Inverse Covariance Matrix Transformation from CAPM

Question

Beginning with the CAPM model we have (with a risk free rate of 0%):

$r_i=\beta_i (r_m)+\varepsilon_i$

with $\varepsilon_i$ the diversifiable risks per assets

The variance matrix:

$\Omega = \beta'\beta \sigma_m^2 + Diag(\sigma_e^2)$

With $\sigma_m$ a constant, $Diag(\sigma_e^2)$ an N $\times$ N matrix, $\beta$ an 1 $\times$ N matrix.

Inverting the matrix we get the following result:

$\Omega^{-1} = Diag(\frac{1}{\sigma_e^2})-\frac{(\frac{\beta}{\sigma_e^2})(\frac{\beta}{\sigma_e^2})'}{\frac{1}{\sigma_m^2}+(\frac{\beta}{\sigma_e^2})'\beta}$

I don't understand how by using the inverse matrix transformation we find this result.

Thank you for your help

Answer 1

This is the result of the Sherman-Morrison inversion for the sum of an invertible matrix and an outer product. You will find this (and many other helpful methods) in the Matrix Cookbook. Specifically, this is equation 160 on p 18:

$$ \left(\boldsymbol{A}+\boldsymbol{bc}^T\right)^{-1}=\boldsymbol{A}^{-1}-\frac{\boldsymbol{A}^{-1}\boldsymbol{bc}^T\boldsymbol{A}^{-1}}{1+\boldsymbol{c^TA}^{-1}b} $$

HTH