Inverse Statistics for Stocks and Markets

Abstract

In recent publications, the authors have considered inverse statistics of the Dow Jones Industrial Averaged (DJIA) [1-3]. Specifically, we argued that the natural candidate for such statistics is the investment horizons distribution. This is the distribution of waiting times needed to achieve a predefined level of return obtained from detrended historic asset prices. Such a distribution typically goes through a maximum at a time coined the {\em optimal investment horizon}, $τ^*_ρ$, which defines the most likely waiting time for obtaining a given return $ρ$. By considering equal positive and negative levels of return, we reported in [2,3] on a quantitative gain/loss asymmetry most pronounced for short horizons. In the present paper, this gain/loss asymmetry is re-visited for 2/3 of the individual stocks presently in the DJIA. We show that this gain/loss asymmetry established for the DJIA surprisingly is {\em not} present in the time series of the individual stocks. The most reasonable explanation for this fact is that the gain/loss asymmetry observed in the DJIA as well as in the SP500 and Nasdaq are due to movements in the market as a whole, {\it i.e.}, cooperative cascade processes (or ``synchronization'') which disappear in the inverse statistics of the individual stocks.