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You have the risk factor $F$ and the asset that it is correlated to $r_m$. You can calculate the variances of each of these, say $\sigma^2_F$ and $\sigma^2_m$. If you do not care about the distribution but just work with variances and correlations then can look at an OLS setting: $$ F = \beta r_m + \epsilon $$ with $\beta = \rho \frac{\sigma_F}{\sigma_m}$ ...



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