Consider an economy with a risk free lending rate \(r_{l}\) lower than the corresponding risk free borrowing rate \(r_{b}\) (i.e., \(r_{b}>r_{l}\) ), reflecting the presence of transaction costs in the risk free market. Suppose that all the agents choose to hold mean-variance efficient portfolios. By relying on the same arguments adopted in Sect. 5.2 and assuming that the net supply of the risk free asset is zero, show that the following Zero \(-\beta\) CAPM relation is obtained:

\[\begin{equation*}\mathbb{E}\left[\tilde{r}_{n}\right]=\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right]+\beta_{n m}\left(\mathbb{E}\left[\tilde{r}^{m}\right]-\mathbb{E}\left[\tilde{r}^{\mathrm{zc}(m)}\right]\right), \quad \text { for all } n=1, \ldots, N \tag{5.37}\end{equation*}\]

with \(\mathbb{E}\left[\tilde{r}^{m}\right]-\mathbb{E}\left[\tilde{r}^{z \mathrm{cc}(m)}\right]>0\) and \(r_{b} \geq \mathbb{E}\left[\tilde{r}^{\mathrm{rc}(m)}\right] \geq r_{l}\).