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In the standard portfolio risk setup, we have

$\sigma_{\Pi} = \sqrt{(w' B (VFV) B' w) + w'Dw}$

where

  • $w$ is our weight vector for N assets
  • $B$ is the Nxm factor beta matrix
  • $V$ is the factor volatility matrix with factor volatilities on the diagonal and zeros elsewhere
  • $F$ is the mxm factor correlation matrix
  • $D$ is the NxN idiosyncratic variance matrix with idio variances on the diagonal and zeros elsewhere

$VFV$ is the factor covariance matrix, but I specify it factored out. A key assumption of course is that the idiosyncratic risk is uncorrelated with the factor risk.

However I have a bit of a different problem statement:

  • I have $N$ idiosyncratic risks with a block diagonal correlation matrix. The blocks represent $k$ groups of assets. The idiosyncratic returns are correlated between assets of the same group, but have zero correlation with assets outside that group.
  • I have $m$ factor risks as usual, but
  • I also have given correlations between the $m$ factors and each of the $k$ groups.
  • As such, I have
    • A top level correlation matrix, $R$ which is (m+k)x(m+k)
    • k little correlation matrices with cumulative dimensions of NxN

I am calculating the portfolio risk in a two stage process:

  • Stage 1, I get the variances within each $k$ groups; this gives me $k$ variances
  • Stage 2,
    • I augment the beta matrix $B$ with indicator variables for group membership; $B$ is now Nx(m+k)
    • I make $V$ such that the diagonal for length $m$ contains the factor variances and stack on $k$ variances
  • I then apply $\sigma_{\Pi} = \sqrt{(w' B (VRV) B' w)}$

My question is How do I formulate the problem so that I do not have to do it in two stages? I am trying to setup a single correlation matrix, beta matrix, and variance matrix that faithfully represents this hierarchical setup.

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