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Calculating an Asset’s Correlation with a Two-Asset Portfolio

Article Quant Q&A · Author: Parallax
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Summary

The document derives the correlation between one asset and a portfolio formed from two other assets. It defines the portfolio as a weighted combination of the two holdings, then applies covariance linearity to express the portfolio’s covariance with the target asset as a weighted sum of pairwise covariances. The denominator uses the target asset’s volatility and the portfolio’s volatility, calculated from both component variances and their covariance.

This gives a direct way to use entries in a variance-covariance matrix and a chosen portfolio weight to compute the desired correlation. The example describes high-yield bonds compared with a mix of equities and U.S. Treasuries, but provides no observed data or numeric result. The formula assumes the portfolio weights sum to one and that the covariance and volatility inputs are measured consistently; it is a calculation method, not a claim that the resulting relationship is stable over time.

Key ideas

  • Represent the two-asset portfolio as a weighted sum of its component assets.
  • Compute its covariance with the target asset using the weighted pairwise covariances.
  • Calculate portfolio variance using component variances and their covariance.
  • Divide the target-portfolio covariance by the product of their standard deviations to obtain correlation.
  • The method requires consistent volatility and covariance estimates and a specified portfolio weight.

Tags

Portfolio constructionStatisticsMulti-asset
Full text
# Correlation of asset X with a portfolio of asset Y and Z


# Correlation of asset X with a portfolio of asset Y and Z












I have three assets and a covariance matrix.

How do I calculate the correlation of asset X with a portfolio that includes assets Y and Z?

For example, assume I want to calculate the correlation of high yield bonds to a portfolio that includes 60% equities and 40% U.S. Treasuries.

Thanks.

## Answer by Daneel Olivaw (score 2, accepted)

https://quant.stackexchange.com/a/34628

This is fairly basic but anyway...

Let us define:

- $X$, $Y$ and $Z$ your assets $-$ without any loss of generality, assume $X$ designates high yield bonds, $Y$ equities and $Z$ US Treasuries $-$ and $P$ your portfolio;

- $\sigma_X$, $\sigma_Y$ and $\sigma_Z$ their respective standard deviation, and $\sigma_P$ the standard deviation of the portfolio;

- $\sigma_{X,Y}$, $\sigma_{X,Z}$ and $\sigma_{Y,Z}$ their pairwise covariances;

- $w$ the proportion of equities in your portfolio $-$ so $60\%$ in your example.

Your portfolio $P$ can be written as:

$$ P = wY+(1-w)Z $$

You are interested in computing the correlation $\mathbb{Corr}(X,P)$ between your portfolio and high yield bonds. By the properties of variance and covariance, we obtain:

$$ \begin{align} \mathbb{Corr}(X,P) & = \frac{\mathbb{Cov}(X,wY+(1-w)Z)}{\sigma_X\sigma_P} \\[9pt] & = \frac{w\,\sigma_{X,Y}+(1-w)\sigma_{X,Z}}{\sigma_X\sqrt{(w^2\sigma_Y^2+(1-w)^2\sigma_Z^2+2w(1-w)\sigma_{Y,Z})}} \end{align} $$

Your are left with $\sigma_X$, $\sigma_Y$, $\sigma_Z$, $\sigma_{X,Y}$, $\sigma_{X,Z}$ and $\sigma_{Y,Z}$, which are the elements of your variance-covariance matrix, and $w$ which you set yourself.

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.