Calculating an Asset’s Correlation with a Portfolio
Summary
This explanation shows how to calculate the correlation between an individual asset and a weighted portfolio from asset volatilities, correlations, and portfolio weights. First, the pairwise correlation matrix is converted to a covariance matrix using the assets’ volatilities. Portfolio variance is then computed as the weight vector multiplied by the covariance matrix and the weight vector’s transpose.
For each asset, its covariance with the portfolio is found by taking the weighted sum of its covariances with all portfolio constituents. Dividing that value by the asset’s standard deviation and the portfolio’s standard deviation gives the correlation. The example uses four assets, equal volatilities, and a simple correlation structure to illustrate the calculation. It reports an approximate result, though the stated example and displayed rounding may account for a small discrepancy. The method applies more broadly when the covariance matrix and weights are known.
Key ideas
- Convert pairwise correlations and asset volatilities into a covariance matrix.
- Compute portfolio variance as the weighted quadratic form of the covariance matrix.
- Find an asset’s covariance with the portfolio by weighting its covariance with each constituent.
- Divide asset-portfolio covariance by the product of their standard deviations to obtain correlation.
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Full text
# Correlation of assets to portfolio of assets
# Correlation of assets to portfolio of assets
How do you calculate the correlation of an asset to a portfolio, when for all assets in the portfolio you know there: correlation to each other, volatility and weight in portfolio.
For example: Assets 1,2,3&4 all have volatility of 15%. Assets 1&2 have a correlation of 1 and all other pairs of assets correlation = 0.
With a portfolio of 16.7% in Assets 1 & 2 and 33.3% in 3 & 4, What I am reading states that all assets (1,2,3,4) have a correlation of 0.577 with the portfolio.
How is this calculated? Is there a formula that can be applied to broader examples with more varied asset volatilities and correlations?
## Answer by phdstudent (score 3, accepted)
https://quant.stackexchange.com/a/39978
This is to basic for this website, but I will answer it anyway as I think it is interesting.
You have a correlation matrix of 4 assets (1, 2, 3, 4). This is how it looks:
$ Correl = \begin{bmatrix} 1 &1& 0& 0 \\ 1 &1& 0& 0 \\ 0 &0& 1& 0 \\ 0 &0& 0& 1 \\ \end{bmatrix}$
Thus the covariance matrix is (show this yourself):
$ Cov= \begin{bmatrix} 0.0225 &0.0225& 0& 0 \\ 0.0225 &0.0225& 0& 0 \\ 0 &0& 0.0225& 0 \\ 0 &0& 0& 0.0225 \\ \end{bmatrix}$
The weights matrix is:
$w= \begin{bmatrix} 0.167 \\ 0.167 \\ 0.333 \\ 0.333 \\ \end{bmatrix}$
Therefore the standard deviation of the whole portfolio is: $Std(P) = \sqrt{w' Cov \text{ }w} = 0.0866$.
Now what is the correlation of the first asset ($A_1$) with the portfolio?
Well it is given by:
$\rho_{P,A_1} = \frac{Cov(A_1,P)}{std(A) std(P)} = \frac{Cov(A_1,P)}{std(A) std(P)} = \frac{w_1 cov(A_1, A_1) + w_2 cov(A_1, A_2) + w_3 cov(A_1, A_3) +w_4 cov(A_1, A_4)}{std(A) std(P)} = 0.57$
Then just repeat the last step for the other assets (2, 3 and 4).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.