There are three horses in the race Redeeming Vices, Gwendolyn, and Ernest. The Odds $($net returns for a winning unit bet$)$ on offer for these runners are $1,2,$ and $4,$ respectively, and the respective 'true' probabilities are $0\mathrm{.}6,0\mathrm{.}3,$ and $0\mathrm{.}1.$

$1.$ Find the optimum allocation for the $3-$asset portfolio of bets, from the Optimal growth perspective

$2.$ Show that the expected returns for the $3$ runners are $($respectively$)0\mathrm{.}2,-0\mathrm{.}1,$ and $-0\mathrm{.}5.$

$3.$ Show that the variances are $($respectively$)0\mathrm{.}96,1\mathrm{.}89,2\mathrm{.}25$ and the covariances are $(1$ with $2)-1\mathrm{.}08,(1$ with $3)-0\mathrm{.}6,$ and $(2$ with $3)-0\mathrm{.}45.$

$4.$ Find the Minimum Variance Portfolio for the $3-$asset portfolio of bets.

$5.$ For each of the possible values of Expected Return $-0\mathrm{.}5,-0\mathrm{.}4,-0\mathrm{.}3,-0\mathrm{.}2,-0\mathrm{.}1,0,0\mathrm{.}1,0\mathrm{.}2$ find the portfolio of minimum variance and plot the corresponding Minimum Variance Set in the Mean$-$SD plane.