Estimated betas and optimal portfolio

Question

I ran a regression on 20 assets to estimate their beta with different methods. I would like to see the differences of these estimation differences in terms of mean-variance optimal portfolio. How can I do that?

The problem is that I do not see clearly the role that the estimated beta plays in portfolio optimization empirically. A mean-variance optimization with N assets should be something like

$min_{\textbf{w}}$ $\textbf{w}^{'}\textbf{V}\textbf{w}$

subject to

$\textbf{w}^{'}E(R)=\bar{R}$,$\textbf{w}^{'}\textbf{I}=1$

Where w are the weights of the N assets, our choice variable. $\textbf{V}$ is the covariance matrix of the returns, $\bar{R}$ is the target return, $\textbf{I}$ is just a vector of ones and $E(R)$ is the vector of expected returns of our assets.

The "unknowns" are V and E(R), should I use historical returns to replace them? If yes, then what is the use of the betas in portfolio optimization? none? or maybe should I use the betas to get the values of E(R) and V?

I do not understand, could someone provide a mini-guide to get from betas to the optimizing portfolio?

Answer 1

Betas aren't traditionally used in creating MV optimal portfolios. Insofar as beta is a proxy for risk, as is vol, there's probably some relationship between your betas and the covariance matrix that IS used in MV optimization, but there's not really any guide to give you that shows how betas are used, because they aren't.

Answer 2

In a factorial model, such as CAPM / APT, you have a linear relationship describing the process generating expected returns.

Therefore, expected returns depend on betas. Therefore, since both the Covariance matrix and expected return vector depend on returns, they will also depend on betas under the assumptions of a factorial model.