Correlations between different baskets of assets

Question

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 1

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.