Bounds on Portfolio Quality
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.