A Century of S&P 500 Returns: The Normal Distribution Doesn't Fit
A Century of S&P 500 Returns: The Normal Distribution Doesn't Fit
Almost a century of daily S&P 500 data (24,746 trading days from December 30, 1927 through July 10, 2026) tells a consistent story. Daily equity returns are not normally distributed, and the ways they deviate matter for anyone sizing risk off a bell curve.
The numbers
| Statistic | Value | |---|---| | Mean daily return | +0.0245% | | Daily volatility (std dev) | 1.19% | | Annualized volatility | 18.95% | | Skewness | −0.47 | | Excess kurtosis | 18.74 | | Jarque-Bera normality test | p ≈ 0 (rejects normality overwhelmingly) |
What the shape actually says
Negative skew (−0.47) means the distribution's left tail is heavier than its right: large down days are more extreme, on average, than large up days of equivalent frequency. This matches the market's well-documented tendency to fall fast and climb slowly. Crashes are sharp, recoveries are gradual.
Excess kurtosis of 18.74 is the more striking number. A normal distribution has excess kurtosis of 0 by definition. A value this large means the actual distribution has dramatically fatter tails than a bell curve would predict, so extreme days in both directions occur far more often than normality assumes. This single number is close to the entire justification for why volatility-clustering models (GARCH and its relatives) exist: a model that assumes constant, normally-distributed daily risk will be blindsided by exactly the kind of tail events that show up disproportionately often in a century of real data.
The Jarque-Bera test formalizes what the skew and kurtosis numbers already suggest: the p-value is effectively zero, meaning the hypothesis that these returns are normally distributed is rejected about as decisively as a statistical test can reject anything.
Why this matters practically
Risk models built on the normal distribution, including many textbook Value-at-Risk calculations, systematically understate the likelihood of large moves. A century of data says the tails are fatter than they look, the downside is worse than the upside, and treating daily returns as a bell curve is a convenient simplification, not a description of what the market actually does.
*This is a Phase 1 finding from an ongoing statistical research project covering stocks, bonds, and indices as far back as available data allows. Data: Yahoo Finance (S&P 500, ^GSPC). Methodology and full reproducibility details available on request.*