<p>This is a one-line calculation under standard CRRA-lognormal assumptions (see, for example, also <a href="https://economics.stackexchange.com/questions/8397/log-normality-assumption-in-consumption-based-asset-pricing?rq=1">this question</a>).</p>
<p>By CRRA and lognormal consumption growth assumptions, the pricing kernel is
<span class="math-container">$$
\frac{\rho u'(c_{t+1})}{u'(c_t)} = \rho (\frac{c_{t+1}}{c_t})^{-\gamma}
$$</span>
where <span class="math-container">$\log \frac{c_{t+1}}{c_t} \stackrel{d}{\sim} \mathcal{N}(\mu, \sigma^2)$</span>. So the risk-free rate <span class="math-container">$r$</span> is given by
<span class="math-container">$$
e^{-r} = E[\rho (\frac{c_{t+1}}{c_t})^{-\gamma}] = \rho e^{-\gamma \mu + \frac12 \gamma^2 \sigma^2},
$$</span>
which implies
<span class="math-container">$$
r = - \log \rho + \gamma \mu - \frac12 \gamma^2 \sigma^2,
$$</span>
with the three terms reflecting time preference, intertemporal substitution, and precautionary saving, respectively.</p>
<p>In your quoted notation, the first difference <span class="math-container">$\Delta \log c_{t+1} = \log c_{t+1} - \log c_t = \log \frac{c_{t+1}}{c_t}$</span>.</p>

CC BY-SA 4.0
2022-03-06T18:44:43.717Z
2022-03-06T18:44:43.717Z
Michael
<p>I am studying consumption CAPM and trying to prove, assuming lognormal consumption growth one can show that the continuously compounded risk-free rate is:
<a href="https://i.sstatic.net/UR6xA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UR6xA.png" alt="enter image description here" /></a></p>

<consumption><risk>
Consumption CAPM and compounded risk free rate
emma smith
# Consumption CAPM and compounded risk free rate

(Asked by: username_0 on 2022-02-27)

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](https://i.sstatic.net/UR6xA.png)](https://i.sstatic.net/UR6xA.png)

**Tags:** consumption, risk


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## 1 answers

### Answer by username_1 on 2022-03-06. Score: 2

This is a one-line calculation under standard CRRA-lognormal assumptions (see, for example, also [this question](https://economics.stackexchange.com/questions/8397/log-normality-assumption-in-consumption-based-asset-pricing?rq=1)).

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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