Consider a strictly increasing and strictly concave utility function \(u\) and suppose that there exist a risk free asset with return \(r_{f}>0\) and \(N\) risky assets whose returns \(\left(\tilde{r}_{1}, \ldots, \tilde{r}_{N}\right)\) satisfy the condition

\[\mathbb{E}\left[\tilde{r}_{n} \mid \tilde{r}_{1}, \ldots, \tilde{r}_{n-1}, \tilde{r}_{n+1}, \ldots, \tilde{r}_{N}\right]=\mathbb{E}\left[\tilde{r}_{n}\right]\]

for all \(n=1, \ldots, N\). Show that \(w_{n}^{*}>0\) if and only if \(\mathbb{E}\left[\tilde{r}_{n}\right]>r_{f}\), for all \(n=\) \(1, \ldots, N\).