Assume that a risk$-$free money market account is added to the market described in Q$2.$ The continuously compounded rate of return on the money market account is $0\%$ per period.

$($i$)$ Use the method of Lagrange multipliers to determine the proportions of wealth invested in the three assets available for the minimum variance portfolio with expected return $.$ Your answer must express the proportions as a function of $.$

$($ii$)$ Recall that the market portfolio has highest Sharpe ratio. Formulate the optimisation problem which characterises the market portfolio. You don't have to solve this optimisation problem.