The example on page $23-25$ in Lecture Notes $4$ illustrated the calculation of EU

portfolio under quadratic utility $u(x)=x-\frac{}{2}{x}^2$ and $x_0=1.$ In this question,

we check whether it leads to a mean$-$variance portfolio using concrete numbers.

Let us take $=0\mathrm{.}6$ and the covariance matrix and expected return vector as

$=\left[\begin{array}{ccc}{}_1^2& 0& 0\\ 0& {}_2^2& 0\\ 0& 0& {}_3^2\end{array}\right]=\left[\begin{array}{ccc}0\mathrm{.}04& 0& 0\\ 0& 0\mathrm{.}16& 0\\ 0& 0& 0\mathrm{.}25\end{array}\right],=([{}_1],[{}_2],[{}_3])=([0\mathrm{.}1],[0\mathrm{.}15],[0\mathrm{.}12]).$

$($a$)$ What is the EU optimal portfolio $w^{**}?$

$($b$)$ What is the expected return of EU portfolio $z={}^{TT}{w}^{**}?$

$($c$)$ What is the mean$-$variance optimal portfolio with the required return equal

to $z$ in $($b$)?$