Asymmetry in distributions of accumulated gains and losses in stock returns

Farahani, Hamed; Serota, Rostislav A. Article Asymmetry in distributions of accumulated gains and losses in stock returns Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Farahani, Hamed; Serota, Rostislav A. (2025) : Asymmetry in distributions of accumulated gains and losses in stock returns, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 6, pp. 1-16, https://doi.org/10.3390/economies13060176 This Version is available at: https://hdl.handle.net/10419/329456 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns. Economies,13(6), 176. https://doi.org/10.3390/ economies13060176 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns Hamed Farahani * and Rostislav A. Serota * Department of Physics, University of Cincinnati, Cincinnati, OH 45221-0011, USA *Correspondence: [email protected] (H.F.); [email protected] (R.A.S.) Abstract: 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. Keywords: accumulated returns; S&P500; power-law tails; outliers; skewness 1. Introduction Research on asymmetry of stock returns has a long and storied history (Albuquerque, 2012;Bekaert & Wu,2000;Braun et al.,1995;Campbell & Hentschel,1992;Chakraborti et al., 2011;Cont,2001;Duffee,1995;French et al.,1987;Glosten et al.,1993;Hong & Stein,2003; Lee & Kang,2023;Neuberger & Payne,2021;Sándor et al.,2016;Sive & Lins,2009;Wu,2001; Załuska-Kotur et al.,2006). Clearly, there are many aspects of asymmetry and approaches to study this phenomenon, such as the first passage time (Sándor et al.,2016;Sive & Lins,2009; Załuska-Kotur et al.,2006), differences between firm-level and overall market performance (Albuquerque,2012;Bekaert & Wu,2000;Braun et al.,1995;Duffee,1995), and many others. The simplest form of asymmetry is that, overall, there is a considerable gain in the stock market: financial advisors like to tell their clients that, on average, there is roughly a 10% annual gain or, more precisely, 12% gain (see the straight line in Figure 1) minus 2% average inflation. Of course, there are periods of market stagnation, decline and rapid growth—such fluctuations around the overall growth trend are attributed to market volatility. Economies 2025,13, 176 https://doi.org/10.3390/economies13060176 Economies 2025,13, 176 2 of 16 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r t log ( S t / S 0 ) with τ = 1 linear fit with slope of μ 1 = 3.0860 × 10 −4 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r t log ( S t / S 0 ) with τ = 20 linear fit with slope of μ 20 = 3.0890 × 10 −4 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r t log ( S t / S 0 ) with τ = 50 linear fit with slope of μ 50 = 3.0968 × 10 −4 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r t log ( S t / S 0 ) with τ = 100 linear fit with slope of μ 100 = 3.0897 × 10 −4 Figure 1. Linear fits of rt=log(St/S0)for t=nτ,n=0, 1, . . ., with τ=1, 20, 50, 100, respectively. A far more interesting question is the asymmetry between gains and losses once the overall growth trends are already accounted for, that is, when the data are de-trended. In this regard, of the otherwise numerous empirical properties of the raw market data (Chakraborti et al.,2011;Cont,2001), our interest is centered mainly on asymmetry as related to heavy tails (Taleb,2007) of the distributions of gains and losses. To this end, we studied distributions of stock returns of the S&P500 index, from daily returns to those accumulated over longer periods of time—a subject largely missing from the literature (for intraday timed trades see (Watorek et al.,2021)). Specifically, we performed linear fits (LFs) of the tails of complementary cumulative distribution functions (CCDFs) of gains and losses on a log–log scale to test for their power-law dependence. We computed confidence intervals (CIs) (Janczura & Weron,2012) of LFs and conducted a statistical U-test (Pisarenko & Sornette,2012) in order to test for possible outliers, such as Dragon Kings (DKs) (Sornette & Ouillon,2012) and negative Dragon Kings (nDKs) (Pisarenko & Sornette,2012). We also performed numerical measures of the full distributions of returns using their probability density functions (PDFs). Specifically, we evaluated dependence on the number of days of accumulation of the mean, variance, Fisher–Pearson coefficient of skewness, and first and second Pearson coefficients of skewness. The key results from those measures 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 observed from PDF and CCDF—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. While the near-linear shift of the mean can be easily accounted for phenomenologically, the current state of theory based on continuous stochastic differential equations (SDEs) does not properly describe statistical measures of the distributions, especially skewness. This paper is organized as follows: In Section 2.1, we explain the de-trending procedure of returns, present the time series of returns for the 1980–2024 period, and discuss the number of data points of gains and losses—all in terms of the number of days of accumula- Economies 2025,13, 176 3 of 16 tion. In Section 2.2, we compare distributions of gains and losses using CCDF, including LF statistical tests for outliers. In Section 2.3, we study full distributions of returns and their statistical measures: mean, variance, and skewness. In Section 3, we address the state of theory vis-a-vis our empirical observations. Section 4summarizes our main results. 2. Empirical Results 2.1. Initial Analysis of Returns With Stbeing the stock price, the linear upward trend of log returns rt=logSt S0(1) is shown in Figure 1for t=nτ , n= 0, 1, ...with τ= 1, 20, 50, 100 and the plot of slopes µτ shown in Figure 2. De-trended log returns (or simply “returns” below) accumulated over time period τ are then given by dxt=xt+τ−xt=rt+τ−rt−µτ =logSt+τ St−µτ (2) where, from now on, we slide t by one day when obtaining distributions as a function of τ and thus use µ=µ1 —the slope of daily log returns, although, clearly, µτ shows only very insignificant dependence on τ. 0 20 40 60 80 100 τ 3.000 3.025 3.050 3.075 3.100 3.125 3.150 3.175 3.200 μ τ ×10 −4 μ τ Figure 2. Slopes of linear fits of log returns rt for t=nτ , n= 0, 1, . . . , as a function of τ . Red dots correspond to τ=1, 20, 50, 100 as in Figure 1. Figure 3shows the time series of returns from 1980 to 2024. Notice the obvious similarity with the time series of realized volatility (J. Liu et al.,2024). Clearly, the largest negative peaks occurred during Black Monday, the Tech Bubble, Financial Crisis, and the COVID pandemic. Not surprisingly, following those drops, the largest positive peaks occurred relatively shortly after. Figure 4shows the number of data points for gains and losses as a functions of τ , as well as their sum—the total number of points in the data set—for the same time period (1980–2024) as the time series in Figure 3. For illustrative purposes, the numbers are explicitly shown for τ= 1, 5, 10, 20 in Table 1. Clearly, the number of gains increases as a function of τ , while the number of losses decreases. The total number of points is given by 11,259 −τ+1, where 11,259 is the size of the data set for daily returns, τ=1. Economies 2025,13, 176 4 of 16 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 −0.3 −0.2 −0.1 0.0 0.1 Return Gains, τ = 1 Losses, τ = 1 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 Return Gains, τ = 20 Losses, τ = 20 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 −0.6 −0.4 −0.2 0.0 0.2 Return Gains, τ = 50 Losses, τ = 50 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 −0.6 −0.4 −0.2 0.0 0.2 0.4 Return Gains, τ = 100 Losses, τ = 100 Figure 3. Time series of daily returns, τ=1, and accumulated returns for τ=20, 50, 100. 1 10 20 30 40 50 60 70 80 90 100 τ 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Number of Points Gains (Line) Losses (Line) Total (Line) Gains (Bars) Losses (Bars) Total (Bars) Figure 4. Number of data points for gains and losses and their sum (the total number of points) as a function of τ. Economies 2025,13, 176 5 of 16 Table 1. Summary of total points, losses, and gains for different τvalues. τTotal Points Losses Gains 1 11,259 5455 5804 5 11,255 5167 6088 10 11,250 5063 6187 20 11,240 4871 6369 2.2. Distributions of Gains and Losses Figures 5–8show CCDF, 1 −Fg(x) and 1 −Fl(x) , of gains and losses on a log–log scale for τ=1, 5, 10, 20. Here, Fg(x) = Zx −∞f(x)dx Fl(x) = Zx ∞f(x)dx(3) are the CDFs of gains and losses, respectively, and f(x) is the PDF of returns (see Figures 9–12 below). Also shown are linear fits of the tails, including their confidence intervals (CIs) and the results of the U-Test to identify outliers, such as DK and nDK. CIs for the fits are evaluated via the inversion of the binomial distribution (Janczura & Weron, 2012); p -values are evaluated in the framework of the U-test, which is based on order statistics (Pisarenko & Sornette,2012), using the following formula: p(xk,n) = 1−B(F(xk,n);k,n−k+1), (4) where xk,nis the k’s member of numbers (SR here) between 1 and nordered by increasing magnitude, F(xk,n) is the assumed CDF (LF here), and B(y ; a , b) is the incomplete Beta function (NIST Digital Library of Mathematical Functions,n.d.). p-values are evaluated in order to test the null hypothesis H0 : all observations of the sample are generated by the same fitting distribution. The p-value (4) is defined as a probability of exceeding the observed value xk,n under the null hypothesis. If among the p-values, there are some small values ( ≤ 0.05 here), then those observed values are identified as DK with probability 1 −p and marked by up triangles. Conversely, large p-values ( ≥ 0.95 here) are identified as nDK with the probability p (Pisarenko & Sornette,2012) and marked by down triangles. While daily returns seem to exhibit rather well-defined linear dependence, for larger τ , the tail behavior is more complex with what might be called a developing shoulder and rapid drop-offs at tail ends. In this regard, instead of thinking of possible DK (pDK) and nDK, perhaps up and down triangles obtained from the U-test and crossing lines of CIs can be simply an indicator of poor goodness of fit. Again, notice the obvious similarities with the tail behavior of the realized volatility (J. Liu et al.,2024). 2.3. Full Distributions of Returns and Their Statistical Measures Figures 9–12 show the PDFs of daily and accumulated returns for τ= 1, 20, 50, 100. Clearly, the PDFs exhibit asymmetry and longer tails for losses versus gains (Palomar, 2018), which are becoming more pronounced with larger τ. Economies 2025,13, 176 6 of 16 2.5 × 10 −2 4 × 10 −2 6 × 10 −2 1 × 10 −1 2 × 10 −1 3 × 10 −1 Return 10 −5 10 −4 10 −3 10 −2 10 −1 CCDF Losses, τ = 1 pDKs for losses nDKs for losses LF for losses LF CI for losses Gains, τ = 1 pDKs for gains nDKs for gains LF for gains LF CI for gains Figure 5. Linear fits of CCDF tails for gains and losses of daily returns, τ= 1, with CIs (dashed lines) and possible DK (pDK), denoted by up triangles, and negative DK (nDK) denoted by down triangles. 6 × 10 −2 1 × 10 −1 2 × 10 −1 3 × 10 −1 4 × 10 −1 Return 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 CCDF Losses, τ = 5 pDKs for losses nDKs for losses LF for losses LF CI for losses Gains, τ = 5 pDKs for gains nDKs for gains LF for gains LF CI for gains Figure 6. Linear fits of CCDF tails for gains and losses of τ= 5 accumulated returns with CIs (dashed lines) and possible DK (pDK), denoted by up triangles, and negative DK (nDK) denoted by down triangles. 6 × 10 −2 1 × 10 −1 2 × 10 −1 3 × 10 −1 4 × 10 −1 Return 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 CCDF Losses, τ = 10 pDKs for losses nDKs for losses LF for losses LF CI for losses Gains, τ = 10 pDKs for gains nDKs for gains LF for gains LF CI for gains Figure 7. Linear fits of CCDF tails for gains and losses of τ= 10 accumulated returns with CIs (dashed lines) and possible DK (pDK), denoted by up triangles, and negative DK (nDK) denoted by down triangles. Economies 2025,13, 176 7 of 16 7 × 10 −2 1 × 10 −1 2 × 10 −1 3 × 10 −1 4 × 10 −1 Return 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 CCDF Losses, τ = 20 pDKs for losses nDKs for losses LF for losses LF CI for losses Gains, τ = 20 pDKs for gains nDKs for gains LF for gains LF CI for gains Figure 8. Linear fits of CCDF tails for gains and losses of τ= 20 accumulated returns with CIs (dashed lines) and possible DK (pDK), denoted by up triangles, and negative DK (nDK) denoted by down triangles. −0.2 −0.1 0.0 0.1 0.2 Return 0 10 20 30 40 50 60 PDF τ = 1 −0.25 −0.20 −0.15 −0.10 −0.05 0.0 0.1 0.2 0.3 0.4 0.5 0.05 0.10 0.15 0.20 0.25 0.0 0.1 0.2 0.3 0.4 0.5 Figure 9. PDF of daily returns, with inserts showing tails of the distribution. −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 Return 0 2 4 6 8 10 12 14 PDF τ = 20 −0.40 −0.35 −0.30 −0.25 −0.20 −0.15 0.0 0.1 0.2 0.3 0.4 0.5 0.15 0.20 0.25 0.30 0.35 0.40 0.0 0.1 0.2 0.3 0.4 0.5 Figure 10. PDF of τ=20 accumulated returns, with inserts showing tails of the distribution. Economies 2025,13, 176 8 of 16 −0.4 −0.2 0.0 0.2 0.4 Return 0 1 2 3 4 5 6 7 8 PDF τ = 50 −0.50 −0.45 −0.40 −0.35 −0.30 −0.25 −0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 Figure 11. τ=50 accumulated returns, with inserts showing tails of the distribution. −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 Return 0 1 2 3 4 5 6 7 PDF τ = 100 −0.6 −0.5 −0.4 −0.3 −0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 Figure 12. τ=100 accumulated returns, with inserts showing tails of the distribution. Next, we address the mean, m1 , variance, m2 , and skewness of distributions in Figures 9–12. For the latter, we employed the Fisher–Pearson coefficient of skewness and the first and second Pearson coefficients of skewness, defined, respectively, as follows: ζ=m3 m3/2 2 (5) ζ1=(m1−m) m1/2 2 (6) ζ2=3(m1−e m) m1/2 2 (7) where m3 is the third central moment of the distribution, m2 is the variance, m1/2 2 is the standard deviation, m is the mode, and e m is the median. The reason for using ζ1 and ζ2 is Economies 2025,13, 176 15 of 16 Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance,1, 223–236. [CrossRef] Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica,3(2), 385–408. [CrossRef] Dashti Moghaddam, M., Liu, Z., & Serota, R. A. (2021). Distributions of historic market data: Relaxation and correlations. The European Physical Journal B,94, 83. [CrossRef] Dashti Moghaddam, M., & Serota, R. A. (2021). Combined mutiplicative-heston model for stochastic volatility. Physica A: Statistical Mechanics and Its Applications,561, 125263. [CrossRef] Dr ˇ agulescu, A. A., & Yakovenko, V. M. (2002). Probability distribution of returns in the Heston model with stochastic volatility. Quantitative Finance,2(6), 443–453. [CrossRef] Duffee, G. R. (1995). Stock returns and volatility A firm-level analysis. Journal of Financial Economics,37, 399–420. [CrossRef] French, K. R., Schwert, G. W., & Stambaugh, R. F. (1987). Expcted stock returns and volatility. Journal of Financial Economics,19, 3-29. [CrossRef] Fuentes, M. A., Gerig, A., & Vicente, J. (2009). Universal behavior of extreme price movements in stock markets. PLoS ONE,4(12), e8243. [CrossRef] [PubMed] Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journl of Fiinance,XLVIII(5), 1779–1801. [CrossRef] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies,6(2), 327–343. [CrossRef] Hong, H., & Stein, J. C. (2003). Differences of opinion, short-sales constraints, and market crashes. The Review of Financial Studies,16(2), 487–525. [CrossRef] Janczura, J., & Weron, R. (2012). Black swans or dragon-kings? A simple test for deviations from the power law. European Physical Journal Special Topics,205, 79–93. [CrossRef] Lee, C. W., & Kang, K. H. (2023). Estimating and testing skewness in a stochastic volatility model. Journal of Empirical Finance,72, 445–467. [CrossRef] Liu, J., Dashti Moghaddam, M., & Serota, R. A. (2024). Are there dragon kings in the stock market? Foundations,4(1), 91–113. [CrossRef] Liu, J., & Serota, R. A. (2023). Rethinking generalized Beta family of distributions. The European Physical Journal B,96(2), 24. [CrossRef] Liu, Z., Dashti Moghaddam, M., & Serota, R. A. (2019). Distributions of historic market data—Stock returns. The European Physical Journal B,92, 60. [CrossRef] Ma, T., & Serota, R. (2014). A model for stock returns and volatility. Physica A: Statistical Mechanics and its Applications,398, 89–115. [CrossRef] Nelson, D. (1990). Arch models as diffusion approximations. Journal of Econometrics,45, 7. [CrossRef] Neuberger, A., & Payne, R. (2021). The skewness of the stock market over long horizons. The Review of Financial Studies,34(3), 1572–1616. [CrossRef] NIST Digital Library of Mathematical Functions. (n.d.). Available online: https://dlmf.nist.gov (accessed on 12 June 2025). Palomar, D. (2018, June 10–13). Financial engineering playground: Signal processing, robust estimation, kalman, HMM, optimization, etc. IEEE SSP 2018, Freiburg, Germany. Available online: https://rc.signalprocessingsociety.org/workshops/spsvid00262 (accessed on 12 June 2025). Perello, J., & Masoliver, J. (2002). Stochastic volatility and leverage effect. arXiv, arXiv:0202203. Pisarenko, V. F., & Sornette, D. (2012). Robust statistical tests of Dragon-Kings beyond power law distribution. The European Physical Journal Special Topics,205, 95–115. [CrossRef] Praetz, P. D. (1972). The distribution of share price changes. Journal of Business,45, 49–55. [CrossRef] Sándor, B., Simonsen, I., Nagy, B. Z., & Néda, Z. (2016). Time-scale effects on the gain-loss asymmetry in stock indices. Physical Review E,94(2), 022311-1–022311-9. [CrossRef] Sive, J. V., & Lins, J. T. (2009). Temporal structure and gain-loss asymmetry for real and artificial stock indices. Physical Review E,80(5), 057102-1–057102-4. Sornette, D., & Ouillon, G. (2012). Dragon-kings: Mechanisms, statistical methods and empirical evidence. The European Physical Journal Special Topics,205, 1–26. [CrossRef] Taleb, N. (2007). The black swan: The impact of the highly improbable. Random House. Watorek, M., Kwapien, J., & Drozdz, S. (2021). Financial return distributions: Past, present, and COVID-19. Entropy,23(7), 884. [CrossRef] [PubMed] Economies 2025,13, 176 16 of 16 Wu, G. (2001). The determinants of asymmetric volatility. The Review of Financial Studies,14(3), 837–859. [CrossRef] Załuska-Kotur, M., Karpio, K., & Orłowski, A. (2006). Comparison of Gain–loss asymmetry behavior for stocks and indexes. Acta Physica Polonica B,37, 3187–3192. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.