<p>For the certainty equivalent and the "risk compensation" (which I am interpreting as probability premium because it's the only thing that intuitively makes sense in this context to me; feel free to correct me), think more intuitively about the concepts. The certainty equivalent is the amount of cold hard cash you'd be indifferent to taking in lieu of the uncertain outcome.</p>

<p>In the good outcome with $\frac{1}{3}$ probability, you'd end up with a wealth of $10 + 12$, and in the the bad state with probability $\frac{2}{3}$ where the lottery does not give you anything, you still end up with wealth of $10$. Thus, we are looking for wealth that I will denote $w^c$ where</p>

<p>$$v(w_c) = \mathbb{E}(v(w)) \implies \frac{w_c - 1}{w_c} = \left(\frac{22-1}{22}\right)\cdot \frac{1}{3} + \left(\frac{10-1}{10}\right) \cdot \frac{2}{3}$$</p>

<p>Solve for $w_c$ and that will be the certainty equivalent. You'll notice that trying to solve this out, we get:</p>

<p>$$\frac{w_c - 1}{w_c} = \frac{101}{110}$$
$$\implies \boxed{w_c = \frac{110}{9} \approx 12.2}$$</p>

<hr>

<p>As for probability premium, it is the shift in probability towards the better lottery outcome that would be needed to make the expected utility of the new lottery equal to the utility of the expected value of the old lottery.</p>

<p>$$\mathbb{E}_{\text{new}}(v(w)) = v(\mathbb{E}_{\text{old}}(w))$$
$$\implies \left(\frac{1}{3} + \pi\right) \cdot \left(\frac{22-1}{22}\right) + \left(\frac{2}{3} - \pi\right) \cdot \left(\frac{10-1}{10}\right) = \left(\frac{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3} -1}{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3}}\right)$$
$$\implies \frac{101}{110} + \frac{6}{110} \pi = \frac{13}{14}$$
$$\implies \pi = \frac{8}{770} \approx 0.01$$</p>

<p>(assuming I calculated my fractions correctly)</p>

<hr>

<p><strong>Edit:</strong> I see that the question was asking for a risk premium, not probability premium. Notice that the expected wealth due to the lottery is 14, but the certainty equivalent is 12.222...</p>

<p>The difference between the two is the risk premium.</p>

CC BY-SA 3.0
2017-01-14T22:10:47.023Z
I am sorry, I had a typo, just corrected it now
Wolfy
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2017-01-14T22:12:18.213Z
*screams internally* I'll edit my answer. :^))))
Kitsune Cavalry
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2017-01-14T22:21:54.890Z
much obliged, sorry for the typo
Wolfy
CC BY-SA 3.0
2017-01-14T22:24:04.123Z
It's fine, I actually made a bigger error when solving for the certainty equivalent. Give me a moment, $\gamma$ equal to 2 is actually fine, I'm just a newb.
Kitsune Cavalry
CC BY-SA 3.0
2017-01-14T22:31:38.373Z
no problem, take your time
Wolfy
CC BY-SA 3.0
2017-01-14T22:56:54.417Z
The certainty equivalent is correct but the solution provided by my professor states the risk premium is 1.78, although he does not show how he got this value
Wolfy
CC BY-SA 3.0
2017-01-14T23:08:29.283Z
Ah, I found the probability premium, not risk premium. That is very simple to look for then.
Kitsune Cavalry
CC BY-SA 3.0
2017-01-15T07:35:00.797Z
@KitsuneCavalry Poor thing could not find [ANYTHING](https://www.google.com/search?sourceid=chrome-psyapi2&ion=1&espv=2&ie=UTF-8&q=certainty%20equivalent&oq=certainty%20&aqs=chrome.2.69i57j0l2j69i60j0l2.3047j0j7) about the certainty equivalent online :( Please help him! Is there any way you could upload your answer in his handwriting? Or, you know, mail him your degree...?
Giskard
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2017-01-15T18:55:44.067Z
@denesp watch your mouth. Standing on top of ones shoulders does not make you tall.
Wolfy
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2017-01-15T21:07:42.997Z
@denesp In my defense his question was more non-trivial before the edit. :^P I have to agree though billy-boy with denesp in different words. Ya gotta learn them expected values (weighted averages fam) or goshdarnit brother you'll be in a pickle. Also my shoulders are a bit too scrawny for denesp imo I'm a lightweight. Anyway Imma be a busy boy for the rest of the week everyone play nice :^))))))))))))))
Kitsune Cavalry
CC BY-SA 3.0
2017-01-14T22:06:16.817Z
2017-01-14T23:14:01.110Z
2017-01-14T23:14:01.110Z
Kitsune Cavalry
<p>I know this is rather easy but I do not have any example or material to really work from that I could find online.</p>
<blockquote>
<p>Consider an agent with a utility index over wealth given by <span class="math-container">$$v(w) = \frac{w^{1 - \gamma} - 1}{1 - \gamma}, \gamma &gt; 0$$</span></p>
<p>Say the agent has <span class="math-container">$10$</span> dollars in wealth and also owns a lottery that pays <span class="math-container">$0$</span> dollars with probability <span class="math-container">$2/3$</span> and pays <span class="math-container">$12$</span> with probability <span class="math-container">$1/3$</span>. What is the certainty equivalent value and risk compensation of this lottery to an agent with risk aversion parameter <span class="math-container">$\gamma = 2$</span>?</p>
</blockquote>
<p>So for <span class="math-container">$\gamma = 2$</span>, we have <span class="math-container">$$v(w) = \frac{w-1}{w}$$</span> so the expectd value of the lottery is <span class="math-container">$4$</span> dollars. My question is how do we calculate the Certainty equivalent and risk premium. I could not find a good example anywhere. It seems like we get at undefined part in the equation when we calculate <span class="math-container">$E[v(L)]$</span> where <span class="math-container">$L$</span> is the lottery, thus I don't see how we can find the Certainty equivalent...</p>

<financial-economics><expected-utility>
Certainty equivalent and risk aversion parameter given utility function
Wolfy
# Certainty equivalent and risk aversion parameter given utility function

(Asked by: username_0 on 2017-01-14)

I know this is rather easy but I do not have any example or material to really work from that I could find online.

> Consider an agent with a utility index over wealth given by $$v(w) = \frac{w^{1 - \gamma} - 1}{1 - \gamma}, \gamma > 0$$
>
> Say the agent has $10$ dollars in wealth and also owns a lottery that pays $0$ dollars with probability $2/3$ and pays $12$ with probability $1/3$. What is the certainty equivalent value and risk compensation of this lottery to an agent with risk aversion parameter $\gamma = 2$?

So for $\gamma = 2$, we have $$v(w) = \frac{w-1}{w}$$ so the expectd value of the lottery is $4$ dollars. My question is how do we calculate the Certainty equivalent and risk premium. I could not find a good example anywhere. It seems like we get at undefined part in the equation when we calculate $E[v(L)]$ where $L$ is the lottery, thus I don't see how we can find the Certainty equivalent...

**Tags:** financial-economics, expected-utility


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## 1 answers

### Answer by username_1 on 2017-01-14. Score: 1 | Accepted answer

For the certainty equivalent and the "risk compensation" (which I am interpreting as probability premium because it's the only thing that intuitively makes sense in this context to me; feel free to correct me), think more intuitively about the concepts. The certainty equivalent is the amount of cold hard cash you'd be indifferent to taking in lieu of the uncertain outcome.

In the good outcome with $\frac{1}{3}$ probability, you'd end up with a wealth of $10 + 12$, and in the the bad state with probability $\frac{2}{3}$ where the lottery does not give you anything, you still end up with wealth of $10$. Thus, we are looking for wealth that I will denote $w^c$ where

$$v(w\_c) = \mathbb{E}(v(w)) \implies \frac{w\_c - 1}{w\_c} = \left(\frac{22-1}{22}\right)\cdot \frac{1}{3} + \left(\frac{10-1}{10}\right) \cdot \frac{2}{3}$$

Solve for $w\_c$ and that will be the certainty equivalent. You'll notice that trying to solve this out, we get:

$$\frac{w\_c - 1}{w\_c} = \frac{101}{110}$$
$$\implies \boxed{w\_c = \frac{110}{9} \approx 12.2}$$

---

As for probability premium, it is the shift in probability towards the better lottery outcome that would be needed to make the expected utility of the new lottery equal to the utility of the expected value of the old lottery.

$$\mathbb{E}\_{\text{new}}(v(w)) = v(\mathbb{E}\_{\text{old}}(w))$$
$$\implies \left(\frac{1}{3} + \pi\right) \cdot \left(\frac{22-1}{22}\right) + \left(\frac{2}{3} - \pi\right) \cdot \left(\frac{10-1}{10}\right) = \left(\frac{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3} -1}{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3}}\right)$$
$$\implies \frac{101}{110} + \frac{6}{110} \pi = \frac{13}{14}$$
$$\implies \pi = \frac{8}{770} \approx 0.01$$

(assuming I calculated my fractions correctly)

---

**Edit:** I see that the question was asking for a risk premium, not probability premium. Notice that the expected wealth due to the lottery is 14, but the certainty equivalent is 12.222...

The difference between the two is the risk premium.

> **Comments:**
> **username_0**: I am sorry, I had a typo, just corrected it now  
> **username_1**: *screams internally* I'll edit my answer. :^))))  
> **username_0**: much obliged, sorry for the typo  
> **username_1**: It's fine, I actually made a bigger error when solving for the certainty equivalent. Give me a moment, $\gamma$ equal to 2 is actually fine, I'm just a newb.  
> **username_0**: no problem, take your time  
> **username_0**: The certainty equivalent is correct but the solution provided by my professor states the risk premium is 1.78, although he does not show how he got this value  
> **username_1**: Ah, I found the probability premium, not risk premium. That is very simple to look for then.  
> **username_2**: @KitsuneCavalry Poor thing could not find [ANYTHING](https://www.google.com/search?sourceid=chrome-psyapi2&ion=1&espv=2&ie=UTF-8&q=certainty%20equivalent&oq=certainty%20&aqs=chrome.2.69i57j0l2j69i60j0l2.3047j0j7) about the certainty equivalent online :( Please help him! Is there any way you could upload your answer in his handwriting? Or, you know, mail him your degree...?  
> **username_0**: @denesp watch your mouth. Standing on top of ones shoulders does not make you tall.  
> **username_1**: @denesp In my defense his question was more non-trivial before the edit. :^P I have to agree though billy-boy with denesp in different words. Ya gotta learn them expected values (weighted averages fam) or goshdarnit brother you'll be in a pickle. Also my shoulders are a bit too scrawny for denesp imo I'm a lightweight. Anyway Imma be a busy boy for the rest of the week everyone play nice :^))))))))))))))  

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