A Sharper Angle on Optimization

Abstract

The classical mean-variance optimization takes expected returns and variances and produces portfolio positions. In this paper we discuss the direction and the magnitude of the positions vector separately, and focus on the former. We quantify the distortions of the mean-variance optimization process by looking at the angle between the vector of expected returns and the vector of optimized portfolio positions. We relate this angle to the condition numbers of the covariance matrix and show how to control it by employing robust optimization techniques. The resulting portfolios are more intuitive and investment-relevant, in particular with lower leverage of the “noise” alphas at the expense of lower ex-ante Sharpe Ratio.

Keywords: mean-variance optimization, covariance matrix, condition number, leverage, Sharpe Ratio

JEL Classification: G11, C61