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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 ?
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.
