Consider an economy with (N) risky assets with random returns (left(tilde{r}_{1}, ldots, tilde{r}_{N} ight)) and a risk

Consider an economy with \(N\) risky assets with random returns \(\left(\tilde{r}_{1}, \ldots, \tilde{r}_{N}\right)\) and a risk free asset with return \(r_{f}\), as in section "The Case of \(N\) Risky Assets and a Risk Free Asset", and suppose that \(r_{f} eq A / C\). Show that any portfolio \(w^{*}\) belonging to the portfolio frontier \(\mathrm{PF}^{*}\) can be expressed as the linear combination of the risk free asset and the tangent portfolio \(w^{\mathrm{e}}\), so that

\[w^{*}=\alpha w^{\mathrm{e}}=\alpha \frac{e-r_{f} \mathbf{1}}{\mathbf{1}^{\top} V^{-1}\left(e-r_{f} \mathbf{1}\right)},\]

for some \(\alpha\), where the second equality follows from the proof of Proposition 3.15