Two investments $X$ and $Y$ give returns as follows (expectation and variance): $E(X)=0.5$,$E(Y)=0.4$ and $V(X)=2$, $V(Y)=1$. The correlation between X and Y is ρ (X,Y) = 0.6.
(a)
Find the expectation and variance of the "combined" investments $U = X + Y$ and $W = 2Y$, and finally find the correlation between $U$ and $W$. Hint: First find $V (U + W)$.
\begin{align}\\ & E(U) = E(X+Y) = E(X) + E(Y) = 0,5+0,4 = 0,9 \\ & E(W) = E(2Y) = (2)E(Y) = (2)(0,4) = 0,8 \\ & \\ & V(U) = V(X+Y) = V(X)+V(Y)+2(p(X,Y))(\sigma_X)\\ & V(U) = 2 + 1 + 2(0,6)(\sqrt{2}) = 3 + 1,2(\sqrt{2}) \\ &\\ & V(W) = V(2Y) = (2^2)(V(Y)) = 2^2(1) = 4 = \sigma_W^2\\ \end{align}
Correlation: \begin{align}\\ & V(U+W) = V(U) + V(W) = 3 + 4 = 7 = \sigma_{U+W}^2 \end{align}
Definition: The correlation coefficient to $X$ and $Y$ $(Corr(X,Y)/p_{X,Y}/p)$ is defined as: $p_{X,Y}=\frac{Cov(X,Y)}{{\sigma_X}*{\sigma_Y}}$
So we have: \begin{align}\\ & \sigma_U = \sqrt{\sigma_U^2} = \sqrt{3} \\ & \sigma_W = \sqrt{\sigma_W^2} = \sqrt{4} = 2 \end{align}
and:
$Cov(U,W) = E[ (U-\mu_U)(W-\mu_W)] = E(UW) - E(U)*E(W)$
This is where I'm stuck, because I don't know how to calculate $Cov(U,W)$ without $E(UW)$, and I don't know if U and W are independent or not. (If U and W are independent, then $E(UW)=E(U)*E(W)$)
First error:
$$\mathbb{V}[X+Y]=2+1+2\cdot0.6\sqrt{2}=3+1.2\sqrt{2}$$
because $X,Y$ are not independent but correlated via $\rho=0.6$
Covariance between $(U,W)$, the following definition will help
$$\mathbb{Cov}[U,W]=\mathbb{E}[UW]-\mathbb{E}[U]\cdot\mathbb{E}[W]$$
substitute $U=X+Y$ and $W=2Y$ and find all the expectations w.r.t X,Y
so you can find
$$\mathbb{E}[UW]=\mathbb{E}[(X+Y)2Y]=2\mathbb{E}[XY]+2\mathbb{E}[Y^2]$$