Unsystematic and systematic risk of a portfolio

Question

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?

Answer 1

Assuming those are arithmetic returns and covariances at the horizon, calculate a $9\times1$ vector containing the betas with respect to the world index using the covariance matrix, call it $\beta$. The covariance resulting from the world index can be described as $\beta\sigma_{world}^{2}\beta'$. The matrix $\Sigma_{residual}\equiv\Omega-\beta\sigma_{world}^{2}\beta'$ will then reflect the residual covariance. Note that this residual covariance matrix is not necessarily a diagonal matrix, as some CAPM-like models would require. To get a measure of the residual risk of the portfolio, you would then calculate $w'\Sigma_{residual}w$.