Consider a single$-$index model economy. The index portfolio M has E$($RM $)=6\%,\backslash$sigma M $=15\%.$

An individual asset $1$ has an estimate of $\backslash$beta $1=1\mathrm{.}5$ and $\backslash$sigma $2$

e$1=0\mathrm{.}04$ using the single index model

R$1=\backslash$alpha $1+\backslash$beta $1$RM $+$ e$1.$ The forecast of asset $1$s return is E$($r$1)=15\%.$ rf $=4\%.$ Total: $30$

marks.

$($a$)$ According to asset $1$s return forecast, calculate $\backslash$alpha $1.(5$ marks$)$

$2$

$($b$)$ Calculate the optimal weight of combining asset $1$ and the index portfolio M $.(15$ marks$)$

$($c$)$ Calculate the Sharpe ratio of the index portfolio M and the portfolio optimally combining

asset $1$ and the index portfolio M $.$