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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.

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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. ...

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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 $r_m$ ...

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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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