Empirical properties of the variety of a financial portfolio and the single-index model

We investigate the variety of a portfolio of stocks in normal and extreme d ays of market activity. We show that tile variety carries information about the market activity which is not present in the single-index model and we observe that the variety time evolution is not time reversal around tile cr ash days, We obtain the theoretical relation between tile square variety an d the mean return of the ensemble return distribution predicted by the sing le-index model. The single-index model is able to mimic the average behavio r of the square variety but fails in describing quantitatively the relation between the square variety and the mean return uf the ensemble distributio n. The difference between empirical data and theoretical description is mor e pronounced for large positive values of the mean return of the ensemble d istribution. Other significant deviations are also observed for extreme neg ative values of the mean return.