# Estimated betas and optimal portfolio

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?

• Betas are not enough to builld a portfolio, you also need the standard deviation of idiosyncratic risk for each stock (and optionally the alphas as well, if you assume they are nonzero). Then it becomes a linear programming problem, as shown by Wm. Sharpe in 1967. ideas.repec.org/a/inm/ormnsc/v13y1967i7p499-510.html Jun 4 '19 at 15:41