Measurement of risk spillover effect based on EV-Copula method

With the rapid development of the global financial market system, international economic cooperation and business exchanges have become increasingly frequent. Meanwhile, with the rapid development of global financial market integration, the openness of financial systems in various countries is also constantly improving, and the linkage effect between cross-border markets is becoming increasingly evident. However, financial capital with different risk characteristics not only expands the scale of financial business and improves the efficiency of financial market operation, but also has an impact on the stability of the market, with the most significant being the continuous intensification of financial risks. In August 2007, the subprime mortgage crisis in the United States swept across the world, led to the rapid deterioration of the global economy, and seriously disrupted the financial system. The cause of the crisis is the lack of risk prediction, supervision, and management of the financial market.

In recent years, problems in the form of a "carbon foam" are putting some fossil energy companies worth hundreds of billions of dollars into trouble. The so-called "carbon foam" means that the current value of fossil fuels is overestimated, and people will have to significantly reduce greenhouse gas emissions in the long run. low-carbon development and climate issues have become a consensus and social responsibility for human social development. Under the urgent situation of global carbon emissions, emission reduction has expanded from the technical level to the financial market level. As a special asset, carbon emission rights have formed a carbon emission trading market through trading and conversion between physical markets. Carbon emissions trading is a mechanism that limits greenhouse gas emissions and promotes sustainable development goals by establishing a carbon market. This mechanism promotes emission reduction and low-carbon investment by setting a total limit on emission quotas and allowing companies to trade emission quotas among themselves. The goal of carbon emissions trading is to encourage enterprises to reduce greenhouse gas emissions through economic incentive mechanisms. Enterprises can manage their emissions by purchasing or selling emission quotas, providing economic incentives for those who can effectively reduce emissions and have excess emission quotas. At the same time, it also provides opportunities for enterprises with higher emission reduction costs to make up for the emission gap. Countries around the world are gradually establishing carbon emission trading markets to promote their low-carbon development through market-oriented means. As regards theory, Azomahou et al. (2006) reveal that carbon emissions come from energy consumption, which is an important factor related to the production and consumption of the world economy. Oberndorfer (2008) shows that the changes in EU carbon emission rights prices are positively correlated with the stock returns of the most important electricity companies in Europe, and the stock price effect of carbon emission rights prices has periodicity and may vary in different countries. Daskalakis and Markellos (2009) find a positive relationship between carbon prices and electricity risk premiums under the EU carbon emission trading system, and the impact of a decrease in carbon price returns on electricity price risks is greater than that of an increase in carbon price returns. Kijima et al. (2010) propose a model and pricing formula for the emission trading license market. Pindyck (2008) finds that carbon emissions can lead to carbon risks. Bushenll et al. (2013) consider that the emission trading system will have an impact on cash flow and expected returns. Fowlie and Reguant (2022) use US energy price variation as a proxy for variation that will be induced by a domestic carbon price and simulate the impacts of a domestic carbon price on US manufacturing with and without these subsidies. With the rapid development of the carbon emissions trading market, there are also certain risks, including market supply and demand, political changes, financial crisis, climate change, and other factors. Therefore, it is necessary to invest more cautiously. A natural question is how to scientifically and reasonably identify and measure market risk, which is conducive to taking effective risk management measures to ensure the implementation of global low-carbon and sustainable development. Undoubtedly, establishing a mathematical model and measuring risk is necessary, which can provide us with some meaningful guidelines in risk measurement.

Risk measurement is mainly obtained by calculating value at risk (VaR). With the globalization of the economy and the liberalization of the financial environment, the connections between different financial institutions are becoming closer and more complex. The mutual influence and risk exposure of financial institutions have gradually increased. Adrian, Brunnermeier (2016) propose the CoVaR, which reveals the risk spillover of one financial institution to another financial institution or to the financial system, thus filling the gap that VaR does not consider risk spillover in risk measurement. It is known that CoVaR may be calculated by quantile regression and DCC-GARCH methods. Among them, the quantile regression method is somewhat rough because it can only measure linear risk spillover effects. Therefore, Girardi and Ergun (2019) propose to use the DCC-GARCH method to calculate CoVaR. Financial markets should generally have complex characteristics such as volatility aggregation and time-varying variance. The linear risk spillover measured by quantile regression is not convincing. Although the DCC-GARCH method improves this shortcoming, tail risk spillovers and non-linear correlation structures in financial markets cannot be fully measured. Mainik and Schaanning (2014) propose to calculate CoVaR by the copula method. For the research and application of the copula model, Wu et al. (2012) use copula-based GARCH models to investigate the economic value of co-movement between oil price and exchange rate (US dollar index). Aloui et al. (2013) apply the copula-GARCH approach to consider the conditional dependence structure between crude oil prices and US dollar exchange rates. Sebai and Naoui (2015) establish the connection between oil prices and the US dollar exchange rate using a copula approach and the DCC-MGARCH model. Hung (2019) investigates the conditional dependence structure between crude oil prices and three US dollar exchange rates (China, India, and South Korea) from a new perspective using a copula-GARCH approach. Hung (2020) studies both the constant and time-varying conditional dependency for crude oil, stock markets, green bonds, and assets by using the conditional copula model. At present, the international environment is complex and changeable, which is bound to have a negative impact on the financial market, especially it may lead to systematic risk in the financial market if some extreme events occur. Therefore, we must consider extreme events such as war and major natural disasters in the study of market risk. Further, we should consider the risk contribution, that is dynamic systematic ACoVaR, which can describe the dynamic variation of systematic risk. Our work attempts to fill this gap, which is the reason why we consider using extreme value theory in the present study. Extreme value theory is a modeling and statistical analysis method for extreme variability that rarely occurs, but once it occurs, it has a significant impact. It provides us with a good robust asymptotic model, which can be used to model the tail of the distribution and assess risk. Research shows that linear models cannot capture the impact of extreme events such as war and major natural disasters on the market. On the contrary, nonlinear models have advantages in this respect. Copula function can effectively describe non-linear relationships and can separate the marginal distribution and the structural relationship between random variables, hence it has many advantages in practical applications. Obviously, the combination of extreme value theory and the copula method can better measure the tail relationship between variables.

There are several reasons why this research selects China as a case study. First, China attaches great importance to climate change and sustainable issues and has made unremitting efforts and positive contributions to addressing climate change from the perspective of global long-term fundamental interests. After the signing of the Paris Agreement, China has proposed phased emission reduction targets for 2020 and 2030. Currently, the annual trading volume of the carbon market is about 50 million tons, and China has become an important carbon emission trading market in the world. Second, the stock market of China has become the second largest stock market in the world, and the future development prospects of the stock market are optimistic. With the advancement of government policies and the adjustment of market structure, the stock market will gradually develop towards a more open, standardized, and transparent direction. Against the backdrop of investors gradually upgrading, the stock market will bring more investment opportunities and space to global investors. Finally, the carbon market can effectively leverage its resource allocation function, and guide social capital to flow to environmental protection enterprises, further promoting industrial structure optimization, therefore, There is a close connection between the carbon market and the stock market.

The main contributions of the work are as follows. First, we use the US Department of Commerce index synthesis method to synthesize the carbon trading price index, it provides guidance for carbon index synthesis. Then we establish EV-Copula CoVaR model and measure the risk spillover effect of the carbon trading market to the stock market, which provides a new idea for the research of risk spillover of the carbon trading market. Finally, we explore the relationship between significance level and risk spillover intensity, it is found that the smaller the significant level, the greater the risk of the carbon market, and the larger ACoVaR, which indicates that there is a positive correlation between the risk of the carbon market and spillover risk.

The remainder of the paper is organized as follows. Section "Literature review" provides the literature review. Section "Methodology" gives the methodology of this article. Section "Empirical results" presents the numerical results. Finally, Section "Conclusion and suggestions" concludes the research and gives suggestions.

In the risk measurement of financial markets, the study of tail dependence risk has attracted increasing attention, and copula functions are used to characterize tail dependence. The copula method compensates for the drawbacks of linear correlation in characterizing the correlation between variables. Therefore, the copula correlation function and the correlation measures are widely used in financial data analysis. Sklar (1959) first proposed the copula theory, which has excellent performance in measuring non-linear and dependent relationships. Embrechts and McNeil (1999) use the copula approach in the financial field. With the widespread application of copula theory in the financial field, many works have been conducted on copula theory and copula linkage method. Embrechts et al. (2003) use the copula function to calculate VaR, and show that the VaR has a better fitting effect with the actual value. Patton (2006) applies the time-varying copula function to the exchange rate problem and studies the linkage of different exchange rate markets. Garcia and Tsafack (2011) propose a copula model that includes symmetric and asymmetric states to study the linkage between the bond market and the stock market. Stöber and Czado (2014) propose a Bayesian method to estimate the parameters of the R-Vine copula model and use this method to study the linkage between the exchange rate of US dollar and nine currencies. Oh and Patton (2017) make the dynamic parameters of the copula function obey the generalized auto-regression model, and use the newly constructed copula function to study the systematic risk level of 100 American companies, finding higher systemic risk during the 2008 crisis. Bensaida (2018) applies a vine copula model with Markov regime-switching to study contagion in the US and Eurozone debt markets. Gomez-Gonzalez and Rojas-Espinosa (2019) use the DCC-GARCH model and copula function to study the linkage of exchange rate markets in 12 Asia-Pacific countries, showing strong linkage during extreme appreciation/depreciation.

For risk spillover measurement, typical approaches include DCC-GARCH, quantile regression, and copula methods. Jiang et al. (2022) show time-varying spillovers among China’s carbon pilots. Han et al. (2022) compile a unified carbon price index and use a TVP-VaR model to document asymmetric, time-varying spillovers among carbon, EUA, and non-ferrous metals. Wang et al. (2022) construct spillover indices among China’s carbon market and sectoral stock markets. Keilbar and Wang (2021) model systemic risk via neural network quantile regression. Yin et al. (2021) propose symmetric and asymmetric ACoVaR methods for oil market risks. Geenens and Dunn (2022) develop a nonparametric CoVaR framework. Zhou et al. (2022) employ a quantile network and GARCHSK for multidimensional spillovers among carbon, energy, and nonferrous metals. However, few studies jointly employ extreme value theory and copulas to study dynamic risk spillovers, motivating the present work.

The study measures risk spillovers from China’s carbon trading market to sectoral stock markets using an EV-Copula CoVaR framework.

Carbon trading price index synthesis: Using the US Department of Commerce index synthesis method, the paper synthesizes a carbon trading price index from five pilot markets (Beijing; Shenzhen; Shanghai, Guangdong, Hubei). Steps: (1) Compute symmetric varying rates C_j(t) for each group and standardize them via A_ij to obtain S_ij(t). (2) Compute standardized average varying rates R_j(t) with equal weights (W_ij=1) and standardize via F_j to obtain V_j(t). (3) Generate the composite index I_j(t) recursively and define CI_j(t) as the chain index with I_j(1)=100.

EV-Copula modeling: The approach separates marginal distributions and dependence structure. For marginals, extreme tails are modeled via Generalized Pareto Distributions (GPD) within a Peaks-Over-Threshold framework; intermediate data use the empirical distribution. Thresholds μ_L and μ_R are set using the 10% exceedance rule (Dumouchel and Waternaux, 1983). Parameters (shape ξ and scale β(μ)) are estimated by maximum likelihood, yielding a piecewise marginal CDF combining empirical middle and GPD tails.

Copula dependence: Among Archimedean copulas (Joe, Clayton, Gumbel, Frank, BB1, BB2, BB6, BB7), the optimal copula is selected by maximum likelihood and AIC. The m-dimensional Sklar theorem is used to construct joint distributions with marginal CDFs F_i; density factorizes via copula density c(u_1,...,u_m).

EV-Copula CoVaR: Given return series X^i (carbon index) and X^j (stock index), the conditional density f_ij(x^j|x^i)=c(F_i(x^i),F_j(x^j)) f_j(x^j). The conditional CDF F_ij(x^j|x^i) integrates this density over x^j. CoVaR^j at probability q conditional on X^i=VaR_q^i is CoVaR^j=F_j^{-1}(q|VaR_q^i). Risk spillover is ΔCoVaR^j=CoVaR^j−CoVaR_{0.5}^j. Dynamic spillovers (ACoVaR) are evaluated across time and significance levels. Backtesting uses the Kupiec (1995) failure rate test.

Data: Daily data from May 11, 2015 to March 31, 2021 (T=1774). Markets: carbon index (synthesized), and five stock sectors: electricity (Huaneng International, 600011), finance (399240), real estate (000006), industry (000004), energy (000032). Returns r_t=100×ln(p_t/p_{t−1}). Analyses are implemented in R.

Descriptive statistics (Table 2):

• Highest volatility (std. dev.): carbon (5.190) and finance (5.250); highest mean returns: industry (0.077) and energy (0.113). Carbon shows the largest maximum daily gain (8.192), finance the most negative minimum (-6.610). All series exhibit excess kurtosis (>3) and non-normality (Jarque–Bera significant at 1%). Finance has the largest kurtosis (8.110), indicating proneness to extreme events.

Copula selection and dependence (Tables 3–4):

• Using carbon and electricity returns as an example, BB7 copula provides the best fit by AIC and significant parameters; overall, BB7 is selected for all pairs. Estimated dependence metrics (Kendall’s tau; upper/lower tail dependence coefficients): • Electricity (HNGJ): tau=0.344; upper tail=0.354; lower tail=0.339. • Finance (FI): tau=0.069; upper=0.077; lower=0.041 (weakest linkage). • Real estate (RE): tau=0.226; upper=0.230; lower=0.193. • Industry (IN): tau=0.201; upper=0.185; lower=0.191. • Energy (TE): tau=0.224; upper=0.233; lower=0.272.
• Lower tail dependence is positive across all sectors, implying that extreme adverse carbon market movements increase downside risk in stock sectors.

Risk spillovers (ACoVaR averages, Table 5):

• Downside spillovers (Carbon → sector): Electricity -0.863 (largest), Energy -0.778, Real Estate -0.650, Industry -0.631, Finance -0.578 (smallest). Larger absolute values indicate stronger spillover contagion.
• Upside spillovers: Electricity 0.274, Energy 0.266, Real Estate 0.173, Industry 0.141, Finance 0.099.
• Even the weakest (finance) shows sizable downside spillover magnitude (57.8%).

Dynamic patterns and validation:

• Spillover curves at 5% significance capture volatility clustering; some extreme points are slightly underestimated but overall trends are well depicted (Fig. 1).
• Backtesting via Kupiec failure rate test shows all five spillover measurements pass (e.g., Carbon→Electricity: N=86, freq=0.048, LR=3.583 with reported p-values indicating significance), validating model adequacy (Table 6).

Significance level relation (Fig. 2):

• As the significance level decreases (stricter tail), the carbon market’s own risk increases and ACoVaR magnitudes grow; risk spillover intensity is positively correlated with carbon market self-risk. Electricity consistently shows the largest ACoVaR across levels, indicating the closest linkage with carbon trading.

The EV-Copula CoVaR analysis demonstrates significant risk spillovers from China’s carbon trading market to major stock sectors, addressing the study’s question on identifying and measuring inter-market contagion under extreme conditions. Positive lower tail dependence across all sectors indicates that adverse carbon market shocks elevate downside risk in equities. The electricity sector exhibits the strongest spillovers, consistent with its direct exposure to carbon costs and policy, while finance shows the weakest linkage, suggesting less immediate sensitivity to carbon price extremes. The finding that spillover intensity rises with the carbon market’s own tail risk (lower significance levels) underscores the systemic nature of carbon-related shocks. Model backtests support the reliability of the spillover estimates. These results are relevant for market participants and regulators in monitoring systemic risk and designing sector-specific risk management strategies in the context of carbon pricing and climate policy.

This paper synthesizes a carbon trading price index for China and employs an EV-Copula CoVaR framework to quantify risk spillovers from the carbon market to sectoral stock markets. Results indicate significant spillovers, with the electricity sector most affected and the financial sector least affected. The analysis across significance levels shows that lower significance (deeper tails) corresponds to higher carbon market self-risk and larger ACoVaR, evidencing a positive relationship between self-risk and spillover intensity. The study provides a methodological contribution by combining extreme value theory with copula-based CoVaR to capture nonlinear, tail-dependent contagion between carbon and equity markets.