# 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 ?

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.

• Thanks a lot for this ! I would guess that is only precisely accurate for geometric baskets, correct ?
– ZRH
May 27 '19 at 12:45