Quadratic Utility Function Before this homework, "Calculate the corresponding premium for a quadratic utility function", we got to solve this example: 

Suppose the insurer has an exponential utility function with parameter $\alpha$. What is the minimum premium $P^-$ to be asked for a risk X if X $ \sim $ Exp(200) and $\alpha$= 0.001. Calculate $P^-$. 

We solved this and we came up with the formula for the minimum premium $P^-= {1\over\alpha} \log   M_x(\alpha)$. Upon simplification, we get $P^-=223.14$ Then after that, my teacher said to "calculate the  corresponding premium for a quadratic utility function." Maybe I'm suppose to find that $P^-$ but this time, using the quadratic utility function. That's what I think the homework was. 
Any help would be appreciated. Thanks for your time. I really want to learn this. 
 A: For an insurer, he should compare the expected utility of taking up a risk in exchange of a premium and not taking up that risk. Suppose his current wealth is $W$. Taking up the risk $X$ in exchange of the premium $P$ gives him a wealth of $W-X+P$, which is a random variable. He'll accept to take that risk if
$$\mathbb{E}[U(W)] \leq \mathbb{E}[U(W-X+P)]$$
and $P^{-}$ is the minimal premium for which he is willing to take this risk. For that premium we'll have
$$\mathbb{E}[U(W)] = \mathbb{E}[U(W-X+P^{-})]$$
Assume a simple quadratic utility function $U(x)=x^2$, then
$$W^2 = \mathbb{E}[(W-X+P^{-})^2]$$
or
$$0 = (P^{-})^2  + 2 P^{-} (W-\mathbb{E}[X]) + \mathbb{E}[X^2] - 2 W \mathbb{E}[X]$$
which is a quadratic equation for $P^{-}$. The solution is
$$P^{-} = (W-\mathbb{E}[X]) \pm\sqrt{(W-\mathbb{E}[X])^2-4(\mathbb{E}[X^2] - 2 W \mathbb{E}[X])} \; .$$
Only one of those solutions is meaningful, presumably the one with the positive root, as in a realistic situation, we can assume that the risk will not be larger than the wealth of the insurer and hence the square root is larger than the first term. The "-" solution would then give a negative premium.
You see that in the case of a quadratic utility function, the result will depend on the wealth of the insurer, which was not the case for an exponential utility function.
