GEOMETRIC INSIGHTS INTO ROBUST PORTFOLIO CONSTRUCTION

We investigate and extend the result that an [Formula: see text]-weight angle from unconstrained quadratic portfolio optimizations has an upper bound dependent on the condition number of the covariance matrix. This is known to imply that better conditioned covariance matrices produce weights from unconstrained mean-variance optimizations that are better aligned with each asset’s expected return. Here, we relate the inequality between the [Formula: see text]-weight angle and the condition number to extend the result to include portfolio optimizations with gearing constraints to provide an extended family of robust optimizations. We use this to argue that the equally weighted portfolio is in general not preferable to the mean-variance portfolio even with poor forecast ability and a badly conditioned covariance matrix.