# Factor Models: uncorrelated errors don't impact covariances of assets

This question stems from time series factor models (e.g., CAPM, Fama-French, etc.), but is a broader idea.

I am trying to comprehend how adding noise to a time series (e.g., error/residual from a regression) doesn't change the covariance between two assets.

For example, let's take two assets that are perfectly correlated with one another. If I add some random noise to each of those two assets (in which the random noise is uncorrelated to the assets and the other random noise), the properties of covariance would say that has no impact on the covariance of the assets, but that just doesn't seem correct.

What am I missing? Is there a more intuitive way of thinking about this?

As it relates to factor models, this essentially shows up in the fact that the asset covariance matrix only receives an addition of the error term variance along the diagonal. As an equation, this shows up in the last term of the following two-factor model:

\begin{eqnarray} \text{Cov}(r_i,r_j) & = & b_{i,1} b_{j,1} \text{Cov}(f_1,f_1) + b_{i,1} b_{j,2} \text{Cov}(f_1,f_2) + b_{i,2} b_{j,1} \text{Cov}(f_2,f_1)\\ & & + b_{i,2} b_{j,2} \text{Cov}(f_2,f_2) + \text{Cov}(e_i,e_j) \end{eqnarray}

Where $\text{Cov}(e_i,e_j) = 0$ unless $i=j$, in which case $\text{Cov}(e_i,e_i) = \text{Var}(e_i)$.