Why is the risk premium always positive for risk averse individuals? - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2019-12-10T21:35:41Z https://economics.stackexchange.com/feeds/question/33060 https://creativecommons.org/licenses/by-sa/4.0/rdf https://economics.stackexchange.com/q/33060 2 Why is the risk premium always positive for risk averse individuals? rickyrichboy https://economics.stackexchange.com/users/25064 2019-12-02T17:42:28Z 2019-12-02T19:02:23Z <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> https://economics.stackexchange.com/questions/33060/-/33064#33064 4 Answer by lunar_props for Why is the risk premium always positive for risk averse individuals? lunar_props https://economics.stackexchange.com/users/24992 2019-12-02T19:02:23Z 2019-12-02T19:02:23Z <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>