Is It Possible to OD on Alpha?

It is well known that combining multiple hedge fund alpha streams yields diversification benefits to the resultant portfolio. Additionally, crossing trades between different alpha streams reduces transaction costs. As the number of alpha streams increases, the relative turnover of the portfolio decreases as more trades are crossed. However, we argue, under reasonable assumptions, that as the number of alphas increases, the turnover does not decrease indefinitely; instead, the turnover approaches a non-vanishing limit related to the correlation structure of the portfolio's alphas. We also point out that, more generally, computational simplifications can arise when the number of alphas is large.