Estimating value at risk of portfolio by conditional copula-GARCH method

Copula functions represent a methodology that describes the dependence structure of a multi-dimension random variable and has become one of the most significant new tools to handle risk factors in finance, such as Value-at Risk (VaR), which is probably the most widely used risk measure in financial institutions. Combining copula and the forecast function of the GARCH model, this paper proposes a new method, called conditional copula-GARCH, to compute the VaR of portfolios. This work presents an application of the copula-GARCH model in the estimation of a portfolio’s VaR, composed of NASDAQ and TAIEX. The empirical results show that, compared with traditional methods, the copula model captures the VaR more successfully. In addition, the Student-t copula describes the dependence structure of the portfolio return series quite well.     

基于Clayton Copula-GARCH模型投资组合VaR的度量

运用Copula方法研究了含股指期货的投资组合的风险度量问题.首先采用不同的GARCH模型对单个资产收益率建模,然后选择Clayton Copula函数来描述投资组合各资产之间的相关结构,建立联合分布模型,进而采用Monte Carlo方法模拟产生各资产的收益率序列,计算出投资组合的VaR.Kupiec检验表明,ClaytonCopula-GARCH模型在投资组合风险度量上具有较高的准确性.     

天然气期货和现货价格间的相关性研究

基于天然气期货价格与现货价格序列间具有强非线性特征,本文将GARCH模型和Copula函数思想进行结合,同时考虑了天然气期货和现货价格间的时变相关结构,构建了时变Copula(GARCH-Normal、GARCH-GED和GARCH-t)模型,利用美国纽约商品交易所(NYMEX)Henry Hub交易中心天然气期货价格和现货价格数据进行实证研究。实证结果表明:GARCH-GED模型能够准确地拟合天然气期货与现货价格时间序列;时变SJC-Copula函数能够更好的描述天然气期货价格与现货价格间的相关性;天然气期货与现货价格间的相关性不是对称的,上尾的相关性小于下尾相的相关性。     

基于半参数多元Copula-GARCH模型的开放式基金投资组合风险分析

利用Copula技术对我国开放式基金市场的投资组合进行了风险分析。为克服传统Copula模型对金融尾部数据刻画能力的不足,建立了半参数的多元Copula-GARCH模型,灵活地对各支基金的边缘分布进行拟合,刻画了开放式基金投资组合的相依结构。并利用基于Copula技术的蒙特卡洛模拟,对投资组合进行了VaR分析,结果证实了所建立模型的可行性和有效性。     

Sensitivity of portfolio VaR and CVaR to portfolio return characteristics

Risk management through marginal rebalancing is important for institutional investors due to the size of their portfolios. We consider the problem of improving marginally portfolio VaR and CVaR through a marginal change in the portfolio return characteristics. We study the relative significance of standard deviation, mean, tail thickness, and skewness in a parametric setting assuming a Student’s t or a stable distribution for portfolio returns. We also carry out an empirical study with the constituents of DAX30, CAC40, and SMI. Our analysis leads to practical implications for institutional investors and regulators.     

A generalized beta copula with applications in modeling multivariate long-tailed data

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases.To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.     

一类重尾风险因子的模拟及其投资高风险值和置信区间的估计

由于金融市场中的日周期或短周期对数回报率的样本数据多数呈现胖尾分布，于是现有的正态或对数正态分布模型都在不同程度上失效，为了准确模拟这种胖尾分布和提高投资风险估计及金融管理，本文引进了一种可根据实际金融市场数据作出调正的蒙特卡洛模拟方法．这个方法可以有效地复制金融产品价格的日周期对数回报率数据的胖尾分布．结合非参数估计方法，利用该模拟方法还得到投资高风险值以及高风险置信区间的准确估计。     

Modeling dependence based on mixture copulas and its application in risk management

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the copula, and the application of copula functions in VaR valuation. After the introduction of the pure copula method and the maximum and minimum mixture copula method, authors present a new algorithm based on the more generalized mixture copula functions and the dependence measure, and apply the method to the portfolio of Shanghai stock composite index and Shenzhen stock component index. Comparing with the results from various methods, one can find that the mixture copula method is better than the pure Gaussian copula method and the maximum and minimum mixture copula method on different VaR level.     

Jump tail dependence in Lévy copula models

This paper investigates the dependence of extreme jumps in multivariate Lévy processes. We introduce a measure called jump tail dependence, defined as the probability of observing a large jump in one component of a process given a concurrent large jump in another component. We show that this measure is determined by the Lévy copula alone and that it is independent of marginal Lévy processes. We derive a consistent nonparametric estimator for jump tail dependence and establish its asymptotic distribution. Regarding the economic relevance of the measure, a simulation study illustrates that jump tail dependence has a substantial impact on financial portfolio distributions and optimal portfolio weights.     

多元Copula-GARCH模型及其在金融风险分析上的应用

针对传统风险分析模型的不足,结合Copula技术和GARCH模型,提出了多元Copula-GARCH模型。指出该模型不仅可以捕捉金融市场间的非线性相关性,还可以得到更灵活的多元分布进而用于资产投资组合VaR分析。在详细探讨了基于Copula技术的资产投资组合的MonteCarlo仿真技术的基础上,运用具有不同边缘分布的多元Copula-GARCH模型,对上海股市进行了研究,结果证实了所提模型和方法的可行性和有效性。     

Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assuming infinitely divisible distributions, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal VaR and the marginal AVaR. We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average, first statistically validating the model based on goodness-of-fit tests and then demonstrating how the marginal VaR and marginal AVaR can be used for portfolio optimization using the model. Based on the empirical evidence presented in this paper, our framework offers more realistic portfolio risk measures and a more tractable method for portfolio optimization.     

Estimating and forecasting portfolio’s Value-at-Risk with wavelet-based extreme value theory: Evidence from crude oil prices and US exchange rates

This article proposes a wavelet-based extreme value theory (W-EVT) approach to estimate and forecast portfolio’s Value-at-Risk (VaR) given the stylized facts and complex structure of financial data. Our empirical application to portfolios of crude oil prices and US dollar exchange rates shows that the W-EVT models provide an effective and powerful tool for gauging extreme moments and improving the accuracy of portfolio’s VaR estimates and forecasts after noise is removed from the original data.     

包含股指期货的投资组合之风险研究——Copula方法在金融风险管理中的应用

本文运用Copula方法研究了含股指期货的投资组合的风险度量问题.由于股指期货和股票现货之间存在很大的相关性,因此在度量组合的风险时,各资产间的相关结构起到了关键作用,但这一相关结构很难用线性的相关系数去刻画,本文采用Copula模型来描述相关结构。而后,我们构建了基于Copula理论的风险度量指标PVaR,并验证了不同Copula模型的拟合效果.我们利用沪深300指数的数据来研究股指期货和现货的相关结构,并使用了多种Copula函数结合不同的边际分布假设进行了模拟,说明了Copula方法在风险度量尤其是包含了股指期货的投资组合的风险度量上具有较高的精确性.     

基于倾斜-t Copula的半参数马尔可夫链

在不指定时间序列结构的情况下,我们的分布模型是基于多变量离散时间的相应马尔可夫族和相关变量一维的边际分布.这样的模型可以同时处理时间序列之间的相互依赖和每个时间序列沿时间方向的依赖.具体的参数copula被指定为倾斜-t. 倾斜-t Copla能够处理不对称,偏斜和粗尾的数据分布.三个股票指数日均收益的实证研究表明,倾斜-t copula的马尔可夫模型要比以下模型更好:倾斜正态Copula马可夫, t-copula马可夫, 倾斜-t copula但无马尔可夫特性.     

Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.     

Copula models for insurance claim numbers with excess zeros and time-dependence

This paper develops two copula models for fitting the insurance claim numbers with excess zeros and time-dependence. The joint distribution of the claims in two successive periods is modeled by a copula with discrete or continuous marginal distributions. The first model fits two successive claims by a bivariate copula with discrete marginal distributions. In the second model, a copula is used to model the random effects of the conjoint numbers of successive claims with continuous marginal distributions. Zero-inflated phenomenon is taken into account in the above copula models. The maximum likelihood is applied to estimate the parameters of the discrete copula model. A two-step procedure is proposed to estimate the parameters in the second model, with the first step to estimate the marginals, followed by the second step to estimate the unobserved random effect variables and the copula parameter. Simulations are performed to assess the proposed models and methodologies.     

Sudden changes in variance and time varying hedge ratios

This paper analyzes the influence of sudden changes in the unconditional volatility on the estimation and forecast of volatility and its impact on futures hedging strategies. We employ several multivariate GARCH models to estimate the optimal hedge ratios for the Spanish stock market including in each one some well-known patterns that may affect volatility forecasts (asymmetry and sudden changes). The main empirical results show that more complex models including sudden changes in volatility outperform the simpler models in hedging effectiveness both with in-sample and out-of-sample analysis. However, the evidence is stronger when the loss distribution tail is used as a measure for the effectiveness (Value at Risk (VaR) and Expected Shortfall (ES)) suggesting that traditional measures based on the variance of the hedged portfolio should be used with caution.     

A copula entropy approach to correlation measurement at the country level

The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.     

Modeling volatility and dependency of agricultural price and production indices of Thailand: Static versus time-varying copulas

Volatility and dependence structure are two main sources of uncertainty in many economic issues, such as exchange rates, future prices and agricultural product prices etc. who fully embody uncertainty among relationship and variation. This paper aims at estimating the dependency between the percentage changes of the agricultural price and agricultural production indices of Thailand and also their conditional volatilities using copula-based GARCH models. The motivation of this paper is twofold. First, the strategic department of agriculture of Thailand would like to have reliable empirical models for the dependency and volatilities for use in policy strategy. Second, this paper provides less restrictive models for dependency and the conditional volatility GARCH. The copula-based multivariate analysis used in this paper nested the traditional multivariate as a special case (Tae-Hwy and Xiangdong, 2009) [13]. Appropriate marginal distributions for both, the percentage changes of the agricultural price and agricultural production indices were selected for their estimation. Static as well as time varying copulas were estimated. The empirical results were found that the suitable margins were skew t distribution and the time varying copula i.e., the time varying rotate Joe copula (270°) was the choice for the policy makers to follow. The one-period ahead forecasted-growth rate of agricultural price index conditional on growth rate of agricultural production index was also provided as an example of forecasting it using the resulted margins and time-varying copula based GARCH model.