Portfolio Weight and Return Angles in Mean-Variance Optimization
Summary
The document asks how to interpret the angle between a vector of expected returns and the portfolio weights produced by mean-variance optimization. It draws on a working paper that uses vector geometry to study Markowitz portfolios and describes robust optimization as a way to control this angle and seek more intuitive allocations. The question focuses on a stated lower bound for the cosine of the angle, expressed using the covariance matrix’s largest and smallest eigenvalue scales.
The proposed bound links the geometry of the return and weight vectors to the conditioning of the covariance matrix: a wider spread between its extreme eigenvalues permits a smaller lower bound on alignment. The document supplies the expression and defines the symbols, but it does not derive the inequality or explain its graphical meaning. It also provides no implementation details or empirical portfolio examples. Readers should treat it as a technical question about a result in a working paper, rather than a complete tutorial or evidence that angle control improves realized investment performance.
Key ideas
- Mean-variance optimization maps an expected return vector and covariance matrix into portfolio weights.
- The document studies the geometric angle between expected returns and resulting weights.
- A stated lower bound on the angle’s cosine depends on the covariance matrix’s extreme eigenvalues.
- The question connects covariance conditioning with the alignment of returns and portfolio weights.
- No derivation, implementation method, or empirical evidence is provided.
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Full text
# Analyzing the angle between vector of weights and vector of returns in mean-variance optimization
# Analyzing the angle between vector of weights and vector of returns in mean-variance optimization
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.