The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management

The t copula is often used in risk management as it allows for modeling the tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, the grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a generalized grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on the finite sample properties of the maximum likelihood estimators and Kendall's tau approximation. This new copula is significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall.     

Tail Dependence in International Real Estate Securities Markets

??Tail dependence?? characterizes the cross market linkages during stressful times. Analyzing tail dependence is of primary interest to portfolio managers who systematically monitor the co-movements of asset markets. However, the relevant literature on real estate securities markets is very thin. Our study extends the literature by using the flexible symmetrized Joe-Clayton (SJC) copula to estimate the tail dependences for six major global markets (U.S., U.K., Japan, Australia, Hong Kong, and Singapore). In implementing the SJC copula, we model the marginal distributions of returns through a semi-parametric method which has never been applied to real estate returns. Our major findings suggest that international markets display different strength and dynamics of tail dependence. We extensively discuss the implications of our findings for financial practices such as portfolio tail diversifications, portfolio selections, portfolio risk management and hedging strategies. Our study also demonstrates that the widely used linear correlation is an inadequate measure of market linkages, especially during periods of crisis.     

Re-examination of international bond market dependence: Evidence from a pair copula approach

The finance literature provides substantial evidence on the dependence between international bond markets across developed and emerging countries. Early works in this area were based on linear models and multivariate GARCH models. However, based on the limitations of these models this paper re-examines the non-linearity, multivariate and tail dependence structure between government bond markets of the US, UK, Japan, Germany, Canada, France, Italy, Australia and the Eurozone, from January 1970 to February 2019 using ARMA-GARCH based pair- copula models. We find that the bond markets in our sample tend to have both upper tail dependence in terms of positive shocks and lower tail dependence in terms of negative shocks. The estimated C-vine shows Eurozone has the highest average dependency. The D-vine, with optimal chain dependency structure shows the best order of connectedness to be the UK, the USA, Italy, Japan, Eurozone, France, Canada, Germany and Australia. The R-vine copula results underline the complex dynamics of bond market relations existing between the selected economies. The estimated R-vine shows Eurozone, Germany and Australia are the most inter-connected nodes. The multivariate distribution structure (interdependency) of bond markets for all countries were modelled with the C-vine, D-vine and R-vine copulas. In this application, the R-vine copula allows for detailed modelling of all bond markets and hence provides a more accurate goodness of fit and mean square error for the interdependency between all markets. In light of the changing volatility in bond markets, we conduct additional tests using time-varying copulas and find that the dependence structure among the bond markets examined is time-varying with the dynamic dependence parameter plots revealing that the nature of the dependence structure is intense during crisis periods.     

Modeling systemic risk and dependence structure between oil and stock markets using a variational mode decomposition-based copula method

This study combines the variational mode decomposition (VMD) method and static and time-varying symmetric and asymmetric copula functions to examine the dependence structure between crude oil prices and major regional developed stock markets (S&P500, stoxx600, DJPI and TSX indexes) during bear, normal and bull markets under different investment horizons. Furthermore, it analyzes the upside and downside short- and long-run risk spillovers between oil and stock markets by quantifying three market risk measures, namely the value at risk (VaR), conditional VaR (CoVaR) and the delta CoVaR (∆CoVaR). The results show that there is a tail dependence between oil and all stock markets for the raw return series. By considering time horizons, we show that there is an average dependence between the considered markets for the short-run horizons. However, the tail dependence is also found for the long-run horizons between the oil and stock markets, with the exception of the S&P500 index which exhibits average dependence with the oil market. Moreover, we find strong evidence of up and down risk asymmetric spillovers from oil to stock markets and vice versa in the short-and long run horizons. Finally, the market risk spillovers are asymmetric over the time and investment horizons.     

Assessing High-Risk Scenarios by Full-Range Tail Dependence Copulas

Copulas with a full-range tail dependence property can cover the widest range of positive dependence in the tail, so that a regression model can be built accounting for dynamic tail dependence patterns between variables. We propose a model that incorporates both regression on each marginal of bivariate response variables and regression on the dependence parameter for the response variables. An ACIG copula that possesses the full-range tail dependence property is implemented in the regression analysis. Comparisons between regression analysis based on ACIG and Gumbel copulas are conducted, showing that the ACIG copula is generally better than the Gumbel copula when there is intermediate upper tail dependence. A simulation study is conducted to illustrate that dynamic tail dependence structures between loss and ALAE can be captured by using the one-parameter ACIG copula. Finally, we apply the ACIG and Gumbel regression models for a dataset from the U.S. Medical Expenditure Panel Survey. The empirical analysis suggests that the regression model with the ACIG copula improves the assessment of high-risk scenarios, especially for aggregated dependent risks.     

Granger-causality in quantiles between financial markets: Using copula approach

This paper considers the Granger-causality in conditional quantile and examines the potential of improving conditional quantile forecasting by accounting for such a causal relationship between financial markets. We consider Granger-causality in distributions by testing whether the copula function of a pair of two financial markets is the independent copula. Among returns on stock markets in the US, Japan and U.K., we find significant Granger-causality in distribution. For a pair of the financial markets where the dependent (conditional) copula is found, we invert the conditional copula to obtain the conditional quantiles. Dependence between returns of two financial markets is modeled using a parametric copula. Different copula functions are compared to test for Granger-causality in distribution and in quantiles. We find significant Granger-causality in the different quantiles of the conditional distributions between foreign stock markets and the US stock market. Granger-causality from foreign stock markets to the US stock market is more significant from UK than from Japan, while causality from the US stock market to UK and Japan stock markets is almost equally significant.     

Global financial crisis, extreme interdependences, and contagion effects: The role of economic structure?

The paper examines the extent of the current global crisis and the contagion effects it induces by conducting an empirical investigation of the extreme financial interdependences of some selected emerging markets with the US. Several copula functions that provide the necessary flexibility to capture the dynamic patterns of fat tail as well as linear and nonlinear interdependences are used to model the degree of cross-market linkages. Using daily return data from Brazil, Russia, India, China (BRIC) and the US, our empirical results show strong evidence of time-varying dependence between each of the BRIC markets and the US markets, but the dependency is stronger for commodity-price dependent markets than for finished-product export-oriented markets. We also observe high levels of dependence persistence for all market pairs during both bullish and bearish markets.     

Diversification evidence from international equity markets using extreme values and stochastic copulas

Tail dependence plays an important role in financial risk management and determination of whether two markets crash or boom together. However, the linear correlation is unable to capture the dependence structure among financial data. Moreover, given the reality of fat-tail or skewed distribution of financial data, normality assumption for risk measure may be misleading in portfolio development. This paper proposes the use of conditional extreme value theory and time-varying copula to capture the tail dependence between the Australian financial market and other selected international stock markets. Conditional extreme value theory enables the model adequacy and the tail behavior of individual financial variable, while the time-varying copula can fully disclose the changes of dependence structure over time. The combination of both proved to be useful in determining the tail dependence. The empirical results show an outperformance of the model in the analysis of tail dependence, which has an important implication in cross-market diversification and asset pricing allocation.     

Financial stress relationships among Euro area countries: an R-vine copula approach

One of the biggest challenges of keeping Euro area financial stability is the negative co-movement between the vulnerability of public finance, the financial sector, security markets stresses as well as economic growth, especially in peripheral economies. This paper utilizes a ARMA-GARCH based R-vine copula method to explore tail dependance between the Financial Stress Indices of 11 euro area countries with an aim of understanding how financial stress are interacting with each other. We find larger economies in the Euro area tend to have closer upper tail dependence in terms of positive shocks, while smaller economies tend to have closer lower tail dependence with respect to negative shocks. The R-vine copula results underline the complex dynamics of financial stress relations existing between Euro Area economies. The estimated R-vine shows Spain, Italy, France and Belgium are the most inter-connected nodes which underlying they might be more efficient targets to treat in order to achieve a quicker stabilizing. Our results relate to the fact that Eurozone is not a unified policy making area, therefore, it needs to follow divergent policies for taming the effects of financial instability to different regions or groups of economies that are more interconnected.     

The distribution of realized variances: Marginal behaviors, asymmetric dependence and contagion effects

We examine the unconditional distribution of the realized variance of three European stock market indexes obtained from intraday transaction prices. We find that they share common distributional features: a significant mass close to zero, a sharp decrease afterwards and a significant right tail. Their important differences, however, compel us to model them non-parametrically through lognormal kernel estimators. We then move to the analysis of their dependence structure and find strong evidence of asymmetry. Hence, unlike common practice, we resort to non-exchangeable copula models. Such a characterization also allows us to assess the direction of greater contamination among stock market variances.     

Studying the properties of the Bitcoin as a diversifying and hedging asset through a copula analysis: Constant and time-varying

In this study, we analyze the properties of Bitcoin as a diversifier asset and hedge asset against the movement of international market stock indices: S&P500 (US), STOXX50 (EU), NIKKEI (Japan), CSI300 (Shanghai), and HSI (Hong Kong). For this, we use several copula models: Gaussian, Student-t, Clayton, Gumbel, and Frank. The analysis period runs from August 18, 2011 to June 31, 2019. We found that the Gaussian and Student-t copulas are best at fitting the structure dependence between markets. Also, these copulas suggest that under normal market conditions, Bitcoin might act as a hedge asset against the stock price movements of all international markets analyzed. However, the dependence on the Shanghai and Hong Kong markets was somewhat higher. Also, under extreme market conditions, the role of Bitcoin might change from hedge to diversifier. In a time-varying copula analysis, given by the Student-t copula, we found that even under normal market conditions, for some markets, the role of Bitcoin as a hedge asset might fail on a high number of days.     

Hedges or safe havens—revisit the role of gold and USD against stock: a multivariate extended skew-t copula approach

Our paper concerns the question of whether there exist hedge assets during extreme market conditions, which has become increasingly important since the recent financial crisis. This paper develops a novel extended skew-t copula model to examine the effectiveness of gold and US dollar (USD) as hedge or safe haven asset against stock prices for seven developed markets over the 2000–2013 period. Our results indicate the existence of skewness and heavy/thin tails in the distributions of all three types of assets in most of the developed markets, lending support to the employment of flexible distributions to evaluate the tail dependences among assets. We find that USD is preferred to gold as a hedge asset during normal market conditions, while both assets can serve as safe haven assets for most countries when stock markets crash. Our simultaneous analysis of the three assets advises against a joint hedge strategy of gold and USD due to the high tail dependence between them during extreme market conditions. This result highlights the importance of simultaneous modelling of multiple assets in financial risk analysis.     

Stock–bond dependence and flight to/from quality

This paper investigates the stock–bond dependence structure using a dependence-switching copula model. The model allows stock–bond dependence to switch between positive dependence regimes (contagions or crashes of the two markets during downturns or booms in both markets during upturns) and negative dependence regimes (flight-to-quality from stock markets to bond markets or flight-from-quality from bond markets to stock markets). Using data from four developed markets including the US, Canada, Germany, and France for the period between January 1985 and August 2022, we find that the within-country stock–bond (extreme) dependence could be both positive and negative. In the positive dependence regimes, the stock–bond dependence is asymmetric with stronger left tail dependence than the right tail dependence, giving evidence of a higher likelihood of joint stock–bond market crashes or contagions during market downturns than the collective stock–bond market booms. Under the negative dependence regimes, we find both flight-from-quality and flight-to-quality, with flight-to-quality being more dominant in the North American markets while flight-from-quality is more prominent in the European markets. Further, the dependence switches between positive and negative regimes over time. Moreover, the dependence is mainly in the positive regimes before 2000 while mostly in the negative regimes after that, indicating contagions mostly before 2000 and flights afterwards. Further, the dependence switches between positive and negative regimes around financial crises and the COVID-19 pandemic. These results greatly enrich the findings in the existing literature on the co-movements of stock–bond markets and are important for risk management and asset pricing.     

Estimating Portfolio Credit Losses in Downturns

This paper suggests formulas able to capture potential strong connection among credit losses in downturns without assuming any specific distribution for the variables involved. We first show that the current model adopted by regulators (Basel) is equivalent to a conditional distribution derived from the Gaussian Copula (which does not identify tail dependence). We then use conditional distributions derived from copulas that express tail dependence (stronger dependence across higher losses) to estimate the probability of credit losses in extreme scenarios (crises). Next, we use data on historical credit losses incurred in American banks to compare the suggested approach to the Basel formula with respect to their performance when predicting the extreme losses observed in 2009 and 2010. Our results indicate that, in general, the copula approach outperforms the Basel method in two of the three credit segments investigated. The proposed method is extendable to other differentiable copula families and this gives flexibility to future practical applications of the model.     

Testing the Gaussian copula hypothesis for financial assets dependences

Abstract

Using one of the key properties of copulas that they remain invariant under an arbitrary monotonic change of variable, we investigate the null hypothesis that the dependence between financial assets can be modelled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student copula, cannot be rejected if it has sufficiently many ‘degrees of freedom’. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the coefficient of correlation between the pairs of assets is too high, such that the tail dependence neglected by the Gaussian copula can became large, leading to the ignoring of extreme events which may occur in unison.     

Nonlinear term structure dependence: Copula functions,empirics, and risk implications

This paper documents nonlinear cross-sectional dependence in the term structure of US-Treasury yields and points out risk management implications. The analysis is based on a Kalman filter estimation of a two-factor affine model which specifies the yield curve dynamics. We then apply a broad class of copula functions for modeling dependence in factors spanning the yield curve. Our sample of monthly yields in the 1982–2001 period provides evidence of upper tail dependence in yield innovations; i.e., large positive interest rate shocks tend to occur under increased dependence. In contrast, the best-fitting copula model coincides with zero lower tail dependence. This asymmetry has substantial risk management implications. We give an example in estimating bond portfolio loss quantiles and report the biases which result from an application of the normal dependence model.     

Dependence structure and extreme comovements in international equity and bond markets

Common negative extreme variations in returns are prevalent in international equity markets. This has been widely documented with statistical tools such as exceedance correlation, extreme value theory, and Gaussian bivariate GARCH or regime-switching models. We point to limits of these tools to characterize extreme dependence and propose an alternative regime-switching copula model that includes one normal regime in which dependence is symmetric and a second regime characterized by asymmetric dependence. We apply this model to international equity and bond markets, to allow for inter-market movements. Empirically, we find that dependence between international assets of the same type is strong in both regimes, especially in the asymmetric one, but weak between equities and bonds, even in the same country.     

Nonparametric testing of the high-frequency efficiency of the 1997 Asian foreign exchange markets

Sherry’s nonparametric pattern tests for neural information processing are used to ascertain if the Asian foreign exchange (FX) rates followed random walks [Sherry, C.J., 1992. The Mathematics of Technical Analysis: Applying Statistics to Trading Stocks, Options and Futuresm Probus, Chicago]. The stationarity and serial independence of the price changes are tested on minute-by-minute data for nine Asian currencies from 1 January 1997 to 30 December 1997. The efficiency of these FX markets before and after the Asian currency ‘regime discontinuity’ are compared. The Thai baht (THB), Malaysian ringgit (MYR), Indonesian rupiah (IDR) and Singapore dollar (SGD) exhibited non-stationary behavior during the entire year, and gave evidence of a trading regime break, while the Phillipines’ peso (PHP), Taiwan dollar (TWD), Japanese yen (JYP) and German deutschmark (DEM) remained stationary, with the US dollar (USD) as numeraire. However, each half-year regime showed stationarity, indicating stable and nonchaotic trading regimes for all currencies, despite their high volatilities, except for the MYR, which exhibited non-stationarity in the second half of 1997. The Thai baht traded nonstationarily in the first half of 1997, but stationarily in the second half. while the TWD reversed that trading pattern. Based on Sherry’s four tests for serial independence, none of the currencies exhibited complete independence. Thus no Asian currency market—including the JYP—exhibited complete efficiency in 1997, in particular when compared with the highly efficient DEM. Remarkably, the PHP remained as efficient as the JYP throughout 1997.     

Analysing financial contagion and asymmetric market dependence with volatility indices via copulas

This paper explores the cross-market dependence between five popular equity indices (S&P 500, NASDAQ 100, DAX 30, FTSE 100, and Nikkei 225), and their corresponding volatility indices (VIX, VXN, VDAX, VFTSE, and VXJ). In particular, we propose a dynamic mixed copula approach which is able to capture the time-varying tail dependence coefficient (TDC). The findings indicate the existence of financial contagion and significant asymmetric TDCs for major international equity markets. In some situations, although contagion cannot be clearly detected by stock index movements, it can be captured by dependence between volatility indices. The results imply that contagion is not only reflected in the first moment of index returns, but also the second moment, i.e. the volatility. Results also show that dependence between volatility indices is more easily influenced by financial shocks and reflects the instantaneous information faster than the stock market indices.     

Volatility Spillovers, Comovements and Contagion in Securitized Real Estate Markets

This paper analyzes the relationships between local and global securitized real estate markets, but also between securitized real estate and common stock markets. First, the volatility transmissions across markets are examined using an asymmetric t-BEKK (Baba-Engle-Kraft-Kroner) specification of their covariance matrix. Second, correlations from that model and tail dependences estimated using a time-varying copula framework are analyzed to assess whether different dynamics underlie the comovements in the whole distribution and those in the tails. Third, we investigate market contagion by testing for structural changes in the tail dependences. We use data for the U.S., the U.K. and Australia for the period 1990–2010 as a basis for our analyses. Spillover effects are found to be the largest in the U.S., both domestically and internationally. Further, comovements in tail distributions between markets appear to be quite important. We also document different dynamics between the conditional tail dependences and correlations. Finally, we find evidence of market contagion between the U.S. and the U.K. markets following the subprime crisis.