<p>There is a typo in the figure that introduces some confusion in the previous answer, which is basically <strong>wrong</strong>.</p>

<p>Based on the numbers and the figure, the utility is such that $$u=\sqrt{x},$$ so
$$E[u]=\frac{1}{2} u(100+125) + \frac{1}{2} u(100−100)= \frac{1}{2} u(225) =\frac{1}{2} \sqrt{225} = 7.5$$.</p>

<p>By definition, the risk premium (R) must satisfy the following condition:
$$ E(u) = u(100 - R)$$ 
$$ \Leftrightarrow 7.5  = \sqrt{100 - R}$$
$$ \Leftrightarrow (7.5)^2 = 100 - R$$
$$ \boxed{\Leftrightarrow R = 43.75}.$$</p>

<p>Notice that this bet is better than a "fair game" because the expected gain is not zero, but positive (0.5∗125+0.5∗(−100)=12.50.5∗125+0.5∗(−100)=12.5). So, despite this very good bet, the risk-averse agent characterized by her concave utility function ($u = \sqrt{x}$), is ready to pay almost half of her initial wealth to avoid risk and get the certainty equivalent amount.</p>

CC BY-SA 3.0
2015-11-06T07:42:05.790Z
2016-05-15T23:14:26.120Z
2016-05-15T23:14:26.120Z
emeryville
<p>The name for the amount $56.25 is <strong>certainty equivalent</strong>.</p>

<p>The expected utility for the individual from taking the bet is calculated as follows:
$$E[U]=\frac12U(100+125)+\frac12U(100-100)=75$$
Suppose the individual can pay an amount of money $x$ so that she can avoid taking the bet (which leads to expected utility $75$). What's the maximum amount of money $x$ she's willing to pay? Well, she would pay up to a point where she's indifferent between taking and not taking the bet.</p>

<p>If she takes the bet, expected utility is $75$. If she pays, her utility is $U(100-x)$. We want her to be indifferent, so that $U(100-x)=75$. Reading off from the blue curve in your graph (the curve describing $U$), we see that 
$$U(56.25)=75$$
which means $100-x=56.25$, or $x=43.75$. </p>

<p>So we can interpret 43.75 as the maximum amount of money that an individual is willing to pay in order to avoid the (risky) bet. </p>

CC BY-SA 4.0
2019-02-09T21:12:02.167Z
It can be negative if they are willing to pay money to take the bet, right?
Christopher King
CC BY-SA 4.0
2019-02-09T21:17:28.127Z
@PyRulez: Yes indeed.
Herr K.
CC BY-SA 3.0
2015-01-22T00:11:08.303Z
2015-01-23T18:15:28.810Z
2015-01-23T18:15:28.810Z
Herr K.
<p>In <a href="http://ocw.mit.edu/courses/economics/14-01sc-principles-of-microeconomics-fall-2011/unit-6-topics-in-intermediate-microeconomics/uncertainty/" rel="noreferrer">Lecture 20</a> of MIT's Microeconomics course, a situation is proposed where a 50/50 bet will either result in losing \$100 or gaining \$125 with a starting wealth of \$100. It is stated that a person would be willing to insure themselves for \$43.75 (the difference between \$100 and \$56.25). What is the intuition behind this?</p>

<p>Thanks in advance!</p>

<p><img src="https://i.sstatic.net/IxebN.jpg" alt="From MIT"></p>

<microeconomics><expected-utility><risk><probability><risk-aversion>
Intuition behind risk premium
Nick
# Intuition behind risk premium

(Asked by: username_0 on 2015-01-21)

In [Lecture 20](http://ocw.mit.edu/courses/economics/14-01sc-principles-of-microeconomics-fall-2011/unit-6-topics-in-intermediate-microeconomics/uncertainty/) of MIT's Microeconomics course, a situation is proposed where a 50/50 bet will either result in losing \$100 or gaining \$125 with a starting wealth of \$100. It is stated that a person would be willing to insure themselves for \$43.75 (the difference between \$100 and \$56.25). What is the intuition behind this?

Thanks in advance!

![From MIT](https://i.sstatic.net/IxebN.jpg)

**Tags:** microeconomics, expected-utility, risk, probability, risk-aversion

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## 2 answers

### Answer by username_2 on 2015-01-22. Score: 9 | Accepted answer

The name for the amount $56.25 is **certainty equivalent**.

The expected utility for the individual from taking the bet is calculated as follows:
$$E[U]=\frac12U(100+125)+\frac12U(100-100)=75$$
Suppose the individual can pay an amount of money $x$ so that she can avoid taking the bet (which leads to expected utility $75$). What's the maximum amount of money $x$ she's willing to pay? Well, she would pay up to a point where she's indifferent between taking and not taking the bet.

If she takes the bet, expected utility is $75$. If she pays, her utility is $U(100-x)$. We want her to be indifferent, so that $U(100-x)=75$. Reading off from the blue curve in your graph (the curve describing $U$), we see that
$$U(56.25)=75$$
which means $100-x=56.25$, or $x=43.75$.

So we can interpret 43.75 as the maximum amount of money that an individual is willing to pay in order to avoid the (risky) bet.

> **Comments:**
> **username_3**: It can be negative if they are willing to pay money to take the bet, right?  
> **username_2**: @PyRulez: Yes indeed.  

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### Answer by username_1 on 2015-11-06. Score: 4

There is a typo in the figure that introduces some confusion in the previous answer, which is basically **wrong**.

Based on the numbers and the figure, the utility is such that $$u=\sqrt{x},$$ so
$$E[u]=\frac{1}{2} u(100+125) + \frac{1}{2} u(100−100)= \frac{1}{2} u(225) =\frac{1}{2} \sqrt{225} = 7.5$$.

By definition, the risk premium (R) must satisfy the following condition:
$$ E(u) = u(100 - R)$$
$$ \Leftrightarrow 7.5 = \sqrt{100 - R}$$
$$ \Leftrightarrow (7.5)^2 = 100 - R$$
$$ \boxed{\Leftrightarrow R = 43.75}.$$

Notice that this bet is better than a "fair game" because the expected gain is not zero, but positive (0.5∗125+0.5∗(−100)=12.50.5∗125+0.5∗(−100)=12.5). So, despite this very good bet, the risk-averse agent characterized by her concave utility function ($u = \sqrt{x}$), is ready to pay almost half of her initial wealth to avoid risk and get the certainty equivalent amount.


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