Why stock beta is not equal to its index weight?

Question

Index is a linear combination of stock prices with known weights. In case index is equally weighted, the weights are fixed. Beta measures stock sensitivity to index - by how much stock moves when index moves by 1%. So we regress one component of that linear combination onto the linear combination itself. Shouldn`t the corresponding regression coefficient be equal to the stock weight in the index ?

My own explanation why this might not be the case:

1. When regressing single stock on index we do not take into account for confounding effects - correlation of this single stock with other stocks. Hence, if we orthogonalize a single stock returns to all other stock returns and then run regression of residuals on index returns than beta should be equal to index weight ?
2. Index is weighted based on prices while beta is calculated on returns. So if we create a returns based index with equal weights and perform orthogonalization as in 1) then beta should be equal to index weight then ?

Answer 1

It's trivial to calculate the betas given the index weights $w$ and the covariance matrix of all stocks $\Sigma$:

The index return is

$$ r_{\rm index} = w^T r $$

The beta of stocks to the index is

$$ \beta = \frac{{\rm cov}(r, r_{\rm index})}{{\rm var}(r_{\rm index})} $$

with

$$ {\rm cov}(r, r_{\rm index}) = {\rm cov}(r, w^T r) = \Sigma w $$

and

$$ {\rm var}(r_{\rm index}) = {\rm cov}(w^T r, w^T r) = w^T \Sigma w $$

so you have

$$ \beta = \frac{\Sigma w}{w^T \Sigma w} $$

So if the covariance matrix is a scalar multiple of the identity (i.e. all stocks uncorrelated and with the same variance) then the beta will be proportional (not equal) to the index weight. But otherwise it will be different.