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I am studying consumption CAPM and trying to prove, assuming lognormal consumption growth one can show that the continuously compounded risk-free rate is: enter image description here

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This is a one-line calculation under standard CRRA-lognormal assumptions (see, for example, also this question).

By CRRA and lognormal consumption growth assumptions, the pricing kernel is $$ \frac{\rho u'(c_{t+1})}{u'(c_t)} = \rho (\frac{c_{t+1}}{c_t})^{-\gamma} $$ where $\log \frac{c_{t+1}}{c_t} \stackrel{d}{\sim} \mathcal{N}(\mu, \sigma^2)$. So the risk-free rate $r$ is given by $$ e^{-r} = E[\rho (\frac{c_{t+1}}{c_t})^{-\gamma}] = \rho e^{-\gamma \mu + \frac12 \gamma^2 \sigma^2}, $$ which implies $$ r = - \log \rho + \gamma \mu - \frac12 \gamma^2 \sigma^2, $$ with the three terms reflecting time preference, intertemporal substitution, and precautionary saving, respectively.

In your quoted notation, the first difference $\Delta \log c_{t+1} = \log c_{t+1} - \log c_t = \log \frac{c_{t+1}}{c_t}$.

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