It's trivial to calculate the betas given the index weights, and the covariance matrix of all stocks.

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.

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.

It's trivial to calculate the betas given the index weights, and the covariance matrix of all stocks.

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.

It's trivial to calculate the betas given the index weights, and the covariance matrix of all stocks.

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.