<p>The equality in question follows from the expression for expectation of a log-normal distribution.
For example, if <span class="math-container">$X \stackrel{d}{\sim} N(\mu, \sigma^2)$</span>, then
<span class="math-container">$$
E[e^{a X}] = e^{a \mu + \frac12 a^2 \sigma^2}.
$$</span>
This is the reason that CARA/normal or CRRA/log-normal (agent utility/asset return distribution pair) set-ups reduce to the mean-variance case, up to certainty equivalence.</p>

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2020-10-16T08:53:27.750Z
You are an absolute legend and life-saver! Thank you so much for a clean, efficient and fast response.
MathyFoxy
CC BY-SA 4.0
2020-10-16T01:53:39.843Z
2020-10-16T01:58:54.620Z
2020-10-16T01:58:54.620Z
Michael
<p>I'm currently writing my thesis in which I compare a series of ESG General Equilibrium models. I fell over this proof in Pastor, Stambaugh, Taylor Sustainable Investing in Equilibrium (2019) page 42. Here, they prove going from a given utility function to the models portfolio weights. These are the calculations of which I simply don't understand how they start on the second line and arrive at the third:
<span class="math-container">\begin{align}
\mathbb{E}( V(\tilde{W}_{1i},X_i)) &amp;=\mathbb{E}(-e^{-A_i(W_{0i}(1+r_f+X'_i \tilde{r})-b'_iX_i)}
\\&amp;= -e^{a_i(1+r_f)} \mathbb{E}( e^{-a_i X_i'(\tilde{r}+\frac{1}{a_i} b_i)})\\
&amp;= -e^{a_i(1+r_f)} ( e^{-a_i X_i'(\mathbb{E}(\tilde{r})+\frac{1}{a_i} b_i)+\frac{1}{2}a^2_i X'_i\Sigma X_i})
\end{align}</span>
We've previously defined <span class="math-container">$\tilde{r}\sim(\mu,\Sigma) $</span>.</p>
<p>I've been staring at this for hours. I hope somebody can help me out with a hint on what is happening between these two steps.</p>

<mathematical-economics><finance><financial-economics><general-equilibrium>
Bothersome Mean/Variance Analysis
MathyFoxy
# Bothersome Mean/Variance Analysis

(Asked by: username_0 on 2020-10-15)

I'm currently writing my thesis in which I compare a series of ESG General Equilibrium models. I fell over this proof in Pastor, Stambaugh, Taylor Sustainable Investing in Equilibrium (2019) page 42. Here, they prove going from a given utility function to the models portfolio weights. These are the calculations of which I simply don't understand how they start on the second line and arrive at the third:
\begin{align}
\mathbb{E}( V(\tilde{W}\_{1i},X\_i)) &=\mathbb{E}(-e^{-A\_i(W\_{0i}(1+r\_f+X'\_i \tilde{r})-b'\_iX\_i)}
\\&= -e^{a\_i(1+r\_f)} \mathbb{E}( e^{-a\_i X\_i'(\tilde{r}+\frac{1}{a\_i} b\_i)})\\
&= -e^{a\_i(1+r\_f)} ( e^{-a\_i X\_i'(\mathbb{E}(\tilde{r})+\frac{1}{a\_i} b\_i)+\frac{1}{2}a^2\_i X'\_i\Sigma X\_i})
\end{align}
We've previously defined $\tilde{r}\sim(\mu,\Sigma) $.

I've been staring at this for hours. I hope somebody can help me out with a hint on what is happening between these two steps.

**Tags:** mathematical-economics, finance, financial-economics, general-equilibrium

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## 1 answers

### Answer by username_1 on 2020-10-16. Score: 3 | Accepted answer

The equality in question follows from the expression for expectation of a log-normal distribution.
For example, if $X \stackrel{d}{\sim} N(\mu, \sigma^2)$, then
$$
E[e^{a X}] = e^{a \mu + \frac12 a^2 \sigma^2}.
$$
This is the reason that CARA/normal or CRRA/log-normal (agent utility/asset return distribution pair) set-ups reduce to the mean-variance case, up to certainty equivalence.

> **Comments:**
> **username_0**: You are an absolute legend and life-saver! Thank you so much for a clean, efficient and fast response.  

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