The possible values assumed by the rates of return Y$1,$ Y$2$ of two financial securities are given by the following table $($which has three states of the world$)$: $,{}_2,{}_3$

$,-4\%,8\%$

$,20\%,40\%$

Probability $,\frac{1}{4},\frac{1}{2}$

$1.$ Compute the means $m_1,m_2,$ the variances ${}_1^2,{}_2^2$ and the covariance

Cov$(y_1,y_2$ of the two rates of return $y_{1,}{y}_2$

$2.$ Determine the ecient frontier of the Markowitzs model made of these two securities$.$

$3.$ 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 of the optimal portfolio which maximizes the investors expected utility. Compute the optimal expected utility.

$4.$ Determine the values assumed by the optimal rate of return in the three states of the world and the certain rate of return equivalent to the optimal random return.