Quantitative Risk Modeling and Portfolio Construction with ARMA-GARCH: An Empirical Study on the S&P 500

Abstract

This study investigates the return dynamics, volatility structure, and risk characteristics of five representative S&P 500 stocks: Johnson & Johnson, Microsoft, NVIDIA, Coca-Cola, and Home Depot, using ARMA-GARCH models. Descriptive statistics and diagnostic tests confirm non-normality, negative skewness, fat tails, and volatility clustering, providing strong justification for conditional mean-variance modelling. Optimal model specifications are selected via the Bayesian Information Criterion, with EGARCH frameworks generally outperforming alternative GARCH variants in capturing asymmetric volatility responses. Rolling-window forecasts for 2024Q1 show that the models generate stable and reliable volatility predictions for low-volatility stocks (JNJ, KO), while performance is weaker for highly volatile stocks (NVDA), highlighting structural limitations under extreme market shifts. To evaluate risk management implications, one percent Value-at-Risk and expected shortfall were computed and backtested. Results indicated conservative tail-risk forecasts, with violation rates well within acceptable thresholds. Portfolio applications are further explored by constructing the Global Minimum Variance Portfolio (GMVP) and the Maximum Sharpe Ratio (Max SR) portfolio using rolling covariance estimates. Out-of-sample backtesting demonstrated that the GMVP delivered low volatility but modest returns, whereas the Max SR portfolio achieved significantly higher performance, consistent with the risk-return trade-off. Overall, the findings confirm that ARMA-GARCH models are effective tools for modelling conditional volatility and informing dynamic asset allocation. However, their limited adaptability to jump risk and nonlinear structural breaks underscores the need for more advanced modelling approaches in high-volatility environments.