An investor forms a portfolio by putting 25% of his money into Google stock and 75% into Amazon stock. He expects a return on investment (ROI) of 8% from Google, and 15% from Amazon. The standard deviation of ROI is 12% for Google and 22% for Amazon. Compute the standard deviation of the portfolio's total ROI assuming that Google and Amazon returns are independent.
The professor has posted the answer to this problem:
Let $G$ be the money put into Google, and $A$ be the money put into Amazon. Then Var$(0.25G+0.75A) = 0.25^2 * 0.12^2 + 0.75^2*0.22^2 = 0.028$.
But I don't quite understand the answer. If $G$ means the money invested into Google, then what does $0.25G$ mean? I get the intuition that 25% of the variance will be based on the variance of $G$, but the formula just doesn't make sense to me.
If you invest $x_1$ in an asset with return $r_1$ and $x_2= 1-x_1$ in another asset with return $r_2$, the overall return on your portfolio is $$x_1r_1+x_2r_2$$
Likewise, $G$ is the r.v. corresponding to the ROI on Google, and $A$ is for the ROI on Amazon. These two variables do not represent the money invested, that is given by the fraction of your investment put in either stock (25% and 75% in your example).