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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.

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    Any help would be astonishing. The question gives too little context for meaningful help to be offered. Can you edit your question to expand on what this is about and what you understand about it? – Joffan Feb 05 '15 at 08:48
  • I'm sorry. I'm going to edit this. I wrote this question before even trying to understand what my teacher wants us to do. –  Feb 05 '15 at 12:50
  • Thanks for updating your Question with more context. However it seems it is still missing details about the quadratic utility function (that replaces the exponential utility function from the earlier exercise). – hardmath Feb 05 '15 at 14:14
  • It's possible, given the informal instruction attributed to your teacher, that you were given latitude to pick a quadratic utility function to complete the exercise. – hardmath Feb 05 '15 at 21:50

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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.

Raskolnikov
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