Consider an economy where all the agents choose to hold meanvariance efficient portfolios but it is not possible to borrow at the risk free rate \(r_{f}\) (i.e., only investing in the risk free asset is allowed). By relying on the same arguments adopted in Sect. 5.2 and assuming that the risk free asset is in strictly positive net supply, show that the following Zero- \(\beta\) CAPM relation is obtained:

\[\begin{equation*}\mathbb{E}\left[\tilde{r}_{n}\right]=\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right]+\beta_{n m}\left(\mathbb{E}\left[\tilde{r}^{m}\right]-\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right]\right), \quad \text { for all } n=1, \ldots, N \tag{5.38}\end{equation*}\]

with \(\mathbb{E}\left[\tilde{r}^{m}\right]-\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right]>0\) and \(\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right] \leq r_{f}\).