Calculating Stock Beta from Index Weights and Covariances
Summary
A stock’s beta to an index is not generally equal to its index weight. The document derives beta from the covariance between each stock’s return and the index return, divided by the index return’s variance. Given index weights and the covariance matrix of constituent returns, the vector of stock betas is the covariance matrix multiplied by the weights, scaled by the variance of the weighted index return.
This relationship shows why weights alone do not determine beta: correlations and variances among the constituents also matter. In the special case where constituent returns are uncorrelated and have identical variance, beta is proportional to index weight, but still need not equal it. The result is an analytical identity under the stated return and covariance setup. The document does not provide empirical data, discuss changing weights or estimation error, or examine how beta behaves when an index includes assets beyond the stocks being analyzed.
Key ideas
- A stock’s beta to an index equals its covariance with the index divided by index variance.
- The index covariance structure and weights jointly determine constituent betas.
- Equal constituent variances and zero cross-correlations make beta proportional to index weight.
- Even in that special case, beta is not necessarily equal to index weight.
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# Why stock beta is not equal to its index weight?
# Why stock beta is not equal to its index weight?
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:
- 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 ?
- 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 by Chris Taylor (score 6, accepted)
https://quant.stackexchange.com/a/77448
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.