Empirical evidence is given for a significant difference in the collective trend of the share prices during the stock index rising and falling periods. Data on the Dow Jones Industrial Average and its stock components are studied between 1991 and 2008. Pearson-type correlations are computed between the stocks and averaged over stock-pairs and time. The results indicate a general trend: whenever the stock index is falling the stock prices are changing in a more correlated manner than in case the stock index is ascending. A thorough statistical analysis of the data shows that the observed difference is significant, suggesting a constant-fear factor among stockholders.