Calculating Correlation Between Two Portfolios

Summary

This discussion gives two ways to calculate the correlation between two portfolios, including when one portfolio contains a subset of the assets in the other. With asset weights and the covariance matrix, portfolio covariance is obtained by applying each portfolio’s weights to the cross-covariances among the assets. Divide this covariance by the product of the portfolios’ standard deviations to get their correlation; each portfolio variance is calculated from its weights and the covariance matrix.

The response also offers an empirical alternative: form time-aligned return series for both portfolios and calculate their sample correlation. The matrix approach is appropriate when asset covariances and portfolio weights are available, while the return-series approach works from observed portfolio returns. The post provides formulas but no worked numerical example, and it does not discuss estimation uncertainty or changing weights over time.

Key ideas

• Portfolio correlation requires the covariance between the portfolios and their individual variances.
• Use the two weight vectors with the asset covariance matrix to calculate portfolio covariance and variance.
• A nested or subset relationship between portfolios does not change the general formula.
• Correlation can also be estimated from aligned historical return series for both portfolios.

Tags

# Correlation Between 2 Portfolios


# Correlation Between 2 Portfolios












I have a set of assets, n. I'm trying to find the correlation between 2 portfolios, say x and y, where x is nested in, or, a sub-set of y. That is, x is a portfolio based on a sub-set of n, while y is a portfolio based on the entire set of assets, n. The weights in each portfolio sum to 1, and I'm using Excel. Thanks for your time!

## Answer by jaamor (score 1)

https://quant.stackexchange.com/a/19011

You will need the covariance matrix to calculate this.

Say you have a collection of $n$ assets. The value of asset $i$ is represented by the random variable $X_i$ and the corresponding portfolio weight is are $w_i$, and $v_i$ for the two portfolios.

The correlation between the two portfolios is: $$ \frac{\sigma(w^TX,v^TX)}{\sqrt{(w^T\Sigma w)(v^T\Sigma v) }} = \frac{w^T\Sigma(X)v}{\sqrt{(w^T\Sigma w)(v^T\Sigma v) }}$$

Where $\Sigma$ is the covariance matrix.

You can arrive to this conclusion by using variance's bilinear property.

And without using vector notation:

$$ \rho = \frac{\Sigma ^n _i \Sigma ^n _k w_iv_k \sigma(X_i,X_k)}{\sqrt{Var( P_1 )Var(P_2) }} $$

Where $Var( P_1 )$ the variance of portfolio 1.

Also take a look at this related question at math stack exchange.

## Answer by Chris (score 0)

https://quant.stackexchange.com/a/19013

If you're in Excel, get the returns of both portfolios into 2 columns, matched up by time. The "correl()" function will get you the correlation coefficient.

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