数收益率的影响。首先,本文通过误差修正模型研究了短期层面投资者情绪对证券市场收益的影响特点,补充了以往在长期层面和整体收益水平上投资者情绪对市场收益影响的研究。由于市场层面的投资者情绪会受到宏观政策影响,之后本文将市场利率作为政策因素,通过分位数回归分析了不同市场收益水平下,市场利率和剔除了宏观政策因素的投资者情绪对市场收益的影响。研究结果表明:投资者情绪和证券市场收益之间的关系在短期层面上更为显著;当我国的证券市场环境处于“牛市”时,市场利率和投资者情绪均会对证券市场指数收益产生显著的影响,且随着市场收益水平的逐步上升,市场利率的反向作用和投资者情绪的正向作用均会逐渐加强。     

GARCH模型在股票市场风险计量中的应用

本文以上证综指的日收益率为研究对象,运用GARCH模型簇分析上海股市日收益率波动的条件异方差性,计算每天的V aR值.实证研究表明,GARCH模型的V aR计算方法对我国股市风险的管理有较好的效果.     

基于动态调整的Copula-非线性分位数回归资产组合优化

为优化资产组合方案,考虑单资产分布的非对称性、异方差性、尖峰厚尾性等特征,资产之间的时变非线性相关性,建立了Copula-非线性分位数回归模型。本文的创新与特色,一是通过构建期望超额收益率与考虑动态损失厌恶效应的VaR比率函数,确定了目标函数的表达式,改变了使用超额收益率标准差度量风险,而实证研究中更关注资产的损失风险而非全部风险,未考虑投资者对于收益与损失非对称偏好的不足;二是通过建立基于支持向量机的非线性分位数回归模型,确定了边缘分布函数表达式,解决了普通模型无法处理非对称、非线性,依赖于分布假设的不足;三是通过构建混合Copula函数,确保能够有效捕捉金融市场中的尾部相关、非对称性,完善了刻画资产之间相关关系的模式;四是通过建立风险非线性叠加的资产总风险评价模型,确定了资产组合总风险的表达式,弥补了现有风险评价模型未考虑资产间的相关性的不足。实证结果表明,本文建立的模型预测性能高于其它模型,该模型有更高的VaR比率值,在单位风险下能够获得更高的资产组合效果。     

Value-at-Risk via mixture distributions reconsidered

Value-at-Risk (VaR) has evolved as one of the most prominent measures of downside risk in financial markets. Zhang and Cheng [M.-H. Zhang, Q.-S. Cheng, An Approach to VaR for capital markets with Gaussian mixture, Applied Mathematics and Computation 168 (2005) 1079–1085] proposed an approach to VaR for daily returns based on Gaussian mixtures, which have become rather popular in empirical economics and finance since the seminal paper of Hamilton [J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (2) (1989) 357–384]. However, they do not conduct tests to assess the accuracy of the mixture-implied VaR measures. Recently, Guidolin and Timmermann [M. Guidolin, A. Timmermann, Term structure of risk under alternative econometric specifications, Journal of Econometrics, 131 (2006) 285–308] showed that Markov mixture models do well in measuring VaR at a monthly frequency, but the results may not hold for daily returns due to their more pronounced non-Gaussian features. This paper provides an extensive application of various Markov mixture models to VaR for daily returns of major European stock markets, including out-of-sample backtesting. To accommodate the properties of daily returns, we consider both Gaussian and Student’s t mixtures, and we compare the performance of both uni- and multivariate models under different parameter updating schemes. We find that a univariate mixture of two Student’s t distributions performs best overall. However, by the example of the recent turmoil in financial markets, we also highlight a weak point of the approach.     

基于ARFIMA-HYGARCH-M-VaR模型的亚洲汇率市场均值和波动过程的双长期记忆性测度研究

作为金融传导机制的一个重要成分,汇率在金融危机的传播中发挥着重要作用.因此本文以亚洲汇率市场的汇率作为研究样本,通过引入skt分布来刻画残差的分布,构建了ARFIMA-HYGARCH-M-VaR模型来测度汇率风险值,并与skt分布下的GARCH及FIGARCH模型的VaR进行失败率回测检验与动态分位数测试.研究结果表明:在不同显著性水平下,skt分布下的各种模型基本都有较好的风险测度能力,且ARFIMA-HYGARCH-M模型的VaR风险测度更加精确与稳定.本研究为我国及亚洲其他国家汇率市场的风险测度与风险管理提供了一定的理论借鉴和方法基础.     

在险风险值估计方法的比较研究

金融数据呈现的厚尾性已达成共识。本文首先基于指数回归模型提出了一种厚尾分布的极值分位数估计方法,得到了在险风险值的估计公式。然后得到了上海上证指数、国债指数和企业债券指数的在险风险值的估计值,比较了他们的极值风险.     

加权分布模型测算个股VaR

本文在同时考虑企业的个性风险和企业将受到整个国家宏观经济形势影响的共性风险的基础上,用一种新的思路定权数,提出了用加权分布测算个股VaR的模型。并以在深圳股市上市的四家企业的股票收益率做实证分析和模型检验。结果表明:加权分布提高了原来的假设收益率分布服从单一分布下测量VaR的准确度,尤其对于个性风险较强的企业而言,加权分布模型是一种形式简单,而又较为精确的模型。     

Momentum Effect Differs Across Stock Performances:Chinese Evidence

Prior empirical studies find positive and negative momentum effect across the global nations, but few focus on explaining the mixed results. In order to address this issue, we apply the quantile regression approach to analyze the momentum effect in the context of Chinese stock market in this paper. The evidence suggests that the momentum effect in Chinese stock is not stable across firms with different levels of performance. We find that negative momentum effect in the short and medium horizon （3 months and 9 months） increases with the quantile of stock returns. And the positive momentum effect is observed in the long horizon （12 months）, which also intensifies for the high performing stocks. According to our study, momentum effect needs to be examined on the basis of stock returns. OLS estimation, which gives an exclusive and biased result, provides misguiding intuitions for momentum effect across the global nations. Based on the empirical results of quantile regression, effective risk control strategies can also be inspired by adjusting the proportion of assets with past performances.     

一类混合序列下分位数估计的一致渐近正态性

本文在混合序列下, 研究了分位数估计的一致渐近正态性. 在一定条件下其收敛速度达到. 所得结果可以应用到风险度量VaR分位数估计.     

基于下界VaR对沪深股市市场风险的实证研究

本文应用Luciano和Marena提出的计算资产组合VaR的上下界的方法,对沪深股市的市场风险做了实证研究,并与传统的正态VaR做了比较。实证分析表明,沪深股市的市场风险确实存在“厚尾”和“波动聚集”现象。本文对国内市场的波动聚集现象进行了详细分析．并讨论了风险控制模型的相关检验,下界VaR通过了两次模型检验。     

A Frisch-Newton Algorithm for Sparse Quantile Regression

Recent experience has shown that interior-point methods using a log barrier approach are far superior to classical simplex methods for computing solutions to large parametric quantile regression problems. In many large empirical applications, the design matrix has a very sparse structure. A typical example is the classical fixed-effect model for panel data where the parametric dimension of the model can be quite large, but the number of non-zero elements is quite small. Adopting recent developments in sparse linear algebra we introduce a modified version of the Prisch-Newton algorithm for quantile regression described in Portnoy and Koenker~([28]). The new algorithm substantially reduces the storage (memory) requirements and increases computational speed. The modified algorithm also facilitates the development of nonparametric quantile regression methods. The pseudo design matrices employed in nonparametric quantile regression smoothing are inherently sparse in both the fidelity and roughness penalty components. Exploiting the sparse structure of these problems opens up a whole range of new possibilities for multivariate smoothing on large data sets via ANOVA-type decomposition and partial linear models.     

Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type

Based on the data-cutoff method,we study quantile regression in linear models,where the noise process is of Ornstein-Uhlenbeck type with possible jumps.In single-level quantile regression,we allow the noise process to be heteroscedastic,while in composite quantile regression,we require that the noise process be homoscedastic so that the slopes are invariant across quantiles.Similar to the independent noise case,the proposed quantile estimators are root-n consistent and asymptotic normal.Furthermore,the adaptive least absolute shrinkage and selection operator(LASSO)is applied for the purpose of variable selection.As a result,the quantile estimators are consistent in variable selection,and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model.Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.