The rates of return of two financial securities have the following vector of expected returns and matrix V of variances$-$covariances:

$=\frac{1}{100}\left[\begin{array}{c}3\\ 6\end{array}\right],$ V$=\frac{1}{100}\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$

$1.$ Compute the ecient frontier of the Markowitzs model.

$2.$ Suppose that short sales are not allowed. Say if the ecient frontier exibits risk reduction and why this happens or does not happen.

$3.$ Assume there is a risk$-$free asset in the market with risk$-$free rate $r_f=2\%.$ Determine the new frontier of ecient portfolios adding the possibility to invest in the risk$-$free asset.

$4.$ Compute the mean, the variance and the composition of the tangency porfolio between the two frontiers.