# Inverse Covariance Matrix Transformation from CAPM

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.

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

• I like that equation 160 is already on page 18 :) Bookmarked. – Bob Jansen Oct 15 '20 at 6:57
• Thank you for your help ! – lays Oct 15 '20 at 8:17