This study examines whether daily S&P 500 returns conform to the normal distribution assumption commonly used in financial modelling and risk management. Daily S&P 500 index data from 2010–2026, comprising 4,030 observations, were analysed using descriptive statistics, empirical distribution analysis, kernel density estimation, empirical cumulative distribution functions, normal Q–Q plots, quantile comparisons, and the Jarque–Bera normality test. The results indicate substantial departures from normality. Returns exhibit negative skewness (-0.402) and pronounced excess kurtosis, with a kurtosis measure of 6.742. The Jarque–Bera test strongly rejects the null hypothesis of normally distributed returns (p < 0.001). Empirical tail quantiles are also more extreme than their normal-theoretical counterparts, demonstrating heavier tails and a greater frequency of extreme market movements. These findings suggest that reliance on normality assumptions may underestimate tail risk and potentially lead to misleading Value-at-Risk and portfolio risk estimates. The study highlights the importance of distributionally robust approaches in financial risk measurement and decision-making.

Key Words: S&P 500; financial returns; non-normality; heavy tails; skewness; kurtosis; Jarque–Bera test; kernel density estimation; empirical distribution; financial risk