Snethemba has identified a risky portfolio which has an expected risk premium of $9\mathrm{.}5\%$ and a standard deviation of $16\%.$ The risk$-$free rate is $1\%.$ Snethemba has a level of risk aversion of $5.$

$($a$)$ How much should Snethemba invest in the risky portfolio in her optimal complete

portfolio? $(3$ marks$)$

$($b$)$ What is the reward$-$to$-$risk ratio of Snethemba$$s optimal complete portfolio? $(4$ marks$)$

$($c$)$ If Snethemba decides to construct a complete portfolio with an expected return of

$12\mathrm{.}4\%,$ how much leverage has she used? $(3$ marks$)$

$($d$)$ What is the reward$-$to$-$risk ratio of the complete portfolio formed in $($c$)?$ How does this compare to the reward$-$to$-$risk ratio of the optimal complete portfolio? $(3$ marks$)$

$($e$)$ What levels of utility does Snethemba gain from the optimal complete portfolio and the complete portfolio with an expected return of $12\mathrm{.}4\%?$ Explain your answer.