Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|improve this question

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 $w'\Sigma_{residual}w$.

share|improve this answer
Are you aware of any papers that try an approach similar to what I've suggested/you've detailed? – user2921 Sep 19 '12 at 0:17
symmys.com/node/196 – John Sep 19 '12 at 3:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.