In a small stock market, there are three assets. The corresponding covariance matrix

and the expected rates of return are respectively

$=\left[\begin{array}{ccc}3& 1& 0\\ 1& 2& 1\\ 0& 1& 1\end{array}\right]\frac{,}{b}ar(r)=\left[\begin{array}{c}0\mathrm{.}5\\ 0\mathrm{.}3\\ 0\mathrm{.}2\end{array}\right].$

$($i$)$ Find the optimal portfolio with an expected rate of return $\frac{?}{\text{b}\text{a}\text{r}}(r)=\frac{23}{60}$ by using two fund

theorem.

$($ii$)$ Assume now there is a risk$-$free asset with rate of return $r_f=0\mathrm{.}1.$ Find the optimal

portfolio with an expected rate of return $\frac{?}{\text{b}\text{a}\text{r}}(r)=\frac{31}{50}$ by using one fund theorem.