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There are some variants to calculate the beta of a stock. If not fully documented at Google, in doubt you have to validate yourself. You will find a help to do this in the linked website. However, the results of the different calculation variants are usually quite similar.


Sounds like PCA is not the approach you're looking for. If you're looking to transform a risk vector in terms of securities V into a risk vector in terms of securities W, then the basic approach would be to perform a linear regression of V against W. The resulting regression coefficients will form a matrix B which will give a change of basis between V and W. ...


I am afraid you are missing something (as a lot of people) about beta computation and meaning: The model you are fitting links the returns of a market factor $r_m$ with your returns $r_g$ thanks to a linear relationship: $$r_g = \beta \;r_m + \epsilon_{m,g}.$$ As you can see, it does not say the volatility of $r_g$ can be deduced from the one of $r_m$ ...


This is an interesting problem. I don't think the problem is set up correctly quite yet. I rewrote it slightly to correspond to how it's generally written as a quadratic program. The optimization problem you write down fixes betas to be a certain value. That could make sense but instead I wondered if we could simply minimize beta across the portfolio while ...

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