<p>Risk aversion is given by the sign of <span class="math-container">$r(w)$</span> according to these cases:</p>
<ul>
<li><span class="math-container">$r(w) &gt; 0 \implies$</span> the individual’s utility function is concave and they are risk averse.</li>
<li><span class="math-container">$r(w) &lt; 0 \implies$</span> the individual’s utility function is convex and they are risk loving.</li>
<li><span class="math-container">$r(w) = 0 \implies$</span> the individual’s utility function is linear and they are risk neutral.</li>
</ul>
<p>We have</p>
<p><span class="math-container">$U(w) = - e^{-w}$</span></p>
<p><span class="math-container">$U’(w) = e^{-w}$</span></p>
<p><span class="math-container">$U’’(w) = - e^{-w}$</span></p>
<p>Therefore,</p>
<p><span class="math-container">$r(w) = - \frac{u’’(w)}{u’(w)} = - \frac{ - e^{-w}}{e^{-w}} = 1$</span>, so your <span class="math-container">$r$</span> value is correct.</p>
<p>Since <span class="math-container">$r(w) = 1 &gt; 0$</span>, the individual in fact has a concave utility function as they are risk averse, so there is no contradiction.</p>

CC BY-SA 4.0
2023-04-09T13:38:23.473Z
Appreciated, thank you! Can't upvote you without signing up fully, so sorry about that.
riskyasset
CC BY-SA 4.0
2023-04-09T13:25:12.623Z
2023-04-09T13:35:49.890Z
2023-04-09T13:35:49.890Z
Nicolas Torres
<blockquote>
<p>Given utility function <span class="math-container">$U(w) = -e^{-w}$</span> of an investor (where <span class="math-container">$w$</span> denotes wealth) and two assets - risky and safe, will the investor's amount of investment into the risky asset be indifferent of his wealth?</p>
</blockquote>
<p>The graph of <span class="math-container">$U(w)$</span> is strictly concave, so I feel like the investor will put in more money into the risky asset as his wealth increases and gain more utility from it. On the other hand, when I test using the <a href="https://en.wikipedia.org/wiki/Risk_aversion#Absolute_risk_aversion" rel="nofollow noreferrer">Arrow-Pratt measure of risk aversion</a>, I find that <span class="math-container">$r(w) = -\frac{u''(w)}{u'(w)} = 1$</span> which tells that the investor will be indifferent.</p>
<p>This is contradictory, so where am I going wrong?</p>

<microeconomics><risk-aversion>
Portfolio choice and risk aversion
riskyasset
# Portfolio choice and risk aversion

(Asked by: username_0 on 2023-04-09)

> Given utility function $U(w) = -e^{-w}$ of an investor (where $w$ denotes wealth) and two assets - risky and safe, will the investor's amount of investment into the risky asset be indifferent of his wealth?

The graph of $U(w)$ is strictly concave, so I feel like the investor will put in more money into the risky asset as his wealth increases and gain more utility from it. On the other hand, when I test using the [Arrow-Pratt measure of risk aversion](https://en.wikipedia.org/wiki/Risk_aversion#Absolute_risk_aversion), I find that $r(w) = -\frac{u''(w)}{u'(w)} = 1$ which tells that the investor will be indifferent.

This is contradictory, so where am I going wrong?

**Tags:** microeconomics, risk-aversion


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## 1 answers

### Answer by username_1 on 2023-04-09. Score: 1 | Accepted answer

Risk aversion is given by the sign of $r(w)$ according to these cases:

* $r(w) > 0 \implies$ the individual’s utility function is concave and they are risk averse.
* $r(w) < 0 \implies$ the individual’s utility function is convex and they are risk loving.
* $r(w) = 0 \implies$ the individual’s utility function is linear and they are risk neutral.

We have

$U(w) = - e^{-w}$

$U’(w) = e^{-w}$

$U’’(w) = - e^{-w}$

Therefore,

$r(w) = - \frac{u’’(w)}{u’(w)} = - \frac{ - e^{-w}}{e^{-w}} = 1$, so your $r$ value is correct.

Since $r(w) = 1 > 0$, the individual in fact has a concave utility function as they are risk averse, so there is no contradiction.

> **Comments:**
> **username_0**: Appreciated, thank you! Can't upvote you without signing up fully, so sorry about that.  

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