  Internal crossing of trades between multiple alpha streams results in
portfolio turnover reduction. Turnover reduction can be modeled using the
correlation structure of the alpha streams. As more and more alphas are added,
generally turnover reduces. In this note we use a factor model approach to
address the question of whether the turnover goes to zero or a finite limit as
the number of alphas N goes to infinity. We argue that the limiting turnover
value is determined by the number of alpha clusters F, not the number of alphas
N. This limiting value behaves according to the "power law" ~ F^(-3/2). So, to
achieve zero limiting turnover, the number of alpha clusters must go to
infinity along with the number of alphas. We further argue on general grounds
that, if the number of underlying tradable instruments is finite, then the
turnover cannot go to zero, which implies that the number of alpha clusters
also appears to be finite.
