I am using the paper "A Sharper Angle on Optimization" by Golts and Jones (2009) as a basis for my (minor) masters thesis in mathematical finance. The paper focuses on the mean-variance analysis of Markowitz but instead turns attention to the vector geometry of the returns vector and vector of resultant portfolio weights. As it is a working paper, most of the concepts are not elaborated on well enough to make sense or for one to implement by him/herself. The paper may be accessed on this link: http://ssrn.com/abstract=1483412.
One of the ideas I am struggling with is the angle between the returns vector and vector of weights and how this angle can be related to the condition number of the covariance matrix. The authors then employ robust optimization techniques to control this angle (i.e. minimize it) to obtain more intuitive investment portfolios.
The authors state that the angle between the returns and positions vector, call it $\omega$, is bounded from below as: $\cos(\omega)=\frac{\alpha^{T}\Sigma^{-1}\alpha}{\sqrt{\alpha^{T}\alpha}\sqrt{\alpha^{T}\Sigma^{-2}\alpha}} \geq \frac{\theta_{\max}\theta_{\min}}{(\theta_{\max}^{2}+\theta_{\min}^{2})/2}$
where $\alpha$ is the vector of returns and $\Sigma$ is the covariance matrix with spectral decomposition given by $\Sigma=Q^{T}\mbox{diag}(\theta_{1}^{2},...,\theta_{n}^{2})Q$ where $\theta_{1}^{2} \geq \theta_{2}^{2} \geq ... \geq \theta_{n}^{2} > 0$ are the eigenvalues in decreasing order and where we let $\theta_{\max}^{2}=\theta_{1}^{2}$ and $\theta_{\min}^{2}=\theta_{n}^{2}$.
If anyone has any ideas on how the authors may have arrived at this, as well as what it means graphically, I would really appreciate it.
Many thanks in advance!
