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**1 - 2**of**2**### Bounds on Portfolio Quality

, 2014

"... The signal-noise ratio of a portfolio of p assets, its expected return di-vided by its risk, is couched as an estimation problem on the sphere Sp−1. When the portfolio is built using noisy data, the expected value of the signal-noise ratio is bounded from above via a Cramér-Rao bound, for the case ..."

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The signal-noise ratio of a portfolio of p assets, its expected return di-vided by its risk, is couched as an estimation problem on the sphere Sp−1. When the portfolio is built using noisy data, the expected value of the signal-noise ratio is bounded from above via a Cramér-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 maximiz-ing 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 popula-tion signal-noise ratio grows slower than the universe size to the 1 4 power, there may be no diversification benefit, rather expected signal-noise ra-tio 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 mod-els 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. 1

### Contents

, 2014

"... Herein is a hodgepodge of facts about the Sharpe ratio, and the Sharpe ratio of the Markowitz portfolio. Connections between the Sharpe ratio and the t-test, and between the Markowitz portfolio and the Hotelling T 2 statistic are explored. Many classical results for testing means can be easily trans ..."

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Herein is a hodgepodge of facts about the Sharpe ratio, and the Sharpe ratio of the Markowitz portfolio. Connections between the Sharpe ratio and the t-test, and between the Markowitz portfolio and the Hotelling T 2 statistic are explored. Many classical results for testing means can be easily translated into tests on assets and portfolios. A ‘unified ’ framework is described which combines the mean and covariance parameters of a multivariate distribution into the uncentered second moment of a related random variable. This trick streamlines some multivariate computations, and gives the asymptotic distribution of the sample Markowitz portfolio.