<p>I am curious how to calculate the following risk premium for a utility function that is not linear in <span class="math-container">$w$</span>. What i'm asking is the following:</p>
<p>Consider an agent with utility function <span class="math-container">$u$</span>, initial wealth <span class="math-container">$\omega_{0}$</span>, and a zero-mean random variable <span class="math-container">$\tilde{x}$</span> that pays <span class="math-container">$+k;-k$</span> with probability <span class="math-container">$\frac{1}{2}$</span> each. Lets further assume that the agents utility function is defined as:</p>
<p><span class="math-container">$$u(w)=a*w \text{   for  } w&lt;w_{0} $$</span> and</p>
<p><span class="math-container">$$ u(w)=a*w_{0}+b(w-w_{0}) \text{   for  } w\geq w_{0}$$</span> where <span class="math-container">$b&lt;a$</span></p>
<p>By definition of the risk premium <span class="math-container">$\pi$</span>, we have</p>
<p><span class="math-container">$$ Eu(w+\tilde{x}) = u(w-\pi). $$</span></p>
<p>By rearranging and given the information above, i can easily solve for <span class="math-container">$\pi=k/2(1-b/a)$</span>. My question is: How to derive <span class="math-container">$\pi$</span> if the utility function is not linear in w, e.g. of the form:</p>
<p><span class="math-container">$$ a(w)^\alpha \text{   for  } w&lt;w_{0} $$</span>
and
<span class="math-container">$$ u(w)=a*(w_{0})^\alpha+b(w-w_{0})^\alpha \text{   for  } w\geq w_{0}$$</span> where <span class="math-container">$b&lt;a$</span> and <span class="math-container">$\alpha&lt;1$</span>? Thanks a lot</p>

CC BY-SA 4.0
2022-03-08T15:39:24.870Z
It would follow the same step as the (piecewise-)linear case, only with an uglier solution.
Herr K.
CC BY-SA 4.0
2022-03-09T08:56:27.793Z
@HerrK. Thanks for your comment on this. What do you mean by "uglier"? If i solve for $\pi$ i get something that looks like this: $\pi=w_{0}-(\frac{{ap(w_{0}-k)^\alpha}-(p-1)(aw_{0}^{\alpha}+bk^{\alpha})}{a})^\frac{1}{\alpha}$ where $p=\frac{1}{2}$. Is this correct?
T123
CC BY-SA 4.0
2022-03-09T15:12:22.097Z
Yes that's what I got
Herr K.
<utility><preferences><risk><risk-aversion>
Risk Premium for Prospect Theory-like value function
T123
# Risk Premium for Prospect Theory-like value function

(Asked by: username_0 on 2022-03-08)

I am curious how to calculate the following risk premium for a utility function that is not linear in $w$. What i'm asking is the following:

Consider an agent with utility function $u$, initial wealth $\omega\_{0}$, and a zero-mean random variable $\tilde{x}$ that pays $+k;-k$ with probability $\frac{1}{2}$ each. Lets further assume that the agents utility function is defined as:

$$u(w)=a\*w \text{ for } w and

$$ u(w)=a\*w\_{0}+b(w-w\_{0}) \text{ for } w\geq w\_{0}$$ where $b

By definition of the risk premium $\pi$, we have

$$ Eu(w+\tilde{x}) = u(w-\pi). $$

By rearranging and given the information above, i can easily solve for $\pi=k/2(1-b/a)$. My question is: How to derive $\pi$ if the utility function is not linear in w, e.g. of the form:

$$ a(w)^\alpha \text{ for } w
and
$$ u(w)=a\*(w\_{0})^\alpha+b(w-w\_{0})^\alpha \text{ for } w\geq w\_{0}$$ where $b and $\alpha<1$? Thanks a lot

**Tags:** utility, preferences, risk, risk-aversion

> **Comments:**
> **username_1**: It would follow the same step as the (piecewise-)linear case, only with an uglier solution.  
> **username_0**: @HerrK. Thanks for your comment on this. What do you mean by "uglier"? If i solve for $\pi$ i get something that looks like this: $\pi=w_{0}-(\frac{{ap(w_{0}-k)^\alpha}-(p-1)(aw_{0}^{\alpha}+bk^{\alpha})}{a})^\frac{1}{\alpha}$ where $p=\frac{1}{2}$. Is this correct?  
> **username_1**: Yes that's what I got  

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## 0 answers

