<p>The gamble:</p>

<p><span class="math-container">$$g = \left(\pi \circ 98, (1-\pi)\circ100\right)$$</span></p>

<p>The expected value of the gamble:</p>

<p><span class="math-container">$$\mathbb{E}[g]= \pi\cdot 98 + (1-\pi)\cdot 100$$</span></p>

<p>Expected utility:</p>

<p><span class="math-container">$$u(g) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$</span></p>

<p>Risk premium is such that:</p>

<p><span class="math-container">$$\ln \left(\mathbb{E}[g]-P\right) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$</span></p>

<p>Where <span class="math-container">$P$</span> is the risk premium. Solve the above equation for <span class="math-container">$P$</span>. You can measure the degree of <strong>absolute risk aversion</strong> as follows:</p>

<p><span class="math-container">$$ARA = -\frac{u''(c_h)}{u'(c_h)}$$</span></p>

<p>And the degree of <strong>relative risk aversion</strong> as:</p>

<p><span class="math-container">$$RRA = - \frac{c_hu''(c_h)}{u'(c_h)}$$</span></p>

CC BY-SA 4.0
2019-12-03T03:43:18.193Z
Thank you! I have one further question I was wondering if you could answer: say there was an offer to buy insurance that fully mitigates the risk of loss for 2 dollars. Would this be an actuarially fair offer?
Biff
CC BY-SA 4.0
2019-12-03T16:16:48.727Z
Being actuarially fair depends on the insurance premium. If the insurance premium is the expected loss (in your case), then yes.
InSung Cho
CC BY-SA 4.0
2019-12-03T19:50:23.073Z
It would be actuarially fair if the insurer has zero expected profit. If you set up the problem of the insurance company, you should find that an actuarially fair offer is one which the cost per dollar of coverage (the premium) is equal to the probability of loss, $\pi$.
lunar_props
CC BY-SA 4.0
2019-12-03T03:06:38.757Z
2019-12-03T03:24:05.283Z
2019-12-03T03:24:05.283Z
lunar_props
<p>To give an example, say we start with 100 dollars and we enter a lottery. With probability <span class="math-container">$\pi$</span>, this 100 dollars is reduced by 2 dollars. Otherwise our endowed 100 dollars does not change. Let's say our consumer's utility function is described by <span class="math-container">$u(c_h)=ln(c_h)$</span>.</p>

<p>It's clear that the consumer is risk-averse, but how would I calculate the degree of risk aversion? How do you find the risk premium here?</p>

<p>Thanks.</p>

CC BY-SA 4.0
2019-12-03T04:54:24.133Z
Are you aware that there are degrees of *absolute* risk aversion and of *relative* risk aversion? Which degree of risk aversion are you asking about?
Herr K.
<microeconomics><risk-aversion>
How to calculate the degree of risk aversion of a consumer in a lottery?
Biff
# How to calculate the degree of risk aversion of a consumer in a lottery?

(Asked by: username_0 on 2019-12-03)

To give an example, say we start with 100 dollars and we enter a lottery. With probability $\pi$, this 100 dollars is reduced by 2 dollars. Otherwise our endowed 100 dollars does not change. Let's say our consumer's utility function is described by $u(c\_h)=ln(c\_h)$.

It's clear that the consumer is risk-averse, but how would I calculate the degree of risk aversion? How do you find the risk premium here?

Thanks.

**Tags:** microeconomics, risk-aversion

> **Comments:**
> **username_1**: Are you aware that there are degrees of *absolute* risk aversion and of *relative* risk aversion? Which degree of risk aversion are you asking about?  

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## 1 answers

### Answer by username_2 on 2019-12-03. Score: 1 | Accepted answer

The gamble:

$$g = \left(\pi \circ 98, (1-\pi)\circ100\right)$$

The expected value of the gamble:

$$\mathbb{E}[g]= \pi\cdot 98 + (1-\pi)\cdot 100$$

Expected utility:

$$u(g) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

Risk premium is such that:

$$\ln \left(\mathbb{E}[g]-P\right) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

Where $P$ is the risk premium. Solve the above equation for $P$. You can measure the degree of **absolute risk aversion** as follows:

$$ARA = -\frac{u''(c\_h)}{u'(c\_h)}$$

And the degree of **relative risk aversion** as:

$$RRA = - \frac{c\_hu''(c\_h)}{u'(c\_h)}$$

> **Comments:**
> **username_0**: Thank you! I have one further question I was wondering if you could answer: say there was an offer to buy insurance that fully mitigates the risk of loss for 2 dollars. Would this be an actuarially fair offer?  
> **username_3**: Being actuarially fair depends on the insurance premium. If the insurance premium is the expected loss (in your case), then yes.  
> **username_2**: It would be actuarially fair if the insurer has zero expected profit. If you set up the problem of the insurance company, you should find that an actuarially fair offer is one which the cost per dollar of coverage (the premium) is equal to the probability of loss, $\pi$.  

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