Energy portfolio risk management using time-varying extreme value copula methods

This work is concerned with the statistical modeling of the dependence structure between three energy commodity markets (WTI crude oil, natural gas and heating oil) using the concept of copulas and proposes a method for estimating the Value at risk (VaR) of energy portfolio based on the combination of time series models with models of the extreme value theory before fitting a copula. Each return series is modeled by AR-(FI) GARCH univariate model. Then, we fit the GPD distribution to the tails of the residuals to model marginal residuals distributions. The extreme value copula to the iid residuals is fitted and we simulate from it to construct N portfolios and estimate VaR. As a first step, the method is applied to a two-dimensional energy portfolio. In second step, we extend method in trivariate context to measure VaR of three-dimensional energy portfolio. Dependences between residuals are modeled using a trivariate nested Gumbel copulas. Methods proposed are compared with various univariate and multivariate conventional VaR methods. The reported results demonstrate that GARCH-t, conditional EVT and FIGARCH extreme value copula methods produce acceptable estimates of risk both for standard and more extreme VaR quantiles. Generally, copula methods are less accurate compared with their predictive performances in the case of portfolio composed of exchange market indices.     

股票资产组合VaR研究基于混合D藤Copula模型

本文在对上证市场五种股票资产组合的风险分析中以VaR作为风险度量指标,采用基于Pair Copula高维建模理论的混合D藤Copula模型,建立了反应多个资产组合相关结构的联合分布模型。该模型对传统D藤Copula建模方法作了进一步的改进,通过一定的选择标准,确定了D藤中每个Pair Copula函数的最优函数族,这样使得所建立的模型不仅考虑到了资产维数的影响,而且还能捕捉到组合内部因子间相关结构的差异性,从而改进后的模型能更好地描述资产组合的相关结构,并且能更精确地反映资产组合收益的实际分布。最后,以混合D藤Copula模型为基础,利用Monte Carlo方法计算了上证市场五种股票资产组合的VaR,并通过实证研究进一步证明了该模型的有效性。     

跳跃随机波动的阈值模型风险值的贝叶斯估计

文章基于一类跳跃随机波动的阈值模型风险值估计贝叶斯分析,在给定先验分布下,以马尔科夫链蒙特卡洛方法估计模型中的未知参数,并给出了MCMC模拟算法,进而讨论了风险值的预测。根据模拟结果,我们得知,如果没有考虑金融时间序列的外生冲击导致的跳跃行为,将会高估风险值,因此考虑跳跃行为后,将增加风险值估计的精度。     

Optimization of Monte Carlo Procedures for Value at Risk Estimates

This paper proposes a methodology which improves the computational efficiency of the Monte Carlo simulation approach of value at risk (VaR) estimates. Principal components analysis is used to reduce the number of relevant sources of risk driving the portfolio dynamics. Moreover, large deviations techniques are used to provide an estimate of the minimum number of price scenarios to be simulated to attain a given accuracy. Numerical examples are provided and show the good performance of the methodolgy proposed.
(J.E.L.: C15, G1).     

Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns

This paper is concerned with linear portfolio value-at-risk (VaR) and expected shortfall (ES) computation when the portfolio risk factors are leptokurtic, imprecise and/or vague. Following Yoshida (2009), the risk factors are modeled as fuzzy random variables in order to handle both their random variability and their vagueness. We discuss and extend the Yoshida model to some non-Gaussian distributions and provide associated ES. Secondly, assuming that the risk factors' degree of imprecision changes over time, original fuzzy portfolio VaR and ES models are introduced. For a given subjectivity level fixed by the investor, these models allow the computation of a pessimistic and an optimistic estimation of the value-at-risk and of the expected shortfall. Finally, some empirical examples carried out on three portfolios constituted by some chosen French stocks, show the effectiveness of the proposed methods.     

Robust estimation of the simplified multivariate GARCH model

In this paper, robust M-estimation of multivariate GARCH models are considered. The simplified GARCH model is chosen that involves the estimation of only univariate GARCH models, and hence easy to estimate, and does not put additional constraints on the model. The results of Monte Carlo simulations showed that accurate estimates of conditional correlations can be obtained using these robust estimators when the errors are heavy-tailed. We also investigate the forecasting performance of the class of robust estimators in predicting value-at-risk using various evaluation measures and collect empirical evidences of the better predictive potential of estimators such as LAD and B-estimator over the widely-used quasi-maximum likelihood estimator for the estimation and prediction of multivariate GARCH models. Applications to real data sets are also presented.     

Conditional volatility and correlations of weekly returns and the VaR analysis of 2008 stock market crash

Modelling of conditional volatilities and correlations across asset returns is an integral part of portfolio decision making and risk management. Over the past three decades there has been a trend towards increased asset return correlations across markets, a trend which has been accentuated during the recent financial crisis. We shall examine the nature of asset return correlations using weekly returns on futures markets and investigate the extent to which multivariate volatility models proposed in the literature can be used to formally characterize and quantify market risk. In particular, we ask how adequate these models are for modelling market risk at times of financial crisis. In doing so we consider a multivariate t version of the Gaussian dynamic conditional correlation (DCC) model proposed by Engle (2002), and show that the t-DCC model passes the usual diagnostic tests based on probability integral transforms, but fails the value at risk (VaR) based diagnostics when applied to the post 2007 period that includes the recent financial crisis.     

基于事件研究法的“大小非”解禁关于股票市场风险研究

通过比较＂大小非＂解禁事件前后不同时期的风险价值VaR,来评价大小非解禁对证券市场风险的影响。首先针对股票收益率序列具有波动聚集以及尖峰、厚尾的分布形态,应用GARCH类模型计算解禁前后各一段时期内沪深两市不同解禁量股票的VaR;其次应用多种定性、定量统计方法对所计算的VaR值进行前后分析比较,分析结果表明,采用的方法能够很好地捕捉到＂大小非＂解禁事件增大股票市场风险趋势这一现象。     

Portfolio optimization using dynamic factor and stochastic volatility: evidence on Fat‐tailed errors and leverage

The portfolio optimization problem is investigated using a multivariate stochastic volatility model with factor dynamics, fat‐tailed errors and leverage effects. The efficient Markov chain Monte Carlo method is used to estimate model parameters, and the Rao–Blackwellized auxiliary particle filter is used to compute the likelihood and to predict conditional means and covariances. The proposed models are applied to sector indices of the Tokyo Stock Price Index (TOPIX), which consists of 33 stock market indices classified by industrial sectors. The portfolio is dynamically optimized under several expected utilities and two additional static strategies are considered as benchmarks. An extensive empirical study indicates that our proposed dynamic factor model with leverage or fat‐tailed errors significantly improves the predictions of the conditional mean and covariances, as well as various measures of portfolio performance.     

标准仓单质押组合的价格风险——基于中国农产品期货规范市场的实证研究

利用基于Copula函数的AR(p)-GARCH(p,q)模型计算的VaR能够对农产品标准仓单的价格风险进行准确度量。对大连商品交易所的典型期货交易品种——黄大豆一号、豆油、豆粕的期货合约日结算价进行了实证研究。研究结果显示:从对价格风险预测的盯市频率来看,时变VaR优于静态VaR,因此重视农产品价格风险的频次预测应替代传统风险判断的单次监测;从对风险因子间相依性结构的刻画来看,基于t-Copula函数计算的VaR优于基于正态Copula函数计算的VaR,因此质押物价格波动间的相关系数是度量组合风险时必须考虑的重要变量。     

Risk assessment of oil price from static and dynamic modelling approaches

The price gap between West Texas Intermediate (WTI) and Brent crude oil markets has been completely changed in the past several years. The price of WTI was always a little larger than that of Brent for a long time. However, the price of WTI has been surpassed by that of Brent since 2011. The new market circumstances and volatility of oil price require a comprehensive re-estimation of risk. Therefore, this study aims to explore an integrated approach to assess the price risk in the two crude oil markets through the value at risk (VaR) model. The VaR is estimated by the extreme value theory (EVT) and GARCH model on the basis of generalized error distribution (GED). The results show that EVT is a powerful approach to capture the risk in the oil markets. On the contrary, the traditional variance–covariance (VC) and Monte Carlo (MC) approaches tend to overestimate risk when the confidence level is 95%, but underestimate risk at the confidence level of 99%. The VaR of WTI returns is larger than that of Brent returns at identical confidence levels. Moreover, the GED-GARCH model can estimate the downside dynamic VaR accurately for WTI and Brent oil returns.     

Bayesian model comparison by Markov chain simulation: Illustration using stock market data

The paper illustrates the computation of marginal likelihoods and Bayes factors when Markov Chain Monte Carlo has been used to produce draws from a model’s posterior distribution. The method is based on Raftery (1996) and does not require that Gibbs sampling is used or conditional posterior distributions are available in closed form. Models used include a normal finite mixture, a GARCH and a Student t -model as alternative models for the Standard and Poor’s stock returns.     

Modelling VaR for foreign-asset portfolios in continuous time

VaR is widely viewed as a measure of market risk of a portfolio. The purpose of this article is to provide a VaR model for foreign-asset portfolios in continuous time. In the VaR model, the VaRs are not only a function of volatilities of asset returns and exchange rate but also a function of correlation coefficient between foreign assets and exchange rate. Moreover, by backtesting, the empirical results show that the new VaR model can efficiently evaluate the market risk of foreign-asset portfolios.     

投资组合VaR分解的应用研究

针对VaR的不足,Garman M．于1997年提出了成分VaR和边际VaR。采用德尔塔——正态法度量投资组合的VaR、边际VaR和成分VaR,使用假设检验法对模型进行回测的研究结果表明,该计算方法下的VaR模型有效,边际VaR和成分VaR能为资产管理者提供更多有关投资组合风险的信息。     

Long range dependence in an emerging stock market’s sectors: volatility modelling and VaR forecasting

This study evaluates the sector risk of the Qatar Stock Exchange (QSE), a recently upgraded emerging stock market, using value-at-risk models for the 7 January 2007–18 October 2015 period. After providing evidence for true long memory in volatility using the log-likelihood profile test of Qu and splitting the sample and dth differentiation tests of Shimotsu, we compare the FIGARCH, HYGARCH and FIAPARCH models under normal, Student-t and skewed-t innovation distributions based on in and out-of-sample VaR forecasts. The empirical results show that the skewed Student-t FIGARCH model generates the most accurate prediction of one-day-VaR forecasts. The policy implications for portfolio managers are also discussed.     

Bayesian asset pricing testing under multivariate t-distribution

The distribution of asset returns has often been proved to be heavy-tailed. In this paper, based on the Fama-French five-factor model with multivariate t-distribution, we develop a convenient and explicit Bayesian approach to test asset pricing. The developed test statistic is only by-product of the Markov Chain Monte Carlo (MCMC) outputs, and hence it is very convenient in practice. Simulation studies demonstrate the effectiveness of the finite sample performance of the proposed approach. Finally, the Fama-French data are used for testing the efficiency of financial markets, and the result shows that the market efficiency is rejected.     

我国国际储备资产的最优结构研究——基于时变Copula及VaR的投资组合模型分析

文章通过构建效用评估函数,使用时变相关T-Copula模型、Monte Carlo模拟和VaR计算方法系统研究了我国国际储备的最优结构。结果发现,我国黄金的最优占比应至少为23%。据此,文章对多个国家储备资产的变化情况进行了动态和静态比较,发现不同类别的演化路径,而以我国和其他金砖四国为代表的类别处于收益递减、风险增大的状态中,亟须加大黄金储备至优化区间。     

马尔可夫调制模型下交换期权的定价

在这篇文章中,假定市场经济状态由一个两状态马尔可夫链描述,风险资产满足一个两状态的马尔可夫调制过程。当市场处于高波动状态时,风险资产的价格满足跳扩散过程;当市场处于稳定状态时,风险资产的价格满足几何布朗运动.通过测度变换的技术,得到了交换期权的定价公式。最后,利用蒙特卡洛方法给出了期权价值的数值结果。     

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.     

Markets liquidity risk under extremal dependence: Analysis with VaRs methods

Value-at-Risk (VaR) is a widely used tool for assessing financial market risk. In practice, the estimation of liquidity extreme risk by VaR generally uses models assuming independence of bid–ask spreads. However, bid–ask spreads tend to occur in clusters with time dependency, particularly during crisis period. Our paper attempts to fill this gap by studying the impact of negligence of dependency in liquidity extreme risk assessment of Tunisian stock market. The main methods which take into account returns dependency to assess market risk is Time series–Extreme Value Theory combination. Therefore we compare VaRs estimated under independency (Variance–Covariance Approach, Historical Simulation and the VaR adjusted to extreme values) relatively to the VaR when dependence is considered. The efficiency of those methods was tested and compared using the backtesting tests. The results confirm the adequacy of the recent extensions of liquidity risk in the VaR estimation. Therefore, we prove a performance improvement of VaR estimates under the assumption of dependency across a significant reduction of the estimation error, particularly with AR (1)-GARCH (1,1)-GPD model.