# 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.

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