Interpreting Implied Average Correlation in an Equity Basket

Summary

The document explains what a quoted basket correlation can mean when a trader gives a single percentage instead of a matrix of pairwise correlations. For a weighted basket, its variance depends on each component's variance and on the weighted contributions of correlations between component pairs. The answer defines implied basket correlation by removing the individual-variance terms from basket variance and normalizing by the cross-term weights and volatilities.

This quantity can be interpreted as an average pairwise correlation for the basket, so a quoted value such as 50% summarizes the combined co-movement implied by the basket's variance. It is not a report of every pair's correlation, which can differ across constituents. The interpretation depends on the basket composition, weights, component volatilities, and the convention used to aggregate correlation. The short exchange gives the variance relationship and interpretation, but does not discuss alternative weighting conventions or provide a numerical worked example.

Key ideas

• Basket variance combines weighted individual variances with covariance contributions from component pairs.
• Implied basket correlation can be derived from basket variance after accounting for individual component variance.
• A single basket correlation summarizes average pairwise co-movement rather than specifying each pairwise value.
• The interpretation depends on constituent weights and volatilities.

Tags

# Correlation basket equities


# Correlation basket equities












One question: when asking for the correlation of a basket, a trader told me 50% whereas I expected him to give me asset pairwise correlations (i.e. the correlation matrix). What does this 50% mean please ?

## Answer by Daneel Olivaw (score 7, accepted)

https://quant.stackexchange.com/a/40128

Let us consider a basket $B$ with components $S_1,\dots,S_n$ : $$B(t) = \sum_{i=1}^nw_iS_i(t)$$ At time $t$, each component has standard deviation $\sigma_i$, $i \in \{1,\dots,n\}$, and pairwise correlations are $\rho_{ij}$, $i \not= j$. Thus: $$\sigma_B^2=\sum_{i=1}^nw_i^2\sigma_i^2+2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j\rho_{ij}$$ The implied basket correlation $\rho_B$ is defined as: $$\rho_B=\frac{\sigma_B^2-\sum_{i=1}^nw_i^2\sigma_i^2}{2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j}$$ It can be interpreted as an "average" pairwise correlation between the components of the basket, and that's what the 50% stands for.

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