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Math Help - Expected utility & Insurance (for risk adverse individuals)?

  1. #1
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    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.

    ====================

    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]
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  2. #2
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    Quote Originally Posted by kingwinner View Post
    [note: this question is posted eariler in other forum, yet nobody has answered it so far]
    Here's your proble, right here...

    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?
    Last edited by TKHunny; October 31st 2009 at 07:48 AM.
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    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
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  4. #4
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    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?
    =====================

    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)
    0.95 sqrt(25-premium) + 0.05 sqrt(25-premium) ≥ 4.8618
    ...
    p ≤ 1.3629
    Answer: 1.3629 million.

    And now I am stuck with part d.
    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.
    Also, assume no progressive tax, agent's fees, etc, and we don't care about how much money he presently has (i.e. assume $0), so wealth here is defined completely in terms of the money that he is going to get after signing the next contract.

    For part d, I beleive that the answer is that he will NOT choose full coverage, but I am not sure how to prove it. Here is my attempt! I think we need 3 cases: full coverage, partial coverage, and the case with no insurace.
    And we need to prove that EU(fully insured) < EU(partially insured), or perhaps EU(fully insured) < EU(no insurance). But how can we prove it?


    Any help is greatly appreciated!
    Last edited by kingwinner; October 31st 2009 at 10:12 AM.
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