Consider an economy with a risk free asset with return \(r_{f}=1.13\) and two risky assets with normally distributed returns with expected values \(\mathbb{E}\left[\tilde{r}_{1}\right]=\) \(1.16, \mathbb{E}\left[\tilde{r}_{2}\right]=1.25\) and variances \(\sigma^{2}\left(\tilde{r}_{1}\right)=2, \sigma^{2}\left(\tilde{r}_{2}\right)=4\) and correlation \(ho\).

(i) Determine the portfolio frontier composed by the risky assets in the two cases \(ho=0.5\) and \(ho=-0.5\).

(ii) Determine the minimum variance portfolio \(w^{\mathrm{MVP}}\) in the two cases \(ho=0.5\) and \(ho=-0.5\).

(iii) Determine the tangent portfolio in the two cases \(ho=0.5\) and \(ho=-0.5\).

(iv) For \(ho=-0.5\), consider an agent with quadratic utility function \(u(x)=x-\frac{b}{2} x^{2}\) and determine his optimal portfolio. Give conditions on \(b\) in order that the optimal investment in the tangent portfolio \(w^{\mathrm{e}}\) is greater than one.