typo...
The above should read
"Suppose that an INSURANCE company offers to insure him at a premium of..."
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 insurface 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 the graph as well.
Your help is very much appreciated!:)
2 answers
I believe what you want do do is calculate John's Expected Utility. Incorporate into the utility function the insurance option.
Let x be the number dollars of insurance coverage, each unit of x costs 0.10. So, the expected utility E(U) is as follows.
E(U) = .95*(25000000 - 0.1*x)^(.5) + .05*(5000000 + 0.9*x)^(.5)
Where the .95 and the .05 are the probabilities of not getting/getting injured, .1 is the insurance policy premium and .9 is the insurance policy payoff less the premium.
From the beginning, you could see the insurance policy was actuarily unfair. If the probability of getting injured was 10% instead of 5%, then the policy would be actuarily fair.
You can graph expected the expected utility function with respect to x. You should get a hump-shaped curve. From the graph, you can tell that John will not fully insure himself. Maximum expected utility occurs when x is between 600,000 and 700,000. (Use calculas to determine the exact amount) From the graph, you should be able to tell that John will NOT fully insure himself. Expected utility at x=20million is actually less than expected utility at x=zero.
Let x be the number dollars of insurance coverage, each unit of x costs 0.10. So, the expected utility E(U) is as follows.
E(U) = .95*(25000000 - 0.1*x)^(.5) + .05*(5000000 + 0.9*x)^(.5)
Where the .95 and the .05 are the probabilities of not getting/getting injured, .1 is the insurance policy premium and .9 is the insurance policy payoff less the premium.
From the beginning, you could see the insurance policy was actuarily unfair. If the probability of getting injured was 10% instead of 5%, then the policy would be actuarily fair.
You can graph expected the expected utility function with respect to x. You should get a hump-shaped curve. From the graph, you can tell that John will not fully insure himself. Maximum expected utility occurs when x is between 600,000 and 700,000. (Use calculas to determine the exact amount) From the graph, you should be able to tell that John will NOT fully insure himself. Expected utility at x=20million is actually less than expected utility at x=zero.