A random Matrix Approach to collective Trends of falling and rising Stock Markets
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- Institutt for fysikk 
An inverse statistics analysis of one minute stock quotes from 492 large Europeancompanies has revealed the existence of a gain-loss asymmetry in thefollowing index. The gain-loss asymmetry differs from that observed for dailyclosure prices of the Dow Jones Industrial Average , as the probability ofthe optimal investment horizon for a gain is higher than that of a loss. Forindividual stocks, the gain-loss asymmetry was observed to only appear forsignificantly larger return-levels. To the best of our knowledge, this is thefirst time such an analysis has been performed on high-frequency data.A principal component analysis was done by performing an eigenvalue decompositionof the correlation matrix from a sliding time-window. The firstprincipal component was observed to describe the market excellently. Its correspondingeigenvalue was observed to be significantly larger than theoreticalpredictions from random matrix theory, implying that the eigenvalue carriesinformation common to all stocks. Using this eigenvalue as an index measuringthe collectivity in the market has revealed the existence of collectivetrends that appear to be stronger during falling than rising markets. Thishas been observed for two different datasets, the above described one minutestock quotes and daily closure prices from 29 stocks composing the DJIA lateFebruary 2008. The observation is in accordance with results of Balogh etal. , and provides further support to the speculation of Johansen et al. that a difference in collective trends is the reason behind the gain-lossasymmetry observed in indexes and not for individual stocks for the samereturn-level.The key idea behind the fear factor model of Donangelo et al.  has beenstrongly supported by the observation that collective trends appear to bestronger during sharp index drops. As the collectivity increment has beenobserved to be dependent on the size of the index drop, it is suggested thatthe model should incorporate also individual fear factors for economic sectors,in addition to the global fear factor governing the market as a whole. Periodsexhibiting a rising index positively correlated to the strength of collectivityhas indicated the presence of an optimism factor that also should be incorporatedin the fear factor model , forcing stocks to rise synchronously.