Modelling the European Carbon Market. Calculating the Value-at-Risk of a portfolio of energy indices

As the global energy sector undergoes a structural transition, the integration of renewable assets with traditional fossil fuel commodities has introduced complex risk dynamics into investment portfolios. This study examines the market risk characteristics of a diversified energy portfolio comprising traditional oil and gas (XLE), natural gas (FCG), and clean energy (ICLN) indices over the volatile period from January 2021 to January 2024. While prior studies typically assess energy risk using a single volatility model in isolation, this paper contributes to the literature by directly comparing two dynamic Value at Risk (VaR) methodologies within a unified multi-asset framework: a parametric Multivariate GARCH(1,1) model with Student-t innovations, and a semi-parametric Historical Simulation ARMA-Forecasting (HSAF) approach. This design isolates the conditions under which theoretical distributional assumptions diverge from empirical return behavior, an aspect largely unaddressed in existing literature.

Preliminary diagnostics reject the hypothesis of normality, revealing significant negative skewness and excess kurtosis across all three indices. The estimated GARCH parameters indicate a high degree of volatility persistence, suggesting a prolonged half-life for market shocks, while diversification is shown to reduce extreme tail risk without eliminating the non-normal character of the portfolio's return distribution.

The results reveal a meaningful divergence between the two risk measures: the parametric Student-t model produces a notably more conservative VaR estimate than the semi-parametric HSAF method, which relies on the empirical distribution of filtered residuals rather than a symmetric theoretical structure. Backtesting via the Kupiec Proportion of Failures test validates the robustness of the semi-parametric approach in capturing idiosyncratic market jumps without overestimating losses. Practically, these findings give portfolio managers and risk officers an evidence-based basis for choosing between competing VaR frameworks when setting capital reserves, showing that hybrid methodologies combining volatility filtering with historical simulation deliver more efficient, defensible tail-risk estimates than standard parametric approaches alone.