What is the minimum Premium to be asked for a risk X?

Question

Suppose that an insurer has an exponential utility function $u(x) =-2e^{-2x}.$ What is the minimum premium $P^{-}$ to be asked for a risk X?

I got some hint for this, but I could not understand this hint.

Answer 1

I hope this can get you started:

What I understood from the picture is that the person who wants to be insured has an exponential utility function with parameter one $$u(x)=-e^{-x},$$ and that the risk $X$ is random with distribution $\text{Exp}(2)$, that is, the expected value of a function $f$ of $X$ is $$\mathbb E(f(X))=\int_0^\infty f(x)\,2e^{-2x}\, dx.$$

Suppose that the person's initial wealth is $W$. He has two options: either take the risk $X$, which I interpret means to lose a random amount of money equal to $X$, or to pay a premium $P^-$ which will keep him safe from the risk.

If he takes the first option his wealth in the future will be random and equal to $W-X,$ whereas if he takes the other one his wealth will be surely $W-P^-.$

In order for the premium to be a fair price to pay the insured wants to expect at least the same utility loss from these two options, which means that the condition for the maximum premium is $$u(W-P^-)=\mathbb E[u(W-X)]. $$ It only remains to compute.