  We analyze empirical data for 4,000 real-life trading portfolios (U.S.
equities) with holding periods of about 0.7-19 trading days. We find a simple
scaling C ~ 1/T, where C is cents-per-share, and T is the portfolio turnover.
Thus, the portfolio return R has no statistically significant dependence on the
turnover T. We also find a scaling R ~ V^X, where V is the portfolio
volatility, and the power X is around 0.8-0.85 for holding periods up to 10
days or so. To our knowledge, this is the only publicly available empirical
study on such a large number of real-life trading portfolios/alphas.
