Deriving Factor-to-Portfolio Covariance from Factor Exposures

Summary

The document explains how to calculate the covariance between a factor return and a portfolio return when the portfolio is a weighted combination of factor returns. Covariance is linear in either argument, so the covariance with the portfolio can be expanded into the sum of each factor exposure multiplied by that factor’s covariance with the factor of interest. The factor’s covariance with itself is its variance; covariance with another factor can be written as the product of their volatilities and correlation.

The example applies this identity to a portfolio with value and momentum exposures, using their volatilities and correlation to compute the value factor’s covariance with portfolio returns. It clarifies that the portfolio’s correlation with value is not required for this calculation. The explanation assumes constant, nonrandom exposures and a portfolio return represented by the stated factor combination; additional residual returns or other sources of portfolio return would also contribute covariance if present.

Key ideas

• Covariance distributes over a weighted sum of random variables.
• A factor’s covariance with itself equals its variance.
• Covariance between two distinct factors equals their correlation times both volatilities.
• The portfolio’s correlation with a factor is unnecessary when factor exposures and factor covariance inputs are known.

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# Covariance and Beta: can anyone explain this calculation?


# Covariance and Beta: can anyone explain this calculation?












Let us consider a simple equity portfolio that has exposures to only two factors: 0.5 exposure to value and 0.8 exposure to momentum. Let us assume that the volatilities of the two factors are 3% for value and 5% for momentum and the correlation between them is 0.2.

I do not understand this calculation, can anyone explain this formula: $$cov(r_{value},r_p) = cov(r_{value}, X_{value}r_{value}+X_{momentum}r_{momentum}=$$ $$X_{value}\sigma_{value}^2+X_{momentum}\sigma_{momentum}\sigma_{value}\rho=0.5\cdot0.03^2+0.8\cdot0.05\cdot0.03\cdot0.2=0.00069$$

I thought I would need the correlation between factor value and the portfolio in order to to find the covariance because $cor(\mathrm{value}, \mathrm{portfolio}) = \frac{cov( \mathrm{value}, \mathrm{portfolio})}{\sigma_{\mathrm{value}}\sigma_{\mathrm{portfolio}}}$?

## Answer by skoestlmeier (score 2, accepted)

https://quant.stackexchange.com/a/42566

If $X$, $Y$, and $Z$ are real-valued random variables and $a$, $b$, $c$, $d$ are constant (i.e. non-random), then the following fact is a consequence of the definition of the covariance: $$cov\left(X, (aY+b)+(cZ+d)\right)=a\cdot cov\left(X,Y\right)+c\cdot cov\left(X,Z\right)$$

For your formula, set $b=d=0$, $X=Y=r_{value}$, $a=X_{value}$ and $c=X_{momentum}$. This straightforward leads to your stated formula, after applying the general statement that $cov(X,Y)=\rho \sigma_X \sigma_Y$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.