Robust Portfolio Optimization under Conditional Value-at-Risk (CVaR) Criterion Based on EGARCH, Extreme Value Theory (EVT), and Copula Approach

Abstract

Objective: Selecting an optimal combination of assets within an investment portfolio has always been one of the fundamental and challenging issues in investment management. In other words, investors continually aim to achieve an efficient and optimal allocation by considering the key factors influencing the decision-making process and aligning the selection with their individual risk–return preferences. However, traditional portfolio theory often relies on simplified statistical assumptions—particularly the assumption of normally distributed returns—that are inconsistent with the empirical characteristics of financial markets. In reality, financial asset returns frequently display heavy tails, negative skewness, volatility clustering, and asymmetric dependence during periods of market stress. These stylized facts imply that extreme losses occur more frequently than predicted by classical models, highlighting the necessity for more sophisticated tools in modern risk management and portfolio construction.

To address these limitations, the present study develops a comprehensive hybrid modeling framework that integrates three advanced components: EGARCH to capture time-varying volatility and leverage effects, Extreme Value Theory (EVT) to model tail behavior and estimate extreme losses more accurately, and t-Copula to model tail dependence and joint downside risk across industries. Additionally, considering that parameter estimates obtained from historical data are subject to uncertainty and may deviate from their true values, a Robust Optimization approach under the Conditional Value-at-Risk (CVaR) criterion is adopted. This framework allows the portfolio to remain resilient against estimation errors and improves decision-making under market uncertainty. Accordingly, the main objective of this research is to present a robust and realistic portfolio optimization model based on the EGARCH–EVT–t-Copula–Robust CVaR approach.









Methods: Daily return data for ten major industry indices of the Tehran Stock Exchange, covering the period from September 2015 to September 2025, were employed. Conditional volatility was modeled using the EGARCH(1,1) specification to account for asymmetry in the volatility response to positive and negative shocks. Standardized residuals obtained from the volatility model were analyzed using EVT under the Peaks-over-Threshold method to estimate the heavy-tail behavior of returns. The dependence structure among industries was then modeled through t-Copula to capture joint extreme movements and tail dependence. Finally, portfolio optimization was conducted under the CVaR measure in both standard and robust formulations, and the performance of the resulting portfolios was compared based on expected return, risk, and Sharpe ratio.

Results: Empirical findings reveal that the integrated EGARCH–EVT–t-Copula–Robust CVaR framework delivers superior accuracy in capturing extreme risks and provides a more effective portfolio allocation in heavy-tailed environments. The EGARCH model successfully captured conditional volatility dynamics, EVT highlighted the presence of heavy left-tail behavior and significant downside risk, and the t-Copula confirmed strong tail dependence among industrial sectors, implying the synchronized occurrence of severe negative shocks. In the optimization step, the Robust CVaR portfolio outperformed the standard CVaR model by yielding higher expected returns and Sharpe ratios, illustrating its enhanced ability to handle parameter uncertainty and asymmetric market volatility.

Conclusion: Overall, the findings confirm that in emerging markets such as Iran—characterized by high kurtosis, negative skewness, and tail dependence—classical risk models based on normality assumptions are inadequate. In contrast, the proposed EGARCH–EVT–t-Copula–Robust CVaR framework provides a more realistic estimation of tail risk and leads to a more stable and efficient capital allocation. Hence, adopting robust CVaR optimization under conditional heteroskedasticity and tail-dependent structures is recommended for professional portfolio management and investment decision-making in high-risk markets.