  The signal-noise ratio of a portfolio of p assets, its expected return
divided by its risk, is couched as an estimation problem on the sphere. When
the portfolio is built using noisy data, the expected value of the signal-noise
ratio is bounded from above via a Cramer-Rao bound, for the case of Gaussian
returns. The bound holds for `biased' estimators, thus there appears to be no
bias-variance tradeoff for the problem of maximizing the signal-noise ratio. An
approximate distribution of the signal-noise ratio for the Markowitz portfolio
is given, and shown to be fairly accurate via Monte Carlo simulations, for
Gaussian returns as well as more exotic returns distributions. These findings
imply that if the maximal population signal-noise ratio grows slower than the
universe size to the 1/4 power, there may be no diversification benefit, rather
expected signal-noise ratio can decrease with additional assets. As a practical
matter, this may explain why the Markowitz portfolio is typically applied to
small asset universes. Finally, the theorem is expanded to cover more general
models of returns and trading schemes, including the conditional expectation
case where mean returns are linear in some observable features, subspace
constraints (i.e., dimensionality reduction), and hedging constraints.
