Given a frontier portfolio (w^{*}), show that the expected return (mathbb{E}left[tilde{r}_{w^{mathrm{zc}}} ight]) of its zero correlation portfolio

Given a frontier portfolio \(w^{*}\), show that the expected return \(\mathbb{E}\left[\tilde{r}_{w^{\mathrm{zc}}}\right]\) of its zero correlation portfolio \(w^{\mathrm{zc}}\) is identified, in the variance-expected return plane, by the intersection of the line connecting the points \(\left(\sigma^{2}\left(\tilde{r}_{w^{*}}\right), \mathbb{E}\left[\tilde{r}_{w^{*}}\right]\right)\) and \(\left(\sigma^{2}\left(\tilde{r}_{w^{\mathrm{MVP}}}\right), \mathbb{E}\left[\tilde{r}_{w^{\mathrm{MVP}}}\right]\right)\) with the vertical axis. Similarly, show that \(\mathbb{E}\left[\tilde{r}_{w^{\mathrm{zc}}}\right]\) is identified, in the standard deviation-expected return plane, by the intersection of the tangent to the portfolio frontier at the point \(\left(\sigma^{2}\left(\tilde{r}_{w^{*}}\right), \mathbb{E}\left[\tilde{r}_{w^{*}}\right]\right)\) with the vertical axis.