I have 8 country stock indexes and 1 world stock index. I do not actually have time series data but I'm given the following data:
- $\mu$, the vector of expected future returns for all 8 country indexes and world index (9 indexes).
- $\Omega$, the variance covariance matrix of all 9 indexes.
I'm forming a MV efficient and Michaud resampling portfolio over the 8 country indexes - the world index is not considered an investable asset class. I want to compare the two portfolios by looking at the systematic risk and unsystematic risk of both portfolios w.r.t. the world market index. So we have the two weights vectors produced by the two methodologies:
- $_1w$ (MV)
- $_2w$ (REF, Resampled Efficient Frontier).
We can calculate the betas of both portfolios by going $_j\beta_p = \sum_{i=1}^8 (_jw_i )\frac{\sigma_{i,world}}{\sigma^2_{world}}$ for $j = 1,2$. Being able to sum the coefficients like this follows from OLS.
How do I get from here to the unsystematic and systematic risk of the portfolios? I can't get the error from the specification that generates the betas so it seems I'm stuck?