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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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If you look at changes of the points on the yield curve, then you probably find something stationary - right? Applying PCA on the covariance of these changes makes sense. E.g. you will find out that on PC describes a parallel shift (a change in the yield curve). Look at this question too: What do eigenvalues/eigenvectors of the yield/forward rates ...

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There is a vast literature on modelling time-series with periodcities. Rob Hyndman is one of the leading reseaerchers in this area. He has published the R package forecast and a free online text book on this subject (with another package and R code in the book). Your task is covered starting here.

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You will need the covariance matrix to calculate this. Say you have a collection of $n$ assets. The value of asset $i$ is represented by the random variable $X_i$ and the corresponding portfolio weight is are $w_i$, and $v_i$ for the two portfolios. The correlation between the two portfolios is:  \frac{\sigma(w^TX,v^TX)}{\sqrt{(w^T\Sigma w)(v^T\Sigma ...

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