Copula Analysis of Risk: A Multivariate Risk Analysis for VaR and CoVaR using Copulas and DCC-GARCH

Introduction

In modern financial markets, risk management is more critical than ever due to the complex interdependencies among assets and sectors. Traditional models like Value-at-Risk (VaR) offer a univariate perspective on tail risks, which often misses how crises propagate across interconnected assets. To address this limitation, this study employs a multivariate approach using copulas and DCC-GARCH models. The research specifically focuses on estimating Conditional Value-at-Risk (CoVaR), which captures systemic risk by assessing the conditional distribution of one asset given distress in another.

Background Theory

This work builds upon foundational statistical concepts like probability distributions and joint modeling. Central to this is the use of copulas—a method to model and understand dependencies between multiple financial variables, irrespective of their marginal distributions. Sklar’s Theorem provides the theoretical underpinning, allowing the separation of marginal behavior from dependence structure. The study explores various families of copulas, including elliptical (Gaussian and Student-t) and Archimedean (Clayton and Gumbel), each offering unique properties for capturing tail behavior and asymmetric risk.

Modeling Framework

The study leverages a hybrid modeling framework combining:

• GARCH models to capture time-varying volatility in individual asset returns.
• Dynamic Conditional Correlation (DCC) to allow the correlations among assets to evolve over time.
• Copula functions to model the dependency structure across multiple assets based on the filtered residuals from GARCH.

Data and Preprocessing

The dataset comprises 5 years of daily stock prices from six multinational companies across three sectors: Energy (Chevron, Exxon), Consumer Staples (Coca-Cola, PepsiCo), and Financial Services (Mastercard, Visa). Preprocessing includes computing daily log returns, aligning data across a common timeline, removing any incomplete records, and validating basic statistical assumptions. The resulting dataset is robust and ready for high-fidelity modeling, consisting of over 1,250 aligned observations.

Model Results and Insights

The model outputs offer granular insights into both individual and systemic risk. DCC-GARCH estimates of VaR and CVaR show that the diversified High-Corr portfolio reduces overall risk exposure. CoVaR analysis reveals significant systemic influence from high-capitalization assets like Visa and Mastercard. Risk metrics such as Pearson correlation and Energy Score are used to validate the quality of each copula family’s fit. The research highlights how copula selection dramatically impacts perceived risk in tail events, supporting tailored modeling strategies for portfolio risk managers.

Stress Testing and Portfolio Analysis

Stress testing is used to simulate extreme scenarios where a particular asset is under financial distress. The study evaluates how shocks in one asset—e.g., Exxon Mobil—propagate throughout the portfolio. The resulting Delta-CoVaR values quantify the systemic impact and reveal how risk amplifies through asset interconnections. Comparisons between individual assets and the High-Corr portfolio further illustrate the effectiveness of diversification in absorbing shocks and reducing overall exposure.

Copula Selection and Goodness-of-Fit

Each copula family is assessed for its ability to model real-world financial dependencies. The Gaussian copula exhibits the lowest AIC/BIC and is optimal for symmetric, low-tail risk environments. The Student-t copula performs better in capturing heavy tails, while the Clayton and Gumbel copulas specialize in lower- and upper-tail dependencies respectively. Fit metrics such as Energy Score and Kendall’s Tau ensure the chosen copula reflects the actual behavior of financial assets, enabling more accurate forecasting and stress scenario analysis.

Conclusion

This study concludes that copula-based multivariate modeling provides superior insights into financial risk compared to traditional approaches. By integrating GARCH for volatility modeling, DCC for evolving correlations, and copulas for dependency structures, it presents a comprehensive risk framework. The results underscore the importance of copula selection, asset correlation analysis, and diversification in designing resilient investment strategies. Overall, this research equips financial analysts and risk managers with a robust methodology for navigating systemic risks.

Research Paper

Explore the full academic paper "Copula Analysis of Risk: A Multivariate Risk Analysis for VaR and CoVaR using Copulas and DCC-GARCH" directly below. This embedded PDF viewer is mobile responsive for easy access across devices.

Developed by Aryan Singh. Explore the full implementation on GitHub.