<p>The notation is a bit confusing but what the author is computing is the <em>expected utility</em> of the investment, not the utility of the expected outcome.</p>

<p>Since the investment results in a wealth of \$120 with probability 0.5 and a wealth of \$90 with probability 0.5, the expected utility equals
\begin{align*}
\mathbb{E}U &amp; = 0.5 \times (-\dfrac{10^6}{90^3}) + 0.5 \times (-\dfrac{10^6}{120^3}) \\
&amp; \sim -0.98 
\end{align*}</p>

<p>Notice that this value is lower than the utility of the expected outcome, which is approximately equal to $-0.86$ as you wrote. This is because the agent is risk-averse and must therefore be compensated for the risk.</p>

CC BY-SA 3.0
2017-04-12T11:23:13.107Z
2017-04-12T11:23:13.107Z
Oliv
<p>Recently I was improving my knowledge of Utility Functions in finances. I stumped upon very nice (yet quite old) article: "<a href="http://www.norstad.org/finance/util.pdf" rel="nofollow noreferrer">An Introduction to Utility Theory</a>" by John Norstad (1999).
What drove my attention was charts in section 4. They represent utility functions of wealth:</p>

<ul>
<li>First chart (figure 2): $- \dfrac{10^6}{w^3}$</li>
<li>Second chart(figure 3): $-\dfrac{10^{10}}{w^5}$</li>
</ul>

<p>(So basically $-\dfrac{10^{2c}}{w^c}$) with c as chosen constant)</p>

<p>Purely for learning purpose I was trying to recreate his example (by using Excel/R) from section 4 (details on page 7). In same page author stated that expected utility (of 105) will be equal to -0.98. It is also shown under figure 2 in table:</p>

<pre><code>  Wealth   Utility
 $90   -1.37   Bad outcome
 $100   -1.00   Current wealth
 $105   -0.98   Expected outcome
 $120   -0.58   Good outcome
</code></pre>

<p>I run test and with those values and for \$105 it does not match. Using $- \frac{10^6}{105^3}$ I get -0.86. What is surprising other stated values are calculated correctly (90, 100, 120). At first I though it is nothing more than some kind of typo, but when I checked the second formula used to draft function in figure 3 the same situation occurred. For table: </p>

<pre><code>Wealth   Utility
 $90   -1.69   Bad outcome
 $100   -1.00   Current wealth
 $105   -1.05   Expected outcome
 $120   -0.40   Good outcome
</code></pre>

<p>when I put value 105 to formula ($-\frac{10^{10}}{105^5}$) it seems to be incorrect (should be -0.78) while rest of them are right. This quite undermines my typo theory.  Additionally this contradicts further conclusions (page 9 under the chart) made by author. </p>

<p>So my question is: did I overlooked something? Or did I just find a mistake? Or I am the one who is mistaken?</p>

<utility>
Confusions about calculations in Utility Theory Paper
user12851
# Confusions about calculations in Utility Theory Paper

(Asked by: username_0 on 2017-04-12)

Recently I was improving my knowledge of Utility Functions in finances. I stumped upon very nice (yet quite old) article: "An Introduction to Utility Theory" by John Norstad (1999).
What drove my attention was charts in section 4. They represent utility functions of wealth:

First chart (figure 2): $- \dfrac{10^6}{w^3}$
Second chart(figure 3): $-\dfrac{10^{10}}{w^5}$

(So basically $-\dfrac{10^{2c}}{w^c}$) with c as chosen constant)
Purely for learning purpose I was trying to recreate his example (by using Excel/R) from section 4 (details on page 7). In same page author stated that expected utility (of 105) will be equal to -0.98. It is also shown under figure 2 in table:
  Wealth   Utility
 $90   -1.37   Bad outcome
 $100   -1.00   Current wealth
 $105   -0.98   Expected outcome
 $120   -0.58   Good outcome

I run test and with those values and for \$105 it does not match. Using $- \frac{10^6}{105^3}$ I get -0.86. What is surprising other stated values are calculated correctly (90, 100, 120). At first I though it is nothing more than some kind of typo, but when I checked the second formula used to draft function in figure 3 the same situation occurred. For table: 
Wealth   Utility
 $90   -1.69   Bad outcome
 $100   -1.00   Current wealth
 $105   -1.05   Expected outcome
 $120   -0.40   Good outcome

when I put value 105 to formula ($-\frac{10^{10}}{105^5}$) it seems to be incorrect (should be -0.78) while rest of them are right. This quite undermines my typo theory.  Additionally this contradicts further conclusions (page 9 under the chart) made by author. 
So my question is: did I overlooked something? Or did I just find a mistake? Or I am the one who is mistaken?

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2017-04-12. Score: 2

The notation is a bit confusing but what the author is computing is the *expected utility* of the investment, not the utility of the expected outcome.

Since the investment results in a wealth of \$120 with probability 0.5 and a wealth of \$90 with probability 0.5, the expected utility equals
\begin{align\*}
\mathbb{E}U & = 0.5 \times (-\dfrac{10^6}{90^3}) + 0.5 \times (-\dfrac{10^6}{120^3}) \\
& \sim -0.98
\end{align\*}

Notice that this value is lower than the utility of the expected outcome, which is approximately equal to $-0.86$ as you wrote. This is because the agent is risk-averse and must therefore be compensated for the risk.


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