Beyond normality: comparative tail-risk analysis of S&P 500 returns

Financial models normally assume that investment returns follow a normal distribution because of the latter’s mathematical simplicity and analytical convenience. However, real market returns often exhibit non-normal characteristics such as heavy tails and skewness, which can lead to underestimation of financial risk. In this study, 31 years of SPY daily return data were analyzed to evaluate the distributional behavior and tail risk of S&P 500 returns. Descriptive statistics, distribution fitting, Quantile-Quantile (Q-Q) plots, and formal normality tests were used to examine deviations of the SPY returns from normality. Additionally, a heavy-tailed t distribution, a two-component Gaussian Mixture Model (GMM-2), and Extreme Value Theory (EVT) were evaluated for tail-risk modeling. Model performance was further assessed using Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Value at Risk (VaR), Expected Shortfall (ES), exceedance backtesting, and rolling-window calibration analyses across different market environments. The results show that, compared to the normal distribution, SPY returns exhibit negative skewness, pronounced central peaks, and heavy tails. The normal model performs reasonably well in moderate-tail regions and during stable market conditions, but increasingly underestimates downside risk at deeper confidence levels and during stressed market periods. Among the models, the t-distribution provides the strongest overall balance between model simplicity and tail-risk representation, while EVT performs best for rare and extreme loss events. Rolling-window and annual analyses further show that model performance varies across different market regimes and volatility conditions. Although heavy-tailed models improve tail-risk calibration relative to the normal distribution, rolling-window analyses show persistent exceedance underestimation across all models, suggesting that time-varying volatility and regime-dependent behavior play a major role in financial tail-risk dynamics. Overall, the findings highlight the limitations of normal-distribution assumptions in financial risk modeling and demonstrate that tail-risk behavior varies substantially across different market regimes, volatility conditions, and tail depths. No single model consistently provides the best performance under all conditions, underscoring the importance of appropriate model selection in financial risk estimation.