Tail Dependence Measures Research Articles

On the tail dependence in bivariate hydrological frequency analysis

Abstract In Bivariate Frequency Analysis (BFA) of hydrological events, the study and quantification of the dependence between several variables of interest is commonly carried out through Pearson’s correlation (r), Kendall’s tau (τ) or Spearman’s rho (ρ). These measures provide an overall evaluation of the dependence. However, in BFA, the focus is on the extreme events which occur on the tail of the distribution. Therefore, these measures are not appropriate to quantify the dependence in the tail distribution. To quantify such a risk, in Extreme Value Analysis (EVA), a number of concepts and methods are available but are not appropriately employed in hydrological BFA. In the present paper, we study the tail dependence measures with their nonparametric estimations. In order to cover a wide range of possible cases, an application dealing with bivariate flood characteristics (peak flow, flood volume and event duration) is carried out on three gauging sites in Canada. Results show that r, τ and ρ are inadequate to quantify the extreme risk and to reflect the dependence characteristics in the tail. In addition, the upper tail dependence measure, commonly employed in hydrology, is shown not to be always appropriate especially when considered alone: it can lead to an overestimation or underestimation of the risk. Therefore, for an effective risk assessment, it is recommended to consider more than one tail dependence measure.

Identification of Asset Price Misalignments on Financial Markets With Extreme Value Theory

ABSTRACTThis article examines the potential for concurrence of crises and asset price misalignments from equilibrium in the foreign exchange, stock, and government bond markets of three Central European countries and the euro area. Concurrence is understood as the joint occurrence of extreme asset changes and is assessed with a measure of asymptotic tail dependence in the distributions studied. The results reveal a significant potential for the co-alignment of crises in the examined markets. Evidence for co-movements in misalignments from equilibrium is found among all examined stock and exchange rate markets; although it is not apparent in some government bond markets.

Interval estimation for a measure of tail dependence

Systemic risk concerns extreme co-movement of several financial variables, which involves characterizing tail dependence. The coefficient of tail dependence was proposed by Ledford and Tawn (1996, 1997) to distinguish asymptotic independence and asymptotic dependence. Recently a new measure based on the conditional Kendall’s tau was proposed by Asimit et al. (2015) to measure the tail dependence and to distinguish asymptotic independence and asymptotic dependence. For effectively constructing a confidence interval for this new measure, this paper proposes a smooth jackknife empirical likelihood method, which does not need to estimate any additional quantities such as asymptotic variance. A simulation study shows that the proposed method has a good finite sample performance.

Mind the Tails

We propose a new measure of country left tail dependence defined as the average pairwise of left tail dependency among major sectors which significantly predicts the equity risk premium in- and out-of-sample. This measure endogenously incorporates asynchronous sectorial shocks which are imperceptible at the aggregate level. Using this measure as a predictor in an asset allocation exercise yields a strategy with positive skewness and an out-of-sample Sharpe ratio of 0.56 from 1998 to 2013, a twofold increase when compared to using either the usual predictors, such as the dividend yield or the Treasury bill rate, or assuming no predictability.

Asymmetry in tail dependence in equity portfolios

The asymmetry in the tail dependence between U.S. equity portfolios and the aggregate U.S. market is a well-established property. Given the limited number of observations in the tails of a joint distribution, standard non-parametric measures of tail dependence have poor finite-sample properties and generally reject the asymmetry in the tail dependence. A parametric model, based on a multivariate noncentral t distribution, is developed to measure and test asymmetry in tail dependence. This model allows different levels of tail dependence to be estimated depending on the distribution’s parameters and accommodates situations in which the volatilities or the correlations across returns are time varying. For most of the size, book-to-market, and momentum portfolios, the tail dependence with the market portfolio is significantly higher on the downside than on the upside.

Nonparametric estimation of general multivariate tail dependence and applications to financial time series

In order to analyse the entire tail dependence structure among random variables in a multidimensional setting, we present and study several nonparametric estimators of general tail dependence functions. These estimators measure tail dependence in different orthants, complementing the commonly studied positive (lower and upper) tail dependence. This approach is in line with the parametric analysis of general tail dependence. Under this unifying approach the different dependencies are analysed using the associated copulas. We generalise estimators of the lower and upper tail dependence coefficient to the general multivariate tail dependence function and study their statistical properties. Tail dependence measures come as a response to the incapability of the correlation coefficient as an extreme dependence measure. We run a Monte Carlo simulation study to assess the performance of the nonparametric estimators. We also employ selected estimators in two empirical applications to detect and measure the general multivariate non-positive tail dependence in financial data, which popular parametric copula models commonly applied in the financial literature fail to capture.

Tail dependence and diversification benefits in emerging market stocks: an extreme value theory approach

This article examines tail dependence, the benefits of diversification and the relation between the two for emerging stock markets. We find most emerging equity markets are independent in limiting joint extremes. However, the dependence in finite levels of extremes is still much stronger than the dependence implied by multivariate normality. Therefore, simple correlation analysis can lead to gross underestimation of the chances of joint crashes in multiple markets. Assuming risk-averse investors guarding against extreme losses, diversification benefits are measured for each two-country optimal portfolio by the reduction in quantile risk measures such as value-at-risk and expected shortfall relative to an undiversified portfolio. It is shown that tail dependence measures developed from multivariate extreme value theory are negatively related to diversification benefits and more importantly can explain diversification benefits better than the correlation coefficient at the most extreme quantiles.

Assessing some stylized facts about financial market indexes: a Markov copula approach

Purpose– The aim of this paper is to measure and evaluate the relationship between returns-volatility and trading volume and returns and volatility of financial market indexes using time-varying copulas.Design/methodology/approach– The time dynamic dependence parameter is allowed to evolve according to a restricted ARMA-type equation which includes a constant term that is driven by a hidden two-state first-order Markov chain.Findings– In using this time dynamics in conjunction with non-elliptical distribution functions and tail dependence measure, the authors are allowing for (and focusing on) non-linearities in the returns-volume-volatility relationship. The results support the assumption that current trading volume provides information about future volatility as well as that there is a negative relationship between returns and their volatilities in financial market indexes.Originality/value– The authors provide an interesting empirical interpretation for the regimes the authors have identified: in the high dependence regime the sequential information arrival hypothesis and/or noise trading hypothesis are valid, consequently future volatility prediction is possible and persistent but does not last indefinitely; in the low dependence regime, the future volatility prediction is more unlikely to occur, since both trading volume and return negatives have a low (near zero) relation with future volatility.

Distributional Bounds for Portfolio Risk with Tail Dependence

The present paper proposes a new method for estimating portfolio risk by applying the concept of bounds to a dependence structure. We introduce four tail dependence measures as partial dependence information and derive bounds on the distribution of a non-decreasing function to obtain bounds on risk measures. We show that bounds on risk measures can be tightened significantly in the probability levels with which we are concerned, those for financial risk management. In the present paper, we provide theorems describing the distributional bounds of the proposed method and prove that these bounds are pointwise best-possible bounds. Furthermore, we calculate risk measures, i.e., value at risk and expected shortfall, from empirical return data and compare the effectiveness of the proposed model with that of typical parametric copula models.

Modeling Dependence of Price Spikes in Australian Electricity Markets

Price spikes are of particular importance due to their severe impacts on consumers, businesses and industry. They constitute a major source of price risk to market participants, e.g., electricity retailers with commitments to meet customers' daily electricity demands. To those trading in several electricity markets simultaneously, the probability of simultaneous price spikes termed as tail dependence is of great importance when computing risks. For this purpose, the problem of modeling joint price spikes in the Australian electricity market is considered. A common measure of tail dependence measure is the so-called tail dependence coefficient (TDC). We present a nonparametric estimator of the tail dependence and further, the point estimation is complemented with an hypotheses test. We find significant tail dependence in electricity prices that cannot be ignored. Accurate characterization of this tail dependence is important for a variety of risk management purposes. These include hedging activities of market participants.

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.

Identifying SIFIs: Toward the Simpler Approach

Systemic risk measures generally aim to identify systemically important financial institutions (SIFIs) that would allow regulators to allocate macro-prudential capital requirements in order to reduce risk stemming from such institutions. Among widely-cited are the measures of tail dependence in financial institutions’ equity returns, such as ΔCoVaR of Adrian and Brunnermeier (2011) and Marginal Expected Shortfall (MES) of Acharya et al. (2010). This paper compares nonlinear and linear approaches to modeling return dependence in the estimation of the ΔCoVaR and MES. Our results show that while the refined and complicated estimation techniques are able to produce more accurate value of institution’s systemic risk contribution they do not greatly improve in terms of identifying SIFIs compared to simpler linear estimation method. Modeling dependence linearly sufficient to identify and rank SIFIs.

Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.

Probabilistic assessment of flood risks using trivariate copulas

In this paper, a copula-based methodology is presented for probabilistic assessment of flood risks and investigated the performance of trivariate copulas in modeling dependence structure of flood properties. The flood is a multi-attribute natural hazard and is characterized by mutually correlated flood properties peak flow, volume, and duration of flood hydrograph. For assessing flood risk, many studies have used bivariate analysis, but a more effective assessment can be possible considering all three mutually correlated flood properties simultaneously. This study adopts trivariate copulas for multivariate analysis of flood risks, and applied to a case study of flood flows of Delaware River basin at Port Jervis, NY, USA. On evaluation of various probability distributions for representation of flood variables, it is found that the flood peak flow and volumes can be best represented by Frechet distribution, whereas flood duration by log-normal distribution. The joint distribution is modeled using four trivariate copulas, namely, three fully nested form of Archimedean copulas: Clayton, Gumbel–Hougaard, Frank copulas; and one elliptical copula: Student’s t copula. Based on distance-based performance measures, graphical tests, and tail-dependence measures, it is found that the Student’s t copula best representing the trivariate dependence structure of flood properties as compared to the other copulas. Similar results are found for bivariate copula modeling of flood variables pairs, where Student’s t copula performed better than the other copulas. The obtained copula-based joint distributions are used for multivariate analysis of flood risks, in terms of primary and secondary return periods. The resultant trivariate return periods are compared with univariate and bivariate return periods, and addressed the necessity of multivariate flood risk analysis. The study concludes that the trivariate copula-based methodology is a viable choice for effective risk assessment of floods.

The simple econometrics of tail dependence

This paper provides a regression approach to estimate tail dependence measures. The estimates coincide with the non-parametric estimates following Extreme Value Theory. The approach can easily be extended to higher dimensional analysis. We provide an example on international stock markets.

Fragility index of block tailed vectors

Financial crises are a recurrent phenomenon with important effects on the real economy. The financial system is inherently fragile and it is therefore of great importance to be able to measure and characterize its systemic stability. Multivariate extreme value theory provide us such a framework through the fragility index (Geluk et al., 2007; Falk and Tichy, to appear-a, to appear-b). Here we generalize this concept and contribute to the modeling of the stability of a stochastic system divided into blocks. We will find several relations with well-known tail dependence measures in the literature, which will provide us immediate estimators. We end with an application to financial data.

Joint Survival Analysis of Hedge Funds and Funds of Funds Using Copulas

This paper uses copulas to model the joint survival within the groups of hedge funds and funds of funds managed by the same manager. Given their skewed distribution, a simple survival analysis based on linearity assumptions may fail to fully capture the dependence caused by extreme events in the tails. The study employs both one and two-parameter families of copulas to capture more than one type of dependence, like upper and lower tail dependence and concordance measures. The paper uses both a one-step approach where the margins and the copula parameters are estimated jointly, and a two-step approach where the margins are fitted first and the copula parameter is estimated thereafter given the fixed margins. The results show that the survivor functions of hedge funds and funds of funds while different from each other are strongly dependent, with the degree of dependence increasing in the tails. The evidence in the paper aligns with previous findings on financial markets contagion and it shows that contagion risk is especially important between hedge funds and funds of funds. In particular, the results suggest that a simple linear approach may provide misleading results regarding the full degree of association between hedge funds and funds of funds.

Extremes and predictability in the European pre-industrial climate of a regional climate model

A high-resolution pre-industrial control simulation with the regional climate model REMO is analyzed in detail for different European subregions. To our knowledge, this is the first long pre-industrial control simulation by a regional climate model as well as at comparable resolution. We assess the ability of the climate model to reproduce the observed climate variability in various parts of the continent. In order to investigate the representation of extreme events in the model under pre-industrial greenhouse gas concentrations, selected seasons are examined with regard to the atmospheric circulation and other climatic characteristics that have contributed to the occurrences. A special focus is dedicated to land-atmosphere interactions. Extreme seasons are simulated by the model under various circumstances, some of them strongly resemble observed periods of extraordinary conditions like the summer 2003 or autumn 2006 in parts of Europe. The regional perspective turns out to be of importance when analyzing events that are constituted by meso-scale atmospheric dynamics. Moreover, the predictability of the European climate on seasonal to decadal time scales is examined by relating the statistics of surface variables to large-scale modes of variability impacting the North Atlantic sector like the Meridional Overturning Circulation, the El Nino Southern Oscillation, and the North Atlantic Oscillation. For this purpose, we introduce a measure of tail dependence that quantifies the correlation between extreme values in two variables that describe the state of the climate system. Significant dependence of extreme events can be detected in various situations.

Asymmetry in Tail Dependence of Equity Portfolios

The asymmetry in the tail dependence between U.S. equity portfolios and the aggregate U.S. market is a well-established property. Given the limited number of observations in the tails of a joint distribution, standard non-parametric measures of tail dependence have poor finite-sample properties and generally reject the asymmetry in the tail dependence. A parametric model, based on a multivariate noncentral t distribution, is developed to measure and test asymmetry in tail dependence. This model allows different levels of tail dependence to be estimated depending on the distribution's parameters and accommodates situations in which the volatilities or the correlations across returns are time varying. For most of the size, book-to-market, and momentum portfolios, the tail dependence with the market portfolio is significantly higher on the downside than on the upside.

Clustering in Emerging Equity Markets

We consider pairwise tail behavior of return series for identifying most important emerging markets clusters. Pairs of markets belonging to the same group present similar type and strength of interdependence during stressful times, represented by a common copula and a statistically equivalent measure of tail dependence. By collapsing data from d markets in a group we overcome the difficult problem of finding their (higher dimensional) d-variate distribution. Results may help portfolio managers to deal out risk due to comovements within clusters. We provide examples on how this can be done. Our study contribute on the discussion about international association among stock markets during turbulent periods, and do not confirm the intuition that the observed association between extremes should be credited to markets drivers.