Consider a financial market with n risky assets whose rates of returns have mean vector $\backslash$mu and

covariance matrix C$.$ Given that a$=1^($T$)$C$^(-1)1,$b$=1^($T$)$C$^(-1)\backslash$mu and c$=\backslash$mu $^($T$)$C$^(-1)\backslash$mu $,$ let $\backslash$mu $\_($x$)$ and $\backslash$sigma $\_($x$)$

respectively denote the mean and standard deviation of the rate of return of a portfolio labeled x$.$

For a given portfolio p$,$ let q denote the efficient portfolio such that $\backslash$sigma $\_($q$)=\backslash$sigma $\_($p$),$r denote the

minimum$-$variance portfolio such that $\backslash$mu $\_($r$)=\backslash$mu $\_($p$)$ and let g denote the global minimum$-$variance

portfolio. Define the relative efficiency of portfolio p by $\backslash$psi $\_($p$)=(\backslash$mu $\_($p$)-\backslash$mu $\_($g$))/(\backslash$mu $\_($q$)-\backslash$mu $\_($g$)).$ Prove that:

$($i$)\backslash$sigma $\_($r$)^(2)=($a$)/($ac$-$b$^(2))(\backslash$mu $\_($r$)-\backslash$mu $\_($g$))^(2)+\backslash$sigma $\_($g$)^(2),[$Hint: Start from the equation of the minimum variance

frontier$]$;

$($ii$)$ Hence show that, $\backslash$psi $\_($p$)^(2)=(\backslash$sigma $\_($r$)^(2)-\backslash$sigma $\_($g$)^(2))/(\backslash$sigma $\_($p$)^(2)-\backslash$sigma $\_($g$)^(2))$