In an economy with \(N+1\) traded assets, suppose that there are no arbitrage opportunities and let \(\tilde{\ell}\) be the likelihood ratio of any risk neutral probability measure. Call the quantity \(\tilde{m}:=\tilde{\ell} / r_{f}\) stochastic discount factor (see Sect. 4.4). Show that

\[\mathbb{E}\left[\log \left(\tilde{r}_{n}\right)\right] \leq-\mathbb{E}[\log (\tilde{m})], \quad \text { for all } n=1, \ldots, N\]