Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns

We studied decades-long (1980 to 2024) historic distributions of accumulated S&P500 returns, from daily returns to those over several weeks. The time series of the returns emphasize major upheavals in the markets—Black Monday, Tech Bubble, Financial Crisis, and the COVID pandemic—which are reflected in the tail ends of the distributions. De-trending the overall gain, we concentrated on comparing distributions of gains and losses. Specifically, we compared the tails of the distributions, which are believed to exhibit a power-law behavior and possibly contain outliers. To this end, we determined confidence intervals of the linear fits of the tails of the complementary cumulative distribution functions on a log–log scale and conducted a statistical U-test in order to detect outliers. We also studied probability density functions of the full distributions of the returns with an emphasis on their asymmetry. The key empirical observations are that the mean of de-trended distributions increases near-linearly with the number of days of accumulation while the overall skew is negative—consistent with the heavier tails of losses—and depends little on the number of days of accumulation. At the same time, the variance of the distributions exhibits near-perfect linear dependence on the number of days of accumulation; that is, it remains constant if scaled to the latter. Finally, we discuss the theoretical framework for understanding accumulated returns. Our main conclusion is that the current state of theory, which predicts symmetric or near-symmetric distributions of returns, cannot explain the aggregate of empirical results.