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If you are investing an amount $M$, split over deals indexed by $i$ and with a weight $w_i$, then your dollar position in each share will be $w_i M$. The exposure to the index will be $\sum \beta_i w_i M$ You should realize that this will not hedge idiosyncratic risks. In general, the more deals you have, the better this type of hedge should work (assuming ...


I think u can hedge using the description given in JC hull.. here he uses index futures. A detailed explanation is given for one stock. I think u can extend it to a portfolio. Also one can hedge by combining two or three stock indices. See page 33 in this link http://www2.fiu.edu/~dupoyetb/Financial_Risk_Mgt/lectures/Ch03.pdf


Speaking from equity quant factor building experience, it is a common practice to build multi-factor models by regressing one component against other(s) and using the residual scores. This is done to avoid bias as you mentioned - these biases could be from the factor itself (in different regimes, Quality / Momentum influencing each other - or earnings, value ...


Bloomberg functions PORT OPT and MARS provide internal factor models for many asset classes + ability to use your own models in optimization.

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