In the Markowitzs model the ecient frontier with only risky assets is given by the following equation

$=\sqrt[\phantom{2}]{10m^2-2m+0\mathrm{.}2}$

$1.$Compute the mean and the standard deviation of the portfolio with minimum variance.

$2.$ Assume there is a risk$-$free asset in the market with risk$-$free rate $r_f=2\%.$ Determine the new frontier of efficient portfolios adding the possibility to invest in the risk$-$free asset. Compute the mean and standard deviation of the tangency porfolio between the two frontiers.

$3.$ Consider an investor whose preferences are represented by the quadratic utility function u$($x$)=$ x$-0\mathrm{.}5x^2.$ Determine the mean, the variance and the composition of the optimal portfolio which maximizes the investors expected utility.