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Purpose – In this chapter, copula theory is used to model dependence structure between hedge fund returns series.

Methodology/approach – Goodness-of-fit tests, based on the Kendall's functions, are applied as selection criteria of the “best” copula. After estimating the parametric copula that best fits the used data, we apply previous results to construct the cumulative distribution functions of the equally weighted portfolios.

Findings – The empirical validation shows that copula clearly allows better estimation of portfolio returns including hedge funds. The three studied portfolios reject the assumption of multivariate normality of returns. The chosen structure is often of Student type when only indices are considered. In the case of portfolios composed by only hedge funds, the dependence structure is of Franck type.

Originality/value of the chapter – Introducing goodness-of-fit bootstrap method to validate the choice of the best structure of dependence is relevant for hedge fund portfolios. Copulas would be introduced to provide better estimations of performance measures.

Portfolio credit derivatives, such as basket credit default swaps (basket CDS), require for their pricing an estimation of the dependence structure of defaults, which is known to exhibit tail dependence as reflected in observed default contagion. A popular model with this property is the (Student's) t-copula; unfortunately there is no fast method to calibrate the degree of freedom parameter.

In this paper, within the framework of Schönbucher's copula-based trigger-variable model for basket CDS pricing, we propose instead to calibrate the full multivariate t distribution. We describe a version of the expectation-maximization algorithm that provides very fast calibration speeds compared to the current copula-based alternatives.

The algorithm generalizes easily to the more flexible skewed t distributions. To our knowledge, we are the first to use the skewed t distribution in this context.

We study the statistical properties of Pearson correlation coefficients of Gaussian ranks, and Gaussian rank regressions – ordinary least-squares (OLS) models applied to those ranks. We show that these procedures are fully efficient when the true copula is Gaussian and the margins are non-parametrically estimated, and remain consistent for their population analogs otherwise. We compare them to Spearman and Pearson correlations and their regression counterparts theoretically and in extensive Monte Carlo simulations. Empirical applications to migration and growth across US states, the augmented Solow growth model and momentum and reversal effects in individual stock returns confirm that Gaussian rank procedures are insensitive to outliers.

Knowledge of the dependence structure between financial assets is crucial to improve the performance in financial risk management. It is known that the copula completely summarizes the dependence structure among multiple variables. We propose a multivariate exponential series estimator (ESE) to estimate copula densities nonparametrically. The ESE has an appealing information-theoretic interpretation and attains the optimal rate of convergence for nonparametric density estimations in Stone (1982). More importantly, it overcomes the boundary bias of conventional nonparametric copula estimators. Our extensive Monte Carlo studies show the proposed estimator outperforms the kernel and the log-spline estimators in copula estimation. It also demonstrates that two-step density estimation through an ESE copula often outperforms direct estimation of joint densities. Finally, the ESE copula provides superior estimates of tail dependence compared to the empirical tail index coefficient. An empirical examination of the Asian financial markets using the proposed method is provided.

This paper aims to analyze the dependence structure between the Green Sukuk Spread in Malaysia and uncertainty factors from January 1, 2017, to May 23, 2023, covering two main periods: the pre-COVID-19 and the COVID-19 periods.

This study contributes to the current literature by explicitly modeling nonlinear dependencies using the Regular vine copula approach to capture asymmetric characteristics of the tail dependence distribution. This study used the Archimedean copula models: Student’s-t, Gumbel, Gaussian, Clayton, Frank and Joe, which exhibit different tail dependence structures.

The empirical results suggest that Green Sukuk and various uncertainty variables have the strongest co-dependency before and during the COVID-19 crisis. Due to external uncertainties (COVID-19), the results reveal that global factors, such as the Infect-EMV-index and the higher financial stress index, significantly affect the spread of Green Sukuk. Interestingly, in times of COVID-19, its dependence on Green Sukuk and the news sentiment seems to be a symmetric tail dependence with a Student’s-t copula. This result is relevant for hedging strategies, as investors can enhance the performance of their portfolio during the COVID-19 crash period.

This study contributes to a better understanding of the dependency structure between Green Sukuk and uncertainty factors. It is relevant for market participants seeking to improve their risk management for Green Sukuk.

This paper examines and forecasts correlations between cryptocurrencies and major fiat currencies using Generalized Autoregressive Score (GAS) time-varying copulas. The authors examine to which extent the multivariate GAS method captures the volatility persistence and the nonlinear interaction effects between cryptocurrencies and major fiat currencies.

The authors model tail dependence between conventional currencies and Bitcoin utilizing a Glosten-Jagannathan-Runkle Generalized Autoregressive Conditional Heteroscedastic model (GJR-GARCH)-GAS copula specification, which allows detecting the leptokurtic feature and clustering effects of currency returns distribution.

The authors' results show evidence of multiple tail dependence regimes, implying the unsuitability of applying static models to entirely describe the extreme dependence between Bitcoin and fiat currencies. Compared to the most common constant copulas, the authors find that the multivariate GAS copulas better forecast the volatility and dependency between cryptocurrencies and foreign exchange markets. Furthermore, based on the value-at-risk (VaR) and expected shortfall (ES) analyses, the authors show that the multivariate GAS models produce accurate risk measures by adding cryptocurrencies to a portfolio of fiat currencies.

This paper has two main contributions to the existing literature on cryptocurrencies. First, the authors empirically examine the tail dependence structure between common conventional currencies and bitcoin using GJR-GARCH GAS copulas which consider the leptokurtic feature and clustering effects of currency returns distribution. Second, by modeling VaR and ES, the authors test the implication of using time-varying models on the performance of currency portfolios, including cryptocurrencies.

What copulas are, their estimation, and use is illustrated using a geographical diversification example. To accomplish this, dependencies between county-level yields are calculated for non-irrigated wheat, upland cotton, and sorghum using Pearson linear correlation and Kendall's tau. The use of Kendall's tau allows the implementation of copulas to estimate the dependency between county-level yields. The paper aims to discuss these issues.

Four parametric copulas, Gaussian, Frank, Clayton, and Gumbel, are used to estimate Kendall's tau. These four estimates of Kendall's tau are compared to Pearson's linear correlation, a more typical measure of dependence. Using this information, functions are estimated to determine the relationships between dependencies and changes in geographic and climate data.

The effect on county-level crop yields based on changes of geographical and climate variables differed among the different dependency measures among the three different crops. Implementing alternative dependency measures changed the statistical significance and the signs of the coefficients in the sorghum and cotton dependence functions. Copula-based elasticities are consistently less than the linear correlation elasticities for wheat and cotton. For sorghum, however, the copula-based elasticities are generally larger. The results indicate that one should not take the issue of measuring dependence as a trivial matter.

This research not only extends the current literature on geographical diversification by taking a more detailed examination of factors impacting yield dependence, but also extends the copula literature by comparing estimation results using linear correlation and copula-based rank correlation.

The purpose of this paper is to analyse the dependence structure in volatility between Shanghai and Shenzhen stock market in China based on high-frequency data.

Using a multiplicative error model (hereinafter MEM) to describe the margins in volatility of China’s Shanghai and Shenzhen stock market, this study adopts static and time-varying copulas, respectively, estimated by maximum likelihood estimation method to describe the dependence structure in volatility between Shanghai and Shenzhen stock market in China.

This paper has identified the asymmetrical dependence structure in financial market volatility more precisely. Gumbel copula could best fit the empirical distribution as it can capture the relatively high dependence degree in the upper tail part corresponding to the period of volatile price fluctuation in both static and dynamic view.

Previous scholars mostly use GARCH model to describe the margins for price volatility. As MEM can efficiently characterize the volatility estimators, this paper uses MEM to model the margins for the market volatility directly based on high-frequency data, and proposes a proper distribution for the innovation in the marginal models. Then we could use copula-MEM other than copula-GARCH model to study on the dependence structure in volatility between Shanghai and Shenzhen stock market in China from a microstructural perspective.

This study aims to use a novel methodology to investigate the performance of several multivariate value at risk (VaR) and expected shortfall (ES) models implemented to assess the risk of an equally weighted portfolio consisting of high-frequency (1-min) observations for five foreign currencies, namely, EUR/USD, GBP/USD, EUR/JPY, USD/JPY and GBP/JPY.

By applying the multiplicative component generalised autoregressive conditional heteroskedasticity (MC-GARCH) model on each return series and by modelling the dependence structure using copulas, the 95 per cent intraday portfolio VaR and ES are forecasted for an out-of-sample set using Monte Carlo simulation.

In terms of VaR forecasting performance, the backtesting results indicated that four out of the five models implemented could not be rejected at 5 per cent level of significance. However, when the models were further evaluated for their ES forecasting power, only the Student’s t and Clayton models could not be rejected. The fact that some ES models were rejected at 5 per cent significance level highlights the importance of selecting an appropriate copula model for the dependence structure.

To the best of the authors’ knowledge, this is the first study to use the MC-GARCH and copula models to forecast, for the next 1 min, the VaR and ES of an equally weighted portfolio of foreign currencies. It is also the first study to analyse the performance of the MC-GARCH model under seven distributional assumptions for the innovation term.