Tail dependence of the Gaussian copula revisited

Tail dependence refers to clustering of extreme events. In the context of financial risk management, the clustering of high-severity risks has a devastating effect on the well-being of firms and is thus of pivotal importance in risk analysis.When it comes to quantifying the extent of tail dependence, it is generally agreed that measures of tail dependence must be independent of the marginal distributions of the risks but rather solely copula-dependent. Indeed, all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copulas’ domain of definition.In this paper we urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. For the Gaussian copula, however, we prove that the classical measures are maximal. The implication of the result is two-fold: On the one hand, it means that in the Gaussian case, the (weak) measures of tail dependence that have been reported and used are of utmost prudence, which must be a reassuring news for practitioners. On the other hand, it further encourages substitution of the Gaussian copula with other copulas that are more tail dependent.     

On a bivariate copula with both upper and lower full-range tail dependence

Copula functions can be useful in accounting for various dependence patterns appearing in joint tails of data. We propose a new two-parameter bivariate copula family that possesses the following features. First, both upper and lower tails are able to explain full-range tail dependence. That is, the dependence in each tail can range among quadrant tail independence, intermediate tail dependence, and usual tail dependence. Second, it can capture upper and lower tail dependence patterns that are either the same or different. We first prove the full-range tail dependence property, and then we obtain the corresponding extreme value copula. There are two applications based on the proposed copula. The first one is modeling pairwise dependence between financial markets. The second one is modeling dynamic tail dependence patterns that appear in upper and lower tails of a loss-and-expense data.     

Bias-reduced estimators for bivariate tail modelling

Ledford and Tawn (1997) introduced a flexible bivariate tail model based on the coefficient of tail dependence and on the dependence of the extreme values of the random variables. In this paper, we extend the concept by specifying the slowly varying part of the model as done by Hall (1982) with the univariate case. Based on Beirlant et al. (2009), we propose a bias-reduced estimator for the coefficient of tail dependence and for the estimation of small tail probabilities. We discuss the properties of these estimators via simulations and a real-life example. Furthermore, we discuss some theoretical asymptotic aspects of this approach.     

尾部风险网络视角下的金融机构系统性风险贡献研究

金融机构的尾部风险关联模式及结构在金融系统性风险的形成演化中发挥重要作用。利用CoVaR指标及分位数回归方法,衡量金融机构之间的尾部风险传染强度,进而建立金融机构尾部风险动态网络。分析全连接网络及阈值法下过滤网络的全局和局部拓扑结构特征及其动态演化规律。建立面板数据回归模型,研究网络拓扑结构特征对金融机构系统性风险贡献的影响。实证研究发现,全连接网络的节点强度,能有效地衡量金融机构尾部风险传染强度及承受强度,并揭示其动态变化规律;各机构的尾部风险传染强度及承受强度排序匹配性存在差异;随着时间推进,各金融机构间的平均尾部风险传染路径缩短、系统性风险更易迅速累积和爆发;滞后一期的节点出度、节点入度及聚集系数越大,相应金融机构的系统性风险贡献越小;滞后一期的节点介数和节点接近中心度越大,相应金融机构的系统性风险贡献越大。研究结果对于金融机构的宏观审慎监管及系统性风险管理,提供了一个全新的基于金融机构尾部风险网络的视角。     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

股票收益率尾部相关性的Copula度量及模拟

股票收益率尾部相关性是研究金融市场关联性的重要内容.由于传统的τ、ρ等相关系数是对随机变量的全局度量,不适合用于收益率分布尾部这种局部特征的相关性度量.因此,在引入左尾(右尾)相关系数的基础上,讨论了它们的Copula度量及其相关性质.最后,通过计算机模拟分析了沪、深股指收益率尾部相关性的变化趋势,有效避免了Copula模型的设定困难,并得到了尾部相关性增强、相关不对称等结论.     

Measuring the subprime crisis contagion: Evidence of change point analysis of copula functions

In this paper, we first determine the existence of structural changes in the dependence between time series of equity index returns of two markets using the change point testing method. The method is based on Archimedean copula functions, which are able to comprehensively describe dependence characteristics of random variables. The degree of financial contagion between markets is subsequently estimated using the tail dependence coefficient of copula functions before and after the change point. We empirically test our method by investigating financial contagion during the subprime crisis between the US S&P 500 index and five Asian markets, namely China, Japan, Korea, Hong Kong and Taiwan. Our results show that a statistically significant change point exists in the dependence between the US market and all Asian stock markets except Taiwan. The upper tail dependence is larger after the time of change, implying the existence of contagion during the banking crisis between the US and the Asian economies. The degree of financial contagion is also estimated and found to be consistent with market events and media reports during that period.     

Tail negative dependence and its applications for aggregate loss modeling

Tail order of copulas can be used to describe the strength of dependence in the tails of a joint distribution. When the value of tail order is larger than the dimension, it may lead to tail negative dependence. First, we prove results on conditions that lead to tail negative dependence for Archimedean copulas. Using the conditions, we construct new parametric copula families that possess upper tail negative dependence. Among them, a copula based on a scale mixture with a generalized gamma random variable (GGS copula) is useful for modeling asymmetric tail negative dependence. We propose mixed copula regression based on the GGS copula for aggregate loss modeling of a medical expenditure panel survey dataset. For this dataset, we find that there exists upper tail negative dependence between loss frequency and loss severity, and the introduction of tail negative dependence structures significantly improves the aggregate loss modeling.     

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.     

Procedure of test to compare the tail indices

We propose a test procedure which compares the extreme value indices of two samples with heavy tail distributions. On a theoretical point of view, we adopt the minimax nonparametric point of view. We exhibit the separating rate between the null hypothesis and the alternative of our procedure. Next, we present a data driven test methodology and we evaluate its performance thanks to an extensive simulation study. As a practical real-life application, we compare the risk behaviors of a panel of different financial data.     

几种Copula函数在沪深股市相关性建模中的应用

研究了Copula函数对沪深股市的相关性建模问题.许多学者用Gaussian Copula建模,但是它无法捕捉到尾部变化,尾部相关系数不存在.用t-Copula度量中国股市的相关性,捕捉到了尾部变化,并计算出了尾部相关系数,克服了Gaussian Copula对相关性建模的不足,并通过AIC准则比较得到t-Copula优于Gaussian Copula.最后对3种Archimedean Copula进行比较,通过比较它们与经验分布函数的距离,说明Gumble Copula更加适用于中国的金融市场.     

沪深股市收益率的尾部相关函数

尾部相关性是相关性分析中重要的一类,利用度量尾部相关性的指标χ,χ-以及尾部相关函数ρ(θ)来分析尾部相关性,并给出ρ(θ)的一种非参数估计方法.通过这两种方法研究上证综合指数和深证成分指数日收盘指数对数收益率在损失情况下的尾部相关性,结果表明两市指数日对数收益率具有很强的尾部相关性.     

Upper comonotonicity

In this article, we study a new notion called upper comonotonicity, which is a generalization of the classical notion of comonotonicity. A random vector is upper-comonotonic if its components are moving in the same direction simultaneously when their values are greater than some thresholds. We provide a characterization of this new notion in terms of both the joint distribution function and the underlying copula. The copula characterization allows us to study the coefficient of upper tail dependence as well as the distributional representation of an upper-comonotonic random vector. As an application to financial economics, we show that the several commonly used risk measures, like the Value-at-Risk, the Tail Value-at-Risk, and the expected shortfall, are additive, not only for sum of comonotonic risks, but also for sum of upper-comonotonic risks, provided that the level of probability is greater than a certain threshold.     

相关系数与相关性度量

研究了度量相关性的两个主要工具:线性相关系数和尾部相关系数.线性相关系数反映了变量间的线性相关性,这对于一般的椭圆型分布是合适的.但如果随机变量具有不对称的尾部变化特征时,要用尾部相关系数描述它们之间的相关性.通过相关函数C opu la,对沪深股市的尾部相关系数进行了定量分析.结果表明:沪深股市具有较强的相关性.     

The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution

The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution.     

Four theorems and a financial crisis

In this paper we give an academic assessment of the financial crisis (crises) from our point of view and discuss where quantitative risk management went wrong. We formulate four mathematical theorems/research areas which have relevance for financial crises in general where the underlying theme is model uncertainty. Related to these theorems, key issues that will be discussed are: financial alchemy on Wall street, risk aggregation and diversification, tail dependence for a portfolio of losses, and the significance of correlation bounds.     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

基于Dirichlet分布有限混合模型的Bayes聚类

本文基于Dirichlet分布有限混合模型,提出了一种用于成分数据的Bayes聚类方法.采用EM算法获得模型参数的估计,用BIC准则确定类数,用类似于Bayes判别的方法对各观测分类.推导了计算公式,编写出程序.模拟研究结果表明,本文提出的方法有较好的聚类效果.     

The bivariate normal copula function is regularly varying

We derive the rate of decay of the tail dependence of the bivariate normal distribution and establish its link with regularly varying functions. This result is an initial step in explaining the discrepancy between a zero asymptotic tail dependence coefficient and mass in the tail of a joint distribution.     

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

In this paper we study the asymptotic behavior of the tail probabilities of sums of dependent and real-valued random variables whose distributions are assumed to be subexponential and not necessarily of dominated variation. We propose two general dependence assumptions under which the asymptotic behavior of the tail probabilities of the sums is the same as that in the independent case. In particular, the two dependence assumptions are satisfied by multivariate Farlie-Gumbel-Morgenstern distributions.