# 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 $$cor(\mathrm{value}, \mathrm{portfolio}) = \frac{cov( \mathrm{value}, \mathrm{portfolio})}{\sigma_{\mathrm{value}}\sigma_{\mathrm{portfolio}}}$$?

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 $$cov(X,Y)=\rho \sigma_X \sigma_Y$$.
• This just follows from the definition of the covariance: $cov(aX,Y)=a \cdot cov(X,Y)$. – skoestlmeier Nov 9 '18 at 17:09