Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns

TL;DR

The paper investigates asymmetry in de-trended S&P500 returns from daily to multi-day accumulations, using CCDF tail analysis with log-log fits, confidence intervals, and outlier tests (including Dragon Kings), alongside full PDF-based moments to characterize skewness and dispersion. It finds that losses exhibit heavier tails than gains and that power-law tail behavior deteriorates as the accumulation window grows, while the mean and variance scale nearly linearly with $\tau$ and skewness remains negative. A theoretical framework based on two mean-reverting stochastic-volatility processes (MM and HM, combined as MHM) yields symmetric return distributions, highlighting a mismatch with empirical observations. The results expose gaps in current continuous-time theories for aggregated returns and underscore the need to account for sustained asymmetry and heavier tails in risk modeling of long-horizon stock returns.

Abstract

We study decades-long 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 Covid Pandemic -- which are reflected in the tail ends of the distributions. De-trending the overall gain, we concentrate on comparing distributions of gains and losses. Specifically, we compare the tails of the distributions, which are believed to exhibit power-law behavior and possibly contain outliers. Towards this end we find confidence intervals of the linear fits of the tails of the complementary cumulative distribution functions on a log-log scale, as well as conduct a statistical U-test in order to detect outliers. We also study probability density functions of the full distributions of the returns with the 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.

Figures (17)

• Figure 1: Linear fits of $r_{t} = \log\left(S_t/S_0\right)$ for $t=n \tau, n=0, 1, ...$, with $\tau=1, 20, 50, 100$ respectively.
• Figure 2: Slopes of linear fits of log returns $r_t$ for $t=n \tau, n=0, 1, ...$, as a function of $\tau$.
• Figure 3: Time series of daily returns, $\tau=1$, and accumulated returns for $\tau=20, 50, 100$.
• Figure 4: Number of data points for gains and losses and their sum (the total number of points) as a function of $\tau$.
• Figure 5: Linear fits of CCDF tails for gains and losses of daily returns, $\tau=1$, with CI (dashed lines) and possible DK (pDK), denoted by up triangles, and negative DK (nDK) denoted by down triangles