1. ## Expected utility & Insurance (for risk adverse individuals)?

John is a professional football player and next year will be able to sign a $25 million contract if he does not get injured this year. If he gets injured this year, his value will be significantly reduced and he will only be able to sign a contract for$5 million. Suppose that the probability he gets injured is 5%. His utility function is of the form U = W^(1/2), where W is his wealth. (therefore he is risk adverse).
Suppose that an insurace company offers to insure him at a premium of $0.10 per$1 of coverage. Would John choose to FULLY insurance himself? (hint: think of the idea of actuarilly fair/unfair game) Explain, and show graphically.

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I am very confused by this question and I don't even know where to begin. Can someone kindly explain the answer to this question. If at all possible, please explain how the graph would look like as well.
Your help is very much appreciated!

[note: this question is posted eariler in other forum, yet nobody has answered it so far]

2. Originally Posted by kingwinner
[note: this question is posted eariler in other forum, yet nobody has answered it so far]

I am very confused by this question and I don't even know where to begin.
Anyone in this position may not be attending class or reading the text book. If you are taking a class and paying ANY attention AT ALL you simply MUST have SOME clue. If you REALLY have no clue after reading your text and attending class, perhaps you are in the wrong class? Perhaps you need to go have a maningful chat with yout academic advisor.

Having said that:

John is a professional football player and next year will be able to sign a $25 million contract if he does not get injured this year. If he gets injured this year, his value will be significantly reduced and he will only be able to sign a contract for$5 million. Suppose that the probability he gets injured is 5%. His utility function is of the form U = W^(1/2), where W is his wealth. (therefore he is risk adverse).
His expected utility, without insurance is simply:

$0.05 \cdot \sqrt{5,000,000} + 0.95 \cdot \sqrt{25,000,000}$

Suppose that an insurace company offers to insure him at a premium of $0.10 per$1 of coverage. Would John choose to FULLY insurance himself?
First, we would have to define what "fully" means. Does it mean $25 MM? Does it mean the Expected Utility from above? Perhaps it means to pay$20 MM if he is injured?

In the last case, he is out $2,000 right from the start. What say you? What, exactly, does "fully" mean? 3. Originally Posted by TKHunny Here's your proble, right here... Anyone in this position may not be attendign class or reading the text book. If you are taking a class and paying ANY attention AT ALL you simply MUST have SOME clue. If you REALLY have no clue after reading your text and attending class, perhaps you are in the wrong class? Perhaps you need to go have a maningful chat with yout academic advisor. Having said that: His expected utility, without insurance is simply: $0.05 \cdot \sqrt{5,000,000} + 0.95 \cdot \sqrt{25,000,000}$ First, we would have to define what "fully" means. Does it mean$25 MM? Does it mean the Expected Utility from above? Perhaps it means to pay $20 MM if he is injured? In the last case, he is out$2,000 right from the start. What say you? What, exactly, does "fully" mean?
There is another problem, in that his utility is defined in terms of his wealth, so surly we need to know how much he presently has and how the value of the contract adds to his wealth (progressive tax, agent's fees, ...)?

CB

4. I have already done parts a, b, and c from the same question (they are all very easy), and the last one is part d which I don't get.

a) What is Jacob’s expected utility? Show graphically.
b) What is the "certainty equivalent" of the risky prospect he faces? Show graphically.
c) What is his reservation price for an insurance policy that offers him FULL coverage in the event of loss?
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All calculations are in millions.

a) EU=0.95(sqrt25)+0.05(sqrt5)=4.8618

b) Set U(CE)=4.8618
=> sqrt(CE)=4.8918
=> CE=23.6371

c) Initial wealth = W_o = 25
W_o - L = 5
Loss = L = 20

Set EU(with insurance) ≥ EU(with no insurance)
...
p ≤ 1.3629
Full coverage means to pay L = $20 (millions) if he is injured. The insurance company is going to compensate for his full loss of$20 (millions) if he is injured, so in any case, his wealth is (25 - premium) if he buys the full coverage insurance.