Let $S_{t}:=\left(S_{t}^{1}, S_{t}^{2}, \ldots, S_{t}^{n}\right)^{\mathrm{T}}$ denote the time- $t$ prices of $n$ risky assets in which an agent may invest, $t=0,1$. He may also invest his money in a bank account, which will return interest at rate $r>0$. At time 0 , he knows $S_{0}$ and $r$, and he knows that $S_{1} \sim N(\mu, V)$. If he chooses at time 0 to invest cash value $\theta_{i}$ in risky asset $i$, express his wealth $w_{1}$ at time 1 in terms of his initial wealth $w_{0}>0$, the choices $\theta:=\left(\theta_{1}, \ldots, \theta_{n}\right)^{\mathrm{T}}$, the value of $S_{1}$, and $r$.

Suppose that his goal is to minimize the variance of $w_{1}$ subject to the requirement that the mean $E\left(w_{1}\right)$ should be at least $m$, where $m \geqslant(1+r) w_{0}$ is given. What portfolio $\theta$ should he choose to achieve this?

Suppose instead that his goal is to minimize $E\left(w_{1}^{2}\right)$ subject to the same constraint. Show that his optimal portfolio is unchanged.