Abstract. -Personal income distribution in the USA has a well-defined two-class structure. The majority of population (97-99%) belongs to the lower class characterized by the exponential Boltzmann-Gibbs ("thermal") distribution, whereas the upper class (1-3% of population) has a Pareto power-law ("superthermal") distribution. By analyzing income data for 1983-2001, we show that the "thermal" part is stationary in time, save for a gradual increase of the effective temperature, whereas the "superthermal" tail swells and shrinks following the stock market. We discuss the concept of equilibrium inequality in a society, based on the principle of maximal entropy, and quantitatively show that it applies to the majority of population.

We study the probability distribution of stock returns at mesoscopic time lags (return horizons) ranging from about an hour to about a month. While at shorter microscopic time lags the distribution has power-law tails, for mesoscopic times the bulk of the distribution (more than 99% of the probability) follows an exponential law. The slope of the exponential function is determined by the variance of returns, which increases proportionally to the time lag. At longer times, the exponential law continuously evolves into Gaussian distribution. The exponential-to-Gaussian crossover is well described by the analytical solution of the Heston model with stochastic volatility.

We compare the probability distribution of returns for the three major stock-market indices (Nasdaq, S&P500, and Dow-Jones) with an analytical formula recently derived by Drȃgulescu and Yakovenko for the Heston model with stochastic variance. For the period of 1982-1999, we find a very good agreement between the theory and the data for a wide range of time lags from 1 to 250 days. On the other hand, deviations start to appear when the data for 2000-2002 are included. We interpret this as a statistical evidence of the major change in the market from a positive growth rate in 1980s and 1990s to a negative rate in 2000s.

We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log-returns) for a given number of trades N is found to be approximately Gaussian. The probability distribution of N for a given time interval ∆t is non-Poissonian and has an exponential tail for large N and a sharp cutoff for small N . Combining these two distributions produces a nontrivial distribution of log-returns for a given time interval ∆t, which has exponential tails and a Gaussian central part, in agreement with empirical observations. PACS numbers: 89.65.Gh, 05.40.Fb,

We present a detailed study of the performance of a trading rule that uses moving average of past returns to predict future returns on stock indexes. Our main goal is to link performance and the stochastic process of the traded asset. Our study reports short, medium and long term effects by looking at the Sharpe ratio (SR). We calculate the Sharpe ratio of our trading rule as a function of the probability distribution function of the underlying traded asset and compare it with data. We show that if the performance is mainly due to presence of autocorrelation in the returns of the traded assets, the SR as a function of the portfolio formation period (look-back) is very different from performance due to the drift (average return). The SR shows that for look-back periods of a few months the investor is more likely to tap into autocorrelation. However, for look-back larger than few months, the drift of the asset becomes progressively more important. Finally, our empirical work reports a new long-term effect, namely oscillation of the SR and propose a non-stationary model to account for such oscillations. * ferfff@usp.br † csilva@idatafactory.com ‡ ju-yi.yen@uc.edu 1 Cross-sectional momentum is not the focus of this article, but it has been studied extensively if compared to time-series momentum or trend-following. Since the work of [41], momentum has been extended to different asset classes, portfolios, and other markets abroad. Momentum has been reported in international equity markets by [22,28,62,65]; in industries by [50,60]; in indexes by [8,2]; and in commodities by [24,61]. Single risky asset momentum is analyzed in [21,5,39,7,44,1,52,66] to cite a few. Recently, momentum has also been studied linking its performance to business cycles and regimes by [16,46,33,35,6]. arXiv:1907.00212v1 [q-fin.ST]