Geometric insights into robust portfolio construction

We investigate and extend the result that an alpha-weight angle from unconstrained quadratic portfolio optimisations 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 optimisations that are better aligned with each assets expected return. Here we relate the inequality between the alpha-weight angle and the condition number to extend the result to include portfolio optimisations with gearing constraints to provide an extended family of robust optimisations. We use this to argue that in general the equally weighted portfolio is not preferable to the mean-variance portfolio even with poor forecast ability and a badly conditioned covariance matrix. We confirm the distribution free theoretical arguments with a simple Gaussian simulation.