<p>Suppose that the vector <span class="math-container">$W=\left(w_1,w_2,\dots,w_n\right)$</span> represents wealth in <span class="math-container">$n$</span> possible states. In addition, assume the probability of each state occurring is represented by the vector <span class="math-container">$\pi=\left(\pi_1,\pi_2,\dots,\pi_n\right)$</span>. We can express this as the simple gamble:</p>

<p><span class="math-container">$$g = \left(\pi_1\circ w_1,\pi_2\circ w_2, \dots, \pi_n\circ w_n\right)$$</span></p>

<p>The expected value of the simple gamble <span class="math-container">$g$</span> is:</p>

<p><span class="math-container">$$\mathbb{E}[g]=\sum_{i=1}^{n}\pi_iw_i$$</span></p>

<p>Suppose a consumer has a sub-utility function <span class="math-container">$u(w)$</span>. Further, suppose that the consumers preferences over gambles are such that they can be represented by a von Neumann Morgenstern utility function (has the expected utility property). The utility of the gamble can be expressed as the expected utility across all states:</p>

<p><span class="math-container">$$u(g)=\sum_{i=1}^{n}\pi_iu(w_i)$$</span></p>

<p>Now, if we assume that the sub-utility function <span class="math-container">$u(w)$</span> is strictly concave (risk averse consumer), by the concavity of functions it must be the case that:</p>

<p><span class="math-container">$$u\left(\mathbb{E}[g]\right) &gt; u(g)$$</span></p>

<p>This implies that the consumer would prefer to receive the expected value of the gamble with certainty rather than the gamble itself. A question we could ask; how much would I need to give to the consumer such that they are indifferent between choosing the gamble and receiving some amount of wealth with certainty (Certainty Equivalent)? The certainty equivalent <span class="math-container">$CE$</span> must satisfy the following:</p>

<p><span class="math-container">$$u\left(CE\right) = u(g) \implies u\left(\mathbb{E}[g]\right) &gt; u\left(CE\right) \implies \mathbb{E}[g] &gt; CE$$</span></p>

<p>Since <span class="math-container">$u(w)$</span> is increasing in wealth, there exists a <span class="math-container">$P&gt;0$</span> such that the following will hold:</p>

<p><span class="math-container">$$u\left(\mathbb{E}[g]-P\right)= u\left(CE\right) \implies \mathbb{E}[g] -P= CE$$</span> </p>

<p><span class="math-container">$P$</span> represents the risk premium. It is the amount you would be willing to pay to get rid of uncertainty (receive the certainty equivalent). The fact that the risk premium is positive does arise from the fact that we assume the sub-utility function is strictly concave. </p>

CC BY-SA 4.0
2019-12-02T19:58:36.797Z
Ahhh that makes a lot of sense. Thanks for your insight!
rickyrichboy
CC BY-SA 4.0
2019-12-02T19:02:23.637Z
2019-12-02T19:02:23.637Z
lunar_props
<p>I think this has to do with the definition of concavity and the fact that a risk averse person has a concave utility function, but I'm not sure how that helps.</p>

<microeconomics><utility><expected-utility><risk><risk-aversion>
Why is the risk premium always positive for risk averse individuals?
rickyrichboy
# Why is the risk premium always positive for risk averse individuals?

(Asked by: username_0 on 2019-12-02)

I think this has to do with the definition of concavity and the fact that a risk averse person has a concave utility function, but I'm not sure how that helps.

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


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## 1 answers

### Answer by username_1 on 2019-12-02. Score: 5 | Accepted answer

Suppose that the vector $W=\left(w\_1,w\_2,\dots,w\_n\right)$ represents wealth in $n$ possible states. In addition, assume the probability of each state occurring is represented by the vector $\pi=\left(\pi\_1,\pi\_2,\dots,\pi\_n\right)$. We can express this as the simple gamble:

$$g = \left(\pi\_1\circ w\_1,\pi\_2\circ w\_2, \dots, \pi\_n\circ w\_n\right)$$

The expected value of the simple gamble $g$ is:

$$\mathbb{E}[g]=\sum\_{i=1}^{n}\pi\_iw\_i$$

Suppose a consumer has a sub-utility function $u(w)$. Further, suppose that the consumers preferences over gambles are such that they can be represented by a von Neumann Morgenstern utility function (has the expected utility property). The utility of the gamble can be expressed as the expected utility across all states:

$$u(g)=\sum\_{i=1}^{n}\pi\_iu(w\_i)$$

Now, if we assume that the sub-utility function $u(w)$ is strictly concave (risk averse consumer), by the concavity of functions it must be the case that:

$$u\left(\mathbb{E}[g]\right) > u(g)$$

This implies that the consumer would prefer to receive the expected value of the gamble with certainty rather than the gamble itself. A question we could ask; how much would I need to give to the consumer such that they are indifferent between choosing the gamble and receiving some amount of wealth with certainty (Certainty Equivalent)? The certainty equivalent $CE$ must satisfy the following:

$$u\left(CE\right) = u(g) \implies u\left(\mathbb{E}[g]\right) > u\left(CE\right) \implies \mathbb{E}[g] > CE$$

Since $u(w)$ is increasing in wealth, there exists a $P>0$ such that the following will hold:

$$u\left(\mathbb{E}[g]-P\right)= u\left(CE\right) \implies \mathbb{E}[g] -P= CE$$

$P$ represents the risk premium. It is the amount you would be willing to pay to get rid of uncertainty (receive the certainty equivalent). The fact that the risk premium is positive does arise from the fact that we assume the sub-utility function is strictly concave.

> **Comments:**
> **username_0**: Ahhh that makes a lot of sense. Thanks for your insight!  

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