# Risk Premium for Prospect Theory-like value function

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

• It would follow the same step as the (piecewise-)linear case, only with an uglier solution. Commented Mar 8, 2022 at 15:39
• @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
Commented Mar 9, 2022 at 8:56
• Yes that's what I got Commented Mar 9, 2022 at 15:12