The Tale of Tail Dependence: Modeling height–diameter relationships with elliptical copulas

Accurately modelling the dependence structure between financial assets in a portfolio optimization framework has attracted growing attention in statistical and financial literature. Since in these assets several types of tail dependence might occur simultaneously, it is fundamental for parametric models to adequately replicate their whole tail dependence structure. This article investigates the effectiveness of Vine copulas in modelling counterdiagonal/nonpositive tail dependence, so far overlooked. We obtain expressions for their corresponding general tail dependence function which accounts for all dependences. This generalises the well-known approach of using the survival copula to study upper tail dependence, rather than using rotations on the data. We prove that, further to the already known flexibility to model asymmetric lower and upper tail dependence, Vine copulas can model all multivariate types of tail dependence simultaneously. In an empirical application, using a D-Vine copula with appropriate choices of bivariate linking copulas, we are able to capture the tail dependence structure of a portfolio of financial data in which different types of tail dependence coexist. Further to this, we test to what extent Vine copulas are able to model different types of tail dependence.

In this study, we measure asymmetric negative tail dependence and discuss their statistical properties. In a simulation study, we show the reliability of nonparametric estimators of tail copula to measure not only the common positive lower and upper tail dependence, but also the negative “lower–upper” and “upper–lower” tail dependence. The use of this new framework is illustrated in an application to financial data. We detect the existence of asymmetric negative tail dependence between stock and volatility indices. Many common parametric copula models used in finance fail to capture this characteristic.

PurposeThis paper examines the extreme dependence and asymmetric risk spillovers between crude oil futures and ten US stock sector indices (consumer discretionary, consumer staples, energy, financials, health care, industrials, information technology, materials, telecommunication and utilities) before and during COVID-19 outbreak. This study is based on the rationale that stock sectors exhibit heterogeneity in their response to oil prices depending on whether they are classified as oil-intensive or non-oil-intensive sectors and the possible time variation in the dependence and risk spillover effects.Design/methodology/approachThe authors employ static and dynamic symmetric and asymmetric copula models as well as Conditional Value at Risk (VaR) (CoVaR). Finally, they use robustness tests to validate their results.FindingsBefore the COVID-19 pandemic, crude oil returns showed an asymmetric tail dependence with all stock sector returns, except health care and industrials (materials), where an average (symmetric tail) dependence is identified. During the COVID-19 pandemic, crude oil returns exhibit a lower tail dependency with the returns of all stock sectors, except financials and consumer discretionary. Furthermore, there is evidence of downside and upside risk asymmetric spillovers from crude oil to stock sectors and vice versa. Finally, the risk spillovers from stock sectors to crude oil are higher than those from crude oil to stock sectors, and they significantly increase during the pandemic.Originality/valueThere is heterogeneity in the linkages and the asymmetric bidirectional systemic risk between crude oil and US economic sectors during bearish and bullish market conditions; this study is the first to investigate the average and extreme tail dependence and asymmetric spillovers between crude oil and US stock sectors.

This paper investigates the dynamic tail dependence risk between BRICS economies and the world energy market, in the context of the COVID-19 financial crisis of 2020, in order to determine optimal investment decisions based on risk metrics. For this purpose, we employ a combination of novel statistical techniques, including Vector Autoregressive (VAR), Markov-switching GJR-GARCH, and vine copula methods. Using a data set consisting of daily stock and world crude oil prices, we find evidence of a structure break in the volatility process, consisting of high and low persistence volatility processes, with a high persistence in the probabilities of transition between lower and higher volatility regimes, as well as the presence of leverage effects. Furthermore, our results based on the C-vine copula confirm the existence of two types of tail dependence: symmetric tail dependence between South Africa and China, South Africa and Russia, and South Africa and India, and asymmetric lower tail dependence between South Africa and Brazil, and South Africa and crude oil. For the purpose of diversification in these markets, we formulate an asset allocation problem using raw returns, MS GARCH returns, and C-vine and R-vine copula-based returns, and optimize it using a Particle Swarm optimization algorithm with a rebalancing strategy. The results demonstrate an inverse relationship between the risk contribution and asset allocation of South Africa and the crude oil market, supporting the existence of a lower tail dependence between them. This suggests that, when South African stocks are in distress, investors tend to shift their holdings in the oil market. Similar results are found between Russia and crude oil, as well as Brazil and crude oil. In the symmetric tail, South African asset allocation is found to have a well-diversified relationship with that of China, Russia, and India, suggesting that these three markets might be good investment destinations when things are not good in South Africa, and vice versa.

Asymmetric Dependence, Tail Dependence, and the Time Interval Over Which the Variables are Measured

Asymmetric Dependence, Tail Dependence, and the Time Interval over Which the Variables Are Measured

<p>This paper investigates the structure and degree of extreme dependence in international equity markets using carefully selected tools from the theory of copulas. We examine both the static and dynamic dependence via unconditional and conditional copulas. We Önd signiÖcant asymmetric tail dependence in equity markets, with the overall larger lower tail dependence than upper tail dependence. Moreover, in Europe and East Asia but not in North America, the extreme dependence is time-varying in both its structure and degree. Our results also indicate a higher intra-continental than inter-continental tail dependence. Our Öndings have important implications in global risk management strategies. </p>

<p>This paper investigates the structure and degree of extreme dependence in international equity markets using carefully selected tools from the theory of copulas. We examine both the static and dynamic dependence via unconditional and conditional copulas. We Önd signiÖcant asymmetric tail dependence in equity markets, with the overall larger lower tail dependence than upper tail dependence. Moreover, in Europe and East Asia but not in North America, the extreme dependence is time-varying in both its structure and degree. Our results also indicate a higher intra-continental than inter-continental tail dependence. Our Öndings have important implications in global risk management strategies. </p>

Bitcoin’s (BT) energy demand due to the proof of work mining algorithm and transaction methods has been criticized in recent literature because of BT’s global energy consumption (EC) being equal to or even greater than that of some industrialized economies. This study explores the contagion and causality dynamics under nonlinearity and heteroskedasticity with Markovian‐type regime switches among BT price volatility, EC from BT, and its effects on the global carbon dioxide (CO2) emissions with a daily sample of January 2, 2012–December 24, 2023. In the empirical methodology, following the preliminary tests that indicated nonlinearity and heteroskedasticity, this study employed novel Markov‐switching‐based nonlinear volatility copula and causality models to capture the marginal distributions of BT, its EC and CO2 emissions, and their joint distributions for contagion and tail dependence and determine nonlinear and asymmetric causality relations in distinct regimes of high and low volatility governed by Markov chains. The empirical results indicate significant and positive copula parameters with high magnitudes demonstrating asymmetric contagion and tail dependence occurring at the extreme levels of BT price volatility in the upper and lower tails during both regimes. Novel Markov‐switching‐based copula causality tests designated unidirectional causality from BT to CO2, as well as from BT’s EC to CO2 emissions during each regime coupled with existent bidirectional causality confirming cyclical feedback effects between BT prices and its EC in both regimes. For robustness and comparison, single‐regime copula models were employed, and their findings confirmed the results concerning significant and positive tail dependence and contagion. However, the single‐regime approach led to a set of inconsistencies in causality results owing to omitting nonlinearity while capturing heteroskedasticity and confirming the nonrejection of causality between BT, EC, and CO2 emissions. Important policy recommendations include green alternatives to cryptocurrency‐mining algorithms.

Global financial investors have been confronted in recent years with an increasing frequency of market shocks and returns’ outliers, until the unprecedented surge of financial risk observed in 2008. From a statistical viewpoint, those market dynamics have shown not only asymmetric returns and fat tails but also a time-varying tail dependence, stimulating the formulation of portfolio selection models based on such assumptions. The concept of tail dependence on upper or lower tails, roughly speaking, focuses on the risk that tail events may occur jointly in different markets. This notion can be given a rigorous probabilistic definition, and it turns out that a distinction between upper and lower tails is relevant in portfolio management. In this paper, relying on a discrete modeling framework, we present a scenario generation algorithm able to capture this time-varying asymmetric tail dependence, and evaluate resulting optimal investment policies based on 4-stages 1-month planning horizons. The scenario tree aims at approximating a stochastic process combining an ARMA-GARCH model and a dynamic Student-t-Clayton copula. From a methodological viewpoint, scenario trees are generated from this model by stage-wisely sampling and clustering and to improve tail fitting with original data, the scenarios’ nodal probabilities are calibrated on the returns’ lower tails for a set of equity indices. The resulting scenario trees are then applied to solve a multiperiod portfolio selection problem. We present a set of empirical results to validate the adopted statistical approach and the optimal portfolio strategies able to capture asymmetric tail returns.

For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.

We examine the asymmetric and extreme tail dependence between five energy markets (crude oil, natural gas, heating oil, gasoline, and coal) and green bonds using a time-varying optimal copula (TVOC) model. The results indicate the existence of multiple tail dependence regimes, implying the unsuitability of applying static or dynamic models to entirely describe the extreme dependence between energy markets and green bonds. There is an extreme negative tail dependence between green bonds and four energy commodities (crude oil, heating oil, gasoline, and coal), whereas an extreme positive tail dependence is shown for green bonds and natural gas. Largely, stressful periods such as the COVID19 outbreak, shape the tail dependence, which is also the case for extreme risk spillovers involving, especially, crude oil. Hedging effectiveness analysis indicates that green bonds can effectively hedge most of the energy commodities. The conditional diversification benefits seem to vary with time, especially for natural gas, and differ across the energy markets. Notably, green bonds provide higher conditional diversification benefits when combined with coal for several portfolio weights.

This study examines the dependence structure and systemic risk concerning Sukuk, Sharia, and the Gulf Cooperation Council (GCC) stock markets. We first use copula functions to investigate the dependence structure between these markets, and subsequently, apply the conditional value‐at‐risk (CoVaR) and delta CoVaR (∆CoVaR) to assess the systemic risk. Results show evidence of time‐varying symmetric tail dependence between Islamic stock markets and GCC stock markets, except for Saudi Arabia in which an average tail dependence is observed. Sukuk has symmetric tail dependence with Bahrain, Oman, and Abu Dhabi; average dependence with Qatar and Saudi Arabia; and asymmetric tail dependence with Dubai and Kuwait. Importantly, no evidence of systemic risk for Islamic (DJIM and Sukuk) and Gulf stock markets was found.

This study proposes AC dynamic skew-t copula with cDCC model to capture the dynamic asymmetric tail dependence structure among multi-asset classes (government bonds, corporate bonds, equities, and REITs). We provide new evidence that lower tail dependence coefficients increased compared to upper ones for all pairs in the COVID-19 crash and the recent high inflation period, indicating that the diversification effect through multi-asset investment decreased. Our empirical analysis also shows that in terms of AIC and BIC, dynamic AC skew-t copula fits data of multi-asset classes better than other dynamic elliptical copulas because it can consider the above dependence structure characteristics. Furthermore, out-of-sample analysis reveals that considering an asymmetry of tail dependence structure at each point with an AC dynamic skew-t copula enhances expected shortfall (ES) estimation accuracy and the performance of a minimum ES portfolio. These results indicate that capturing dynamic asymmetric tail dependence is crucial for multi-asset portfolio management.

This paper develops a new time-varying mixture copula, in which the dynamic weights of four distinct copulas are determined by a two-stratum process, to investigate the magnitude of tail dependence in four independent quadrants. In the two-stratum process, the weight of each copula is determined firstly by the relative importance of positive and negative dependence structures, and then by its own past values and adjustment processes. The weighting mechanism is time-varying in each stratum. This new specification is applied to analyze the asymmetric tail dependencies between the stock and exchange rate markets. Empirical results show four interesting findings. First, the quasi-maximum likelihood estimation (QMLE) has a better fitting ability than does the inference function for margins. The relative efficiency of the QMLE is irrespective of marginal specifications. Second, the goodness-of-fit tests of the new time-varying mixture copula are crucially affected by the marginal specifications. Third, estimation methods impact mixture weights. Four distinct tail dependencies are observed, revealing the importance of considering all four tails concurrently, and not just parts of the four tails. Fourth, the asymmetric positive and negative dependencies are significant. Each country shows a similar pattern of asymmetric negative dependence, but a different pattern of asymmetric positive dependence. These empirical findings provide important portfolio allocation implications.