There are a few online resources available to help with log-linearization (e.g., here or here). However, log-linearization where an expectation is involved is a little tricky because the log can't simply "pass through" the expectation operator. Could somebody help with the algebra in this example?
I have the Euler equation (equation 1) $$ 1 = E_t \left [ \left \{ \delta \left (\frac{C_{t+1}}{C_t} \right )^{-1/\psi} \right \}^\theta \left \{ \frac{1}{1 + R_{m,t+1}} \right \}^{1 - \theta} 1 + R_{i, t+1} \right ] $$ where $\theta = ( 1 -\gamma)/(1 - 1/\psi)$. I'm trying to derive an expression for the risk-free rate and an expression for the equity premium. How should I go about doing this?
It seems from the second link above that I should start by replacing the variables of interest like so $C_t = c e^{\tilde C_t}$. Then following the steps given, it seems like I should arrive at (equation 2)
\begin{align} 1 = E_t \left [ \left \{ \delta \left (\frac{\tilde C_{t+1} + 1}{\tilde C_t + 1} \right )^{-1/\psi} \right \}^\theta {} \left \{ \frac{1}{(1 + R_m)[\widetilde{(1 + R_{m,t+1})} + 1]} \right \}^{1 - \theta} \cdot \\ \cdot [(1 + R_i)[\widetilde{(1 + R_{i, t+1})} + 1]] \right ]. \end{align}
But where do I go from here?
EDIT:
I have copied equation 1 directly from the notes that I have. It is probably the case that the term on the right, $1 + R_{i,t+1}$, should be in parentheses, $(1 + R_{i,t+1})$. In my initial attempt at log-linearization I have treated it this way.
In equation 2, I have followed the steps in the instruction that can be found in the second link at the beginning. So, $R_i$ and $R_m$ without time subscripts are these values in the steady state.
$R_m$ is the return on the market portfolio and $R_i$ is the return on asset $i$.
EDIT 2:
Thanks for the useful comments. So, from what I have gathered so far, I should get something like this:
\begin{align} 1 &= E_t \left [ \delta^\theta (1 - \frac \theta \psi (\tilde C_{t+1} - \tilde C_t ) (1 + R_m)^{\theta - 1} (\theta - 1) \left ( 1 + \tilde R_{m,t} \frac{R_m}{1 + R_m} \right ) \right .\\ & \left . \, \cdot (1 + R_i) \left ( (1 + \tilde R_{i,t} \frac{ R_i}{1 + R_i} \right ) \right ] \end{align}
Then this would imply that the risk-free rate is found as follows:
\begin{align} 1 &= E_t \left [ \delta^\theta (1 - \frac \theta \psi (\tilde C_{t+1} - \tilde C_t ) (1 + R_m)^{\theta - 1} (\theta - 1) \left ( 1 + \tilde R_{m,t} \frac{R_m}{1 + R_m} \right ) (1 + R_f) \right ] \\ 1 &= E_t[ m_{t+1} (1 + R_f)] \\ \frac{1}{E_t[m_{t+1}]} &= 1 + R_f. \end{align}
Is this correct? And now, to finish the question, how would I find the equity premium?