Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns

Paper: arXiv 2503.24241

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.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced statistical methods like linear fits on log-log scales, U-tests for outlier detection, and computes multiple skewness coefficients, indicating moderate to high mathematical complexity. Empirical rigor is strong due to the extensive use of decades-long historical S&P500 data, explicit statistical tests, and analysis of distributions across multiple accumulation periods, making it backtest-ready despite lacking explicit code or portfolio implementation.

Research Flowchart

  flowchart TD
  A["<b>Research Goal</b><br/>Explain asymmetry in accumulated<br/>S&P 500 return distributions"] --> B["<b>Data Source</b><br/>Historic S&P 500 returns<br/>Daily to multi-week intervals"]
  B --> C["<b>Methodology</b><br/>1. De-trend overall gain<br/>2. Analyze tails (power-law)<br/>3. U-test for outliers<br/>4. PDF & asymmetry analysis"]
  C --> D["<b>Computational Processes</b><br/>- Log-log linear fits of CCDF tails<br/>- Confidence intervals on fits<br/>- Statistical testing<br/>- Moment calculations"]
  D --> E["<b>Key Findings</b><br/>1. Mean scales near-linearly<br/>2. Variance scales perfectly linearly<br/>3. Skew remains negative (asymmetric)<br/>4. Theory fails to explain results"]

Cited References (34)

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2. NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov
3. On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks (1993)
4. Distributions of historic market data: relaxation and correlations (2021)
5. Stochastic volatility and leverage effect (2002)
6. A theory of the term structure of interest rates (1985)
7. A model for stock returns and volatility (2014)
8. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)

0 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.