Calculating an Asset’s Correlation and Beta to a Portfolio
Summary
The document derives the correlation between an asset and a weighted portfolio from constituent weights, volatilities, and pairwise correlations. The covariance between the asset and portfolio is the weighted sum of its covariances with each constituent. Portfolio variance is obtained by summing all weighted pairwise covariance terms, and dividing the covariance by the product of the asset and portfolio standard deviations gives the desired correlation.
It also connects this result to beta: beta is the asset’s correlation with the portfolio multiplied by the ratio of the asset’s volatility to portfolio volatility. The supplied example includes the asset within the portfolio, so its own variance contributes to covariance as well as portfolio variance. The formulas assume compatible return definitions and a valid covariance matrix; they provide a direct calculation from summary statistics without requiring historical price series, but do not address estimation uncertainty or changing correlations.
Key ideas
- Asset-portfolio covariance is the weighted sum of covariances between the asset and each holding.
- Portfolio variance uses all weighted pairwise constituent covariances.
- Divide covariance by the product of asset and portfolio volatility to obtain correlation.
- Beta equals correlation multiplied by the ratio of asset volatility to portfolio volatility.
- When the asset is held in the portfolio, its own variance contributes to the calculation.
Tags
Full text
# Correlation between asset A and Portfolio X (which contains A)
# Correlation between asset A and Portfolio X (which contains A)
After a few hours trying to solve this I give up! I need help.
I need to calculate the BETA of an asset with respect to a portfolio that contains this asset. I have the volatility and correlations for all the portfolio, and the allocation.
I could use a formula to calculate the correlation between asset A and the portfolio. Because I could get Beta with the usual formula.
I need a formula, not a method to calculate this based on historical prices.
So this is what I have:
Correlation:
```
A B C
A 1 0.85 0.78
B 0.85 1 0.84
C 0.78 0.84 1
```
S.D.
```
A B C
19.74% 25.76% 31.19%
```
Allocation:
```
A B C
25.00% 25.00% 50.00%
```
Hope you can help me!!
## Answer by Gordon (score 2)
https://quant.stackexchange.com/a/26212
You only need to note the following \begin{align*} corr\left(X_1, \sum_{i=1}^nw_i X_i\right) &= \frac{cov\big(X_1, \, \sum_{i=1}^nw_i X_i \big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ &= \frac{E\Big(\big(X_1-E(X_1)\big)\big(\sum_{i=1}^n w_i X_i - E(\sum_{i=1}^n w_i X_i) \big)\Big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ &= \frac{\sum_{i+1}^n w_i E\big((X_1-E(X_1))(X_i - E( X_i) )\big)}{\sqrt{var(X_1)} \sqrt{var(\sum_{i=1}^n w_i X_i)}}\\ &= \frac{\sum_{i=1}^n w_i\rho_{1, i} \sigma_1 \sigma_i}{\sigma_1 \sqrt{\sum_{i, j=1}^n w_iw_j\rho_{i, j} \sigma_i \sigma_j}}\\ &= \frac{\sum_{i=1}^n w_i\rho_{1, i} \, \sigma_i}{\sqrt{\sum_{i, j=1}^n w_iw_j\rho_{i, j} \sigma_i \sigma_j}}. \end{align*}
## Answer by Nicolas Galarza Ricci (score 1)
https://quant.stackexchange.com/a/26244
Thank you gordon! So in addition to the solution you posted, here´s what I actually used in the script where I needed this formula. In the case anyone else can use it:
Cov(X,A) = Cov(0.25A+0.25B+0.5C,A) = 0.25Var(A) + 0.25Cov(B,A) + 0.5 Cov(C,A)
Corr(X,A) = Cov(X,A) / sqrt( Var(X)*Var(A) )
beta[A] = ( vol[A] / vol[X] ) * Corr[X,A]
have a Good day!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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