Hey guys I'm pretty new here, not sure how to code my question so I'll include a picture reference instead. I'm a bit confused on how the standard deviation of F (commodity price) would affect the already existing matrix. Is each vector's standard deviation a combination of both risk and F? How would the covariance matrix of the returns look like? 
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$\begingroup$ Why are people downvoting this question? This person has literally just said they’re new here, downvoting and not saying why doesn’t help them. $\endgroup$– Hamish GibsonMar 19, 2021 at 23:22
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Forget the matrix ;-) It makes it look like the regression problem from hell! Given each term in the matrix is just a function of E123 (which are all random and independent of each other, and independent of Fhat), each term can be (tediously) calculated on its own.
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$\begingroup$ I see. In that case since Fhat and e1,2,3 are both factors in the vector that can cause variance within the return, should I just add them (i.e 0.01+0.09=0.1 for r1) to get the variance of each return? What would the return matrix covariances look like? $\endgroup$ Mar 19, 2021 at 22:07
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$\begingroup$ Since this is clearly an exam or interview situation, I'll leave it to you to revsisit how to calculate covariances... which, I guess, was the original point of the question... sorry ;-) $\endgroup$– demullyMar 19, 2021 at 22:14
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$\begingroup$ the covariance of $F_3$ and $F_2$ is $\frac{1}{2} \times \sigma^2_f$. Other than that, the other covariances are only due to the covariance matrix of the $\epsilon$ 's. $\endgroup$ Mar 20, 2021 at 1:22