Consider an economy with three traded assets, whose returns are generated by the following linear model with respect to two risk factors \(\tilde{f}_{1}\) and \(\tilde{f}_{2}\) (with \(\mathbb{E}\left[\tilde{f}_{1}\right]=\mathbb{E}\left[\tilde{f}_{2}\right]=0\) ):

\[\left\{\begin{array}{l}\tilde{r}_{1}=0.1+0.3 \tilde{f}_{1}-0.2 \tilde{f}_{2}, \\\tilde{r}_{2}=0.5-0.4 \tilde{f}_{1}+0.3 \tilde{f}_{2}, \\\tilde{r}_{3}=0.2-0.2 \tilde{f}_{1}+0.4 \tilde{f}_{2} .\end{array}\right.\]

(i) Determine the portfolios of the three traded assets which have unitary exposure to one risk factor and zero exposure to the other risk factor.

(ii) Determine the risk free rate implicit in the economy.

(iii) Consider an additional asset with return \(\tilde{r}_{4}=\alpha+0.1 \tilde{f}_{1}+0.3 \tilde{f}_{2}\). Determine the constant \(\alpha\) such that there is no arbitrage opportunity in the economy extended with this fourth asset.

(iv) Verify that the APT is satisfied in exact form.

(v) Verify that an asset with return \(\tilde{r}_{5}=0.2-0.1 \tilde{f}_{1}+0.1 \tilde{f}_{2}\) would generate an arbitrage opportunity.

(vi) Assuming that the first three assets are available for trading at unitary price, determine the value of a consumption plan equal to \(0.4-0.2 \tilde{f}_{1}+0.4 \tilde{f}_{2}\).