The daily price change in commodity 1 is distributed N(0,0.152) and the daily price change in commodity 2 is distributed N(0,0.32)
. The two commodities are 100% correlated.
1) Does the relative value of commodity 1 vs commodity 2 change over the next year?
I would have thought no as the relative value is distributed N(0−0,0.152+0.32)
but a quick sketch of the problem suggests otherwise.
2) Is the change of value of commodity 1 the same as the change of value of 2x commodity 2? or is the change of value of 2x commodity 1 the same as the change of value of commodity 2?
My first thought here is that this is a trick question as this may not always be true in either case. This may be a trick question as the standard deviations are related by a factor of 2, but the moves may not be exactly double for all moves. However, as they are perfectly correlated maybe there is another trick. Can anyone answer this better?
Being perfectly correlated and having mean $0$ means $E(XY)^2 = E(X^2)E(Y^2)$, which by the Cauchy-Schwarz inequality means $X = aY$ for some $a \in \mathbb{R}$. The answer to your first question is therefore no.
To find what this constant $a$ is, we may look at the variances of $X$ and $Y$. $Var(X) = Var(aY) = a^2Var(Y)$, so $a = \sqrt{\frac{Var(X)}{Var(Y)}} = \frac{1}{\sqrt{2}}$. Hence the change in commodity $2$ relative to the change in commodity $1$ will always be $\sqrt{2}$.