Calculating Correlation Between Arithmetic Asset Baskets
Summary
The document explains how to derive the correlation between two arithmetic baskets from the volatilities and pairwise correlations of their component assets. For two two-asset baskets, it first adds the four cross-basket covariances, expressing each covariance as correlation multiplied by the assets’ standard deviations. It then calculates each basket’s variance using its component variances and within-basket covariance, and divides the combined covariance by the product of the baskets’ standard deviations.
The example shows why pairwise correlations alone are not enough: the individual asset volatilities are also needed. Assets with larger standard deviations can dominate the basket correlation. The explanation is limited to the arithmetic case and gives no treatment of geometric baskets, weighting choices beyond equal addition, or empirical validation. Its general covariance approach extends to larger subsets, provided the required volatilities and pairwise correlations are known.
Key ideas
- Basket covariance is the sum of the covariances between every asset pair spanning the two baskets.
- Each component covariance can be calculated from its correlation and the two assets’ standard deviations.
- Basket variances depend on component variances and the correlations within each basket.
- The final basket correlation divides their covariance by the product of their standard deviations.
- Higher-volatility components can have greater influence on the resulting correlation.
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Full text
# Correlations between different baskets of assets
# Correlations between different baskets of assets
Struggling to see the answer to the following problem - Assume you have $N$ different assets, and all pair-correlation coefficients $\rho_{ij}$ between them are known. If you now form two arithmetic baskets, from said assets (any subset of the assets in them), is there a simple way to tell how those two baskets would correlate ?
Note that I put $arithmetic$ in Italics, because that means there will likely not be an analytical result. Failing good answers for the arithmetic case, is there a simple result for the geometric case ?
## Answer by Attack68 (score 1, accepted)
https://quant.stackexchange.com/a/45811
Just looking at the basic properties of RVs in terms of correlation and covariance:
Suppose 4 assets; $A,B,C,D$ with $\rho_{X,Y}$ known $\forall X,Y \in \{A,B,C,D\}$.
Let, $U=A+B$, and $V=C+D$.
Then $\rho_{U,V} = \frac{Cov(U,V)}{\sigma_U \sigma_V} $,
where $Cov(U,V) = Cov(A+B, C+D)= Cov(A,C) + Cov(A,D) + Cov(B,C) + Cov(B,D)$,
where $Cov(X, Y) = \rho_{X,Y} \sigma_X \sigma_Y$.
and $\sigma_U = \sqrt{Var(U)} = \sqrt{\sigma_A^2 + \sigma_B^2 + 2 \sigma_A \sigma_B \rho_{AB}}$
and this boils down to:
$$\rho_{U,V} = \frac{ \rho_{A,C} \sigma_A \sigma_C + \rho_{A,D} \sigma_A \sigma_D + + \rho_{B,C} \sigma_B \sigma_C + \rho_{B,D} \sigma_B \sigma_D } { \sqrt{\sigma_A^2 + \sigma_B^2 + 2 \sigma_A \sigma_B \rho_{AB}} \sqrt{\sigma_C^2 + \sigma_D^2 + 2 \sigma_C \sigma_D \rho_{CD}}} $$
I think it is somewhat intuitive that if say asset A and asset C had much greater standard deviations than assets B and D then the result for U and V should converge to the same correlation as that for just A and C. Indeed the below formula is dominated by the terms with larger volatility, so in general I would state that you cannot tell the correlation of your baskets unless you know by which assets (which asset combinations) are dominant.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.