Allocating Portfolio Correlation Contributions by Asset Beta
Summary
The note derives one possible way to attribute an index portfolio’s relationship with itself across its constituents. Starting with a weighted sum of asset returns, it expands the covariance of the portfolio with itself into weighted constituent covariances. Dividing by the portfolio variance produces contributions that sum to one, or 100 percent, by construction.
Each contribution can be expressed as the asset weight times the asset’s covariance with the portfolio divided by portfolio variance. Equivalently, it is the weight multiplied by the asset’s volatility relative to the portfolio’s volatility and its correlation with the portfolio, or simply the weight times the asset’s beta to the portfolio. This decomposition is useful for interpreting constituent roles, but the note addresses correlation of a portfolio with itself; it does not directly derive contributions to correlation between a distinct portfolio and an external index. The attribution convention should therefore be matched to the question being analyzed.
Key ideas
- Portfolio variance can be written as the sum of weighted constituent covariances with the portfolio.
- Dividing each weighted covariance by portfolio variance gives contributions that sum to one.
- A constituent’s contribution equals its portfolio weight multiplied by its beta to the portfolio.
- The derivation concerns portfolio self-correlation and may not directly answer attribution against a separate index.
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Full text
# How to calculate the contribution (%) of an asset to the global correlation of the portfolio?
# How to calculate the contribution (%) of an asset to the global correlation of the portfolio?
I have a portfolio X with weights $w_i$. I am trying to find the contribution $\xi_i$ of asset $i$ to the total correlation $\rho_{XM}$ of the portfolio X to an index M. I can't find these contributions when I develop the correlation portfolio/index formula.
Obviously, the sum of all contributions must be equal to 100%, but the sum over all $i$ of the contributions times the asset/index correlation $\rho_{iM}$ , must be equal to the portfolio/index correlation $\rho_{XM}$.
Any ideas ?
## Answer by Chris Taylor (score 2)
https://quant.stackexchange.com/a/36128
Say the index is
$$ X = \sum_{i=1}^n w_iX_i $$
and the variance is ${\rm Var}(X) = \sigma^2$ then obviously the covariance of the index with itself is $\sigma^2$ so
$$ {\rm Cov}\left(\sum_{i=1}^n w_i X_i, X\right) = \sum_{i=1}^n w_i{\rm Cov}(X_i,X) = \sigma^2 $$
The correlation is the covariance divided by the square root of the produce of the variances, so dividing everything through by $\sigma^2$ you have
$$ \sum_{i=1}^n w_i \frac{{\rm Cov}(X_i,X)}{\sigma^2} = 1 $$
You might choose to consider the components of this sum the "contribution to the correlation" from each member of the index, i.e.
$$ \xi_i = \frac{w_i}{\sigma^2}{\rm Cov}(X_i,X) $$
Noting that the covariance can be expressed in terms of the correlation, you also have
$$ \xi_i = \frac{w_i\sigma_i}{\sigma} {\rm Corr}(X_i, X) $$
and finally, you could note that this is $w_i$ times the coefficient of regression of $X_i$ on $X$, commonly known as the beta, so you have
$$ \xi_i = w_i\beta_i $$
It then becomes obvious that the $\xi_i$ must sum to one, since the average beta of the constituents to an index must be one.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.