Consider two risky assets with random returns \(\tilde{r}_{1}\) and \(\tilde{r}_{2}\), with expected values \(e_{1}\) and \(e_{2}\), respectively, and variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) (with \(\sigma_{1}^{2} \leq \sigma_{2}^{2}\) ), respectively, and correlation \(ho\). For a portfolio \((w, 1-w)\), denote by \(\mathbb{E}\left[\tilde{r}_{w}\right]\) and \(\sigma^{2}\left(\tilde{r}_{w}\right)\) the expectation and the variance, respectively, of the corresponding random return \(\tilde{r}_{w}\). Verify the following claims.

(i) If \(ho<\sigma_{1} / \sigma_{2}\) then there exists \(w \in(0,1)\) such that \(\sigma\left(\tilde{r}_{w}\right)<\sigma_{1}\) and, for all \(w otin[0,1]\), it holds that \(\sigma\left(\tilde{r}_{w}\right) \geq \sigma_{1}\).

(ii) If \(ho=\sigma_{1} / \sigma_{2}\) then \(\sigma\left(\tilde{r}_{w}\right)>\sigma_{1}\) for every \(w \in \mathbb{R}\).

(iii) If \(ho>\sigma_{1} / \sigma_{2}\) then there exists \(w otin(0,1)\) such that \(\sigma\left(\tilde{r}_{w}\right)<\sigma_{1}\) and, for all \(w \in(0,1)\), it holds that \(\sigma\left(\tilde{r}_{w}\right)>\sigma_{1}\).

(iv) Show that, if \(\sigma_{1}=\sigma_{2}=\) : \(\sigma\), then the minimum variance portfolio is given by \(w^{\mathrm{MVP}}=(1 / 2,1 / 2)^{\top}\), independently of the value of the correlation coefficient \(ho\). What happens in the case \(ho=1\) ?