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

$=\frac{1}{100}\left[\begin{array}{c}5\\ 20\end{array}\right],$ V$=\frac{1}{100}\left[\begin{array}{cc}2& 1\\ 1& 8\end{array}\right]$

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

$2.$ 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 and the composition of the tangency porfolio between the two frontiers.

$3.$ Consider an investor whose preferences on the rates of return are represented by the quadratic utility function u$(x)=x-x^2.$ Determine the mean, the variance and the composition in the three dierent assets of the optimal portfolio which maximizes the investors expected utility.