Variety Of Behavior Of Equity Returns In Financial Markets

We investigate a statistical ensemble of daily returns of n equities traded in United States financial markets. For each trading day of our database, we study the ensemble return distribution. We find that a typical ensemble return distribution exists in most of the trading days 1 with the exception of crash and rally days 2 . We analyze each ensemble return distribution by extracting its first two central moments. We call the second moment of the ensemble return distribution the variety of the market. We choose this term because high variety implies a variated behavior of the equities returns in the considered day. We observe that the mean return and the variety are fluctuating in time and are stochastic processes themselves. The variety is a long-range correlated stochastic process. Customary time-averaged statistical properties of time series of stock returns are also considered. In general, time-averaged and portfolio-averaged returns have different statistical properties 1 . We infer from these differences information about the relative strength of correlation between equities and between different trading days. We also compare our empirical results with those predicted by the single-index model and we conclude that this simple model is unable to explain the statistical properties of the second moment of the ensemble return distribution. In our study, a special emphasis is devoted to the differences observed in the statistical properties of the ensemble return distribution both during typical trading days and during extreme market changes. We find that the symmetry properties of the ensemble return distribution drastically change in crash and rally days of the market. Specifically, in crash and rally days, the distribution becomes asymmetric. In particular for crashes the positive tail is steeper than the negative one whereas the reverse is observed in rally days 2 . References 1 Fabrizio Lillo and Rosario N. Mantegna, Variety and volatility in financial markets, Phys. Rev. E 62, 6126-6134 (2000). 2 Fabrizio Lillo and Rosario N. Mantegna, Symmetry alteration of ensemble return distribution in crash and rally days of financial market, Eur. Phys. J. B {\\bf 15}, 603-606 (2000).