How Signed Weights Affect Long-Short Portfolio Correlation

Summary

The note explains how to include long and short stock positions when calculating a portfolio correlation statistic based on pairwise asset correlations. The positions enter through signed portfolio weights, so a pair’s contribution changes sign when one weight is negative. For two assets with correlation 0.5 and equal-sized positions, the example shows that the statistic has equal magnitude but opposite signs for two same-direction positions versus a long and a short.

This answer applies to the particular formula cited in the question, which normalizes weighted pairwise correlations by a function of squared weights. It does not establish that this statistic is the right measure for every definition of portfolio correlation, nor does it discuss choices such as gross versus net exposure normalization. The result is a concise illustration of how position direction affects the formula, rather than a general treatment of portfolio risk or realized return correlation.

Key ideas

• Use signed portfolio weights when applying the stated pairwise correlation formula.
• A negative weight reverses the sign of that asset pair’s weighted correlation contribution.
• With two equally weighted assets, switching one position from long to short flips the example statistic’s sign.
• The explanation is specific to the formula under discussion.

Tags

# Portfolio correlation of a long-short portfolio


# Portfolio correlation of a long-short portfolio












I have a portfolio of long/short positions in stocks. I would like to calculate the portfolio correlation. Should I somehow account for the short position while calculating the portfolio correlation? I am using formula a) from here.

Would be grateful for help!

## Answer by lehalle (score 2, accepted)

https://quant.stackexchange.com/a/40489

If you follow the methodology on the page you point to, you will take these correlations into account because it will be in the sign of the weights $w_i$:

For instance, the first method: $$C := {2\sum_i \sum_{j> i} w_i \rho_{i,j} w_j \over 1 - \sum_i w_i^2}$$

Say you have only two stocks in the portfolio correlated at 0.5:

- Say $w_i=w_j=1/2$, $$C^+={1/4 \over 1 - 1/2}$$

- And now say $w_i=-w_j=1/2$, $$C^-=-{1/4 \over 1 - 1/2}=-C^+$$

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.