$($d$)$ Assume there are $N$ risky assets with an expected return vector $$ and covariance matrix $.$ The risk$-$free asset has return $r_f,$ and let $l$ denote a column vector of $N$ ones. Assume the investor can borrow or lend at the risk$-$free rate so we assume the investor invests $w$ in risky assets and $1-l^{\text{'}}w$ in the risk$-$free asset.

i$)$ Write an expression for the expected return on the investor's portfolio.

$[2]$

ii$)$ Write an expression for the Lagrangian that the investor will optimise to find the tangent portfolio $($one of the two usual constraints is no longer needed in the case of the tangent portfolio$).$

$2$

iii$)$ Show the weights on the tangent portfolio are given by the following formula

$w=(\frac{{}_P-r_f}{H}){}^{-1}$

where $H=(-l{r}_f{)}^{\text{'}}{}^{-1}(-l{r}_f).$

$5$